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In the wild nonabelian Hodge correspondence on curves, filtered Stokes G-local systems are regarded as the objects on the Betti side. In this paper, we demonstrate a construction of the moduli space of them, called the Betti moduli space, and it reduces to the wild character variety when the Betti weights are trivial. We study some particular examples including Eguch-Hanson space and the Airy equation together with the corresponding moduli spaces. Furthermore, we provide a proof of the correspondence among irregular singular G-connections, Stokes G-local systems, and Stokes G-representations. This correspondence can be viewed as the G-version of irregular Rieman-Hilbert correspondence on curves., Comment: 27 pages

Solving real-world complex tasks using reinforcement learning (RL) without high-fidelity simulation environments or large amounts of offline data can be quite challenging. Online RL agents trained in imperfect simulation environments can suffer from severe sim-to-real issues. Offline RL approaches although bypass the need for simulators, often pose demanding requirements on the size and quality of the offline datasets. The recently emerged hybrid offline-and-online RL provides an attractive framework that enables joint use of limited offline data and imperfect simulator for transferable policy learning. In this paper, we develop a new algorithm, called H2O+, which offers great flexibility to bridge various choices of offline and online learning methods, while also accounting for dynamics gaps between the real and simulation environment. Through extensive simulation and real-world robotics experiments, we demonstrate superior performance and flexibility over advanced cross-domain online and offline RL algorithms.

The computation of the ground states of special multi-component Bose-Einstein condensates (BECs) can be formulated as an energy functional minimization problem with spherical constraints. It leads to a nonconvex quartic-quadratic optimization problem after suitable discretizations. First, we generalize the Newton-based methods for single-component BECs to the alternating minimization scheme for multi-component BECs. Second, the global convergent alternating Newton-Noda iteration (ANNI) is proposed. In particular, we prove the positivity preserving property of ANNI under mild conditions. Finally, our analysis is applied to a class of more general "multi-block" optimization problems with spherical constraints. Numerical experiments are performed to evaluate the performance of proposed methods for different multi-component BECs, including pseudo spin-1/2, anti-ferromagnetic spin-1 and spin-2 BECs. These results support our theory and demonstrate the efficiency of our algorithms.

In this paper, we construct the moduli spaces of filtered $G$-local systems on curves for an arbitrary reductive group $G$ over an algebraically closed field of characteristic zero. This provides an algebraic construction for the Betti moduli spaces in the tame nonabelian Hodge correspondence for vector bundles/principal bundles on noncompact curves. As a direct application, the tame nonabelian Hodge correspondence on noncompact curves holds not only for the relevant categories, but also for the moduli spaces., Comment: Some gaps fixed, and the part on the construction of the moduli spaces of filtered Stokes G-local systems will appear in a separate forthcoming project. Comments Welcome!

In this paper, we consider the wild nonabelian Hodge correspondence for principal $G$-bundles on curves, where $G$ is a connected complex reductive group. We establish the correspondence under a ``very good" condition introduced by Boalch, and thus confirm one of his conjectures. We first give a version of Kobayashi--Hitchin correspondence, which induces a one-to-one correspondence between stable meromorphic parahoric Higgs torsors of degree zero (Dolbeault side) and stable meromorphic parahoric connections of degree zero (de Rham side). Then, by introducing a notion of stability condition on filtered Stokes local systems, we prove a one-to-one correspondence between stable meromorphic parahoric connections of degree zero (de Rham side) and stable filtered Stokes $G$-local systems of degree zero (Betti side). When $G={\rm GL}_n(\mathbb{C})$, the main result in this paper reduces to Biquad--Boalch's result., Comment: 27 pages

The nonabelian Hodge correspondence for vector bundles over noncompact curves is adequately described by implementing a weighted filtration on the objects involved. In order to establish a full correspondence between a Dolbeault and a de Rham space for a general complex reductive group $G$, we introduce torsors given by parahoric group schemes in the sense of Bruhat--Tits. Combined with existing results on the Riemann--Hilbert correspondence for logarithmic parahoric connections, this gives a full nonabelian Hodge correspondence from Higgs bundles to fundamental group representations over a noncompact curve beyond the $\text{GL}_n(\mathbb{C})$-case.

Shaping codes are used to generate code sequences in which the symbols obey a prescribed probability distribution. They arise naturally in the context of source coding for noiseless channels with unequal symbol costs. Recently, shaping codes have been proposed to extend the lifetime of flash memory and reduce DNA synthesis time. In this paper, we study a general class of shaping codes for noiseless finite-state channels with cost and i.i.d. sources. We establish a relationship between the code rate and minimum average symbol cost. We then determine the rate that minimizes the average cost per source symbol (total cost). An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. source.

