arXiv subjects

Michael Lau

Publications and source records attributed to Michael Lau.

At least 19 recordsLinked to original sources

arXiv2026

Analyzing Performance and Scalability of Benders Decomposition for Generation and Transmission Expansion Planning Models

Generation and Transmission Expansion Planning (GTEP) problems co-optimize generation and transmission expansion, enabling them to provide better planning decisions than traditional Generation Expansion Planning or Transmission Expansion Planning problems, but GTEPs can be computationally complex or intractable. Benders Decomposition (BD) has been applied to expansion planning problems, with various methods applied to accelerate convergence. In this work, we test strategies for improving the performance of BD on GTEP models with nodal resolution and DCOPF constraints. We also present an alternative approach for handling the bilinear constraints that can result in these problems. These tests included combinations of using generalized Benders decomposition (GBD), hot-starting via a transport constrained model, using linear relaxations of the master problem, and using regularization. We test these methods on mixed-integer linear programming GTEP models with up to 146 buses (10 million continuous variables and 400 mixed-integer decisions). With selected accelerated Benders decomposition approaches, the problems can be solved to under a 1\% gap in as little as 5 hours where they were otherwise intractable. Results also suggest that using regularization on these initial hot-starting and relaxation steps and turning it off after they are complete was generally the best combination of strategies.

math.OC↗

arXiv2026

Dynamics of antiskyrmion shrinking

Antiskyrmions are unstable in ferromagnetic systems with isotropic bulk or interfacial Dzyaloshinskii-Moriya interaction (DMI). We develop a continuum model for the shrinking dynamics of antiskyrmions in bulk DMI systems, using the Landau-Lifshitz-Gilbert equation for the time derivative of the magnetization field. Owing to the structure of their azimuthal angle, or helicity, elliptic antiskyrmions are energetically favored over circular ones. To capture this feature, we parametrize the magnetization field with a triangular radial profile and an elliptic in-plane shape. This ansatz yields four coupled dynamical equations governing time evolution of the semi-axes, helicities, and rotation angles. In the absence of the DMI, circular antiskyrmions shrink isotropically, exhibiting a crossover from exponential decay to square-root collapse. Initially elliptic antiskyrmions are driven towards circularity. For finite DMI, the semi-axes dynamics couples to the helicity and rotation, where the theory predicts a rotation angle following by half of the slope of the helicity evolution which is linear in time. Only at small semi-axes a cross-over to a logarithmic divergence occurs. The shrinking dynamics of the antiskyrmion size is found to be accompanied by quadrupole-like oscillations. Numerical simulations on the lattice support the predictions from the continuum model.

cond-mat.mes-hall↗

arXiv2025

A Parallelized Cutting-Plane Algorithm for Computationally Efficient Modelling to Generate Alternatives

Contemporary macro energy systems modelling is characterized by the need to represent strategic and operational decisions with high temporal and spatial resolution and represent discrete investment and retirement decisions. This drive towards greater fidelity, however, conflicts with a simultaneous push towards greater model representation of inherent complexity in decision making, including methods like Modelling to Generate Alternatives (MGA). MGA aims to map the feasible space of a model within a cost slack by varying investment parameters without changing the operational constraints, a process which frequently requires hundreds of solutions. For large, detailed energy system models this is impossible with traditional methods, leading researchers to reduce complexity with linearized investments and zonal or temporal aggregation. This research presents a new solution method for MGA type problems using cutting-plane methods based on a tailored reformulation of Benders Decomposition. We accelerate the algorithm by sharing cuts between MGA master problems and grouping MGA objectives. We find that our new solution method consistently solves MGA problems times faster and requires less memory than existing monolithic Modelling to Generate Alternatives solution methods on linear problems, enabling rapid computation of a greater number of solutions to highly resolved models. We also show that our novel cutting-plane algorithm enables the solution of very large MGA problems with integer investment decisions.

