Modulation Theory Research Articles

On Top Local Homology Modules

In this article, we can to relate the theory of local cohomology, with respect to an ideal, to the theory of local homology modules. With the results of the article, we show the importance of local homology theory as a study tool within of the commutative algebra theory.

Hydrodynamics of a discrete conservation law

AbstractThe Riemann problem for the discrete conservation law is classified using Whitham modulation theory, a quasi‐continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.

Whitham modulation theory and the classification of solutions to the Riemann problem of the Fokas–Lenells equation

AbstractIn this work, we explore the Riemann problem of the Fokas–Lenells (FL) equation given initial data in the form of a step discontinuity by employing the Whitham modulation theory. The periodic wave solutions of the FL equation are characterized by elliptic functions along with the Whitham modulation equations. Moreover, we find that the signs for the velocities of the periodic wave solutions remain unchanged during propagation. Thus, when analyzing the propagation behavior of solutions, it is necessary to separately consider the clockwise (negative velocity) and counterclockwise (positive velocity) cases. In this regard, we present the classification of the solutions to the Riemann problem of the FL equation in both clockwise and counterclockwise cases for the first time.

SparseAuto: An Auto-scheduler for Sparse Tensor Computations using Recursive Loop Nest Restructuring

Automated code generation and performance enhancements for sparse tensor algebra have become essential in many real-world applications, such as quantum computing, physical simulations, computational chemistry, and machine learning. General sparse tensor algebra compilers are not always versatile enough to generate asymptotically optimal code for sparse tensor contractions. This paper shows how to generate asymptotically better schedules for complex sparse tensor expressions using kernel fission and fusion. We present generalized loop restructuring transformations to reduce asymptotic time complexity and memory footprint. Furthermore, we present an auto-scheduler that uses a partially ordered set (poset)-based cost model that uses both time and auxiliary memory complexities to prune the search space of schedules. In addition, we highlight the use of Satisfiability Module Theory (SMT) solvers in sparse auto-schedulers to approximate the Pareto frontier of better schedules to the smallest number of possible schedules, with user-defined constraints available at compile-time. Finally, we show that our auto-scheduler can select better-performing schedules and generate code for them. Our results show that the auto-scheduler provided schedules achieve orders-of-magnitude speedup compared to the code generated by the Tensor Algebra Compiler (TACO) for several computations on different real-world tensors.

Mechanisms of Reappeared Period Shifts of Internal Solitary Waves Based on Mooring Array Observations in the Northern South China Sea

AbstractMooring observations have revealed that the daily reappeared time (DRt) of internal solitary waves (ISWs) in the northern South China Sea (SCS) manifests systematic shifts. This study aims to reveal the dynamic mechanisms behind this phenomenon using the theory of modulation of shear flow to the internal tide wave (IT) propagation. Approximate solutions for the frequency modulation function (FMF) and the amplitude modulation function (AMF) of ITs are derived and validated by the mooring array measurements on the northern SCS continental shelf in 2019 and 2020. Namely, the mechanisms of the ISW reappeared period shift (RPS derived from DRt) and amplitude variation are attributed to the modulation of low‐frequency flow with a period of about 7 days to ITs. The FMF is induced by the vorticity of the low‐frequency flow (FS1) and the Doppler shift (FS2). Thus, the sign of the FMF value may be positive (blue shift) and negative (red shift), depending on the algebraic summation of the two terms. The FS2 derived from mooring observed velocities confirms the modulation effects of mean current on the frequency shift and the amplitude variation of ITs and ISWs.

Mechanism and Characteristics of Cogging Torque in Surface-Mounted PMSM: A General Airgap Field Modulation Theory Approach

Mechanism and Characteristics of Cogging Torque in Surface-Mounted PMSM: A General Airgap Field Modulation Theory Approach

Wave breaking, dispersive shock wave, and phase shift for the defocusing complex modified KdV equation.

We study the problem of wave breaking for a simple wave propagating to a quiescent medium in the framework of the defocusing complex modified KdV (cmKdV) equation. It is assumed that a cubic root singularity is formed at the wave-breaking point. The dispersive regularization of wave breaking leads to the generation of a dispersive shock wave (DSW). We describe the DSW as a modulated periodic wave in the framework of the Gurevich-Pitaevskii approach based on the Whitham modulation theory. The generalized hodograph method is used to solve the Whitham equations, and the boundaries of the DSW are found. Most importantly, we determine the correct phase shift for the DSW from the generalized phase relationships and the modified Gurevich-Pitaevskii matching conditions, so that a complete description of the DSW is obtained rather than just its envelope. All of our analytical predictions agree well with the numerical simulations.

Evolution of water wave packets by wind in shallow water

We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

A novel study on the structure of left almost hypermodules

The concept of left almost hypermodule evolves as a novel extension in the field of abstract algebra, specifically within the broader framework of hypermodules. The left almost hypermodule is characterized by a set endowed with two operations, evincing properties that extends across traditional module theory and hypermodules. This abstract intents to provide a succinct overview of salient attributes and prospective implications of the left almost hypermodule, stimulating further exploration of its properties and applications. The paper provides a new definition of hypermodule that acts on the left almost hyperring, referred to as left almost hypermodule (abbreviated as LA-hypermodule), and provides some examples of this new structure. We further examine the variations between hypermodules and left almost hypermodules. By using the concept of left almost polygroups, we explore the transition from left almost polygroup to a left almost hypermodule over left almost hyperring. Lastly, we observe the outcomes in connection to homomorphism and regular relations on left almost hypermodules.