Conjugate partial-symmetric (CPS) tensor is a generalization of Hermitian matrices. For the CPS tensor decomposition some properties are presented. For real CPS tensors in particular, we note the subtle difference from the complex case of the decomposition. In addition to traditional decompositions in the form of the sum of rank-one tensors, we focus on the orthogonal matrix decomposition of CPS tensors, which inherits nice properties from decomposition of matrices. It then induces a procedure that reobtain the CPS decomposable property of CPS tensors. We also discuss the nonnegativity of the quartic real-valued symmetric conjugate form corresponding to fourth-order CPS tensors in real and complex cases, and establish its relationship to different positive semidefiniteness based on different decompositions. Finally, we give some examples to illustrate the applications of presented propositions.

This paper considers the moduli spaces (stacks) of parabolic bundles (parabolic logarithmic flat bundles with given spectrum, parabolic regular Higgs bundles) with rank 2 and degree 1 over $\mathbb{P}^1$ with five marked points. The foliation and stratification structures on these moduli spaces (stacks) are investigated. In particular, we confirm Simpson's conjecture for moduli space of parabolic logarithmic flat bundles with certain non-special weight system., Comment: Minor corrections, comments welcome!

The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). In this paper, we use two Newton-based methods to compute the positive ground state of GPE. The first method comes from the Newton-Noda iteration for saturable nonlinear Schr\"odinger equations proposed by Liu, which can be transferred to GPE naturally. The second method combines the idea of the Bisection method and the idea of Newton method, in which, each subproblem involving block tridiagonal linear systems can be solved easily. We give an explicit convergence and computational complexity analysis for it. Numerical experiments are provided to support the theoretical results.

In this paper, we investigate the geometry of the base complex manifold of an effectively parametrized holomorphic family of stable Higgs bundles over a fixed compact K\"{a}hler manifold. The starting point of our study is Schumacher-Toma/Biswas-Schumacher's curvature formulas for Weil-Petersson-type metrics, in Sect. 2, we give some applications of their formulas on the geometric properties of the base manifold. In Sect. 3, we calculate the curvature on the higher direct image bundle, which recovers Biswas-Schumacher's curvature formula. In Sect. 4, we construct a smooth and strongly pseudo-convex complex Finsler metric for the base manifold, the corresponding holomorphic sectional curvature is calculated explicitly., Comment: Some minor corrections. Comments are welcome! To appear in Communications in Mathematical Physics

In this paper, we generalize the construction of Deligne-Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate such generalized Deligne-Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections with weight-one property and semi-negative energy, and it carries a balanced metric, and its holomorphic tangent bundle (for the case of rank one) is stable. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne-Hitchin twistor space., Comment: An update of the revised version with some gaps fixed in 2022, comments welcome!

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

We consider a special nonconvex quartic minimization problem over a single spherical constraint, which includes the discretized energy functional minimization problem of non-rotating Bose-Einstein condensates (BECs) as one of the important applications. Such a problem is studied by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv), which admits a unique nonnegative eigenvector, and this eigenvector is exactly the global minimizer to the quartic minimization. With these properties, any algorithm converging to the nonnegative stationary point of this optimization problem finds its global minimum, such as the regularized Newton (RN) method. In particular, we obtain the global convergence to global optimum of the inexact alternating direction method of multipliers (ADMM) for this problem. Numerical experiments for applications in non-rotating BEC validate our theories.

Shaping codes are used to encode information for use on channels with cost constraints. Applications include data transmission with a power constraint and, more recently, data storage on flash memories with a constraint on memory cell wear. In the latter application, system requirements often impose a rate constraint. In this paper, we study rate-constrained fixed-to-variable length shaping codes for noiseless, memoryless costly channels and general i.i.d. sources. The analysis relies on the theory of word-valued sources. We establish a relationship between the code expansion factor and minimum average symbol cost. We then determine the expansion factor that minimizes the average cost per source symbol (total cost), corresponding to a conventional optimal source code with cost. An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. source. Shaping codes often incorporate, either explicitly or implicitly, some form of non-equiprobable signaling. We use our results to further explore the connections between shaping codes and codes that map a sequence of i.i.d. source symbols into an output sequence of symbols that are approximately independent and distributed according to a specified target distribution, such as distribution matching (DM) codes. Optimal DM codes are characterized in terms of a new performance measure - generalized expansion factor (GEF) - motivated by the costly channel perspective. The GEF is used to study DM codes that minimize informational divergence and normalized informational divergence.