Singular Metrics Research Articles

Reinforcement Learning-Based Network Dismantling by Targeting Maximum-Degree Nodes in the Giant Connected Component

Tackling the intricacies of network dismantling in complex systems poses significant challenges. This task has relevance across various practical domains, yet traditional approaches focus primarily on singular metrics, such as the number of nodes in the Giant Connected Component (GCC) or the average pairwise connectivity. In contrast, we propose a unique metric that concurrently targets nodes with the highest degree and reduces the GCC size. Given the NP-hard nature of optimizing this metric, we introduce MaxShot, an innovative end-to-end solution that leverages graph representation learning and reinforcement learning. Through comprehensive evaluations on both synthetic and real-world datasets, our method consistently outperforms leading benchmarks in accuracy and efficiency. These results highlight MaxShot’s potential as a superior approach to effectively addressing the network dismantling problem.

Ding stability and Kähler–Einstein metrics on manifolds with big anticanonical class

Abstract We introduce a notion of uniform Ding stability for a projective manifold with big anticanonical class, and prove that the existence of a unique Kähler–Einstein metric on such a manifold implies uniform Ding stability. The main new techniques are to develop a general theory of Deligne functionals – and corresponding slope formulas – for singular metrics, and to prove a slope formula for the Ding functional in the big setting. This extends work of Berman in the Fano situation, when the anticanonical class is actually ample, and proves one direction of the analogue of the Yau–Tian–Donaldson conjecture in this setting. We also speculate about the relevance of uniform Ding stability and K-stability to moduli in the big setting.

New characterization of plurisubharmonic functions and positivity of direct image sheaves

abstract: We discover a new characterization of plurisubharmonic functions in terms of $L^p$ extension from one point and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics in terms of $L^p$ extensions. As applications, we give a stronger result or new proof of some well-known theorems on the Griffiths positivity of the holomorphic vector bundles and their direct image sheaves associated to certain holomorphic fibrations.

Characterizations of Griffiths positivity, pluriharmonicity and flatness

Deng-Ning-Wang-Zhou showed that a Hermitian holomorphic vector bundle is Griffiths semi-positive if it satisfies the optimal L2-extension condition. As a generalization, we present a quantitative characterization of Griffiths positivity in terms of certain L2-extension conditions. We also show that a R-valued measurable function is pluriharmonic if and only if it satisfies the equality part of the optimal Lp-extension condition. This answers a conjecture of Inayama affirmatively. Moreover, the flatness of a possibly singular Hermitian metric is also equivalent to the equality part of the optimal Lp-extension condition.

Thurston’s asymmetric metric on the space of singular flat metrics with a fixed quadrangulation

Consider a compact surface equipped with a fixed quadrangulation. One may identify each quadrangle on the surface with a Euclidean rectangle to obtain a singular flat metric on the surface with conical singularities. We call such a singular flat metric a rectangular structure. We study a metric on the space of unit area rectangular structures which is analogous to Thurston’s asymmetric metric on the Teichmüller space of a surface of finite type. We prove that the distance between two rectangular structures is equal to the logarithm of the maximum of ratios of edges of these rectangular structures. We give a sufficient condition for a path between two points of the this Teichmüller space to be geodesic and we prove that any two points of this space can be joined by a geodesic. We also prove that this metric is Finsler and give a formula for the infinitesimal weak norm on the tangent space of each point. We identify the space of unit area rectangular structures with a submanifold of a Euclidean space and we show that the subspace topology and the topology induced by the metric we introduced coincide. We show that the space of unit area rectangular structures on a surface with a fixed quadrangulation is in general not complete.

The complex hyperbolic form as a Weil–Petersson form

For the moduli space of the punctured spheres, we find a new equality between two symplectic forms defined on it. Namely, by treating the elements of this moduli space as the singular Euclidean metrics on a sphere, we give an interpretation of the complex hyperbolic form, i.e. the Kähler form of the complex hyperbolic structure on the moduli space, as a kind of Weil–Petersson form.

On the symmetries of singular limits of spacetimes

We consider spacetime metrics with a given (but quite generic) dependence on a dimensionful parameter such that in the 0 and ∞ limits of that parameter the metric becomes singular. We study the isometry groups of the original spacetime metrics and of the singular metrics that arise in the limits and the corresponding symmetries of the motion of p-branes evolving in them, showing how the Killing vectors and their Lie algebras can be found in general. We illustrate our general results with several examples which include limits of anti-de Sitter spacetime in which the holographic screen is one of the singular metrics and of pp-waves.

