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Introduction Interpolation Shape Functions Choice of Interpolation Shape Functions Available Interpolation Functions for Classical Finite Strips Longitudinal Functions for Spline Finite Strips Computed Shape Function Strips Constant Strain Conditions Plate Bending Problems Introduction Formulation for Thin Plate Bending Classical Plate Strips Spline Bending Strips Computed Shape Function (COMSFUN) Strips Curved Plate Strips Skew Plates Analysis of Moderately Thick Plates Least Square Formulation of Plate Bending Problems Plane Stress Analysis Introduction Classical Plane Stress Strips Spline Plane Stress Strips Computed Shape Function Strip (COMSFUN) Curved Plane Stress Strips Examples Mixed Formulation of Plane Stress Analysis Vibration and Stability Matrix Theory of Free Vibration Derivation of Consistent Mass Matrix of Classical Strips Derivation of Consistent Mass Matrix of Spline Strip Examples Linear Instability and Geometric (Stress) Stiffness Matrix Vibration Analysis of Moderately Thick Plates Modelling of Three Dimensional Solids: Finite Prism and Finite Layer Methods Introduction Finite Prism Method Finite Layer Method Domain Transformation: Treatments for Arbitrarily Shaped Structures Introduction Parametric Representation of Arbitrary Plane Curves Transformation for Arbitrarily Shaped Quadrilateral Plates Transformation of Arbitrarily Shaped Shells Transformation for Longitudinally Arbitrarily Curved Box Girders Applications to Shell Structures and Bridges Shell Structures Bridges Applications to Tall Buildings General Consideration and Structural Idealization Strip Elements Line Elements Discrete Beam Model Examples Applications to Layered Systems in Structural and Geotechnical Engineering Sandwich and Laminated Plates Geotechnical Problems Non-Linear Analysis Introduction Solution for Non-Linear System Material Non-Linearity Geometric Non-Linearity Non-Linear Vibration Transformation Approach: Fourier and Hankel Transforms Introduction Plane Strain Analysis Circular Loading Three-Dimensional Problem Viscoelastic Material (Creep) Consolidation Analysis Transport Problems Appendix: Program for the Analysis of Folded Plates and Box Girder Bridges by Classical Finite Strip Index

Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than $2$, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves., 24 pages, 20 figures

International audience; We introduce the concept of a good Saito basis for a plane curve and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we give a lower bound for the Tjurina number in terms of the Milnor number that allow us to present a positive answer for a question of Dimca and Greuel.

Maximal functions and singular integrals along a class of flat variable plane curves are studied, and Lp estimates for 1 < p < ∞ are obtained.

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^2$ of degree $n$. This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$, we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in Pic$(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz-Harris about whether the canonical class of the Severi variety is effective., 6 pages. Comments or suggestions welcome

We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral complex and compute its dimension. When the genus is at most 1, we show that all the tropical divisors that move in the expected dimension are realizable. As part of the proof, we introduce a combinatorial tool for explicitly constructing large families of realizable tropical divisors., 17 pages, 6 figures. Major revisions from previous version

In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and the topology of plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve inflectional tangent lines.

In this paper, we study the geometry of two-torsion points of elliptic curves in order to distinguish the embedded topology of reducible plane curves consisting of a smooth cubic and its tangent lines. As a result, we obtain a new family of Zariski tuples consisting of such curves.

The purpose of these notes is to explain why a generic projection to a plane of a reduced germ of complex analytic space curve is a bi-Lipschitz homeomorphism for the outer metric. This is related to the fact that all topologically equivalent germs of plane curves are exactly the generic projections of a single germ of a space curve. The analytic algebra of this germ is the algebra of Lipschitz meromorphic functions on any of its generic projections. An application to the Lipschitz geometry of polar curves is given.

This text is about the algebra of germs of Lipschitz meromorphic functions on a germ of reduced complex analytic space (X, 0). It is shown to be an analytic algebra, the Lipschitz saturation of the algebra of (X, 0), which in some important cases coincides with Zariski’s algebraic saturation. In the case of reduced germs of plane curves, the results in Sect. 10.6 imply that two such germs are topologically equivalent if and only if their Lipschitz saturations are analytically isomorphic. Applications to bi-Lipschitz equisingularity are given.

