Holomorphic Flow Research Articles

Generators of C0-semigroups of weighted composition operators

We prove that in a large class of Banach spaces of analytic functions in the unit disc ⅅ an (unbounded) operator Af = G · f′ + g · f with G, g analytic in ⅅ generates a C0-semigroup of weighted composition operators if and only if it generates a C0-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces X of analytic functions in ⅅ contained in the Bloch space, we prove that no non-trivial holomorphic flow induces a C0-semigroup of weighted composition operators on X. This generalizes previous results in [7] and [1] regarding C0-semigroup of (unweighted) composition operators.

Complex flows, escape to infinity and a question of Rubel

Let \(f\) be a transcendental entire function. It was shown in a previous paper (2017) that the holomorphic flow \(\dot z = f(z)\) always has infinitely many trajectories tending to infinity in finite time. It will be proved here that such trajectories are in a certain sense rare, although an example will be given to show that there can be uncountably many. In contrast, for the classical antiholomorphic flow \(\dot z = \bar f(z)\), such trajectories need not exist at all, although they must if \(f\) belongs to the Eremenko-Lyubich class \(\mathcal{B}\). It is also shown that for transcendental entire \(f\) in \(\mathcal{B}\) there exists a path tending to infinity on which \(f\) and all its derivatives tend to infinity, thus affirming a conjecture of Rubel for this class.

Some Aspects of Positive Kernel Method of Quantization

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups tau _t in text{ Aut }(P,vartheta ) of automorphisms of a G-principal bundle P(G,pi ,M) with a fixed connection form vartheta on its total space P. We show that the generator {hat{F}} of the unitary flow U_t = e^{it {hat{F}}} being the quantization of tau _t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when G=hbox {GL}(N,{mathbb {C}}) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow tau _t^{mathrm{hol}} in text{ Aut }(P,vartheta ). For the above case, we present the integral decompositions of the positive kernels on Ptimes P invariant with respect to the flows tau _t^{mathrm{hol}} in terms of the spectral measure of {hat{F}}. These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on {mathbb {C}} times {mathbb {C}} invariant with respect to the one-parameter groups of translations of complex plane.

Stratified Control Applied to a Three-Phase Unbalanced Low Voltage Distribution Grid in a Local Peer-to-Peer Energy Community

This paper presents control relationships between the low voltage distribution grid and flexibilities in a peer-to-peer local energy community using a stratified control strategy. With the increase in a diverse set of distributed energy resources and the next generation of loads such as electric storage, vehicles and heat pumps, it is paramount to maintain them optimally to guarantee grid security and supply continuity. Local energy communities are being introduced and gaining traction in recent years to drive the local production, distribution, consumption and trading of energy. The control scheme presented in this paper involves a stratified controller with grid and flexibility layers. The grid controller consists of a three-phase unbalanced optimal power flow using the holomorphic embedding load flow method wrapped around a genetic algorithm and various flexibility controllers, using three-phase unbalanced model predictive control. The control scheme generates active and reactive power set-points at points of common couplings where flexibilities are connected. The grid controller’s optimal power flow can introduce additional grid support functionalities to further increase grid stability. Flexibility controllers are recommended to actively track the obtained set-points from the grid controller, to ensure system-level optimization. Blockchain enables this control scheme by providing appropriate data exchange between the layers. This scheme is applied to a real low voltage rural grid in Austria, and the result analysis is presented.

Machine learning to alleviate Hubbard-model sign problems

Lattice Monte Carlo calculations of interacting systems on nonbipartite lattices exhibit an oscillatory imaginary phase known as the phase or sign problem, even at zero chemical potential. One method to alleviate the sign problem is to analytically continue the integration region of the state variables into the complex plane via holomorphic flow equations. For asymptotically large flow times, the state variables approach manifolds of constant imaginary phase known as Lefschetz thimbles. However, flowing such variables and calculating the ensuing Jacobian is a computationally demanding procedure. In this paper, we demonstrate that neural networks can be trained to parametrize suitable manifolds for this class of sign problem and drastically reduce the computational cost for different severely afflicted small volume systems. In particular, we apply our method to the Hubbard model on the triangle and tetrahedron, both of which are nonbipartite. At strong interaction strengths and modest temperatures, the tetrahedron suffers from a severe sign problem that cannot be overcome with standard reweighting techniques, while it quickly yields to our method. We benchmark our results with exact calculations and comment on future directions of this work.

