Simplex Research Articles

NCMP-02. UTILITY OF ICE PACK TEST FOR EARLY RECOGNITION OF IMMUNE CHECKPOINT INHIBITOR MEDIATED OCULAR WEAKNESS

Abstract INTRODUCTION Immune checkpoint inhibitors (ICI) are associated with neurological immune-related adverse events (NiArE) such as myasthenia gravis (MG) and myositis. These conditions often present with overlapping symptoms, notably ptosis, dyspnea, and extremity weakness. Early recognition and treatment are warranted to prevent disease progression. OBJECTIVE To test the utility of simple bedside ice pack test (IPT) in early recognition and distinction of ICI-mediated MG and myositis ocular manifestations. METHODS We administered IPT in 14 patients who had received ICI in the past 6 months and presented with ptosis. Improvement in ptosis was considered positive IPT (non-fixed, fatigable ptosis) that is suggestive of Immune-mediated MG-predominant process. Work up included myasthenia antibodies, EMG, serum markers of muscle injury (creatine kinase, aldolase), myositis panel and muscle biopsy. Statistical analyses employed median for continuous and frequencies for categorical variables. RESULTS Positive IPT was found in 2 (14%) patients with positive myasthenia antibodies and neuromuscular junction dysfunction on EMG in one patient with repetitive nerve stimulation. These patients had mild elevation in creatine kinase (CK). Muscle biopsy did not reveal a myositic process. Twelve (86%) patients had negative IPT (fixed and non-fatigable ptosis) with significantly elevated CK (>4000), positive myositis panel and muscle biopsy consistent with myositic process. Six of these patients (42%) also had positive myasthenia antibodies. Concomitant myocarditis was observed in 5 (36%) patients. CONCLUSION Clinical overlap between myasthenia and myositis is observed in immune mediated toxicity that may be due to cross-reactivity of antibodies, however, IPT was able to distinguish MG vs myositis-predominant process, thus helping guide management.

Indications of a changing winter through the lens of lake mixing in Earth’s largest freshwater system

Abstract Global surface freshwater primarily resides in lakes, with the overwhelming majority found in Earth’s largest lakes, thus understanding potential climate change effects in these large lakes is critical. In dimictic lakes, climate change has extended the duration of summer thermal stratification and reduced the length of the ice season. These changes are relatively straightforward to evaluate in smaller, inland lakes. However in large lakes, such as the North American Great Lakes, temporally intermittent and spatially heterogeneous ice cover, and spatial thermal heterogeneity limit the utility of simple ice on-off or mixing classifications; therefore, assessing how climate change is impacting winter conditions in large lakes is challenging. Here, we use in-situ and satellite-derived surface water temperature observations from the North American Great Lakes to overcome these limitations and show that warming air temperatures are driving reductions in the number of winter days, collectively those with either ice cover or inverse thermal stratification, in favor of increases in isothermal conditions for the period 1995 to 2023. We find that on average the Great Lakes are experiencing a loss of 14 winter days per decade. Our results demonstrate how climate change has yielded disproportionate changes in the annual thermal cycle and mixing conditions of Earth’s largest freshwater system and signals the potential for fundamental ecosystem shifts due to a loss of winter.

Graded Lie algebras, compactified Jacobians and arithmetic statistics

A simply laced Dynkin diagram gives rise to a family of curves over \mathbb{Q} and a coregular representation, using deformations of simple singularities and Vinberg theory, respectively. Thorne conjectured and partially proved a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne’s picture and show that 2-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck–Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives.

