Singular Locus Research Articles

Sweeping Surfaces according to Type-3 Bishop Frames in Euclidean 3-Space

The aim of this work is to investigate sweeping surfaces and their local singularities due to type-3 Bishop frames in Euclidean 3-space, E3. A sweeping surface a is surface that is designed from a section curve positioned along a path, which acts as the vertebral column or spine curve, and it has symmetrical characteristics. In this work, we have specified a sweeping surface and have examined its geometry and singularity. Thereafter, we deduced the circumstances required for this surface to be a developable surface. In great detail, we concentrated on the fundamental discussion on whether the resulting developable surface is a cylindrical, cone or tangent surface. Meanwhile, examples are detailed to explain the applications of the notional outcomes.

The collective intelligence of evolution and development

Collective intelligence and individual intelligence are usually considered to be fundamentally different. Individual intelligence is uncontroversial. It occurs in organisms with special neural machinery, evolved by natural selection to enable cognitive and learning functions that serve the fitness benefit of the organism, and then trained through lifetime experience to maximise individual rewards. Whilst the mechanisms of individual intelligence are not fully understood, good models exist for many aspects of individual cognition and learning. Collective intelligence, in contrast, is a much more ambiguous idea. What exactly constitutes collective intelligence is often vague, and the mechanisms that might enable it are frequently domain-specific. These cannot be mechanisms selected specifically for the purpose of collective intelligence because collectives are not (except in special circumstances) evolutionary units, and it is not clear that collectives can learn the way individual intelligences do since they are not a singular locus of rewards and benefits. Here, we use examples from evolution and developmental morphogenesis to argue that these apparent distinctions are not as categorical as they appear. Breaking down such distinctions enables us to borrow from and expand existing models of individual cognition and learning as a framework for collective intelligence, in particular connectionist models familiar in the context of neural networks. We discuss how specific features of these models inform the necessary and sufficient conditions for collective intelligence, and identify current knowledge gaps as opportunities for future research.

A note on the smooth blowups of in torus-invariant subvarieties

AbstractThis papers classifies toric Fano threefolds with singular locus $\{ \frac {1}{k}(1,1,1) \}$ for $k \in \mathbb {Z}_{\geq 1}$ building on the work of Batyrev (1981, Nauk SSSR Ser. Mat. 45, 704–717) and Watanabe–Watanabe (1982, Tokyo J. Math. 5, 37–48). This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore, we identify birational relationships between entries of the classification. For a fixed value $k \geq 4$ , there are exactly two such toric Fano threefolds linked by a blowup in a torus-invariant line.

The Development and Validation of Qualitative Value Indicators of Region-Based Community Dance for Cultural Urban Regeneration

The purpose of this study was to develop and validate qualitative value indicators of region-based community dance for cultural urban regeneration. To this end, the conceptual structure of local-based community dance was explored and evaluation indicators were developed by deriving questions related to this. The specific research procedures and results are as follows: First, a literature review and Delphi survey were conducted to obtain factors and questions regarding qualitative value indicators. Consequently, four factors of local singularity, convergent creativity, sustainability, and community culture were obtained with 43 questions. Next, a preliminary survey targeting 122 people who have participated in representative local dance festivals (the ‘Wonju Dynamic Dancing Carnival’, ‘Cheonan World Dance Festival’, and ‘Andong Maskdance Festival’) in Korea was conducted to determine the relationship between the factors and questions. Multi-dimensional value indicators with 30 questions in four factors were obtained from an exploratory factor analysis using SPSS version 25.0. Then, the main survey, targeting 341 people who have participated in three representative local dance festivals in Korea, was conducted to verify the validity of the value indicators. Finally, evaluation indicators of 26 questions (seven questions for regional uniqueness, five questions for convergent creativity, seven questions for sustainability, and seven questions for community culture) were developed from a confirmatory factor analysis using AMOS version 24.0. Additionally, the contents evaluated by each factor are as follows: Firstly, ‘Local Singularity’ is a factor in whether the evaluation target sufficiently contains the unique characteristics of the region. Secondly, ‘Convergent Creativity’ is a factor in whether the evaluation target converges the indigenous culture (traditional culture) and acceptance culture (modern culture) of the region well. Thirdly, ‘Sustainability’ is a factor in whether the evaluation target contributes to the sustainable development of the region. Fourthly, ‘Community Culture’ is a factor in whether the evaluation target contributes to the formation of community culture in the region. This study is significant in that it emphasises the potential possibility of community dance, which can contribute to cultural urban regeneration based on the perspectives of cultural democracy. Therefore, the results of this study are expected to be extended and applied to various types of community dance in the future. Moreover, it is further forecasted that this study will suggest the meaning of the qualitative valuation of art and culture for a variety of subjects, such as policy experts and administrators, as well as for directors, choreographers, and participants in community dance.

