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In this review, we begin with a historical survey of some singular solutions in the theory of gravitation, as well as a very brief discussion of how black holes could physically form. Some possible scenarios which could perhaps eliminate these singularities are then reviewed and discussed. Due to the vastness of the field, its coverage is not exhaustive; instead, the concentration is on a small subset of topics such as possible quantum gravity effects, non-commutative geometry, and gravastars. A simple singularity theorem is also reviewed. Although parts of the manuscript assume some familiarity with relativistic gravitation or differential geometry, the aim is for the broad picture to be accessible to non-specialists of other physical sciences and mathematics.

We study the existence and properties of wormhole throats in modified f (R) gravity theory. Specifically, we concentrate on the cases where the lapse is not necessarily constant, and hence are not limited to the zero tidal force scenarios. In the class of theories whose actions are generated by Lagrangians of the form f (R) = ∑ αn Rn we find parameters which allow for the existence of energy condition respecting throats, which do not exist in Einstein gravity. We also consider the effect of the modified action on the anisotropy of the models, and find that modified gravity can minimize the amount of anisotropy required to support the existence of a throat. In both these respects, the sector containing theories with positive n is more promising than the negative n sector in comparison to Einstein gravity alone, with large n being most favorable.

Several states and countries have adopted targets for deep reductions in greenhouse gas emissions by 2050, but there has been little physically realistic modeling of the energy and economic transformations required. We analyzed the infrastructure and technology path required to meet California's goal of an 80% reduction below 1990 levels, using detailed modeling of infrastructure stocks, resource constraints, and electricity system operability. We found that technically feasible levels of energy efficiency and decarbonized energy supply alone are not sufficient; widespread electrification of transportation and other sectors is required. Decarbonized electricity would become the dominant form of energy supply, posing challenges and opportunities for economic growth and climate policy. This transformation demands technologies that are not yet commercialized, as well as coordination of investment, technology development, and infrastructure deployment.

The director field adopted by a confined liquid crystal is controlled by a balance between the externally imposed interactions and the liquid’s internal orientational elasticity. While the latter is usually considered to resist all deformations, liquid crystals actually have an intrinsic propensity to adopt saddle-splay arrangements, characterised by the elastic constant K 24 . In most realisations, dominant surface anchoring treatments suppress such deformations, rendering K 24 immeasurable. Here we identify regimes where more subtle, patterned surfaces enable saddle-splay effects to be both observed and exploited. Utilising theory and continuum calculations, we determine experimental regimes where generic, achiral liquid crystals exhibit spontaneously broken surface symmetries. These provide a new route to measuring K 24 . We further demonstrate a multistable device in which weak, but directional, fields switch between saddle-splay-motivated, spontaneously-polar surface states. Generalising beyond simple confinement, our highly scalable approach offers exciting opportunities for low-field, fast-switching optoelectronic devices which go beyond current technologies. Liquid crystals are used for devices with increasingly complex geometries which makes it important to understand the influence of deformation. Xia et al. measure the poorly investigated saddle-splay elastic constant by means of surface patterning and simulation for programming emergent symmetries.

This paper develops a linear regression model for using actively traded NYMEX natural gas futures as a cross-hedge against electricity spot-price risk in the Pacific Northwest and for pricing the forward contracts in the presence of temperature and hydro risks. Our approach comports with reality and provides power purchasers with an effective instrument through which they can hedge their electricity bets through natural gas futures. It also demonstrates the sharp month-to-month variations in the natural gas futures' optimal hedge ratios and hedge effectiveness. Finally, it finds significant risk premiums in the Pacific Northwest forward prices, supporting the hypothesis that forward-contract buyers are relatively more risk-averse than sellers.

This is a brief review article of a seminar given at the International Conference on Gravitation, Astrophysics and Cosmology 2024 (ICGAC-2024), Mathura, India. We begin with a historical survey of some singular solutions in the theory of gravitation, as well as a very brief discussion of how black holes could physically form. Some possible scenarios which could perhaps eliminate these singularities are then reviewed and discussed. Due to the vastness of the field the coverage is not exhaustive, but instead the concentration is on a small subset of topics such as possible quantum gravity effects, non-commutative geometry, and gravastars. A simple singularity theorem is also presented in the appendix. Although parts of the manuscript assume some familiarity with relativistic gravitation or differential geometry, the aim is for the broad picture to be accessible to non-specialists of other physical sciences and mathematics.