The Action of a Plane Singular Holomorphic Flow on a Non-invariant Branch

AbstractWe study the dynamics of a singular holomorphic vector field at $(\mathbb{C}^{2},0)$. Using the associated flow and its pullback to the blow-up manifold, we provide invariants relating the vector field, a non-invariant analytic branch of curve, and the deformation of this branch by the flow. This leads us to study the conjugacy classes of singular branches under the action of holomorphic flows. In particular, we show that there exists an analytic class that is not complete, meaning that there are two elements of the class that are not analytically conjugated by a local biholomorphism embedded in a one-parameter flow. Our techniques are new and offer an approach dual to the one used classically to study singularities of holomorphic vector fields.

GIS Integrated Automation of a Near Real-Time Power-Flow Service for Electrical Grids

Electrical grid's modernization and the need for sustainable sources of energy lead to changes in physical infrastructure and subsequent operations of the grid, such as penetration of renewables and local power generation from them. There is a shift in paradigm from remotely concentrated generation toward distributed generations that are local to the loads. To meet the objective of sustainability, the local generations must include renewable energy conversions. This requires power-flow management of power variability arising from the spatial and temporal characteristics of the renewables like solar energy. Such management must require least human intervention, and it is achievable through fast and reliable automations. This paper presents geographic information system integrated automation of a noniterative power-flow method known as holomorphic embedded power flow (HEPF), thereby providing local grids the ability to manage themselves. Since HEPF reliably guarantees the solution and reaches it very fast through constructing the solution, it is an ideal candidate for near real-time updates of the electrical states of a power system. The role of the HEPF as a key enabler for the services of a power-flow manager, that performs optimizations for a cluster of buildings, is then illustrated.

Unification of the complex Langevin method and the Lefschetzthimble method

Recently there has been remarkable progress in solving the sign problem, which occurs in investigating statistical systems with a complex weight. The two promising methods, the complex Langevin method and the Lefschetz thimble method, share the idea of complexifying the dynamical variables, but their relationship has not been clear. Here we propose a unified formulation, in which the sign problem is taken care of by both the Langevin dynamics and the holomorphic gradient flow. We apply our formulation to a simple model in three different ways and show that one of them interpolates the two methods by changing the flow time.

Deep learning beyond Lefschetz thimbles

The generalized thimble method to treat field theories with sign problems requires repeatedly solving the computationally-expensive holomorphic flow equations. We present a machine learning technique to bypass this problem. The central idea is to obtain a few field configurations via the flow equations to train a feed-forward neural network. The trained network defines a new manifold of integration which reduces the sign problem and can be rapidly sampled. We present results for the $1+1$ dimensional Thirring model with Wilson fermions on sizable lattices. In addition to the gain in speed, the parameterization of the integration manifold we use avoids the "trapping" of Monte Carlo chains which plagues large-flow calculations, a considerable shortcoming of the previous attempts.

Slit Holomorphic Stochastic Flows and Gaussian Free Field

It was realized recently that the chordal, radial and dipolar Schramm–Lowner evolution (SLEs) are special cases of a general slit holomorphic stochastic flow. We characterize those slit holomorphic stochastic flows which generate level lines of the Gaussian free field. In particular, we describe the modifications of the Gaussian free field (GFF) corresponding to the chordal and dipolar SLE with drifts. Finally, we develop a version of conformal field theory based on the background charge and Dirichlet boundary condition modifications of GFF and present martingale-observables for these types of SLEs.

A Splitting Method for Diffeomorphism Optimization Problem Using Beltrami Coefficients

Finding a meaningful 1---1 correspondence between different data, such as images or surface data, has important applications in various fields. It involves the optimization of certain energy functionals over the space of all diffeomorphisms. This type of optimization problems (called the diffeomorphism optimization problems, DOPs) is especially challenging, since the bijectivity of the mapping has to be ensured. Recently, a method, called the Beltrami holomorphic flow (BHF), has been proposed to solve the DOP using quasi-conformal theories (Lui et al. in J Sci Comput 50(3):557---585, 2012). The optimization problem is formulated over the space of Beltrami coefficients (BCs), instead of the space of all diffeomorphisms. BHF iteratively finds a sequence of BCs associated with a sequence of diffeomorphisms, using the gradient descent method, to minimize the energy functional. The use of BCs effectively controls the smoothness and bijectivity of the mapping, and hence makes it easier to handle the constrained optimization problem. However, the algorithm is computationally expensive. In this paper, we propose an efficient splitting algorithm, based on the classical alternating direction method of multiplier (ADMM), to solve the DOP. The basic idea is to split the energy functional into two energy terms: one involves the BC whereas the other involves the quasi-conformal map. Alternating minimization scheme can then be applied to minimize the energy functional. The proposed method significantly speeds up the previous BHF approach. It also extends the previous BHF algorithm to Riemann surfaces of arbitrary topologies, such as multiply-connected shapes. Experiments have been carried out on synthetic together with real surface data, which demonstrate the efficiency and efficacy of the proposed algorithm to solve the DOP.