The fundamental group of the complement of the Hesse configuration

This note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let [Formula: see text] be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane [Formula: see text]. Such [Formula: see text] has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of [Formula: see text] [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. [Formula: see text] has a 9-fold Galois covering space [Formula: see text] given by the complement of the Hesse configuration of 12 lines in [Formula: see text]. Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of [Formula: see text], which we carry out in this note.

Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle

AbstractLetXXbe a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that ifXXisdd-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus ofXX, and the first author’s existence result of holomorphic volume forms on global smoothings ofXX. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples ofdd-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, andK3K3surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable toK3K3surfaces.

K-th singular locus moduli algebras of singularities and their derivation Lie algebras

In this paper, we introduce a series of new invariants for singularities. A new conjecture about the non-existence of negative weight derivations of the new k-th singular locus moduli algebras for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture in some cases.

Experimental Safety Analysis of R-Min, an Underactuated Parallel Robot

AbstractThe R-Min robot is an intrinsically safe parallel manipulator dedicated to pick-and-place operations. The proposed architecture is based on a five-bar mechanism, with additional passive joints in order to obtain a planar seven-bar mechanism with two degrees of underactuation, allowing the robot to reconfigure in case of a collision. A preload bar is added between the base and the end-effector to constrain the additional degrees-of-freedom. This article presents an analysis of the workspace and of the safety performances of the R-Min robot, and it compares them with those of the five-bar mechanism, in order to evaluate the benefits of introducing underactuation in a parallel architecture to obtain intrinsically safer robots. The geometrico-static model of the R-Min robot is formulated as an optimization problem. The direct and inverse kinemato-static models are derived from the geometrico-static model and they allow to express the singularity conditions of the R-Min robot. An analysis of the singularity loci is carried out among the robot’s workspace. A controller based on the dynamic model is proposed and experimentally validated on a prototype of the R-Min robot. Finally, the safety performances of the R-Min robot are evaluated experimentally and they are compared with that of an equivalent five-bar mechanism, using the maximum impact force as a safety criteria in accordance with recent international standards.

Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups

AbstractWe study the mirror symmetry on the singular locus of the Hitchin system at two levels. First, by covering it by (supports of) (BBB)‐branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup P, whose conjectural dual (BAA)‐branes we describe. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of P. Second, when P is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the (BBB)‐brane and study the variation of the dual under this choice. We give evidence of both families of branes being dual under mirror symmetry via an integral functor induced by Fourier–Mukai in the moduli stack of Higgs bundles.

Rational Singularities of Nested Hilbert Schemes

Abstract The Hilbert scheme of points $\textrm {Hilb}^{n}(S)$ of a smooth surface $S$ is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that $\textrm {Hilb}^{n}(S)$ is a smooth variety of dimension $2n$. In recent years there has been growing interest in a natural generalization of $\textrm {Hilb}^{n}(S)$, the nested Hilbert scheme$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$, which parametrizes nested pairs of zero-dimensional subschemes $Z_{1} \supseteq Z_{2}$ of $S$ with $\deg Z_{i}=n_{i}$. In contrast to Fogarty’s theorem, $\textrm {Hilb}^{(n_{1}, n_{2})}(S)$ is almost always singular, and very little is known about its singularities. In this paper, we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70’s shows that $\textrm {Hilb}^{(n,1)}(S)$ is a normal Cohen–Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme $\textrm {Hilb}^{(n,2)}(S)$ has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between $\textrm {Hilb}^{(n,2)}(S)$ and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including the Kempf–Lascoux–Weyman technique of calculating syzygies, square-free Gröbner degenerations, and the Stanley–Reisner correspondence. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and $F$-singularities in positive characteristic.