math.OC↗

arXiv2025

The dynamics of skyrmion shrinking

When magnetic skyrmions decay, their size in real space decreases in a finite time before they eventually collapse. We construct an effective continuum model and use its dynamics to describe the shrinking behavior of skyrmions before they collapse. Using the Landau-Lifshitz-Gilbert equation and the time derivative of the vector field, we find a set of coupled nonlinear ordinary differential equation for the time dependent effective skyrmion radius and its helicity. In particular, we use a triangular-shaped skyrmion profile of its polar angle. Contrary to the commonly expected simple exponential decrease in size, we reveal a more complicated time dependence, in which the time-dependent radius crosses over from an exponential decay towards a square root decrease, $\sim (t - t_c)^{1/2}$, near a critical time $t_c$ at which it collapses. This critical time is found to depend logarithmically on the lattice constant. In addition, we examine the interplay between the shrinking dynamics and an accompanying transformation through different skyrmion configurations, depending on the various system parameters. The findings are verified by numerical studies on the lattice, supporting the predictions from the theoretical continuum model.

cond-mat.mes-hall↗

arXiv2025

Free Jordan Algebras and Representations of $\widehat{\mathfrak{sl}}_2(J)$

Let $J$ be a unital Jordan algebra, and let $\widehat{\mathfrak{sl}}_2(J)$ be the universal central extension of its Tits-Kantor-Koecher Lie algebra. In Part A, we study the category of $(\widehat{\mathfrak{sl}}_2(J), SL_2(K))$-modules. We characterize the dominant $J$-spaces, which are analogous to the dominant highest weights appearing in classical settings. A family of universal envelopes $\mathcal{U}_n(J)$ associated to such modules is introduced and studied. We also prove some finiteness theorems. In Part C, we define the notion of smooth $\widehat{\mathfrak{sl}}_2(J)$-modules for augmented Jordan algebras $J$, and investigate the category of smooth modules in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules of this category are finite dimensional when $J$ is finitely generated. The free unital Jordan algebra $J(D)$ over $D$ variables is an elusive object, but finiteness and Ext-vanishing properties suggest that the smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues might form a generalized highest weight category. However, we prove that such an assertion would contradict recently obtained information about the growth of free Jordan algebras. See [24] and [13] for more details. It then follows that the category of smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues is not a generalized highest weight category when $D\geq 2$. Surprisingly, the proofs of most of these results make use of deep theorems of E. Zelmanov.

math.RT↗

arXiv2024

Modelling to Generate Continuous Alternatives: Enabling Real-Time Feasible Portfolio Generation in Convex Planning Models

Decarbonization provides new opportunities to plan energy systems for improved health, resilience, equity, and environmental outcomes, but challenges in siting and social acceptance of transition goals and targets threaten progress. Modelling to Generate Alternatives (MGA) provides an optimization method for capturing many near-cost-optimal system configurations, and can provide insights into the tradeoffs between objectives and flexibility available in the system. However, MGA is currently limited in interactive applicability to these problems due to a lack of methods for allowing users to explore near-optimal feasible spaces. In this work we describe Modelling to Generate Continuous Alternatives (MGCA), a novel post-processing algorithm for convex planning problems which enables users to rapidly generate new interior solutions, incorporate new constraints, and solve within the space with convex objectives. MGCA begins with a dimensionality reduction to capacity decisions and metric values. We then take advantage of convex combinations to generate interior points by allowing user weight specification and encoding convex combinations in an optimization problem with user-defined additional constraints and objective. Dimensionality reduction enables this problem to solve in tenths of a second, suitable for analysis in interactive settings. We discuss the interpolation of capacity and operational metric values, finding capacity metrics can be perfectly interpolated while operational metrics remain within the feasible range of the points used to create them. We demonstrate interpolated solutions can be exported and re-solved with an economic dispatch model to provide operational metric values consistent with least-cost decision-making and show interpolated metric values are generally within 10% of the optimal value.

math.OC↗

arXiv2024

Moving skyrmions in Antiferromagnets by Sublattice Displacements

The texture in antiferromagnets hosting a topologically protected skyrmion can be viewed as two effectively coupled ferromagnetic skyrmions. Assuming rigid magnetic configurations, we show that this coupling results in an effective mass of the antiferromagnetic skyrmion and that its dynamics can be viewed as due to a relative displacement of the two sublattice ferromagnetic skyrmions. The theory holds for different antiferromagnetic systems and includes effects from dissipation and external forces caused by electric currents. We verify our analytical results by micromagnetic simulations.

cond-mat.mes-hall↗

arXiv2024

Measuring Exploration: Review and Systematic Evaluation of Modelling to Generate Alternatives Methods in Macro-Energy Systems Planning Models

