More Like this

1. CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON ℝ2

   https://doi.org/10.1142/S0218348X19500993  

   DE LEO, ROBERTO (September 2019, Fractals)

   We collect from several sources some of the most important results on the forward and backward limits of points under real and complex rational functions, and in particular real and complex Newton maps, in one variable and we provide numerical evidence that the dynamics of Newton maps [Formula: see text] associated to real polynomial maps [Formula: see text] with no complex roots has a complexity comparable with that of complex Newton maps in one variable. In particular such a map [Formula: see text] has no wandering domain, almost every point under [Formula: see text] is asymptotic to a fixed point and there is some non-empty open set of points whose [Formula: see text]-limit equals the set of non-regular points of the Julia set of [Formula: see text]. The first two points were proved by B. Barna in the real one-dimensional case. 

   more » « less
2. Tri-plane diagrams for simple surfaces in S4

   https://doi.org/10.1142/S0218216523500414  

   Allred, Wolfgang; Aragón, Manuel; Dooley, Zack; Goldman, Alexander; Lei, Yucong; Martinez, Isaiah; Meyer, Nicholas; Peters, Devon; Warrander, Scott; Wright, Ana; et al (June 2023, Journal of Knot Theory and Its Ramifications)

   Meier and Zupan proved that an orientable surface [Formula: see text] in [Formula: see text] admits a tri-plane diagram with zero crossings if and only if [Formula: see text] is unknotted, so that the crossing number of [Formula: see text] is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in [Formula: see text], proving that [Formula: see text], where [Formula: see text] denotes the connected sum of [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text] and [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text]. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers. 

   more » « less
3. Algorithms for Covering Barrier Points by Mobile Sensors with Line Constraint

   https://doi.org/10.1142/S0218195924500018  

   Jain, Princy; Wang, Haitao (February 2024, International Journal of Computational Geometry & Applications)

   We study the problem of covering barrier points by mobile sensors. Each sensor is represented by a point in the plane with the same covering range [Formula: see text] so that any point within distance [Formula: see text] from the sensor can be covered by the sensor. Given a set [Formula: see text] of [Formula: see text] points (called “barrier points”) and a set [Formula: see text] of [Formula: see text] points (representing the “sensors”) in the plane, the problem is to move the sensors so that each barrier point is covered by at least one sensor and the maximum movement of all sensors is minimized. The problem is NP-hard. In this paper, we consider two line-constrained variations of the problem and present efficient algorithms that improve the previous work. In the first problem, all sensors are given on a line [Formula: see text] and are required to move on [Formula: see text] only while the barrier points can be anywhere in the plane. We propose an [Formula: see text] time algorithm for the problem. We also consider the weighted case where each sensor has a weight; we give an [Formula: see text] time algorithm for this case. In the second problem, all barrier points are on [Formula: see text] while all sensors are in the plane but are required to move onto [Formula: see text] to cover all barrier points. We also solve the weighted case in [Formula: see text] time. 

   more » « less
4. On maps preserving square roots of idempotent and rank-one nilpotent matrices

   https://doi.org/10.1142/S0219498822501237  

   Borisov, Nikita; Julius, Hayden; Sikora, Martha (January 2020, Journal of Algebra and Its Applications)

   null (Ed.)

   We characterize bijective linear maps on [Formula: see text] that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next, we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar linear preserver problems. 

   more » « less
5. K-theoretic Gromov–Witten invariants of line degrees on flag varieties

   https://doi.org/10.1142/S0217751X24460138  

   Buch, Anders S; Chen, Linda; Xu, Weihong (November 2024, International Journal of Modern Physics A)

   A homology class [Formula: see text] of a complex flag variety [Formula: see text] is called a line degree if the moduli space [Formula: see text] of 0-pointed stable maps to X of degree d is also a flag variety [Formula: see text]. We prove a quantum equals classical formula stating that any n-pointed (equivariant, [Formula: see text]-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety [Formula: see text]. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags [Formula: see text]. Our formulas make it straightforward to compute the big quantum [Formula: see text]-theory ring [Formula: see text] modulo the ideal [Formula: see text] generated by degrees d larger than line degrees. 

   more » « less