Modeling two-dimensional ice shape based on fractal interpolation

The ice accretion data obtained from ice wind tunnel tests reveal a multiscale structure and rough surface. In the follow-up aerodynamic evaluation of icing airfoils, simplified two-dimensional ice shapes are generally used as substitutes, but this simplification changes the aerodynamic effects of the original ice shape. Therefore, finding a simple and effective two-dimensional ice shape simulation method is urgent. Due to the self-similarity characteristics of ice shape surfaces, fractal interpolation is proposed for ice shape simulation. First, the geometric characteristics of the ice shapes are analyzed to determine interpolation points, and an iterative function system is constructed for interpolation simulation. Considering the influence of various characteristics of ice shapes on aerodynamics, interpolation parameters are limited to simulating more realistic ice shapes. High-order numerical simulation methods were utilized to numerically simulate and analyze the aerodynamic characteristics of icing airfoil while also verifying the feasibility of fractal interpolation for simulating ice shapes. The analysis revealed that this method could effectively simulate ice profiles of various feature scales with minimal ice shape data. These simulated shapes closely resemble real ice formations and maintain the original aerodynamic characteristics of the icing airfoil. This method can be used to improve the computational accuracy of ice accretion codes and provide improvement strategies for complex ice shape prediction; thus, this method has great application prospects in engineering.

JWST observations of 13CO2 ice

The structure and composition of simple ices can be severely modified during stellar evolution by protostellar heating. Key to understanding the involved processes are thermal and chemical tracers that can be used to diagnose the history and environment of the ice. The 15.2 µm bending mode of 12CO2 in particular has proven to be a valuable tracer of ice heating events but suffers from grain shape and size effects. A viable alternative tracer is the weaker 13CO2 isotopologue band at 4.39 µm, which has now become accessible at high S/N with the James Webb Space Telescope (JWST). In this study, we present JWST NIRSpec observations of 13CO2 ice in five deeply embedded Class 0 sources that span a wide range in masses and luminosities (0.2–104 L⊙) taken as part of the Investigating Protostellar Accretion Across the Mass Spectrum (IPA) program. The band profiles vary significantly depending on the source, with the most luminous sources showing a distinct narrow peak at 4.38 µm. We first applied a phenomenological approach with which we demonstrate that a minimum of three to four Gaussian profiles are needed to fit the absorption feature of 13CO2. We then combined these findings with laboratory data and show that a 15.2 µm 12CO2 bending-mode-inspired five-component decomposition can be applied to the isotopologue band, with each component representative of CO2 ice in a specific molecular environment. The final solution consists of cold mixtures of CO2 with CH3OH, H2O, and CO as well as segregated heated pure CO2 ice at 80 K. Our results are in agreement with previous studies of the 12CO2 ice band, further confirming that 13CO2 is a useful alternative tracer of protostellar heating and ice composition. We also propose an alternative solution consisting only of heated mixtures of CO2:CH3OH and CO2:H2O ices and warm pure CO2 ice at 80 K (i.e., no cold CO2 ices) for decomposing the ice profiles of HOPS 370 and IRAS 20126, the two most luminous sources in our sample that show strong evidence of ice heating resulting in ice segregation.

A simple scoring algorithm based on intrinsic capacity for functional ability in community-dwelling older adults in Taiwan

BackgroundIntrinsic capacity (IC) is a comprehensive indicator of the overall well-being of older adults, and assessing of IC can help identify early stage of disability and tailor intervention to individual needs. However, there is a lack of effective and simple IC assessment tools. This study aimed to establish predictive scoring algorithms of IC to identify older adults at high risk of impaired functional ability.MethodsWe conducted a cross-sectional study in Southern Taiwan, measuring IC using 7 subitems: cognition, locomotion, vitality, vision, hearing, psychological well-being, and medication usage were measured. Functional ability outcomes included frailty, basic activities of daily living, and instrumental activities of daily living (IADL). The capability of 7 domains of IC in predicting functional ability was assessed by multivariable logistic regression. The prediction of capability of scoring algorithms was indicated by receiver operating characteristic (AUC) curves and measures of sensitivity and specificity.ResultsA total of 1,152 older adults were recruited and analyzed. Locomotion emerged as a significant predictor of IADL disability and worsening frailty. The IC-based weighted scoring algorism for predicting IADL demonstrated satisfactory capability (AUC: 0.80), as did the algorithm for predicting worsening frailty (AUC: 0.90). The optimal cutoff points for predicting IADL disability and frailty worse were estimated respectively at 13 and 16, with sensitivity/specificity values of 0.74/0.75 for the IADL prediction algorithm and 0.92/0.77 for the frailty prediction algorithm.ConclusionOur 7-domain IC screening tool proves to be sensitive and practical for early identification of functional disability and frailty among community-dwelling older adults in Taiwan.

