Classification Of Singularities Research Articles

Local models for smooth vector fields of the line

We present the local classification of singularities of smooth vector fields on the line, with respect to the equivalence relation of C1–conjugacy. Along the way, we recall the analogous classification, up to C0 and C∞ conjugacy. We also give the transversal unfoldings of the corresponding normal forms and treat the case where the changes of coordinates are tangent to the identity. Thus, a fairly complete description of the 1–d case is achieved.

Pedal and contrapedal curves in equi-affine plane

In this paper, we define equi-affine pedal and contrapedal curves in equi-affine plane. Then, we also consider the relationships among equi-affine evolutes, involutes, parallels, pedal and contrapedal curves. In addition, we investigate the classifications of singularities of equi-affine pedal and contrapedal curves.

Cell envelope structural and functional contributions to antibiotic resistance in Burkholderia cenocepacia.

Antibiotic activity is limited by the physical construction of the Gram-negative cell envelope. Species of the Burkholderia cepacia complex (Bcc) are known as intrinsically multidrug-resistant opportunistic pathogens with low permeability cell envelopes. Here, we re-examined a previously performed chemical-genetic screen of barcoded transposon mutants in B. cenocepacia K56-2, focusing on cell envelope structural and functional processes. We identified structures mechanistically important for resistance to singular and multiple antibiotic classes. For example, susceptibility to novobiocin, avibactam, and the LpxC inhibitor, PF-04753299, was linked to the BpeAB-OprB efflux pump, suggesting these drugs are substrates for this pump in B. cenocepacia. Defects in peptidoglycan precursor synthesis specifically increased susceptibility to cycloserine and revealed a new putative amino acid racemase, while defects in divisome accessory proteins increased susceptibility to multiple β-lactams. Additionally, disruption of the periplasmic disulfide bond formation system caused pleiotropic defects on outer membrane integrity and β-lactamase activity. Our findings highlight the layering of resistance mechanisms in the structure and function of the cell envelope. Consequently, we point out processes that can be targeted for developing antibiotic potentiators.IMPORTANCEThe Gram-negative cell envelope is a double-layered physical barrier that protects cells from extracellular stressors, such as antibiotics. The Burkholderia cell envelope is known to contain additional modifications that reduce permeability. We investigated Burkholderia cell envelope factors contributing to antibiotic resistance from a genome-wide view by re-examining data from a transposon mutant library exposed to an antibiotic panel. We identified susceptible phenotypes for defects in structures and functions in the outer membrane, periplasm, and cytoplasm. Overall, we show that resistance linked to the cell envelope is multifaceted and provides new targets for the development of antibiotic potentiators.

Abrogation and homeostatic restoration of IgE responses by a universal IgE allergy CTL vaccine-The three signal self/non-self/self (S/NS/S) theory.

Natural IgE cytotoxic peptides (nECPs), which are derived from the constant domain of the heavy chain of human IgE producing B cells via endoplasmic reticulum (ER) stress, are decorated onto MHC class 1a molecules (MHCIa) as unique biomarkers for CTL (cytotoxic T lymphocyte)-mediated immune surveillance. Human IgE exhibits only one isotype and lacks polymorphisms; IgE is pivotal in mediating diverse, allergen-specific allergies. Therefore, by disrupting self-IgE tolerance via costimulation, the CTLs induced by nECPs can serve as universal allergy vaccines (UAVs) in humans to dampen IgE production mediated by diverse allergen-specific IgE-secreting B cells and plasma cells expressing surface nECP-MHCIa as targets. The study herein has enabled the identification of nECPs, A32 and SP-1/SP-2 nonameric natural peptidesproduced through the correspondence principle. Vaccination using nECP induced nECP-specific CTL that profoundly suppressed human IgE production in vitro as well as chimeric human IgE production in human IgE/HLA-A2.01/HLA-B7.02 triple transgenic rodents. Furthermore, nECP-tetramer-specific CTLs were found to be converted into CD4 Tregs that restored IgE competence via the homeostatic principle, mediatepred by SREBP-1c suppressed DCs. Thus, nECPs showed causal efficacy and safety as UAVs for treating categorically type I hypersensitivity IgE-mediated allergies. The applied vaccination concept presented provides the foundation to unify, integrate through a singular class of tetramer-specific TCR clonotypes for regulaing human IgE production. The three signal theory pertains to mechanisms of three cells underlying central tolerance (S), breaking self tolerance (NS) and regaining peripheral tolerance (S) via homeostasisconcerning nECP as an efficacious and safe UAV to treat type I IgE-mediated hypersensitivity. The three signal theory impirically extended, may be heuritic for immuno-regulation of adaptive immune repertoire in general.

Bertini theorems admitting base changes

Given a base point free linear system on an algebraic variety, many classes of singularities are stable under taking suitable members after enlarging the base field. We establish analogous results when the base ring is an excellent ring.