We determine the distribution of Galois points for plane curves over a finite field of $q$ elements, which are Frobenius nonclassical for different powers of $q$. This family is an important class of plane curves with many remarkable properties. It contains the Dickson-Guralnick-Zieve curve, which has been recently studied by Giulietti, Korchmaros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified., Comment: 8 pages

Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the M&ouml, bius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper.

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs (C, o) of complex analytic curves contained in a smooth complex analytic surface S. The embedded topological type of such a pair (S, C) is usually defined to be that of the oriented link obtained by intersecting C with a sufficiently small oriented Euclidean sphere centered at the point o, defined once a system of local coordinates (x, y) was chosen on the germ (S, o). If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of (S, C). One may define it by looking either at the Newton-Puiseux series associated to C relative to a generic local coordinate system (x, y), or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ (C, o) or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of (C, o) by successive toric modifications.

In this paper, we aim to get the general formulation of evolving generalized plane curves for motion by an inelastic and an elastic flow in the Galilean plane. We get the linear and nonlinear partial differential equations. Then the new classes of symmetry reductions of these type equations are designated and exact solutions are investigated. Finally, we give applications of evolutions.

We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed., v2. Terminology and references improved following very useful comments by Gert-Martin Greuel

The traveling waves for area-preserving geometric flows will be discussed. For an evolving plane curve, which is governed by the area-preserving curvature flow and has two endpoints moving freely on the $x$-axis with generating constant contact angles, we prove that there exists only a convex traveling wave. In the proof, a boundary value problem for an angle function associated with the Gauss map of the profile curve is analyzed by means of a shooting argument.

Özgür, Nihal (Balikesir Author), We present a new type of activation functions for a complex-valued neural network (CVNN). A proposed activation function is constructed such that it fixes a given ellipse. We obtain an application to a complex-valued Hopfield neural network (CVHNN) using a special form of the introduced complex functions as an activation function. Considering the interesting geometric properties of the plane curve ellipse such as focusing property, we emphasize that these properties may have possible applications in various neural networks.

The Poincare series of a ring associated to a plane curve was defined by Campillo, Delgado, and Gusein-Zade. This series, defined through the value semigroup of the curve, encodes the topological information of the curve. In this paper we extend the definition of Poincare series to the class of good semigroup ideals, to which value semigroups of curves belong. Using this definition we generalize a result of Pol: under suitable assumptions, given good semigroup ideals E and K, with K canonical, the Poincare series of K − E is symmetric to the Poincare series of E.

It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree $q^2+q+1$ for which the Gauss map is inseparable with inseparable degree $q$. As a special case, we show a relation between the dual curve of the Fermat curve of degree $q^2+q+1$ and the Ballico-Hefez curve., Comment: 12 pages

Motivated by the classical Euler elastic curves and the Bernoulli's lemniscate, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. The aim of this talk is the analysis of the authors' contribution to the above problem when the curvature is prescribed in terms of the distance to a line.

We demonstrate the possibility to create optical beams with phase singularities engraved into exotic intensity landscapes imitating the shapes of a large variety of diverse plane curves. To achieve this aim, we have developed a method for directly encoding the geometric properties of some selected curve into a single azimuthal phase factor without passing through indirect encryption methods based on lengthy numerical procedures. The outcome is utilized to mould the optic axis distribution of a liquid-crystal-based inhomogeneous waveplate. The latter is finally used to sculpt the wavefront of an input optical gaussian beam via Pancharatnam-Berry phase., Comment: 11 pages, 6 figures, Published on 21 April 2020 Front. Phys

The present work is motivated by the problem of mathematical handwriting recognition where symbols are represented as parametric plane curves in a Legendre-Sobolev basis. An early work showed that approximating the coordinate functions as truncated series in a Legendre-Sobolev basis yields fast and effective recognition rates. Furthermore, this representation allows one to study the geometrical features of handwritten characters as a whole. These geometrical features are equivalent to baselines, bounding boxes, loops, and cusps appearing in handwritten characters. The study of these features becomes a crucial task when dealing with two-dimensional math formulas and the large set of math characters with different variations in style and size.

