Exact solutions in general relativity

In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field. Background and definition These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor T^{\alpha\beta}. (A field is described by a Lagrangian, varying with respect to the field should give the field equations and varying with respect to th

Optimal Co-Planning of ESSs and Line Reinforcement Considering the Dispatchability of Active Distribution Networks

Katarina Knezovic, Mario Paolone, Ji Hyun Yi

The paper presents a method for the co-optimization of energy storage systems allocation and line reinforcement in active distribution networks. The objective is to guarantee the capability of an active distribution network to follow a dispatch plan by appropriately coping with the high uncertainties of loads and stochastic renewable generation while ensuring the secure operation of the grid and minimizing the power grid losses. The proposed formulation relies on a modified formulation of the so-called Augmented Relaxed Optimal Power Flow (AR-OPF) method, which considers the exact (i.e., non-approximated) AC-OPF in a convexified way (the AR-OPF is proven to provide the global optimal and exact solution of the OPF problem for radial power grids). To tackle the complexity and computational burden of the proposed planning problem, the Benders decomposition algorithm is used and, in order to enhance the convergence speed of the numerical solution of the proposed problem, the Benders decomposition has been suitably modified to determine the energy storage systems site and size sequentially. To assess the performance of the proposed method, simulations are conducted on a real Swiss distribution network composed of 55 nodes and hosting a large amount of stochastic photovoltaic generation. The sensitivity analysis with respect to the photovoltaic capacity is also carried out to assess the effectiveness of the proposed method.

2022

Spinon-like excitations in spin-1/2 quantum antiferromagnets

Irina Safiulina

Quantum magnetism remains a hot topic in condensed matter physics due to its complexity and possible powerful and significant applications in data storage and memory. To understand how the materials can achieve these goals, one should have a clear idea about the fundamentals behind it. In this thesis, we focus on three examples that can help us deepen the knowledge in many-body effects, which stand to be crucial for quantum magnetism.(1) The well-known \textbf{CuSO

_4\cdot

_2

O} material has already demonstrated the model behaviour as one-dimensional Heisenberg antiferromagnet in zero and high (

H>H_{sat}

) fields. The fully-polarized magnetic ground state is described by linear spin-wave theory with magnons, whereas at zero field, the excitations are pairs of topological excitations called spinons. In an intermediate field, the dynamic properties are even more complicated. The inelastic spectrum cannot be reproduced without considering exotic elementary excitations and bound states such as psinons and Bethe strings. Although Bethe in 1931 provided an exact solution for 1D Heisenberg systems, there is still no quantitative comparison between theory and experiment.(2) The magnetic ground state and hence the dynamic properties of the gemstone mineral green dioptase, \textbf{Cu

_6[

_6

_{18}]\cdot6

_2

O} are under debate: starting from controversial theories and continuing with non-explained experimental observations. Dioptase is a quasi-one-dimensional spin chain with dominant antiferromagnetic interactions. Recent studies claim the classical spin chain behaviour and absence of any quantum fluctuations in the system. In contrast, our experimental findings indicate the presence of continuous excitations above and below T

_N

.(3) Newly synthesized material \textbf{CuSb

_2

_6

} of rosiaite-type structure tends to become a quantum spin-liquid (QSL) candidate since the magnetic cations Cu

^{2+}

are arranged in trigonal layers, and no long-range order is observed down to 2~K. The idea of QSL on the triangular lattice was proposed by P. Anderson in 1973. Since his work, a lot of efforts have been made to explore deeper, both theoretically and experimentally, this state. As for now, there are several requirements needed to be met -- small spin number, absence of long-range order or spin freezing, long-range entanglement, and the associated fractional spin excitations. We aim to establish whether or not CuSb

_2

_6

can be considered as a potential quantum spin-liquid candidate employing different techniques suitable for a powder sample.

EPFL2022

Convergence Analysis For Spectral Approximation To A Scalar Transport Equation With A Random Wave Speed

This paper is concerned with the initial-boundary value problems of scalar transport equations with uncertain transport velocities. It was demonstrated in our earlier works that regularity of the exact solutions in the random spaces (or the parametric spaces) can be determined by the given data. In turn, these regularity results are crucial to convergence analysis for high order numerical methods. In this work, we will prove the spectral convergence of the stochastic Galerkin and collocation methods under some regularity results or assumptions. As our primary goal is to investigate the errors introduced by discretizations in the random space, the errors for solving the corresponding deterministic problems will be neglected.

Vsp Bv2012