Energy condition and cosmic censorship conjecture in the perfect fluid collapse

The hypothesis of cosmic censorship plays a crucial role in classical general relativity, as an aspect of the strong version it expresses that naked singularities would never occur from the black hole singularity. In this paper, we will present how energy conditions prohibit forming the local naked singularity in the spherical perfect fluid collapse, and thus we get one step closer to the strong cosmic censorship conjecture proof. We also show that this result can be extended to the cosmological constant background.

Geochemical multifractal modeling of soil and stream sediment data applied to gold prospectivity mapping of the Pitangui Greenstone Belt, northwest of Quadrilátero Ferrífero, Brazil

Geochemical mapping has become essential to prospective assessments in both brownfield and greenfield exploration targeting. During the last two decades, several techniques have been applied to the identification of mineralization's geochemical signature. Among these techniques, Hierarchical Clustering (HC) and Principal Component Analysis (PCA) have been widely used. Other techniques, like Multifractal Spectrum (MS) and Local Singularity Mapping (LSM), are efficient in the generation of models suitable to determine the type of database statistical distribution and mapping zones of geochemical enrichment and depletion. We combined these techniques for soil and stream sediment geochemistry and generated two prospective geochemical models for the gold deposits from the Pitangui Greenstone Belt (PGB), located at northwest of Quadrilátero Ferrífero – Brazil. The results of PCA and HC showed a geochemical association of As-Bi-Mo-Sb with gold mineralization in the PGB. According with descriptive statistical analysis, these elements do not display normal or log-normal statistical distribution, which allowed the application of multifractal models on the geochemical mapping. MS technique revealed that all elements associated with gold have a multifractal distribution. In addition, geochemistry data shows that high contents of elements measured in the stream sediment analysis displays greater dispersion when compared to elements of soil geochemistry data, due to their mobility in environmental conditions. We used LSM technique to map geochemical enrichment of the elements associated with gold. The maps were evaluated using the Ground Truth Analysis (GTA) and the results showed a good performance in the identification of prospective zones. The anomalies identified in both geochemical maps showed a strong correlation between geochemical enrichment zones and geological factors that favors gold mineralization. These results bring a new perspective on the distribution of geochemical anomalies using, for the first time, the multifractal geochemical modeling on the PGB. The geochemical maps point to new targets for mineral exploration and contribute significantly for the prospective models of the region.

On parabolic subgroups of symplectic reflection groups

AbstractUsing Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups.Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open.Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

New Explicit Propagating Solitary Waves Formation and Sensitive Visualization of the Dynamical System

This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ6-model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature.

SOLITONS AND DOMAIN-WALL-ARRAY SOLUTIONS OF THE SCHRÖDINGER FLOW AND LANDAU-LIFSHITZ EQUATION

We obtained some solitons and domain-wall-array solutions of the multidimensional Schrödinger flow and the Landau-Lifshitz equation using the homogeneous balance principle and general Jacobi elliptic-function method. These solutions include bright solitons and periodic solutions in terms of elliptic functions. We excluded several special types of solutions, such as kink profile solutions and dark solitons. The total phase profile of the solitons have two components: the kinematic origin, and the self-steepening effect. For the domain-wall-array solutions, the total phase profile consists of the kinematic origin, kinematic chirping, and self-steepening effect. In certain parameter domains, fundamental domain wall-array-solutions are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. For ODEs deduced from Schrödinger flow that have no analytical solution, phase analysis is used to identify and classify the typical evolutionary pattern. Furthermore, the existence of limit cycles is verified, and the locations of singularities are precisely estimated.

Geometric singularities and regularity of solution of the Stokes system in nonsmooth domains

This paper deals with the geometrical singularities of the weak solution of the mixed boundary value problem governed by the stationary Stokes system in two-dimensional nonsmooth domains with corner points and points at which the type of boundary conditions changes. The presence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. Moreover, the asymptotic singular representations for the solution which inherently depend on the zeros of certain transcendental functions are presented.

Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots

<abstract><p>We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives in a different discrete space that resolves the local singularities of the analytical solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least-squares or an interpolatory approach, yielding a function-valued version of the the standard rational interpolation method ($ \mathcal{V} $-SRI) and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the real axis), the spatially adaptive $ \mathcal{V} $-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the $ \mathcal{V} $-SRI method seems to be the best-performing one.</p></abstract>

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.