In this paper, we give a definition of $\mathbb{Z}$-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by $\sum_i \alpha_i x_i$. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the chord diagrams with at most $l$ chords, called relators of Type (I) ((SII), (WII), (SIII), or (WIII), resp.), and introduce another function $\sum_i \alpha_i \tilde{x}_i$ derived from $\sum_i \alpha_i x_i$. The main result (Theorem~1) shows that if $\sum_i \alpha_i \tilde{x}_i$ vanishes for the relators of Type (I) ((SII), (WII), (SIII), or (WIII), resp.), then $\sum_i \alpha_i x_i$ is invariant under the Reidemeister move of type RI (strong RII, weak RII, strong RIII, or weak RIII, resp.) that is defined in [Ito-Takimura (2013), J. Knot Theory Ramifications]., Comment: 21 pages, 12 figures, 1 table. arXiv admin note: text overlap with arXiv:1908.06085

We consider the geometry of regular holomorphic curves in C 2 \mathbb C^2 viewed as surfaces in the affine space R 4 \mathbb R^4 . We study the A \mathcal A -singularities of parallel projections of generic such surfaces along planes to transverse planes. We show that at any point on the surface which is not an inflection point of the curve there are two tangent directions determining two planes along which the projection has singularities of type butterfly or worse. The integral curves of these directions form a pair of foliations on the surface defined by a binary differential equation (BDE). The singularities of this BDE are the inflection points of the curve together with other points that we call butterfly umbilic points. We determine the configurations of the solution curves of the BDE at its singularities. Finally, we prove that an affine view of an algebraic curve of degree d ≥ 2 d\ge 2 in C P 2 \mathbb CP^2 has 8 d ( d − 2 ) 8d(d-2) butterfly umbilic points.

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., Comment: Mathematics of Computation, American Mathematical Society, In press

In the note we study the multipoint Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial., Comment: 12 pages, accepted in Electronic Research Archive

We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points defined over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree., Comment: 17 pages

Aiming at the problem of low efficiency of the 3D reconstruction, a fast method based on piecewise quadratic curve is proposed. Firstly, a multi-line structured light binocular vision measurement system is used to obtain the two-dimensional image. Through the two-dimensional image, the center line and feature points of the laser strip are quickly acquired. Then, the acquired feature points are used as the given data points of curve interpolation, and NURBS curves are used to interpolate the feature lines of the image. According to the perspective invariance principle of plane curve, the zero curvature point and cusp point on the curve are detected and used as segmentation points. Using the segmentation results, a number of points on each NURBS curve segment are taken to fit as the quadratic curve. Finally, by constructing the right and left space straight taper surfaces, the three-dimensional reconstruction of the feature lines is completed by the method of surface intersection. Taking the black shoe sole as the experimental object, the results show that the method proposed in this paper can quickly complete the three-dimensional reconstruction of the inner and outer feature lines of the shoe sole, which can provide the basis and guarantee for the follow-up shoe sole process monitoring and quality inspection.

In this paper, the bounded properties of oscillatory hyper-Hilbert transformalong certain plane curves γ(t),are studied. For general curves, these operators are bounded in L2() if β ≥ 3α. Their boundedness in Lp() is also obtained, whenever β > 3α and .

For finding global approximate solutions to an algebraic equation in n unknowns, the Hadamard open polygon for the case n = 1 and Hadamard polyhedron for the case n = 2 are used. The solutions thus found are transformed to the coordinate space by a translation (for n = 1) and by a change of coordinates that uses the curve uniformization (for n = 2). Next, algorithms for the local solution of the algebraic equation in the vicinity of its singular (critical) point for obtaining asymptotic expansions of one-dimensional and two-dimensional branches are presented for n = 2 and n = 3. Using the Newton polygon (for n = 2), the Newton polyhedron (for n = 3), and power transformations, this problem is reduced to situations similar to those occurring in the implicit function theorem. In particular, the local analysis of solutions to the equation in three unknowns leads to the uniformization problem of a plane curve and its transformation to the coordinate axis. Then, an asymptotic expansion of a part of the surface under examination can be obtained in the vicinity of this axis. Examples of such calculations are presented.