As decarbonization agendas mature, macro-energy systems modelling studies have increasingly focused on enhanced decision support methods that move beyond least-cost modelling to improve consideration of additional objectives and tradeoffs. One candidate is Modeling to Generate Alternatives (MGA), which systematically explores new objectives without explicit stakeholder elicitation. Previous literature lacks both a comprehensive review of MGA vector selection methods in large-scale energy system models and comparative testing of their relative efficacies in this setting. To fill this gap, this paper provides a comprehensive review of the MGA literature, identifying at least seven MGA vector selection methodologies and carrying out a systematic evaluation of four: Hop-Skip-Jump, Random Vector, Variable Min/Max, and Modelling All Alternatives. We examine each method's runtime, parallelizability, new solution discovery efficiency, and spatial exploration in lower dimensional (N <= 100) spaces, as well as spatial exploration in a three-zone, 8760-hour capacity expansion model case. Through these tests, we find Random Vector provides the broadest exploration of the near-optimal feasible region and Variable Min/Max provides the most extreme results, while the two tie on computational speed. We thus propose a new Hybrid vector selection approach combining the two methods to take advantage of the strengths of each. Additional analysis is provided on MGA variable selection, in which we demonstrate MGA problems formulated over generation variables fail to retain cost-optimal dispatch and are thus not reflective of real operations of equivalent hypothetical capacity choices. As such, we recommend future studies utilize a parallelized combined vector approach over the set of capacity variables for best results in computational speed and spatial exploration while retaining optimal dispatch.

math.OC↗

arXiv2023

Jordan algebras and weight modules

We consider bounded weight modules for the universal central extension ${\mathfrak{sl}}_2(J)$ of the Tits-Kantor-Koecher algebra of a unital Jordan algebra $J$. Universal objects called Weyl modules are introduced and studied, and a combinatorial dominance criterion is given for analogues of highest weights. Specializing $J$ to the free Jordan algebra $J(r)$ of rank $r$, the category $\mathcal{C}^{fin}$ of finite-dimensional $\mathbb{Z}$-graded ${\mathfrak{sl}}_2(J)$-modules shares many properties with the representation theory of algebraic groups. Using a deep result of Zelmanov, we show that this subcategory admits Weyl modules. By analogy, we conjecture that $\mathcal{C}^{fin}$ is a highest weight category. The resulting homological properties would then imply cohomological vanishing results previously conjectured as a way of determining graded dimensions of free Jordan algebras.

math.RT↗

arXiv2023

Whittaker vectors in singular Whittaker modules

Let ${\mathfrak{g}}$ be a complex semisimple Lie algebra with Borel subalgebra ${\mathfrak{b}}$ and corresponding nilradical ${\mathfrak{n}}$. We show that singular Whittaker modules $M$ are simple if and only if the space $\hbox{Wh}\,M$ of Whittaker vectors is $1$-dimensional. For arbitrary locally ${\mathfrak{n}}$-finite ${\mathfrak{g}}$-modules $V$, an immediate corollary is that the dimension of $\hbox{Wh}\,V$ is bounded by the composition length of $V$.

math.RT↗

arXiv2023

Spin Wave Driven Skyrmions in a Bipartite Antiferromagnetic Lattice

We show that a Skyrmion in a classical bipartite antiferromagnetic lattice can be spatially displaced in a controlled manner by externally applied spin waves. We reveal the relation between the Skyrmion motion and the spin wave properties. To this end, we derive a classical spin wave formalism which is tailored to the antiferromagnetic two-dimensional square lattice. The antiferromagnetic spin waves can be classified into two types with respect to their polarization, with two modes each. The circularly polarized spin waves oscillate with different amplitudes in the respective sublattices and induce a Skyrmion Hall effect. The two modes are symmetric under sublattices exchange and determine the overall sign of the Hall angle. Linearly polarized spin waves oscillate elliptically, however, with the same amplitude on each sublattice. These accelerate the Skyrmion solely into their own propagation direction. The two modes are symmetric under component x-y exchange and impact Bloch- or N\'eel Skyrmions differently. Our results indicate possible technical applications of spin-wave driven Skyrmion motion. As one example we propose a racetrack where spin waves pump Skyrmions along the track in antiferromagnets.

