A Class of Sextic Trigonometric Bézier Curve with Two Shape Parameters

In this paper, we present a new class of sextic trigonometric Bernstein (ST-Bernstein, for short) basis functions with two shape parameters along with their geometric properties which are similar to the classical Bernstein basis functions. A sextic trigonometric Bézier (ST-Bézier, for short) curve with two shape parameters and their geometric characteristics is also constructed. The continuity constraints for the connection of two adjacent ST-Bézier curves segments are discussed. Shape control parameters can provide an opportunity to modify the shape of curve as designer desired. Some open and closed curves are also part of this study.

Ready-made short basis for GLV+GLS on high degree twisted curves

<abstract><p>The crucial step in elliptic curve scalar multiplication based on scalar decompositions using efficient endomorphisms—such as GLV, GLS or GLV+GLS—is to produce a short basis of a lattice involving the eigenvalues of the endomorphisms, which usually is obtained by lattice basis reduction algorithms or even more specialized algorithms. Recently, lattice basis reduction is found to be unnecessary. Benjamin Smith (AMS 2015) was able to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS of quadratic twists using elementary facts about quadratic rings. Certainly it is always more convenient to use a ready-made short basis than to compute a new one by some algorithm. In this paper, we extend Smith's method on GLV+GLS for quadratic twists to quartic and sextic twists, and give ready-made short bases for $ 4 $-dimensional decompositions on these high degree twisted curves. In particular, our method gives a unified short basis compared with Hu et al.'s method (DCC 2012) for $ 4 $-dimensional decompositions on sextic twisted curves.</p></abstract>

Relevance of Leverage and Profitability in Risk-Return Paradox: A Study from Pakistan Stock Exchange

The purpose of this investigation has to investigate the degree of relevance of these components in the domain of asset pricing by augmenting the basic FF-5 factor model with leverage and profitability that whether leverage-profitability dumping portfolios capturing the return by this model. We performed the five-factor model of Fama-French to capture the joint effect of variables and augmented it with leverage and profitability for the Pakistani market utilizing current periodic month-to-month data from the year 1998 to the year 2019. Construction of factors (MKT, SMB, HML, HLMLL, and RMW) is done for the PSX market and scrutinize that how much this five-factor augmented model detentions the returns premium in the Pakistani equity market. To analyze the model experimentally that consolidate subgroups of its factors that explain the portfolio's normal returns are outlined to make tremendous spreads in size, B/M, leverage and profitability and to capture the pattern for observing their imperativeness, 2X2X2X2 sorted version is adopted to construct the factors. Investigation documents the significant evidence on the leverage and profitability premium. Taking a glimpse at the variables alone, the recently presented leverage and profitability factors show alluring, measurably critical returns for a long/short basis for the Pakistani equity market.