Rational Plane Research Articles

Quatroids and rational plane cubics

Quatroids and rational plane cubics

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

Detecting similarities of rational plane curves using complex differential invariants

We present a new and efficient method to detect whether or not two given rational plane curves are similar. If both curves are the same, the method finds the symmetries of the curve. The method relies on the introduction of a complex differential invariant that has a nice behavior with respect to Möbius transformations, which are the mappings lying behind the similarities in the parameter space. From a computational point of view, our algorithm only requires bivariate gcds and factoring. The algorithm is implemented in Maple™ [Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario (2021)]. An extensive study of its practical performance is also provided.

Chapitre 5. L’impact de la pandémie de COVID-19 sur l’évolution du paradigme éthique de la médecine clinique en Russie (« à la guerre comme à la guerre »)

The CОVID-19 pandemic became a catalyst for the transformation of the ideas of the existence of global ethics in favor of real moral pluralism, exposed the problem of abandoning the principles of personalized medicine in favor of public health ethics (collective ethics of civil society). The authors sequentially analyze the objective factors that influenced the change in the moral paradigm of the clinical medicine in Russia: the peculiarities of the course of infection, the lack of resources in the health-care sector, the inability to use advanced treatment methods in different groups of patients, the protection of medical workers, the provision of emergency and planned surgical interventions, and the prevention of further spread of infection. In addition, the moral consequences of the use of administrative measures to limit the spread of the pandemic are given: restriction of social contacts, the use of personal protective equipment, retraining of specialists, re-profiling of the bed fund, leveling problems of communication with colleagues, patients and students. Special attention is paid to the problem of “anti-vaxxers”, representing a significant part of society and hindering the implementation of the vaccination program of the population. We believe that active and passive protest against vaccinations lie not in a rational plane, but in an emotional one, in an immanent distrust of the state and its institutions. Thus arises a secondary ethical problem of the state’s responsibility for the life and health of every citizen, regardless of their beliefs. Contradictions in the moral principles of individual strata of the population (including groups like those who agree to get vaccinated, those who doubt, the indifferent ones, “active anti-vaccinators”) currently seem irreconcilable due to the state’s detachment from solving moral problems.Ethic divergence in the context of the pandemic has become real, stable and difficult to overcome, while the claims of a single, global, global bioethics are being questioned. The COVID-19 pandemic has formulated the ethical task of the 21st century, which will have to be the development of public policy and practice of clinical medicine in the context of serious moral contradictions and significant bioethical differences.

Remarks on double points of plane curves

We study the relation between the type of a double point of a plane curve and the curvilinear 0-dimensional subschemes of the curve at the point. An Algorithm related to a classical procedure for the study of double points via osculating curves is described and proved. Eventually we look for a way to create examples of rational plane curves with given singularities A_s.

The concept of axial weak twins

It is generally assumed that the parent-twin interface is a fully invariant plane. The calculations of the rational interfaces of the type I twins and irrational interfaces of the type II twins are based on this hypothesis. Recently, some asymmetric facets and unconventional twins have been observed in magnesium that do not agree with this paradigm. Here, we show that these features can be explained by admitting that some of the facets of the interface or the interface itself may be a rational plane that can be slightly distorted and transformed into a new rational plane with non-equivalent Miller indices. Relaxing the invariant plane assumption permits to broaden the usual lattice theory of twinning and include these unconventional "weak" interfaces and twins. A computer program was written to calculate the type I, type II and weak twins. It explains the asymmetric facets, the unconventional deformation twins, and the “deformation graining” phenomenon recently reported in magnesium.

Planes in four-space and four associated CM points

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian Gr(2,4) and four shapes of lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement, and the second two essentially arise from the accidental local isomorphism between SO(4) and SO(3)×SO(3). As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings, we prove simultaneous equidistribution of all of these objects under two splitting conditions.

Dirichlet is not just bad and singular

It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of Davenport & Schmidt (1969) which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main “continuum” result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dirichlet improvable points.

Public inquiry in the periods of social turbulence (based on the materials of sociological research in Tomsk)

This article discusses the questions associated with the impact of social turbulence and exacerbation regimes upon the society. The topic is viewed in the context of the effect of behavioral deviations upon public administration from the perspective of the need for changing its content in the conditions of escalation of crisis situation. The author verifies the hypotheses on the shift of value orientations of the population from the rational plane to the irrational, as well as the formation of public inquiry for the “strong” state. The article aims to identify the attitude of Tomsk population towards the global and local agenda of rapid processes through assessing their own behavior and appeals to the government institutions on the examples of situations caused by outbreak of coronavirus pandemic and transformation of real estate market in the city. Research methodology employs documentary analysis, formalized interview, statistical data processing, and logical inductive-deductive inference. The article obtains new empirical data related to the behavioral assessment of the population in the conditions of social turbulence and its effect upon the formation of public inquiry. The main result consists in indicating the need for transformation of the system of public administration in the periods with nonlinear dynamics and high speeds in implementation of the policy of strengthening the regulatory impact upon maintaining the spheres most affected by the crisis. The presented materials are valuable for the employees of government and administrative branches, as well as researchers dealing with the problematic of social turbulence.

Counting planar curves in [formula omitted] with degenerate singularities

In this paper, we consider the following question: how many degree d curves are there in P3 (passing through the right number of generic lines and points), whose image lies inside a P2, having δ nodes and one singularity of codimension k? We obtain an explicit formula for this number when δ+k≤4 (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in the paper by A. Zinger [30] and which is further pursued by S. Basu and the second author in [1], [2] and [3]. Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by S. Kleiman and R. Piene (in [17]) and by T. Laarakker (in [19]). We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author (in [22]), where they compute the characteristic number of rational planar curves in P3 with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande in [23], regarding the enumerativity of BPS numbers for P3.