Whitham modulation theory and Riemann problem for the Kundu–Eckhaus equation

In this paper, the Riemann problem for the defocusing Kundu–Eckhaus equation is investigated by Whitham modulation theory. First, we study the dispersion relation for linear waves. Then, the zero-phase and one-phase periodic solutions of the Kundu–Eckhaus equation along with the corresponding Whitham modulation equations are derived by the finite-gap integration method. Further, employing the Whitham equations parametrized by the Riemann invariants, the main fundamental wave structures induced by the discontinuous initial data are found. Analytical and graphic methods are utilized to provide the wave structures of rarefaction waves and dispersive shock waves, and thus for a complete classification of solutions under general step-like conditions of initial discontinuity.

About Supports of Local Cohomology Modules

The article provides the relation between the theory of local cohomology modules, and vanishing results, and also about the theory of support of such modules. Here, we put results about the theory, and also we provide a relation of local cohomology in the theory of commutative algebra and homological algebra.

An entropy modulation theory of creative exploration.

An entropy modulation theory of creative exploration.

Coarse nodal count and topological persistence

Courant’s theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, there have been various attempts to find an appropriate generalization of this statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant’s theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal sets. We also obtain a coarse version of the Bézout estimate for common zeros of linear combinations of eigenfunctions. We show that our results are essentially sharp and that the coarse count is necessary, since these extensions fail in general for the standard count. Our approach combines multiscale polynomial approximation in Sobolev spaces with new results in the theory of persistence modules and barcodes.

An algebraic approach to circulant column parity mixers

Circulant Column Parity Mixers (CCPMs) are a particular type of linear maps, used as the mixing layer in permutation-based cryptographic primitives like Keccak-f (SHA3) and Xoodoo. Although being successfully applied, not much is known regarding their algebraic properties. They are limited to invertibility of CCPMs, and that the set of invertible CCPMs forms a group. A possible explanation is due to the complexity of describing CCPMs in terms of linear algebra. In this paper, we introduce a new approach to studying CCPMs using module theory from commutative algebra. We show that many interesting algebraic properties can be deduced using this approach, and that known results regarding CCPMs resurface as trivial consequences of module theoretic concepts. We also show how this approach can be used to study the linear layer of Xoodoo, and other linear maps with a similar structure which we call DCD-compositions. Using this approach, we prove that every DCD-composition where the underlying vector space with the same dimension as that of Xoodoo has a low order. This provides a solid mathematical explanation for the low order of the linear layer of Xoodoo, which equals 32. We design a DCD-composition using this module-theoretic approach, but with a higher order using a different dimension.

Vanishing of local cohomology with applications to Hodge theory

Let H=((H,F•),L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U. By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M∈HM(U). Let X be a compactification of U, and j:U↪X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j+M∈DbMHM(X). In particular, we study the graded pieces of the de Rham complex GrpFDR(j+M)∈Dcohb(X), and the Hodge structure of Hi(U,L) for i in sufficiently low degrees.

On the Auslander–Bridger–Yoshino theory for complexes of finitely generated projective modules

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable “complex” theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as ⁎torsionfreeness and ⁎reflexivity. Also, we prove the Auslander–Bridger approximation theorem for complexes of finitely generated projective R-modules.

Reduction of losses in dual‐permanent‐magnet‐excited Vernier machine by segmented stator for electric aircraft

AbstractThe authors present a low‐loss dual‐permanent‐magnet‐excited vernier (DPMEV) machine with segmented stator design to meet the requirements of low iron loss of electric aircraft based on the field modulation theory. The stator topology features and losses of both the original and segmented DPMEV machines are comparatively investigated. Then, the armature air‐gap flux density is deduced by the magnetic motive force‐permeance model, and the influence of each harmonic on the losses are analysed. It is found that the harmonic which produces losses in the original machine reduced greatly after introducing the segmented structure. Furthermore, the electromagnetic performances of two DPMEV machines are comparatively analysed by finite element analysis. Finally, two prototypes of the original and segmented DPMEV machines are built and tested to verify the theoretical analysis.

Whitham Modulation Theory and Two-Phase Instabilities for Generalized Nonlinear Schrödinger Equations with Full Dispersion

Whitham Modulation Theory and Two-Phase Instabilities for Generalized Nonlinear Schrödinger Equations with Full Dispersion

The single-phase solution and Whitham modulation equations of the defocusing self-induced transparency system

Under investigation in this paper is the defocusing self-induced transparency system which can describe propagation characters of short pulses in Erbium doped fibers with a two-level medium. The single-phase periodic solution and Whitham modulation equations will be derived via finite-gap integration method and Whitham modulation theory. In addition, the oscillatory structure of dispersive shock waves will be constructed and analyzed.

Integrable approximations of dispersive shock waves of the granular chain

In the present work we revisit the shock wave dynamics in a granular chain with precompression. By approximating the model by an α-Fermi–Pasta–Ulam–Tsingou chain, we leverage the connection of the latter in the strain variable formulation to two separate integrable models, one continuum, namely the KdV equation, and one discrete, namely the Toda lattice. We bring to bear the Whitham modulation theory analysis of such integrable systems and the analytical approximation of their dispersive shock waves in order to provide, through the lens of the reductive connection to the granular crystal, an approximation to the shock wave of the granular problem. A detailed numerical comparison of the original granular chain and its approximate integrable-system-based dispersive shocks proves very favorable in a wide parametric range. The gradual deviations between (approximate) theory and numerical computation, as amplitude parameters of the solution increase are quantified and discussed.