Singular Locus Research Articles

Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

Algorithmic local monomialization of a binomial: A comparison of different approaches

We investigate different approaches to transform a given binomial into a monomial via blowing up appropriate centers. In particular, we develop explicit implementations in Singular, which allow to make a comparison on the basis of numerous examples. We focus on a local variant, where centers are not required to be chosen globally. Moreover, we do not necessarily demand that centers are contained in the singular locus. Despite these restrictions, the techniques are connected to the computation of [Formula: see text]-adic integral whose data is given by finitely many binomials.

Data loci in algebraic optimization

We consider parametric optimization problems from an algebraic viewpoint. The idea is to find all of the critical points of an objective function thereby determining a global optimum. For generic parameters (data) in the objective function the number of critical points remains constant. This number is known as the algebraic degree of an optimization problem. In this article, we go further by considering the inverse problem of finding parameters of the objective function so it gives rise to critical points exhibiting a special structure. For example if the critical point is in the singular locus, has some symmetry, or satisfies some other algebraic property. Our main result is a theorem describing such parameters.

Local singularity and S–A methods for analyzing ore-producing anomalies in the Jianbiannongchang area of Heilongjiang, China

Local singularity and S–A methods for analyzing ore-producing anomalies in the Jianbiannongchang area of Heilongjiang, China

International Fascism and the Allure of the ‘Third Way’ in Interwar Greece

Abstract The rise and victory of Italian Fascism in the first half of the 1920s passed Greece by. Yet soon afterwards the international experience of ‘fascism’ found more receptive audiences within the prodigious dissident ‘third spaces’ where more and more mainstream Greek political actors chose to operate in the interwar period. This article explores the dynamics of the ideological and political formation of ‘third ways’ in interwar Greece, paying attention to the interplay between international stimuli and local contextual singularities. In these thirding spaces ‘fascism’ was understood and operationalised in very different, subjective, and ever-shifting ways by each of these actors. It was regarded mostly as a potential component of diverse thirding processes/solutions and rarely as the desired outcome thereof. This explains why fascism came to inform a range of very different thirding projects in interwar Greece—from pursuing rupture and renewal to aspiring to status quo-affirmation; from liberal to conservative to authoritarian visions; from searching for a short-term ‘remedy’ to envisioning a long-term radical transformation.

0-form, 1-form, and 2-group symmetries via cutting and gluing of orbifolds

Orbifold singularities of M-theory constitute the building blocks of a broad class of supersymmetric quantum field theories (SQFTs). In this paper we show how the local data of these geometries determines global data on the resulting higher symmetries of these systems. In particular, via a process of cutting and gluing, we show how local orbifold singularities encode the 0-form, 1-form and 2-group symmetries of the resulting SQFTs. Geometrically, this is obtained from the possible singularities which extend to the boundary of the non-compact geometry. The resulting category of boundary conditions then captures these symmetries, and is equivalently specified by the orbifold homology of the boundary geometry. We illustrate these general points in the context of a number of examples, including 5D superconformal field theories engineered via orbifold singularities, 5D gauge theories engineered via singular elliptically fibered Calabi-Yau threefolds, as well as 4D SQCD-like theories engineered via M-theory on non-compact $G_2$ spaces.

Algebraic foliations and derived geometry: the Riemann–Hilbert correspondence

This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations defined in terms of differential ideals in the algebra of forms. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension ge 2, its analytification is a locally integrable singular foliation on the associated complex manifold X^h. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.

On a question of Moshe Roitman and Euler class of stably free module

Let A be a ring of dimension d containing an infinite field k, T1,…,Tr be variables over A and P be a projective A[T1,…,Tr]-module of rank n. Assume one of the following conditions holds.(1)2n≥d+3 and P is extended from A.(2)2n≥d+2, A is an affine F‾p-algebra and P is extended from A.(3)2n≥d+3 and singular locus of Spec(A) is a closed set V(J) with ht J≥d−n+2. Assume Um(Pf)≠∅ for some monic polynomial f(Tr)∈A[T1,…,Tr]. Then Um(P)≠∅ (see 6.1).

Moduli of G-covers of curves: geometry and singularities

In a recent paper Chiodo and Farkas described the singular locus and the locus of non-canonical singularities of the moduli space of level curves. In this work we generalize their results to the moduli space $\overline{\mathcal R}_{g,G}$ of curves with a $G$-cover for any finite group $G$. We show that non-canonical singularities are of two types: $T$-curves, that is singularities lifted from the moduli space $\overline{\mathcal M}_g$ of stable curves, and $J$-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case $G=S_3$, the $J$-locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of $\overline{\mathcal R}_{g,S_3}$.