Ice chemistry modeling of active phase comets: Hale–Bopp

We present a chemical kinetics model of the solid-phase chemical evolution of a comet, beginning with a long period of cold-storage in the Oort Cloud, followed by five orbits that bring the comet close to the Sun. Current orbital parameters for Hale–Bopp are used for all orbits. The chemical model is based on an earlier treatment (Garrod 2019) that considered only the cold-storage phase, and which was based on the interstellar ice chemical kinetics model MAGICKAL. The comet is treated as 25 chemically distinct layers, whose thicknesses increase with depth; each layer has a dust component, and the chemical network includes a further 200 atomic/molecular species. The new model includes several key updates to the previous treatment: (i) Time- and depth-dependent temperature profiles based on heat transfer according to heliocentric distance; (ii) a rigorous treatment of back-diffusion (random walk) for species capable of diffusing through the thick bulk-ice layers; (iii) adoption of recent improvements in the kinetic treatment of nondiffusive chemical reaction rates in the ice and on the ice surface. Starting from an initially simple ice composition, interstellar UV photons drive a rapid chemistry in the upper micron of material, but diminished by absorption of the UV by the dust component. Galactic cosmic rays (GCRs) drive a much slower chemistry in the deeper ices over the long cold-storage period, to depths on the order of 10 m. The first solar approach drives off the upper layers of ice material via thermal desorption and/or dissociation (the model does not include a treatment for the more complex cometary outbursts), bringing closer to the surface the deeper material that previously underwent long-term processing by GCRs. Subsequent orbits are more uniform in their chemical behavior after the upper layers are lost. Loss of molecular material leads to concentration of the dust in the upper layers, with a large dust fraction extending to depths on the order of 10 cm after all orbits are complete. Substantial quantities of complex organic molecules (COMs) are formed in the upper 10 m during the cold storage phase, with some of this material released during solar approach; however, their abundances with respect to water appear too low to account for the observed gas-phase values for comet Hale–Bopp, indicating that the majority of complex molecular material observed, at least in comet Hale–Bopp, is an inheritance of primordial material.

On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities

Lie algebra plays an important role in the study of singularity theory and other fields of sciences. Finding numerous invariants linked with isolated singularities has always been a primary interest in the field of classification theory of isolated singularities. Any Lie algebra that characterizes simple singularity produces a natural question. The study of properties such as to find the dimensions of newly defined algebra is a remarkable work. Hussain, Yau and Zuo have found a new class of Lie algebra Lk(V), k≥1, i.e., Der (Mk(V),Mk(V)) and proposed a conjecture over its dimension δk(V) for k≥0. Later, they proved it true for k up to k=1,2,3,4,5. In this work, the main concern is whether it is true for a higher value of k. According to this, we first calculate the dimension of Lie algebra Lk(V) for k=6 and then compute the upper estimate conjecture of fewnomial isolated singularities. Additionally, we also justify the inequality conjecture δk+1(V)<δk(V) for k=6. Our calculated results are innovative and serve as a new addition to the study of singularity theory.

Passive Aggressive Ensemble for Online Portfolio Selection

Developing effective trend estimators is the main method to solve the online portfolio selection problem. Although the existing portfolio strategies have demonstrated good performance through the development of various trend estimators, it is still challenging to determine in advance which estimator will yield the maximum final cumulative wealth in online portfolio selection tasks. This paper studies an online ensemble approach for online portfolio selection by leveraging the strengths of multiple trend estimators. Specifically, a return-based loss function and a cross-entropy-based loss function are first designed to evaluate the adaptiveness of different trend estimators in a financial environment. On this basis, a passive aggressive ensemble model is proposed to weigh these trend estimators within a unit simplex according to their adaptiveness. Extensive experiments are conducted on benchmark datasets from various real-world stock markets to evaluate their performance. The results show that the proposed strategy achieves state-of-the-art performance, including efficiency and cumulative return.