“There is More to It than Recidivism” – Outcome Scores among Released Prisoners who participated in Prison-Based “Employment World” Programmes

AbstractIn the quest for effective prisoner rehabilitation, while there is existing literature that predominantly emphasizes recidivism rates, it is important to recognize that this singular focus may sometimes overshadow the broader dimensions crucial for successful societal reintegration. This study, therefore, investigates the multifaceted outcomes of an employment training programme offered by the Israel Prison Service. Employing propensity score matching, we compared outcomes between programme participants and non-participants. Our analysis encompassed recidivism rates, employment stability, income levels, tax-paying behaviour and welfare service engagement. While recidivism rates remained largely unaffected, our findings unveiled promising results in other domains. Programme participants exhibited enhanced employment stability, realized higher incomes, demonstrated increased engagement in tax-paying and accessed welfare services more frequently. This study underscores the need for a comprehensive approach to assessing rehabilitation effectiveness, extending beyond the singular metric of recidivism. While acknowledging certain limitations, our research highlights that employment training programmes may not have a direct impact on recidivism but can significantly contribute to labour market reintegration and promote more productive and socially responsible behaviours. It advocates further exploring diverse outcome measures to develop holistic rehabilitation strategies.

Harmonic maps between 2-dimensional simplicial complexes: conformal and singular metrics

Harmonic maps between 2-dimensional simplicial complexes: conformal and singular metrics

Аналіз методології оцінки економічних показників відновлення авіаційної техніки

The article delves into the complexities of evaluating the economic aspects of maintaining and restoring aviation equipment, highlighting the challenges associated with assessing the quality of these processes. It critically examines the use of both singular and multiple performance metrics, emphasizing the need for a comprehensive evaluation system that considers various industry-specific factors. Drawing from methodologies utilized in the chemical and shipbuilding sectors, the study underscores the significance of incorporating diverse parameters, including equipment com-plexity, labor costs, and technical infrastructure, in the evaluation framework. The article investi-gates the methodologies for assessing the quality of technical maintenance and restoration of avia-tion equipment, focusing on the complexities associated with evaluating economic indicators. It dis-cusses the limitations of using singular versus multiple performance metrics in assessing the effec-tiveness of maintenance processes. Furthermore, the article emphasizes the significance of not only quantifying the costs but also analyzing the economic outcomes in relation to the operational efficiency of the aviation fleet. It un-derlines the necessity of establishing a direct correlation between the incurred expenses and the re-sultant operational status of the aviation equipment. In light of the research findings, the article ad-vocates for the implementation of a holistic approach that integrates both quantitative expenditure analysis and qualitative assessments of the fleet's functional state. The study concludes by proposing a sequential research framework for investigating the eco-nomic dimensions of aviation equipment maintenance, emphasizing the interconnected nature of maintenance quality, operational status, and equipment utilization efficiency. It calls for further ex-ploration of these interconnected factors to develop a more nuanced understanding of the economic dynamics within the realm of aviation equipment maintenance.

Weyl's law for singular Riemannian manifolds

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl's asymptotics. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function υ we construct a singular Riemannian structure whose spectrum is discrete and satisfiesN(λ)∼ωn(2π)nλn/2υ(λ). Examples of slowly varying functions are log⁡λ, its iterations logk⁡λ=logk−1⁡log⁡λ, any rational function with positive coefficients of logk⁡λ, and functions with non-logarithmic growth such as exp⁡((log⁡λ)α1…(logk⁡λ)αk) for αi∈(0,1). A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.

Beyond force and injuries: Examining alternative (and important) outcomes for police De-escalation training

PurposeThe evidence on de-escalation training is limited and has focused almost exclusively on use of force as the primary outcome. We evaluate a de-escalation training program in Tempe, Arizona, delivered via a squad-based randomized controlled trial, using methods that tap into alternative outcomes in encounters that did not involve use of force. MethodsWe conducted phone surveys with 282 randomly selected Tempe community members who had a recent encounter with Treatment and Control officers post-training. We also examined body-worn camera footage of 476 randomly selected encounters that occurred pre- (n = 230) and post-training (n = 246) with Treatment and Control officers. ResultsCommunity members reported that trained officers were significantly more likely to use procedural justice and de-escalation; and were more likely to leave them satisfied with their treatment and how their encounter was resolved. Results from BWC footage were less persuasive, though trained officers were more likely to use some de-escalation tactics. ConclusionsUse of force as a singular metric of a training effect is too narrow. A comprehensive assessment of de-escalation training should consider multiple outcomes, including the potential effects in non-force encounters. Random review of BWC footage and community member surveys are two viable methods for achieving this objective.

