Multiple Base Points Research Articles

Secant non-defectivity via collisions of fat points

The computation of dimensions of secant varieties of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. Non-defectivity can be proved via degenerations. In this paper, we use a technique that allows some of the base points to collapse together in order to deduce a new general criterion for non-defectivity. We apply this criterion to prove a conjecture by Abo and Brambilla: for c≥3 and d≥3, the Segre-Veronese embedding of Pm×Pn in bidegree (c,d) is non-defective.

Boundary value problems of the nonlinear multiple base points impulsive fractional differential equations with constant coefficients

In this paper, the nonlinear multiple base points boundary value problems of the impulsive fractional differential equations are studied. By using the fixed point theorem and the Mittag-Leffler functions, the sufficient conditions for the existence of the solutions to the given problems are formulated. An example is presented to illustrate the result.

Existence of Solutions of Impulsive Anti-periodic Type Boundary Value Problems for Singular Fractional Differential Systems

Results on the existence of solutions to a new class of impulsive singular fractional differential systems with multiple base points are established. The assumptions imposed on the nonlinearities, see (D1) and (D2) in Theorem 3.1, are weaker than known ones, (i.e., (A) in Introduction section). The analysis relies on a well known fixed point theorem. An example is given to illustrate the efficiency of the main theorems.

Existence of solutions for the nonlinear multiple base points impulsive fractional differential equations with the three-point boundary conditions

In this paper, we study a class of nonlinear multiple base points impulsive fractional differential equations involving the three-point boundary conditions. A new result on the existence of a solution is established by using fixed point theorems. An example is presented to illustrate the result.

Khovanov homology and knot Floer homology for pointed links

A well-known conjecture states that for any [Formula: see text]-component link [Formula: see text] in [Formula: see text], the rank of the knot Floer homology of [Formula: see text] (over any field) is less than or equal to [Formula: see text] times the rank of the reduced Khovanov homology of [Formula: see text]. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose [Formula: see text] page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field [Formula: see text].

New results on the existence of solutions of boundary value problems for singular fractional differential systems with impulse effects

Results on the existence of solutions to a new class of impulsive singular fractional differential systems with multiple base points are established. The assumptions imposed on the nonlinearities, see ((C) and (D) in Theorem 3.1), are weaker than known ones, (i.e., (A) in Introduction

Studies on impulsive differential models with multi-term Riemann–Liouville fractional derivatives

In this article, two impulsive models for fractional differential equations are proposed. By converting the impulsive fractional differential equations into integral equations technically, some existence results on solutions of two classes of initial value problems for nonlinear singular multi-term fractional differential models with impulses involving Riemann–Liouville fractional derivatives are established. The first one has a single base point and the second one has multiple base points. In both cases, the corresponding fractional derivative of lower order is involved in the fractional differential equations. Our results improve some known theorems see Remarks 4.1 and 5.1. Examples are presented to illustrate the main theorems. Finally we give a conclusions section in which some basic errors in a recent paper involving impulsive fractional equations with the Riemann–Liouville derivative are addressed, see Claim 7.1 and Remark 7.2.

A Study of Impulsive Multiterm Fractional Differential Equations with Single and Multiple Base Points and Applications

We discuss the existence and uniqueness of solutions for initial value problems of nonlinear singular multiterm impulsive Caputo type fractional differential equations on the half line. Our study includes the cases for a single base point fractional differential equation as well as multiple base points fractional differential equation. The asymptotic behavior of solutions for the problems is also investigated. We demonstrate the utility of our work by applying the main results to fractional-order logistic models.

IVPs for singular multi-term fractional differential equations with multiple base points and applications

IVPs for singular multi-term fractional differential equations with multiple base points and applications

Study on Automatic Threshold Selection Algorithm of Sensor Images

Abstract The threshold selection of sensor images has great influences on the detection accuracy of the mass center, while which will directly affect the recovery of the wave-front detection. Through introducing several principles and algorithms of the threshold segmentation, based on the detection accuracy of the mass center with relatively multiple base points of images, it has been demonstrated that for lattice images with large contrast, there is no much difference in the standard deviations of solving mass centers by four algorithms, however, the OTSU Algorithm based on gray extension obtain the smallest deviation, therefore, for bitmap images, regarding to the binary threshold selection, in this paper, it adopts the OTSU Algorithm based on gray extension, and then solves the center of mass through conducting the binarization. In the end of this paper, it points out the OTSU Algorithm based on gray extension is a better method for the threshold segmentation.