Twin boundary structure and mobility

Twinning is an important mechanism of deformation in various crystalline materials, and in particular in shape memory alloys, where it is inherent to the shape memory and super-elasticity effects. This paper presents a generalized methodological approach for analyzing and modeling twin boundary dynamics with particular relevance for shape memory alloys. This approach combines the topological model description of the interface structure at the atomistic/lattice scale with analytical analysis of energy barriers and mechanisms of motion that provide macro-scale kinetic laws for the twin boundary motion. We emphasize the main differences between the topological structures of different types of twin interfaces and their implications for the mobilities of the different twin types. In particular, we elaborate on the relaxed topological structure of type II twin boundaries that contains a coherently facetted structure, where the facets are rational planes that accommodate misfit strain. Then, we clarify the lattice barriers’ role in determining the different regimes of the kinetics of twin boundary motion. Further, we develop models leading to analytical expressions for the activation energies of various nucleation processes that dictate the overall kinetics of twin boundary motion, and identify of the rate-limiting process for the different twin types. In the case of compound and type I twins, the analysis leads to an explicit expression for the magnitude of the twinning stress, revealing a strong dependency on the shear modulus and the twinning shear, which is in excellent quantitative agreement with experimental values reported for BaTiO3, Ni-Ti, Cu-Al-Ni, and 10M and NM Ni-Mn-Ga. Moreover, our analysis reasons the different temperature dependencies of the twinning stress exhibited by the different twin types, and in particular the very low temperature sensitivity of type II twins.

Exact and approximate similarities of non-necessarily rational planar, parametrized curves, using centers of gravity and inertia tensors

We provide an algorithm to detect whether two bounded, planar parametrized curves are similar, i.e. whether there exists a similarity transforming one of the curves onto the other. The algorithm is valid for completely general parametrizations, and can be adapted to the case when the input is given with finite precision, using the notion of approximate [Formula: see text]. The algorithm is based on the computation of centers of gravity and inertia tensors of the considered curves or of the planar regions enclosed by the curves, which have nice properties when a similarity transformation is applied. In more detail, the centers of gravity are mapped onto each other, and the matrices representing the inertia tensors satisfy a simple relationship: when the similarity is a congruence (i.e. distances are preserved) the matrices are congruent, and in the more general case the relationship is analogous, but involves the square of the scaling constant. Using both properties, and except for certain pathological cases, the similarities can be found. Additional ideas are presented for the case of closed, i.e. compact, curves.

Sheaf Theoretic Compactifications of the Space of Rational Quartic Plane Curves

Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $\alpha$-semistable pairs on $\mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincare polynomial of the compactified space.

Inversion, degree, reparametrization and implicitization of improperly parametrized planar curves using μ-basis

The μ-basis of a rational curve/surface is a new algebraic tool which plays an important role in connecting the rational parametric form and the implicit form of a rational curve/surface. However, most results for μ-bases are presented for proper rational parametrizations. In this paper we consider the μ-basis for an improper rational planar curve. Based on the known properties and new results, we design two new proper reparametrization algorithms using μ-basis. The inversion, degree of the induced rational map and implicitization formulas are also derived.

Rational points on three mutually orthogonal great circles

Let [Formula: see text] be a prime of the form [Formula: see text]. Then for any three mutually orthogonal great circles of the sphere [Formula: see text] lying on rational planes, at least one of these circles does not contain any rational points.

Freeness and invariants of rational plane curves

Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

ISSAC 2019 software presentations communicated by Yue Ren

Implicit is a package for implicitizing rational planar curves and rational tensor product surfaces, developed on Maplesoft based on the state-of-the-art implicitization techniques. The main functions ImpCurve and ImpSurface return the implicit equation of a rational planar curve or a rational tensor product surface. Other popularly used functions, such as ImpDegree, ImpMatrix and ImpRuled that are used for deciding the implicit degree and the implicit matrix of a general rational surface, and computing the implicit equation of a rational ruled surface in a more efficient way, are also proposed.

Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom–Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

Miejsce Hipokratesa i Samarytanina we współczesnej medycynie, czyli etyczna formacja pracownika służby zdrowia w świetle Nowej Karty Pracowników Służby Zdrowia

Wydana przez Papieską Radę ds. Służby Zdrowia Nowa Karta Pracowników Służby Zdrowia przywołuje we wstępie i w zakończeniu Przysięgę Hipokratesa. Podobnie we wstępie przypomina o ewangelicznej przypowieści o dobrym Samarytaninie. Niniejszy artykuł wykazuje, że w przywołaniach tych nie chodzi o zwykłe przypomnienie tych ważnych dla etyki lekarskiej starożytnych tekstów. Zacytowanie Przysięgi Hipokratesa oraz przypowieści o dobrym Samarytaninie wskazuje na ich aktualność. Te powszechnie znane teksty syntetyzują wiele szczegółowych zagadnień poruszanych przez bioetykę i etykę lekarską, wnikliwie omówionych w Nowej Karcie, a także, co chyba jest najważniejsze, wytyczają drogę formacji sumienia pracowników służby zdrowia, która winna przebiegać dwutorowo: poprzez refleksję etyczną, dokonywaną na płaszczyźnie rozumowej oraz poprzez zgłębianie na płaszczyźnie wiary przesłania moralnego, płynącego z objawienia chrześcijańskiego, a wyjaśnianego i głoszonego przez nauczanie Kościoła.

Representing rational curve segments and surface patches using semi-algebraic sets

We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the implicit equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional equations and inequalities associated to the implicit equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such as voxels for representing surface patches.