Harnessing multiple data sources and emerging technologies for comprehensive urban green space evaluation

Urban greening is vital for environmental sustainability and human welfare, but assessing its impact can be challenging due to traditional methodologies' limitations. These methods usually focus on singular metrics, giving a constrained understanding of urban green spaces. This study presents an innovative methodology that overcomes traditional research constraints, providing a comprehensive view of urban green spaces. The approach integrates longitudinal and cross-sectional assessments, remote sensing images, and Google Street View data. It leverages geodata processing and semantic segmentation models for key metrics—green coverage rate and green view index—while classifying various landscape types through an image model. Osaka City has been selected as the study's exemplar to integrate topographic variables and validate this novel methodology. The study results showed that Osaka has conditions such as uneven distribution of green spaces and different quality of green spaces. Targeted suggestions such as focusing on areas with low green visibility and improving the quality of existing green spaces need to be developed accordingly based on the results, paving the way for strengthening the city's urban greening initiatives. Subsequent research must tackle limitations like data quality and model accuracy, spotlighting the crucial role of green spaces in sustainable urban development.

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

We prove that a torsion-free sheaf F \mathcal {F} endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal extension property admits a direct-sum decomposition F ≃ U ⊕ A \mathcal {F}\simeq \mathcal {U}\oplus \mathcal {A} where U \mathcal {U} is a hermitian flat bundle and A \mathcal {A} is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles f ∗ ω X / Y ⊗ m f_* \omega _{X/Y}^{\otimes m} under a surjective morphism f : X → Y f\colon X \to Y of smooth projective varieties with m ≥ 2 m\geq 2 . This extends previous results of Fujita, Catanese–Kawamata, and Iwai.

Trivialization of the bottom line and losing relevance of losses

The purpose of this article is to illustrate the growing limitations of the current methods of calculating earnings, particularly when earnings is a negative number. Earnings, presumably the most important output of a financial reporting system, is not a singular metric. It is obtained by subtracting numerous expense line items from revenues, both of which are calculated after applying a diverse, and often inconsistent, set of accounting conventions. Despite this apparent deficiency, earnings could be informative of recurring profits, if revenues are measured correctly and expenses are traced to revenues. However, both principles are increasingly violated for the cohorts of firms listed in the last 30 years, which now constitute over 80% of the set of listed firms. Revenues of recent cohorts do not capture many events that create recurring cash flows. Their operating expenses are dominated by intangible outlays that are unmatched to current revenues. As a result, newer cohorts’ profits and profit margins, especially when negative, offer little to inform future profits. Given that revenue and expense recognition rules are unlikely to change anytime soon, the current developments raise a question: Should the reporting of the summary measure of earnings be voluntary instead of mandatory?

Transforming singular black holes into regular black holes sourced by nonlinear electrodynamics

We present a way to generate regular black hole spacetimes that are a solution of General Relativity with nonlinear electrodynamics (NLED) as a source, out of a particular metric that inherits the “mass function” of a singular black hole geometry. The singular metric may not be a vacuum, or electro-vacuum, solution of General Relativity.

Singular Griffiths semi-positivity of higher direct images

We show that the higher direct image sheaves of the relative canonical bundle tensorized with a Nakano semi-positive vector bundle admit singular Hermitian metrics with Griffiths semi-positive curvature and the minimal extension property.

On the theory of Bott-Chern secondary characteristic classes with applications to singular metrics

On the theory of Bott-Chern secondary characteristic classes with applications to singular metrics

Bogomolov–Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics

In this article, we obtain the Bogomolov–Sommese type vanishing theorem involving multiplier ideal sheaves for big line bundles. We define a dual Nakano semi-positivity of singular Hermitian metrics with L^2-estimates and prove a vanishing theorem which is a generalization of the Bogomolov–Sommese type vanishing theorem to holomorphic vector bundles.

Nonradial singular metrics of constant Q-curvature

In this paper, we study the existence of solutions to the problem(−Δ)n/2u=enuonRn﹨{0},Λ=∫Rnenudx<∞ for n≥3, which corresponds to finding singular conformal metrics gu:=e2u|dx|2 on Rn﹨{0} with constant Q-curvature equal to one and finite volume Λ. We prove that under suitable conditions, the above equation admits a (nonradial) solution with prescribed polynomial behaviors at zero and infinity. The mass Λ and asymptotic behaviors of solution u can be simultaneously prescribed. This extends results of König-Laurain and Hyder-Mancini-Martinazzi.