A health promotion intervention for vulnerable schools (BeE-school): a cluster-randomized trial

Abstract Background Noncommunicable diseases (NCDs) share key determinants like unhealthy diet, unhealthy 24h- movement behaviour (sleep, sedentary behaviour and physical activity), leading to other risks including overweight, obesity and raised blood pressure. Although the manifest of the NCDs in childhood is rare, risk behaviours, obesity, and raised blood pressure that accelerates their development begin during childhood. The current study aims to analyze the effectiveness of the intervention program based on the promotion of health literacy and lifestyles, on children’s health literacy, lifestyles (e.g. dietary intake, 24hmovement behaviour) and overweight and obesity. Methods 478 children (6 schools) aged 6-12years old will participate in this cluster-randomized trial, having schools as the unit of randomization, assigned into intervention (239-3schools) and the control arm (239-3schools). This project is currently performing social listening (online and offline) and stakeholders’ involvement. Data collection includes sociodemographics, health literacy and infodemic resilience, dietary intake and children’s 24-h movement behaviour (e.g. accelerometry), anthropometry (e.g. weight, height and waist circumference) and blood pressure. It will occur at baseline and after the intervention (follow-up, 6 months after the beginning of the intervention). Results Expected outputs and outcomes include the 1-creation of a model for characterizing NCDs and health topics based on artificial intelligence techniques (e.g. deep learning, social network analysis methods).2- improved health literacy and infodemic resilience of children, families and teachers.3- enhanced children’s lifestyles. 4- reduce NCDs’ physical risk factors (e.g. overweight, raised blood pressure). Conclusions A feasible intervention program for school-aged children with vulnerabilities enhances tailored policies about health promotion and NCDs’ prevention, respecting the contextś singularities. Key messages • Health promotion and NCDs prevention are crucial for the well-being of our societies. • Feasible intervention programs advocates for evidence-based policies that respect local singularities.

Bivariant Class of Degree One

Let f:Xrightarrow Y be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class theta in H^0(X{mathop {rightarrow }limits ^{f}}Y)cong Hom_{D^{b}_{c}(Y)}(Rf_*{mathbb {A}}_X, {mathbb {A}}_Y) (here {mathbb {A}} is a Noetherian commutative ring with identity, and {mathbb {A}}_X and {mathbb {A}}_Y denote the constant sheaves). Let theta _0:H^0(X)rightarrow H^0(Y) be the induced Gysin morphism. We say that theta has degree one if theta _0(1_X)= 1_Yin H^0(Y). This is equivalent to say that theta is a section of the pull-back f^*: {mathbb {A}}_Yrightarrow Rf_*{mathbb {A}}_X, i.e. theta circ f^*={text {id}}_{{mathbb {A}}_Y}, and it is also equivalent to say that {mathbb {A}}_Y is a direct summand of Rf_*{mathbb {A}}_X. We investigate the consequences of the existence of a bivariant class of degree one. We prove explicit formulas relating the (co)homology of X and Y, which extend the classic formulas of the blowing-up. These formulas are compatible with the duality morphism. Using which, we prove that the existence of a bivariant class theta of degree one for a resolution of singularities, is equivalent to require that Y is an {mathbb {A}}-homology manifold. In this case theta is unique, and the Betti numbers of the singular locus {text {Sing}}(Y) of Y are related with the ones of f^{-1}({text {Sing}}(Y)).

Singularity Computation for Rational Parametric Surfaces Using Moving Planes

Singularity computation is a fundamental problem in Computer Graphics and Computer Aided Geometric Design, since it is closely related to topology determination, intersection, mesh generation, rendering, simulation, and modeling of curves and surfaces. In this article, we present an efficient and robust algorithm for computing all the singularities (including their orders) of rational parametric surfaces using the technique of moving planes. The main approach is first to construct a representation matrix whose columns correspond to moving planes following the parametric surface. Then, by substituting the parametric equation of the rational surface into this representation matrix, one can extract the singularity information from the corresponding matrix and return all the singular loci including self-intersection curves, cusp curves, and isolated singular points of the rational surface, together with the order of each singular locus. We present some examples to compare our algorithm with state-of-the-art methods from different perspectives including robustness, efficiency, order computation, and numerical stability, and the experimental results show that our method outperforms existing methods in all these aspects. Furthermore, applications of our algorithm in surface rendering, mesh generation and surface/surface intersections are provided to demonstrate that correctly computing the self-intersection curves of a surface is essential to generate high quality results for these applications.

Singularity Properties of Timelike Sweeping Surface in Minkowski 3-Space

In this paper, we give the parametric equation of the Bishop frame for a timelike sweeping surface with a unit speed timelike curve in Minkowski 3-space. We introduce a new geometric invariant to explain the geometric properties and local singularities of this timelike surface. We derive the sufficient and necessary conditions for this timelike surface to be a timelike developable ruled surface. Afterwards, we take advantage of singularity theory to give the classification of singularities of this timelike developable surface. Furthermore, we give some representative examples to show the applications of the theoretical results.