Computing Extremal Teichmüller Map of Multiply-Connected Domains Via Beltrami Holomorphic Flow

A numerical method for computing the extremal Teichmuller map between multiply-connected domains is presented. Given two multiply-connected domains, there exists a unique Teichmuller map (T-Map) between them minimizing the conformality distortion. The extremal T-Map can be considered as the `most conformal' map between multiply-connected domains. In this paper, we propose an iterative algorithm to compute the extremal T-Map using the Beltrami holomorphic flow (BHF). The BHF procedure iteratively adjusts the initial map based on a sequence of Beltrami coefficients, which are complex-valued functions defined on the source domain. It produces a sequence of quasi-conformal maps, which converges to the T-Map minimizing the conformality distortion. We test our method on synthetic data together with real human face data. Results show that our algorithm computes the extremal T-Map between two multiply-connected domains of the same topology accurately and efficiently.

Erratum to: Optimization of Surface Registrations Using Beltrami Holomorphic Flow

Erratum to: Optimization of Surface Registrations Using Beltrami Holomorphic Flow

Optimization of Surface Registrations Using Beltrami Holomorphic Flow

In shape analysis, finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings is of great importance. Such process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making exhaustive search for the best mapping challenging. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs), which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map can be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

Making big steps in trajectories

We consider the solution of initial value problems within the context of hybrid systems and emphasise the use of high precision approximations (in software for exact real arithmetic). We propose a novel algorithm for the computation of trajectories up to the area where discontinuous jumps appear, applicable for holomorphic flow functions. Examples with a prototypical implementation illustrate that the algorithm might provide results with higher precision than well-known ODE solvers at a similar computation time.

Locally conformal Kähler manifolds with potential

A locally conformally Kahler (LCK) manifold M is one which is covered by a Kahler manifold \({\widetilde M}\) with the deck transformation group acting conformally on \({\widetilde M}\). If M admits a holomorphic flow, acting on \({\widetilde M}\) conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M ≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky (Math Ann 332:121–143, 2005).

Brain Surface Conformal Parameterization with Holomorphic Flow Method and Its Application to HIV/AIDS

Brain Surface Conformal Parameterization with Holomorphic Flow Method and Its Application to HIV/AIDS

Normal almost contact structures and non-K\"ahler compact complex manifolds

We construct some families of complex structures on compact manifolds by means of normal almost contact structures (nacs) so that each complex manifold in the family has a non-singular holomorphic flow. These families include as particular cases the Hopf and Calabi-Eckmann manifolds and the complex structures on the product of two normal almost contact manifolds constructed by Morimoto. We prove that every compact Kahler manifold admitting a non-vanishing holomorphic vector field belongs to one of these families, and is a complexification of a normal almost contact manifold. Finally, we show that if a complex manifold obtained by our constructions is Kahlerian, then the Euler class of the nacs (a cohomological invariant associated to the structure) is zero. Under extra hypotheses, we give necessary and sufficient conditions for the complex manifolds so obtained to be Kahlerian.

Dicritical holomorphic flows on Stein manifolds

We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical singularity of the form \(\sum^{n}_{j=1}\lambda_{j}z_{j}\frac{\partial}{\partial z_{j}}+\ldots, \lambda_{j}\in \mathbb{Q}_{+},\forall j \in \{1,\ldots,n\}\) on a Stein manifold \(M^n, n \geq 2\) with \({\mathop{H}\limits^{\vee}}{^{2}}(M^{n}, {{{\mathbb{Z}}}})=0\), is globally analytically linearizable; in particular M is biholomorphic to \({\mathbb{C}}^{n}\). A complete stability result for periodic orbits is also obtained.

Corrigendum to ``The holomorphic flow of the Riemann zeta function''

Theorem 4.5 of [2], describing the topological type of the zeros of the flow ṡ = ζ(s) at reflected points off the critical line, claiming they were the same, contains an error. We gratefully acknowledge Professor Cevat Gokcek for pointing out the error to us. In some circumstances the types will never be the same. Indeed, it appears to be unlikely that a fixed formal relationship will exist between the types of flow for ζ(s) at reflected points off the critical line. Given that such points are not at hand (and expected not to be found), examples are currently impossible to come by. The revised statement and proof are given below. Some proof reading errors picked up by a helpful referee are also noted. Let Φ(s) = π−s/2Γ( s 2 ),