Due to their anisotropic shape, liquid crystal molecules exert elastic forces that attempt to align themselves with their neighbors. Perfect alignment may be attainable for liquid crystals in a trivial geometry with harmonious boundary conditions, but antagonistic anchoring conditions or complex geometries introduce geometric frustration. When a system is geometrically frustrated, the preferred ordering of the liquid crystals is incompatible with the space in which the molecules are confined. This frustration results in distortions of the liquid crystal elastic field, and in soft systems, deformations of the enclosing boundary. On surfaces patterned with chemical and/or topographical features, liquid crystals are often unable to satisfy their own alignment preference while meeting boundary conditions imposed by the pattern. These patterns can allow for control over liquid crystal orientation, as well as defect occurrence and location, all of which are important considerations in liquid crystal display and sensing applications. When the container housing the liquid crystal is non-rigid, the complexity of the problem increases. These systems exhibit geometric frustration, but allow shape deformation to alleviate some or all of this frustration at the expense of increasing surface tension. Understanding this interplay between shape and order provides insight into biological systems that exhibit shape deformation and guides the design of certain self-assembled systems. In this thesis, we study shape deformation problems in the context of liquid crystal systems. We begin by examining patterned surfaces of flat circles and micro-posts in square arrays to develop an understanding of liquid crystal behavior in complex geometries. Next we develop a locally refined moving mesh finite element simulation model capable of solving shape deformation problems with evolving domains. This model is used to simulate nematic tactoids and the effect of isotropic inclusions in these tactoid systems. Then we determine the properties of nanoparticle shells that self assemble in a quenched nematic background, and look at the effect of shell morphology on their alignment angle. Finally, we consider thin films of smectic liquid crystals in an attempt to see the layer structure given a variety of anchoring conditions.

In this paper we study gravitationally bound compact objects sourced by a string theory inspired Born-Infeld scalar field. Unlike many of their canonical scalar field counterparts, these ``boson stars'' do not have to extend out to infinity and may generate compact bodies. We analyze in detail both the junction conditions at the surface as well as the boundary conditions at the center which are required in order to have a smooth structure throughout the object and into the exterior vacuum region. These junction conditions, although involved, turn out to be relatively easy to satisfy. Analysis reveals that these compact objects have a richer structure than the canonical boson stars and some of these properties turn out to be physically peculiar: There are several branches of solutions depending on how the junction conditions are realized. Further analysis illustrates that in practice the junction conditions tend to require interior geometries reminiscent of ``bag of gold'' spacetimes, and also hide the star behind an event horizon in its exterior. The surface compactness of such objects, defined here as the ratio \\(2M/r\\), can be made arbitrarily close to unity indicating the absence of a Buchdahl bound. Some comments on the stability of these objects is provided to find possible stable and unstable regimes. However, we argue that even in the possibly stable regime the event horizon in the vacuum region shielding the object is potentially unstable, and would cut off the star from the rest of the universe.

Within the paradigm of non-perturbative Einstein gravity we study continuous curvature manifolds which possess de Sitter interiors and Kerr exteriors. These manifolds could represent the spacetime of rotating gravastars or other similar black hole mimickers. The scheme presented here allows for a \\(C^n\\) metric transition from the exactly de Sitter interior to the exactly Kerr exterior, with \\(n\\) arbitrarily large. Generic properties that such models must possess are discussed, such as the changing of the topology of the ergosphere from \\(S^2\\) to \\(S^1 S^1\\). It is shown how in the outer layers of the transition region (the ``atmosphere'' as it is often called in astrophysics) the dominant/weak and strong energy conditions can be respected. However, much like in the case of its static spherically symmetric gravastar counterpart, there must be some assumptions imposed in the atmosphere for the energy conditions to hold. These assumptions turn out to not be severe. The class of manifolds presented here are expected to possess all the salient features of the fully generic case. Strictly speaking, a number of the results are also applicable to the locally anti-de Sitter core scenario, although we focus on the case of a positive cosmological constant.

Within the paradigm of non-perturbative Einstein gravity we study continuous curvature manifolds which possess de Sitter interiors and Kerr exteriors. These manifolds could represent the spacetime of rotating gravastars or other similar black hole mimickers. The scheme presented here allows for a \\(C^n\\) metric transition from the exactly de Sitter interior to the exactly Kerr exterior, with \\(n\\) arbitrarily large. Generic properties that such models must possess are discussed, such as the changing of the topology of the ergosphere from \\(S^2\\) to \\(S^1 S^1\\). It is shown how in the outer layers of the transition region (the ``atmosphere'' as it is often called in astrophysics) the dominant/weak and strong energy conditions can be respected. However, much like in the case of its static spherically symmetric gravastar counterpart, there must be some assumptions imposed in the atmosphere for the energy conditions to hold. These assumptions turn out to not be severe. The class of manifolds presented here are expected to possess all the salient features of the fully generic case. Strictly speaking, a number of the results are also applicable to the locally anti-de Sitter core scenario, although we focus on the case of a positive cosmological constant.