Specializations of partial differential equations for Feynman integrals

Starting from the Mellin–Barnes integral representation of a Feynman integral depending on a set of kinematic variables zi, we derive a system of partial differential equations w.r.t. new variables xj, which parameterize the differentiable constraints zi=yi(xj). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

Inclusion of Biological and Chemical Self-Heating Processes in Compost Piles Model: A Semenov Formulation

In this study, a uniformly distributed mathematical model for the self-heating process within compost piles is formulated and investigated. It consists of mass balance equations for oxygen and energy as well as the heat-generation processes due to both biological and chemical activities. This model is an extension of the study of Luangwilai et al. [2013] which consists of only biological activity. The singularity and degenerate Hopf bifurcation theories are used to determine the loci of different singularities: the isola, cusp, double-limit points and boundary limit set as well as the double-Hopf, generalized Hopf (Bautin) and Bogdanov–Takens bifurcations. In this investigation it is found that these loci separate the secondary parameter plane into twenty-two regions of different steady-state solution behaviors, whereas the model in the earlier study reported only eight regions. With more topological detail in the parameter space, a clearer understanding of the thermal behavior of a compost pile can be obtained, thus assisting compost operators in controlling the temperature within the pile more effectively: for example, achieving desirable elevated temperature range for ideal composting conditions via periodic solutions and S-shaped solution branch, or alternatively, understanding conditions, if not monitored carefully, that can also increase the likelihood of spontaneous ignition within the pile.

Vanishing cohomology and Betti bounds for complex projective hypersurfaces

We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tib\u{a}r in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.

Anisotropy of geochemical distributions characterized by irregular window-based local singularity analysis

Supporting the discovery of new resources to maintain the continuous development of human society is a long-term goal in mineral exploration. Geochemical data as an important geo-information carrier played significant roles in geographical information system (GIS)-based mineral prospectivity. Among the various analytical methodologies, local singularity analysis has been broadly discussed due to its high efficiency in the characterization of non-linear mineralization. Focusing on this topic, the current study first reviews local singularity analysis from its original proposition to recent progresses. Meanwhile, limitations in square and rectangular window-based singularity index α estimation algorithms are discussed. Based on this, analytical window parameters in U -statistics and algorithms to quantify the power-law relationship between the physical quantity and the measuring scale in local singularity are integrated. At each scale, optical elliptical windows along all azimuth angles are determined and further merged, which defines an irregular analytical window. Irregular windows at all scales are further applied to local singularity analysis that can consequently propose an irregular window-based singularity algorithm. Practical application in the Duolong mineral district, northern Tibet, China demonstrates well the comparable efficiency of weak anomaly identification with that of the original algorithm. More importantly, multiscale irregular windows can supplement delicate patterns to characterize anisotropic natures associated with mineralization. Thematic collection: This article is part of the Applications of Innovations in Geochemical Data Analysis collection available at: https://www.lyellcollection.org/cc/applications-of-innovations-in-geochemical-data-analysis

Landau discriminants

Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics. The algorithms used in this work are implemented in the open-source Julia package Landau.jl available at https://mathrepo.mis.mpg.de/Landau/.

Strength-based concurrent shape and fiber path optimization of continuous fiber composites

This paper introduces a novel strength-based structural optimization technique capable of concurrent design of shape and fiber path in continuous fiber-reinforced composites. To this end, a higher order function, i.e., the level-set function, is employed to update both shape and fiber placement. The shape evolution relies on a shape sensitivity analysis based on the Tsai–Wu failure criterion. The fiber path evolution depends on the level-set function determined from the shape boundary. A scheme combining the classical level-set method using fixed mesh and a level-set-based mesh evolution method is developed to ensure the efficiency and accuracy of the process. A revision step over fiber orientations along the ridge of the level-set function is introduced to alleviate the local failure index singularity. Manufacturability of the design is further ensured by incorporating a thickness control term, avoiding the appearance of ultra-thin members in the optimized design. The developed scheme is applied to three benchmark examples, showing that the failure index concentration is largely reduced compared with the initial shape, the compliance-based optimized design, and the design with optimized shape but unidirectional fibers.

A compact non‐formal closed G2 manifold with b1=1$b_1=1$

AbstractWe construct a compact manifold with a closed G2 structure not admitting any torsion‐free G2 structure, which is non‐formal and has first Betti number . We develop a method of resolution for orbifolds that arise as a quotient with M a closed G2 manifold under the assumption that the singular locus carries a nowhere‐vanishing closed 1‐form.