Numerical simulation of rime ice accretion on airfoil using rigid sphere model

A 2D/3D numerical model for rime ice shape prediction is developed by combining the rigid sphere model (RSM) and computational fluid dynamics (CFD), and its characteristics are investigated. In the RSM, the ice accretion is modeled based on first principles with little physical modeling. The ice particle on an airfoil is described as a rigid sphere, and the ice shape is represented as a collection of rigid spheres. We use a non-body-fitted hierarchical Cartesian grid-based CFD solver so that complex ice shape represented by RSM can be handled. The tree data of the hierarchical Cartesian grid is also used for fast search of droplet positions. In addition, multiple ice particles are modeled as one large particle to reduce the number of trajectory calculations. The sensitivity of several parameters in the calculation is studied in 2D. Subsequently, 3D ice accretion analysis on NACA0012 airfoil using the RSM is performed for the first time. The simulation results are compared with those of the wind-tunnel experiments conducted at NASA. Although the RSM is a simple ice accretion model, reliable results are obtained in both 2D and 3D calculations.

Hydromechanical Transmission IC2OC: Component Sizing and Optimization

The IC2OC transmission is a continuous transmission whose layout can change from simple IC to simple OC configuration and vice versa. It was proposed to cover a wider range of vehicle speeds without adding gears. Its sizing can lead to higher efficiencies than those of the IC and OC layouts. Therefore, this work deals with the sizing methodologies of this transmission. Two methodologies are proposed and discussed: the first uses the functional and constitutive equations of the transmission; the second is based on a mathematical programming problem. Both methodologies start from the choice of the full mechanical point speeds. The comparison between the two methods is carried out on the transmission of a commercially available 230 kW reach stacker. The comparison shows that the functional method, leaner and faster, can provide results very close to those obtained with the heavy and time-consuming optimization, provided that the values of the two full mechanical point speeds are the optimal ones for the two basic transmissions taken individually.

A rapid interactive chitosan-based medium with antioxidant and pro-vascularization properties for infected burn wound healing

Large-pore hydrogels are better suited to meet the management needs of nutrient transportation and gas exchange between infected burn wounds and normal tissues. However, better construction strategies are required to balance the pore size and mechanical strength of hydrogels to construct a faster substance/gas interaction medium between tissues. Herein, we developed spongy large pore size hydrogel (CS-TA@Lys) with good mechanical properties using a simple ice crystal-assisted method based on chitosan (CS), incorporating tannic acid (TA) and ε-polylysine (Lys). A large-pore and mechanically robust hydrogel medium was constructed based on hydrogen bonding between CS molecules. On this basis, a pro-restorative functional platform with antioxidation and pro-vascularization was constructed using TA and Lys. In vitro experiments displayed that the CS-TA@Lys hydrogel possessed favorable mechanical properties and fast interaction performances. In addition, the CS-TA@Lys hydrogel possessed the capacity to remove intra/extracellular reactive oxygen species (ROS) and possessed antimicrobial and pro-angiogenic properties. In vivo experiments displayed that the CS-TA@Lys hydrogel inhibited wound inflammation and promoted wound vascularization. In addition, the CS-TA@Lys hydrogel showed the potential for rapid hemostasis. This study provides a potential functional wound dressing with rapid interaction properties for skin wound repair.

A Spectral Method for Identifiable Grade of Membership Analysis with Binary Responses.

Grade of membership (GoM) models are popular individual-level mixture models for multivariate categorical data. GoM allows each subject to have mixed memberships in multiple extreme latent profiles. Therefore, GoM models have a richer modeling capacity than latent class models that restrict each subject to belong to a single profile. The flexibility of GoM comes at the cost of more challenging identifiability and estimation problems. In this work, we propose a singular value decomposition (SVD)-based spectral approach to GoM analysis with multivariate binary responses. Our approach hinges on the observation that the expectation of the data matrix has a low-rank decomposition under a GoM model. For identifiability, we develop sufficient and almost necessary conditions for a notion of expectation identifiability. For estimation, we extract only a few leading singular vectors of the observed data matrix and exploit the simplex geometry of these vectors to estimate the mixed membership scores and other parameters. We also establish the consistency of our estimator in the double-asymptotic regime where both the number of subjects and the number of items grow to infinity. Our spectral method has a huge computational advantage over Bayesian or likelihood-based methods and is scalable to large-scale and high-dimensional data. Extensive simulation studies demonstrate the superior efficiency and accuracy of our method. We also illustrate our method by applying it to a personality test dataset.

