Choice Of Curve Research Articles

Curve-lifted codes for local recovery using lines

AbstractIn this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over $$\mathbb {F}_{2^r}$$ F 2 r . These codes are designed for locality and availability, and their particular parameters depend on the choice of curve and its properties. Due to the construction, the numbers of rational points of intersection between curves and lines play a key role. To demonstrate that and generate new families of locally recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted codes.

The valuative tree is the projective limit of Eggers-Wall trees

Consider a germ $C$ of reduced curve on a smooth germ $S$ of complex analytic surface. Assume that $C$ contains a smooth branch $L$. Using the Newton-Puiseux series of $C$ relative to any coordinate system $(x,y)$ on $S$ such that $L$ is the $y$-axis, one may define the {\em Eggers-Wall tree} $\Theta_L(C)$ of $C$ relative to $L$. Its ends are labeled by the branches of $C$ and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically $\Theta_L(C)$ into Favre and Jonsson's valuative tree $\mathbb{P}(\mathcal{V})$ of real-valued semivaluations of $S$ up to scalar multiplication, and to show that this embedding identifies the three natural functions on $\Theta_L(C)$ as pullbacks of other naturally defined functions on $\mathbb{P}(\mathcal{V})$. As a consequence, we prove an inversion theorem generalizing the well-known Abhyankar-Zariski inversion theorem concerning one branch: if $L'$ is a second smooth branch of $C$, then the valuative embeddings of the Eggers-Wall trees $\Theta_{L'}(C)$ and $\Theta_L(C)$ identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space $\mathbb{P}(\mathcal{V})$ is the projective limit of Eggers-Wall trees over all choices of curves $C$. As a supplementary result, we explain how to pass from $\Theta_L(C)$ to an associated splice diagram.

Considerations of “Combined Probability of Injury” in the Next-Generation USA Frontal NCAP

Objective: The numerical basis for assigning star ratings in the next-generation USA New Car Assessment Program (NCAP) for frontal impacts was assessed. That basis, the Combined Probability of Injury, or CPI, is the probability of an occupant sustaining an injury to any of the specified body regions. For an NCAP test, a CPI value is computed by (a) using risk curves to convert body-region responses from a test dummy into body-region risks and (b) using a theoretical, overarching CPI equation to convert those separate body-region risks into a single CPI value. Though the general concept of applying a CPI equation to assign star ratings has existed since 1994, there will be numerous changes to the 2011 frontal NCAP: there will be two additional body regions (n = 4 vs. 2), the injury probabilities will be evaluated for lower-severity (more likely) injury levels, and some of the occupant responses will change. These changes could yield more disperse CPIs that could yield more disperse ratings. However, the reasons for this increased dispersion should be consistent with real-world findings. Related assessments were the topic of this two-part study, focused on drivers. Methods: In Part 1, the CPI equation was assessed without applying risk curves. Specifically, field injury probabilities for the four body regions were used as inputs to the CPI equation, and the resulting equation-produced CPIs were compared with the field CPIs. In Part 2, subject to analyses of test dummy responses from recent NCAP tests, the effect of risk curve choice on CPIs was assessed. Specifically, dispersion statistics were compared for CPIs based on various underlying risk curves applied to data from 2001–2005 model year vehicles (n = 183). Results and Conclusions: From Part 1, the theoretical CPI equation for four body regions demonstrated acceptable fidelity when provided field injury rates (R2= 0.92), with the equation-based CPIs being approximately 12 percent lower than those of ideal correlation. From Part 2, the 2011 NCAP protocol (i.e., application of a four-body-region CPI equation whose inputs were from risk curves) generally increased both the CPIs and their dispersion relative to the current NCAP protocol. However, the CPIs generally increased due to an emphasis on neck injury—an emphasis not observed in real-world crashes. Subject to alternative risk curves for the neck and chest, again there was increased dispersion of the CPIs, but the unrealistic emphasis on the neck was eliminated. However, risk estimates for the knee/thigh/hip (KTH) for NCAP-type events remained understated and did not fall within the confidence bands of the field data. Accordingly, KTH risk estimation is an area for future research.

Isoparametric and Spiral Toolpath for Free-form Surfaces Machining

By analysis of the validity of tool path, tool path planning is divided into two processes, the choice of curves on surface and the design of effective and rational path arrangement. On this basis, a spiral topology toolpath for continuous cutfing is developed in order to reduce the number of interruptions and rapid traversals during high-speed machining (HSM) process. Based on the isoparametric toolpath, the contour-parallel isoparametric path is firstly created, and then a spiral path is generated as a kind of diagonal curve between the isoparametric offset curves. Experimental results demonstrate that the proposed method is straightforward and avoids the self-intersection of the toolpath, especially suitable for HSM.

Evolutionary optimisation in chute design

Chute design is a complex process that takes into account many competing objectives and relies heavily on the intuition of an experienced designer. In this paper, we examine one small part of chute design, which is the choice of curve or profile that the chute follows. Overall, this choice relies on minimising wear in the chute/product and impact on the belt. The optimisation problem is a complex one with nonlinearities in both the governing equation and the objective function. In this paper, we solve the optimisation problem by writing it as an evolutionary system, which is subject to Darwinian natural selection. The evolutionary method is detailed, and optimal loading chute profiles are generated using an evolutionary strategy.

Orbits of Curves on Certain K3 Surfaces

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in \(\mathbb{P}^{2} { \times }\mathbb{P}^{2} \) defined over \(\mathbb{C}\) and with Picard number 3. We describe the group of automorphisms \(\mathcal{A} = Aut (V / {\mathbb{C}})\) on V. For an ample divisor D and an arbitrary curve C0 on V, we investigate the asymptotic behavior of the quantity \(N_{\mathcal{A}{\text{(}}C_0 {\text{)}}} (t) = \# \{ C \in \mathcal{A}{\text{(}}C_0 {\text{)}}\;{\text{:}}\;C \cdot D < t\} \). We show that the limit $$\mathop {\lim }\limits_{t \to \infty } \frac{{\log N_{\mathcal{A}(C)} (t)}}{{\log t}} = \alpha $$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.

The Static and the Dynamic in Statistical Demand Curves

Introduction, 36. — I. Controversy concerning the nature of statistical demand curves, 37; Distinction between “time” and “dynamics,” 39.— II. Different types of “demand” relationships, 42.— III. Statistical determination of demand curves, static conditions, 43; Simple dynamic conditions, 43; First and second order dynamics, 44; Trend-ratios, 45; Link relatives, 46; Different phenomena measured, 46; Distortion by link relatives, 47.— IV. The scatter-diagram — choice of curves, 49; Demand curves for individual years, and shifts, 53.— V. Correlated changes of demand and supply, 54; Dealers' and speculators' demands, 55; Pigou's method, 56; Douglas' method, 56; Leontief's method, 57; Mrs. Gilboy's method, 58; Multiple and partial correlation methods, 59.— Conclusions, 61.