A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described., 7 pages; Changed the title, Added the reference [10], Made some minor changes

The paper is devoted to diminishing of the edge effect in three nonstandard dished heads of a cylindrical pressure vessel subjected to internal uniform pressure. The problem of the edge effect diminishing in the joint of the dished head with the cylindrical shell is analytically and numerically studied. The meridians of the analysed dished heads as the shells of revolution are plane curves in the Cassini oval, Booth lemniscate and clothoid forms. Geometrical relationships of the middle surfaces of the dished heads are formulated. The stress state of these dished heads are analytically and numerically studied using finite element method in Ansys system. The results of the studies are compared and presented in Tables and Figures.

This paper studies the shape similarity evaluation of free-form surfaces expressed by B-spline with single curvature feature and proposes a similarity evaluation algorithm based on curvature feature. Firstly, we calculate the normal vector direction of the two surfaces compared, and use it as the Z axis, so that the two surfaces are aligned on the Z axis. Then, the two surfaces are cut with planes that all perpendicular to the Z axis, and the intersection sets of two surfaces are obtained respectively. Finally, we design the similarity algorithm of plane curves to realize the similarity comparison of corresponding curves in the two sets of intersection, and which is used as the basis for evaluating the similarity between two surfaces. The algorithm transforms the problem of similarity comparison between 3D surfaces into two dimensional space by plane cutting method, and reduces the complexity of the problem effectively. The algorithm only needs to align one coordinate axis in the process of posture adjustment, so it is easy to implement. In order to test the effect of the algorithm, simulation experiments on different type of single curvature feature B-spline surfaces are carried out. The results show that the proposed similarity comparison algorithm of free-form surfaces is feasible and effective.

Let $$\Sigma _1$$ be a Hirzebruch surface of degree one. Let C be a smooth curve on $$\Sigma _1$$ which is given as the proper transform of a smooth plane curve $$C_0$$ of degree $$d\ge 4$$ by the blow up at one point on $$C_0$$ , and let P be a point on C. Let $$\pi :\tilde{C}\longrightarrow C$$ be a double covering of C, and assume that P is a branch point of $$\pi $$ . In this paper, we investigate the Weierstrass semigroup at the ramification point of $$\pi $$ over P and a necessary and sufficient condition for $$\pi $$ to extend to a double covering $$\tilde{\pi }:X\longrightarrow \Sigma _1$$ branched along a reduced divisor belonging to $$|-2K_{\Sigma _1}|$$ , where $$K_{\Sigma _1}$$ is the canonical bundle of $$\Sigma _1$$ .

The splitting number of a plane irreducible curve for a Galois cover is effective to distinguish the embedded topologies of plane curves. In this paper, we define a connected number of any plane curve for a Galois cover whose branch divisor has no common components with the plane curve, which is similar to the splitting number. We classify the embedded topology of Artal arrangements of degree $b\geq 4$ by the connected number, where an Artal arrangement of degree $b$ is a plane curve consisting of one smooth curve of dgree $b$ and three total inflectional tangents., Comment: 12 pages

For a relative effective divisor $$\mathcal {C}$$ on a smooth projective family of surfaces $$q:\mathcal {S}\rightarrow B$$ , we consider the locus in B over which the fibres of $$\mathcal {C}$$ are $$\delta $$ -nodal curves. We prove a conjecture by Kleiman and Piene on the universality of an enumerating cycle on this locus. We propose a bivariant class $$\gamma (\mathcal {C})\in A^*(B)$$ motivated by the BPS calculus of Pandharipande and Thomas, and show that it can be expressed universally as a polynomial in classes of the form $$q_*(c_1(\mathcal {O}(\mathcal {C}))^a c_1(T_{\mathcal {S}/B})^b c_2(T_{\mathcal {S}/B})^c)$$ . Under an ampleness assumption, we show that $$\gamma (\mathcal {C})\cap [B]$$ is the class of a natural effective cycle with support equal to the closure of the locus of $$\delta $$ -nodal curves. Finally, we apply our method to calculate node polynomials for plane curves intersecting general lines in $$\mathbb {P}^3$$ . We verify our results using nineteenth century geometry of Schubert.