Aldaz-Kounchev-Render operators on simplices

We introduce the Aldaz-Kounchev-Render operators on a multidimensional simplex. In the case of the unit simplex of Rm these operators preserve the functions 1,x1j,…,xmj, j a positive integer. The Voronovskaja formula, the behaviour with respect to the convex function, and the limit of iterates of the operators are investigated.

Evaluation of four calving laws for Antarctic ice shelves

Abstract. Many floating ice shelves in Antarctica buttress the ice streams feeding them, thereby reducing the discharge of icebergs into the ocean. The rate at which ice shelves calve icebergs and how fast they flow determine whether they advance, retreat, or remain stable, exerting a first-order control on ice discharge. To parameterize calving within ice sheet models, several empirical and physical calving “laws” have been proposed in the past few decades. Such laws emphasize dissimilar features, including along- and across-flow strain rates (the eigencalving law), a fracture yield criterion (the von Mises law), longitudinal stretching (the crevasse depth law), and a simple ice thickness threshold (the minimum thickness law), among others. Despite the multitude of established calving laws, these laws remain largely unvalidated for the Antarctic Ice Sheet, rendering it difficult to assess the broad applicability of any given law in Antarctica. We address this shortcoming through a set of numerical experiments that evaluate existing calving laws for 10 ice shelves around the Antarctic Ice Sheet. We utilize the Ice-sheet and Sea-level System Model (ISSM) and implement four calving laws under constant external forcing, calibrating the free parameter of each of these calving laws for each ice shelf by assuming that the current position of the ice front is in steady state and finding the set of parameters that best achieves this position over a simulation of 200 years. We find that, in general, the eigencalving and von Mises laws best reproduce observed calving front positions under the steady-state position assumption. These results will streamline future modeling efforts of Antarctic ice shelves by better informing the relevant physics of Antarctic-style calving on a shelf-by-shelf basis.

Disentangling Dynamic from Thermodynamic Summer Ice Area Loss from Observations (1979–2021): A Potential Mechanism for a “First-Time” Ice-Free Arctic

Abstract In recent decades, the Arctic minimum sea ice extent has transitioned from a predominantly thick multiyear ice cover to a thinner seasonal ice cover. We partition the total (observed) Arctic summer area loss into thermodynamic and dynamic (convergence, ridging, and export) sea ice area loss during the satellite era from 1979 to 2021 using a Lagrangian sea ice tracking model driven by satellite-derived sea ice velocities. Results show that the thermodynamic signal dominates the total summer ice area loss and the dynamic signal remains small (∼20%) even in 2007 when dynamic loss was largest. Sea ice loss by compaction (within pack ice convergence) dominates the dynamic area loss, even in years when the export is largest. Results from a simple (Ekman) free-drift sea ice model, supported by results from the Lagrangian model, suggest that nonlinear effects between dynamic and thermodynamic area loss can be important for large negative anomalies in sea ice extent, in accord with previous modeling studies. A detailed analysis of two all-time record minimum years (2007 and 2012)—one with a semipermanent high in the southern Beaufort Sea and the other with a short-lived but extreme storm in the Pacific sector of the Arctic in late summer—shows that compaction by Ekman convergence together with large thermodynamic melt in the marginal ice zone dominated the sea ice area loss in 2007 whereas, in 2012, it was dominated by Ekman divergence amplified by sea–ice albedo feedback—together with an early melt onset. We argue that Ekman divergence from more intense summer storms when the sun is high above the horizon is a more likely mechanism for a “first-time” ice-free Arctic.