Quasi-homogeneous linear systems on P2with base points of multiplicity 7, 8, 9, 10

We prove that the Segre–Gimigliano–Harbourne–Hirschowitz conjecture holds for quasi-homogeneous linear systems on $\mathbb P^2$ for $m=7$, 8, 9, 10, i.e. systems of curves of a given degree passing through points in general position with

Linear systems over P1×P1with base points of multiplicity bounded by three

We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general position. As an application, we obtain a classification of special linear systems on ${\mathbb P}^1\times{\mathbb P}^1$ with multiplici

Special Effect Varieties and $(-1)$-Curves

Here we introduce the concept of special effect curve which permits to study, from a different point of view, special linear systems in P^2, i.e. linear system with general multiple base points whose effective dimension is strictly greater than the expected one. In particular we study two different kinds of special effect: the \alpha-special effect is defined by requiring some numerical conditions, while the definition of h^1-special effect concerns cohomology groups. We state two new conjectures for the characterization of special linear systems and we prove they are equivalent to the Segre and the Harbourne-Hirschowitz ones.

A counterexample to a conjecture on linear systems on ℙ3

In his paper (1) Ciliberto proposes a conjecture in order to characterize special linear systems of P n through multiple base points. In this note we give a counterexample to this con- jecture by showing that there is a substantial dierence between the speciality of linear systems on P 2 and those of P 3 . Let us take the projective space P n and let us consider the linear system of hyper- surfaces of degree d having some points of fixed multiplicity. The virtual dimension of such systems is the dimension of the space of degree d polynomials minus the con- ditions imposed by the multiple points and the expected dimension is the maximum between the virtual one and � 1. The systems whose dimension is bigger than the ex- pected one are called special systems. There exists a conjecture due to Hirschowitz (see (5)), characterizing special linear systems on P 2 , which has been proved in some special cases (2), (3), (7), (6). Concerning linear systems on P n , in (1) Ciliberto gives a conjecture based on the classification of special linear systems through double points. In this note we describe a linear system on P 3 that we found in a list of special systems generated with the help of Singular (4) and which turns out to be a counterexample to that conjecture. The paper is organized as follows: in Section 1 we fix some notation and state Ciliberto's conjecture, while Section 2 is devoted to the counterexample. In Section 3 we try to explain speciality of some systems by the Riemann-Roch formula, and we conclude the note with an appendix containing some computations.

Linear systems with multiple base points in ℙ2

Conjectures for the Hilbert function hðn;mÞ and minimal free resolution of the mth symbolic power Iðn;mÞ of the ideal of n general points of P are verified for a broad range of values of m and n where both m and n can be large, including (in the case of the Hilbert function) for infinitely many m for each square n > 9 and (in the case of resolutions) for infinitely many m for each even square n > 9. All previous results require either that n be small or be a square of a special form, or that m be small compared to n. Our results are based on a new approach for bounding the least degree among curves passing through n general points of P with given minimum multiplicities at each point and for bounding the regularity of the linear system of all such curves. For simplicity, we work over the complex numbers.

Quasi-Homogeneous Linear Systems on ℙ2 with Base Points of Multiplicity 5

AbstractIn this paper we consider linear systems of ℙ2 with all but one of the base points of multiplicity 5. We give an explicit way to evaluate the dimensions of such systems.

On the geometric structure of the class of planar quadratic differential systems

In this work we are interested in the global theory of planar quadratic differential systems and more precisely in the geometry of this whole class. We want to clarify some results and methods such as the isocline method or the role of rotation parameters. To this end, we recall how to associate a pencil of isoclines to each quadratic differential equation. We discuss the parameterization of the space of regular pencils of isoclines by the space of its multiple base points and the equivariant action of the affine group on the fibration of the space of regular quadratic differential equations over the space of regular pencils of isoclines. This fibration is principal, with a projective group as structural group, and we prove that there exits an open cone in its Lie algebra whose elements generate rotation parameter families. Finally we use this geometric approach to construct specific families of quadratic differential equations depending in a nonlinear way of parameters which have a geometric meaning: they parameterize the set of singular points or are rotation parameters leaving fixed this set.

On Linear Systems of Curves on Rational Scrolls

In this paper we prove a conjecture on the dimension of linear systems, with base points of multiplicity 2 and 3, on an Hirzebruck surface.