Tubular surface generated by a curve lying on a regular surface and its characterizations

<abstract><p>In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.</p></abstract>

Deformations of Calabi–Yau varieties with k-liminal singularities

Abstract The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least $4$ . For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with $1$ -liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

The question of Arnold on classification of co-artin subalgebras in singularity theory

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: ‘Classification of singularities of curves can be interpreted in dual terms as a description of “co-artin” subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)’. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let [Formula: see text] be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras [Formula: see text] of the polynomial algebra [Formula: see text] that contains the ideal [Formula: see text] for some [Formula: see text]. It is proven that the set [Formula: see text] is a disjoint union of affine algebraic varieties (where [Formula: see text] is the semigroup of the singularity and [Formula: see text] is the Frobenius number). It is proven that each set [Formula: see text] is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on [Formula: see text]. An isomorphism criterion is given for the algebras in [Formula: see text]. For each algebra [Formula: see text], explicit sets of generators and defining relations are given and the automorphism group [Formula: see text] is explicitly described. The automorphism group of the algebra [Formula: see text] is finite if and only if the algebra [Formula: see text] is not isomorphic to a monomial algebra, and in this case [Formula: see text] where [Formula: see text] is the conductor of [Formula: see text]. The set of orders of the automorphism groups of the algebras in [Formula: see text] is explicitly described.

Crystal structure and solution state of the C-terminal head region of the narmovirus receptor binding protein.

Genetically diverse paramyxoviruses are united in their presentation of a receptor-binding protein (RBP), which works in concert with the fusion protein to facilitate host-cell entry. The C-terminal head region of the paramyxoviral RBP, a primary determinant of host-cell tropism and inter-species transmission potential, forms structurally distinct classes dependent upon protein and glycan receptor specificity. Here, we reveal the architecture of the C-terminal head region of the RBPs from Nariva virus (NarV) and Mossman virus (MosV), two archetypal rodent-borne paramyxoviruses within the recently established genus Narmovirus, family Paramyxoviridae. Our analysis reveals that while narmoviruses retain the general architectural features associated with paramyxoviral RBPs, namely, a six-bladed β-propeller fold, they lack the structural motifs associated with known receptor-mediated host-cell entry pathways. This investigation indicates that the RBPs of narmoviruses exhibit pathobiological features that are distinct from those of other paramyxoviruses.

Variational Iteration Approach for Solving Fractional Integro-Differential Equations with Conformable Differintegrals

Fractional integro-differential equations involving conformable differintegrals have promising applications but lack solution methods. This work develops a variational iteration approach for such equations. Conformable fractional derivatives and integrals that generalize integer-order equivalents are defined. Existence and uniqueness results are established for a class of nonlinear fractional integro-differential equations. These exist under less restrictive smoothness assumptions than integer-order cases. The variational iteration method (VIM) is then applied, providing an analytical approximate technique for integro-differential problems of fractional order. Convergence analysis demonstrates the efficiency and accuracy of the VIM solutions. Several test cases validate the VIM, matching analytical solutions available for simpler fractional differential sub-cases. The proposed technique advances available methods for this emerging class of fractional integro-differential equations. Significantly, it enables application of such models by allowing accurate solution of associated mathematical system representations. Extensions to include more singular equation classes, comparisons with other methods, and real-world applications are suggested as future work.
 The abstract summarizes the key points: significance of problem, fractional models used, existence/uniqueness proofs, VIM approach and analysis, test case validations, implications for applications

On multifractal analysis and large deviations of singular hyperbolic attractors

In this paper we study the multifractal analysis and large derivations for singular hyperbolic attractors, including the geometric Lorenz attractors. For each singular hyperbolic homoclinic class whose periodic orbits are all homoclinically related and such that the space of ergodic probability measures is connected, we prove that: (i) level sets associated to continuous observables are dense in the homoclinic class and satisfy a variational principle; (ii) irregular sets are either empty or are Baire generic and carry full topological entropy. The assumptions are satisfied by C 1-generic singular hyperbolic attractors and Cr -generic geometric Lorenz attractors . Finally we prove level-2 large deviations bounds for weak Gibbs measures, which provide a large deviations principle in the special case of Gibbs measures. The main technique we apply is the horseshoe approximation property.

Sliding stability and uniqueness for the set Y × Y

This article is dedicated to discuss the sliding stability and the uniqueness property for the 2-dimensional minimal cone Y×Y in R4. This problem is motivated by the classification of singularities for Almgren minimal sets, a model for Plateau's problem in the setting of sets. Minimal cones are blow up limits of Almgren minimal sets, thus the list of all minimal cones gives all possible types of singularities that can occur for minimal sets.As proved in [16], when several 2-dimensional Almgren (resp. topological) minimal cones are Almgren (resp. topological) sliding stable, and Almgren (resp. topological) unique, the almost orthogonal union of them stays minimal. Hence if several minimal cones admit sliding stability and uniqueness properties, then we can use their almost orthogonal unions to generate new families of minimal cones. One then naturally ask which minimal cones admit these two properties.This list of known 2-dimensional minimal cones in arbitrary ambient dimension is not long, and the stability and uniqueness properties for all the known 2-dimensional minimal cones, except for Y×Y, have already been established in the previous works [18,17]. Among all the known 2-dimensional minimal cones, Y×Y is the only one whose stability and uniqueness properties were left unsolved. This is due to two main reasons: 1) Y×Y is the only known minimal cone which is essentially of codimension larger than 1—that is, we cannot decompose it into transversal unions of minimal cones of codimension 1. 2) Y×Y lives in R4, where we know very little about which types of singularities can occur in a minimal set. This makes it difficult to control and estimate the measures of all possible competitors. Due to the above two issues, new ideas are required here for solving the problem.We give affirmative answers to this problem for the stability and uniqueness properties for Y×Y in this paper: we prove that the set Y×Y is both Almgren sliding stable, and Almgren unique; for the topological case, we prove its topological sliding stability and topological uniqueness for the coefficient group Z2. This result, along with the results in [16,18,17], allows us to use all the known 2-dimensional minimal cones to generate new 2-dimensional minimal cones by taking almost orthogonal unions.