cond-mat.mes-hall↗

arXiv2022

Toda systems for Takiff algebras

We study completely integrable systems attached to Takiff algebras $\mathfrak{g}_N$, extending open Toda systems of split simple Lie algebras $\mathfrak{g}$. With respect to Darboux coordinates on coadjoint orbits $\mathcal{O}$, the potentials of the hamiltonians are products of polynomial and exponential functions. General solutions for equations of motion for $\mathfrak{g}_N$ are obtained using differential operators called jet transformations. These results are applied to a $3$-body problem based on $\mathfrak{sl}(2)$, and to an extension of soliton solutions for $A_\infty$ to associated Takiff algebras. The new classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, solving a Vinberg problem and quantizing the Poisson algebra of functions on $\mathcal{O}$.

math-ph↗

arXiv2017

Extensions of modules for twisted current algebras

Twisted current algebras are fixed point subalgebras of current algebras under a finite group action. Special cases include equivariant map algebras and twisted forms of current algebras. Their finite-dimensional simple modules fall into two categories, those which factor through an evaluation map and those which do not. We show that there are no nontrivial extensions between finite-dimensional simple evaluation and non-evaluation modules. We then compute extensions between any pair of finite-dimensional simple modules for twisted current algebras, and use this information to determine the block decomposition for the category. In the special case of twisted forms, this decomposition can be described in terms of maps to the fundamental group of the underlying root system.

math.RT↗

arXiv2017

Classification of Harish-Chandra modules for current algebras

For any reductive Lie algebra $\mathfrak{g}$ and commutative, associative, unital algebra $S$, we give a complete classification of the simple weight modules of $\mathfrak{g}\otimes S $ with finite weight multiplicities. In particular, any such module is parabolically induced from a simple admissible module for a Levi subalgebra. Conversely, all modules obtained in this way have finite weight multiplicities. These modules are isomorphic to tensor products of evaluation modules at distinct maximal ideals of $S$. Our results also classify simple Harish-Chandra modules up to isomorphism for all central extensions of current algebras.

math.RT↗

arXiv2014

Weight modules for current algebras

For any finite-dimensional simple Lie algebra $\mathfrak{g}$ and commutative associative algebra $S$ of finite type, we give a complete classification of the simple weight modules of $\mathfrak{g}\otimes S$ with bounded weight multiplicities.

math.RT↗

arXiv2013

Lie-Poisson theory for direct limit Lie algebras

In this paper, we develop the fundamentals of Lie-Poisson theory for direct limits $G=\dirlim G_{n}$ of complex algebraic groups $G_{n}$ and their Lie algebras $\fg=\dirlim \fg_{n}$. We show that $\fg^{*}=\invlim\fg_{n}^{*}$ has the structure of a Poisson provariety and that each coadjoint orbit of $G$ on $\fg^{*}$ has the structure of an ind-variety. We construct a weak symplectic form on every coadjoint orbit and prove that the coadjoint orbits form a weak symplectic foliation of the Poisson provariety $\fg^{*}$. We apply our results to the specific setting of $G=GL(\infty)=\dirlim GL(n,\C)$ and $\fg^{*}= M(\infty)=\invlim \fgl(n,\C)$, the space of infinite complex matrices with arbitrary entries. We construct a Gelfand-Zeitlin integrable system on $M(\infty)$, which generalizes the one constructed by Kostant and Wallach on $\fgl(n,\C)$. The system integrates to an action of a direct limit group $A(\infty)$ on $M(\infty)$, whose generic orbits are Lagrangian ind-subvarieties of the corresponding coadjoint orbit of $GL(\infty)$ on $M(\infty)$.

math.RT↗

arXiv2013

Maximal ideals and representations of twisted forms of algebras

Given a central simple algebra $\mathfrak{g}$ and a Galois extension of base rings $S/R$, we show that the maximal ideals of twisted $S/R$-forms of the algebra of currents $\mathfrak{g}(R)$ are in natural bijection with the maximal ideals of $R$. When $\mathfrak{g}$ is a Lie algebra, we use this to give a complete classification of the finite-dimensional simple modules over twisted forms of $\mathfrak{g}(R)$.

math.RT↗

arXiv2013

Representations of twisted current algebras

We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.

math.RT↗