We study bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves., Comment: 8 pages, 3 figures. Initiated at the Fields Undergraduate Summer Research Program

A generalisation is found of previously known quantised spiral beams in the form of closed non-self-intersecting plane curves. An analytic expression is obtained for the distributions of the complex amplitudes of the fields of these beams. It is shown that the intensity of such light fields can reach zeros on the generating curve and the generalisation allows the number of zeros to be controlled inside the closed curves, which is important for applied problems. The issue of the extremal properties of the orbital angular momentum of generalised quantised spiral beams is investigated.

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ⊂ C be finite sets of size n. We show that F vanishes on at most O(n8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabo theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.

We study the boundedness problem for maximal operators $${\mathbb {M}}^{\sigma }$$ associated with flat plane curves with mitigating factors, defined by $$\begin{aligned} {\mathbb {M}}^{\sigma }f(x) \, := \, \sup _{1 \le t \le 2} \left| \int _{0}^{1} f(x-t\Gamma (s)) \, \kappa (s)^{\sigma } \, \mathrm{d}s\right| , \end{aligned}$$ where $$\kappa (s)$$ denotes the curvature of the curve $$\Gamma (s)=(s, g(s)+1), ~g(s) \in C^5[0,1]$$ in $${\mathbb {R}}^2$$ . Let $$\triangle $$ be the closed triangle with vertices $$P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).$$ In this paper, we prove that for $$ (\frac{1}{p}, \frac{1}{q}) \in \left( \triangle {\setminus } \{P, Q\} \right) \cap \{(\frac{1}{p}, \frac{1}{q}) :q > \text{ max }\{\sigma ^{-1},2\} \}$$ , there is a constant B such that $$ \Vert {\mathbb {M}}^{\sigma }f\Vert _{L^q({\mathbb {R}}^2)} \le \, B \, \Vert f\Vert _{L^p({\mathbb {R}}^2)}.$$

The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularity f. Assuming that f is irreducible, one can write $$ f = \sum _{q\alpha + p\beta \ \ge \ pq}c_{\alpha \beta }\ x^{\alpha }y^{\beta } $$ where $$c_{p0}c_{0q}\ne 0$$ , $$2\le p

For a smooth complex curve C ⊂ ℂ2 we consider the link Lr = C ∩ ∂Br, where Br denotes an Euclidean ball of radius r > 0. We prove that the diagram Dr obtained from Lr by a complex stereographic projection satisfies χ(C ∩Br) = rot(Dr)−wr(Dr). As a consequence we show that if Dr has no negative Seifert circles and Lr is strongly quasipositive and fibered, then the Yamada–Vogel algorithm applied to Dr yields a quasipositive braid.

Let (X,L) be a polarized manifold of dimension n. Its Hilbert curve is an affine algebraic plane curve of degree n encoding properties related to fibrations of X, defined by suitable adjoint ...

Let X be a smooth irreducible algebraic curve of genus g . The projective normality of a complete embedding of X is determined by only its quadratic normality in case the embedding is of degree at least g . This means that the complete embedding fails to be projectively normal if and only if it admits an effective divisor which fails to impose independent conditions on quadrics in the embedded projective space. Thus if X admits a net, then it is interesting to compare the conditions for the projective normality of an embedding of X with properties of conic sections of the plane curve given by the net. For such a curve X with a net, we show that the projectively normal embeddings are closely related to properties of conic sections.

An invariant submodel of gas dynamics equations constructed on a three-dimensional subalgebra with a projective operator for the case of monatomic gas is under consideration. The submodel is reduced to an Abel equation, with integral curves constructed for it. For a separatrix of a saddle, an approximate solution is studied. Such solutions describe the vortex scattering of gas along plane curves placed on the surface of revolution.