Singularity classes of special multi-flags, I

The construction of the geometric singularity classes of special multi-flags was exhaustively done in [P. Mormul, SIGMA, 5:Paper 102, 22 pages, 2009] for special 2-flags, i.e. when flag's width, typically denoted by m, was 2. Now analogous singularity classes are being constructed for special flags of all widths m≥2, compatible with and extending by far the construction in the mentioned paper.

Natural boundaries for scattering amplitudes

Singularities, such as poles and branch points, play a crucial role in investigating the analytic properties of scattering amplitudes that inform new computational techniques. In this note, we point out that scattering amplitudes can also have another class of singularities called natural boundaries of analyticity. They create a barrier beyond which analytic continuation cannot be performed. More concretely, we use unitarity to show that 2 \to 22→2 scattering amplitudes in theories with a mass gap can have a natural boundary on the second sheet of the lightest threshold cut. There, an infinite number of ladder-type Landau singularities densely accumulates on the real axis in the center-of-mass energy plane. We argue that natural boundaries are generic features of higher-multiplicity scattering amplitudes in gapped theories.

Anomalies in the Noun Class Systems of Niger-Congo: Towards a Typology of the Atypical

The paper focuses on minor noun classes and minor genders in Niger-Congo languages, which are usually regarded as anomalies in class systems. It takes into consideration data from 243 languages, and shows that such anomalies are most characteristic for quite specific meanings, namely those traditionally regarded as prototypical meanings of noun classes: ‘person’, ‘thing’, ‘foot’, ‘tree’, ‘eye’, ‘place’, etc. Thus, the particular patterns of class agreement observed, or particular correlations between singular and plural noun classes, often do not represent accidental exceptions, but are better thought of as a kind of marker of noun class paradigmatic semantics.

Classification of Singularities of Smooth Functions with a Finite Cyclic Symmetry Group

Classification of Singularities of Smooth Functions with a Finite Cyclic Symmetry Group

Many Faces of Singularities in Robotics

In this survey paper some issues concerning a singularity concept in robotics are addressed. Singularities are analyzed in the scope of inverse kinematics for serial manipulator, a motion planning task of nonholonomic systems and the optimal control covering a large area of practical robotic systems. An attempt has been made to define the term singularity, which is independent on a specific task. A few classifications of singularities with respect to different criteria are proposed and illustrated on simple examples. Singularities are analyzed from a numerical and physical point of view. Generally, singularities pose some problems in motion planning and/or control of robots. However, as illustrated on the example on force/momenta transformation in serial manipulators, they can also be desirable is some cases. Singularity detection techniques and some methods to cope with them are also provided. The paper is intended to be didactic and to help robotic researchers to get a general view on the singularity issue.

DERIVATIONS OF LOCAL k-TH HESSIAN ALGEBRAS OF SINGULARITIES

In our previous work, we introduced a series of new derivation Lie algebras Lk(V) associated to an isolated hypersurface singularity (V,0). These are new analytic invariants of singularities. Here, we investigate L2(V) for fewnomial isolated singularities and obtain the formula of λk(V) (i.e., the dimension of Lk(V)) for trinomial singularities. Furthermore, we prove the sharp upper estimate conjecture for L2(V). This is a continuation of our previous work (Math. Z. 298:3-4 (2021), 1813–1829). We proposed two new conjectures for τk(V) and λk(V) and we prove these conjectures for a large class of singularities.

Cobordism, singularities and the Ricci flow conjecture

In the following work, an attempt to conciliate the Ricci flow conjecture and the Cobordism conjecture, stated as refinements of the Swampland distance conjecture and of the No global symmetries conjecture respectively, is presented. This is done by starting from a suitable manifold with trivial cobordism class, applying surgery techniques to Ricci flow singularities and trivialising the cobordism class of one of the resulting connected components via the introduction of appropriate defects. The specific example of {varOmega}_4^{SO} is studied in detail. A connection between the process of blowing up a point of a manifold and that of taking the connected sum of such with ℂℙn is explored. Hence, the problem of studying the Ricci flow of a K3 whose cobordism class is trivialised by the addition of 16 copies of ℂℙ2 is tackled by applying both the techniques developed in the previous sections and the classification of singularities in terms of ADE groups.

Singularities for Timelike Developable Surfaces in Minkowski 3-Space

In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.