Search

In the present paper we investigate conditions for the non-existence of a smoothable limit linear series on a curve of compact type such that two smooth curves are bridged by a chain of two elliptic curves. Combining this work with results on the existence of a smoothable limit linear series on such a curve, we show relations among Brill–Noether loci of codimension at most two in the moduli space of complex curves. Specifically, Brill–Noether loci of codimension two have mutually distinct supports. [ABSTRACT FROM AUTHOR]

In the present paper, we give an upper bound for the generic degree of the generalized Verschiebung between the moduli spaces of rank two stable bundles with trivial determinant. [ABSTRACT FROM AUTHOR]

By taking full advantage of the structure of complex algebraic curves and by using compactness arguments, in this paper we give a self-contained proof that holomorphic algebraic functions verify a uniform Bernstein–Remez inequality. Namely, their growth over a bounded, open, complex set is uniformly controlled by their size on a compact complex subset of sufficiently high cardinality. Up to our knowledge, the first known demonstration on the existence of such an inequality for a specific subset of algebraic functions is contained in Nekhoroshev's 1973 breakthrough on the genericity of close-to-integrable Hamiltonian systems that are stable over long time. Despite its pivotal rôle, this passage of Nekhoroshev's proof has remained unnoticed so far. This work aims at extending and generalizing Nekhoroshev's arguments to a modern framework. We stress the fact that our proof is different from the one contained in Roytwarf and Yomdin's seminal work (1998), where Bernstein-type inequalities are proved for several classes of functions. [ABSTRACT FROM AUTHOR]

Abstract We study a generalization of genus-zero r -spin theory in which exactly one insertion has a negative-one twist, which we refer to as the "closed extended" theory, and which is closely related to the open r -spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the genus-zero part of a special solution to the system of differential equations for the wave function of the r th Gelfand–Dickey hierarchy. This parallels an analogous result for the open r -spin generating function in the companion paper Buryak et al. (2018) to this work. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we provide a method that allows to construct parametric curves having (or not) non-ordinary singularities and having (or not) neighboring points. This method is based on a characterization of the non-ordinary singularities and neighboring points by means of linear equations involving the given parametrization. As a consequence, we obtain an algorithm that constructs a parametrization which contains a given point, P , as a singularity as well as some additional information as for instance, the order of P , parameters corresponding to P , multiplicity of each parameter and the singularities in the first neighborhood of the singularity P. Highlights • We provide a method that allows to construct parametric curves having (or not) non-ordinary singularities and having (or not) neighboring points. • The method is based on a characterization of the non-ordinary singularities and neighboring points by means of linear equations. • We obtain an algorithm that outputs a parametrization of a rational curve having singularities at some given input points. • In the algorithm, the singularity, P , the order of P , the parameters (and their multiplicity) corresponding to P , and the first neighborhood of P , are fixed. • We translate every detail of the definitions and resolutions into the language of parametric equations, which are quite helpful to CAGD. [ABSTRACT FROM AUTHOR]

• A new 7R unit is proposed to design pyramid and frustum mechanism. • A new RRR chain is proposed to fold the frustum into a bundle. • A general frame is proposed to design 1 DOF deployable mechanism. • Two methods are proposed to fold a polyhedron into a bundle. In this paper, a series of deployable polygon mechanisms (DGMs) and deployable polyhedron mechanisms (DPMs) are proposed. Firstly, we propose a new plane-symmetric 7R unit with folded lateral edges. Kinematics and suitable parameter values are calculated to realize full expansion. N-pyramid and N-frustum and a larger DPM are designed. Secondly, we propose a new RRR chain to connect existing deployable units. Threefold Bricard mechanism and alternative Bennett mechanism are used as two examples to fold triangular and square frustum into a bundle. Thirdly, we propose a general frame to design 1 DOF deployable mechanism based on the continuation method. Three improvements in accuracy, speed, and generality are made. Feasible points of general and special N-side DGMs are calculated (3≤N≤7) based on plane-symmetric hexagon 6R and heptagon 7R mechanisms. Our results not only unify most deployable units but also design many new ones and reconfigurable DGMs. Fourthly, we propose the polyhedron net method and the least joint angle method to fold a polyhedron into a bundle and reduce the number of links, joints, and DOF. Many DPMs are designed. [ABSTRACT FROM AUTHOR]

In this paper, a general theoretical study, from the perspective of the algebraic geometry, of the untrimmed bisector of two real algebraic plane curves is presented. The curves are considered in C 2 , and the real bisector is obtained by restriction to R 2 . If the implicit equations of the curves are given, the equation of the bisector is obtained by projection from a variety contained in C 7 , called the incidence variety, into C 2 . It is proved that all the components of the bisector have dimension 1. A similar method is used when the curves are given by parametrizations, but in this case, the incidence variety is in C 5 . In addition, a parametric representation of the bisector is introduced, as well as a method for its computation. Our parametric representation extends the representation in Farouki and Johnstone (1994b) to the case of rational curves. [ABSTRACT FROM AUTHOR]

In this paper, we aim for the theta function representation of quasi-periodic solution and related crucial quantities for a two-component generalization of Burgers' equation. Our tools include the theory of algebraic curves, meromorphic functions, Baker—Akhiezer functions and the Dubrovin-type equations for auxiliary divisor. Eith these tools, the explicit representations for above quantities are obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two. [ABSTRACT FROM AUTHOR]

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in F q n × n the idealizers have been proved to be isomorphic to finite fields of size at most q n. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in F q n × n for n ≤ 9 with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealizers for n = 7 , q odd and for n = 8 , q ≡ 1 (mod 3). These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for n ≥ 9. [ABSTRACT FROM AUTHOR]

In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus 4 over F 11. Our algorithm determining automorphism groups works for any nonhyperelliptic curve of genus 4 over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus 4 over F 11 obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory. [ABSTRACT FROM AUTHOR]