On $\mu$-Zariski pairs of links

The notion of Zariski pairs for projective curves in $\mathbb{P}^{2}$ is known since the pioneer paper of Zariski. In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski) pair of curves $C = \{f(x,y,z) = 0\}$ and $C' = \{g(x, y, z) = 0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. They have a same zeta function and a Milnor number. We give new examples of weak Zariski pairs which have same $\mu^{*}$ sequences and same zeta functions but two functions belong to different connected components of $\mu$-constant strata (Theorem 14). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology, which implies the Jordan forms of their monodromies are different (Theorem 24). We start from weak Zariski pairs of projective curves to construct new Zariski pairs of surfaces which have non-diffeomorphic link 3-folds. We also prove that the hypersurface pair constructed from a Zariski pair of irreducible plane curves with simple singularities give a diffeomorphic links (Theorem 25).

Real-time microseismic evaluation of coalbed methane reservoir stimulation based on improved metaheuristic inversion strategy

For the enhancement of coalbed methane production, the microseismic source location plays a crucial role in evaluating fracture information during hydraulic fracturing (HF). A swift and precise microseismic source location is essential for optimizing HF schemes. This optimization necessitates an inversion strategy characterized by low computational costs and high optimization accuracy. In this study, we propose an SM-MFO inversion strategy that functions in parallel with the simplex (SM) algorithm and is founded on the original moth flame optimizer (MFO) algorithm. This strategy was validated through gas blast microseismic monitoring field tests to assess the source location performance of the SM-MFO inversion strategy, followed by its application in real-time evaluation of staged HF for the Xinji 1# horizontal well. The results reveal that within the SM-MFO inversion strategy, SM can prevent local convergence by leveraging the capabilities of MFO. Conversely, MFO can bolster local search abilities with the assistance of SM. As a result, both the convergence speed and global optimization abilities of the strategy are enhanced. In field tests, the average location error of the SM-MFO algorithm was 18.11 m. This figure is more favorable compared to the errors in the MFO algorithm (22.15 m) and the SM algorithm (35.83 m). Moreover, the SM-MFO inversion strategy facilitates real-time evaluation of HF with a computational cost of just 1.01 s for a single microseismic event. Upon analysis of the overall incidence across the nine stages in the radial direction, a span of 270 m is attained, with fewer frequency of fractures observed in the southeast direction at each respective stage Additionally, the geometric parameters of the fractures and the stimulated reservoir volume (SRV) exhibit a positive correlation with the total fluid volume. The injected fluid volume peaks in the second and sixth stages, leading to fracture mesh dimensions of 312 m and 316 m, respectively. In summary, the total SRV of the Xinji 1# horizontal well is 201.7 × 105 m³, and the reservoir stimulation effect aligns with the anticipated results. This work highlights the efficacy of the SM-MFO inversion strategy in enhancing the efficiency and accuracy of microseismic source location for hydraulic fracturing, contributing valuable insights to the field.

On the adaptation of reference sets using niching and pair-potential energy functions for multi-objective optimization

Multi-objective evolutionary algorithms (MOEAs) have shown a good capability to approximate the Pareto front (PF) of complex multi-objective optimization problems (MOPs). Recently, several decomposition-based, indicator-based, and reference set-based MOEAs have been proposed to tackle MOPs with four or more objective functions, known as many-objective optimization problems (MaOPs). Some well-known MOEAs from these design strategies use a predefined reference set generated with uniformly distributed weight vectors on a unit simplex. These MOEAs have demonstrated good convergence, coverage, and uniformity of solutions on MOPs with regular PF shapes, i.e., simplex-like shapes. However, it has been shown that their performance degrades on MOPs with irregular PF shapes. In this paper, we proposed a pluggable reference set adaptation method based on niching and pair-potential energy functions (PPFs), called AdaK, to enhance the performance of these MOEAs on MOPs with irregular PF shapes while maintaining their good behavior on MOPs with regular PF shapes. Our adaptation method was validated by plugging it into three well-known MOEAs that use a predefined weight vector-based reference set. We perform an empirical study using different PPFs for our adaptation method on the DTLZ, WFG, Minus-DTLZ, Minus-WFG, IMOP, and VNT test suites. Our experimental results show the capability of our adaptation method to promote an invariant performance regardless of the PF shape. Additionally, we show that MOEAs with our adaptation method have a competitive performance against state-of-the-art MOEAs.

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.