Strange Curves Research Articles

Separable Degree of the Gauss Map and Strict Dual Curves Over Finite Fields

Let $$\mathcal X$$ be a non-degenerate projective algebraic curve and denote by $$\mathcal X^{'}$$ its strict dual curve. The map $$\gamma :\mathcal X\longrightarrow \mathcal X^{'}$$ is called (strict) Gauss map of $$\mathcal X$$ . In this manuscript, we study the separable degree of the Gauss map of curves defined over finite fields. In particular, we give a generalization of a known result on the separable degree of the Gauss map of plane Frobenius nonclassical curves. We also obtain a characterization of certain plane strange curves.

PROJECTIVE CURVES SUCH THAT A GENERAL POINT OF THE AMBIENT PROJECTIVE IS CONTAINED IN A UNIQUE OSCULATING HYPERPLANE OF THE CURVE

We study the non-degenerate integral curves X ⊂ P n such that a general point of P n is contained in a unique osculating hyperplane of X (they are a generalization of the strange curves to the case n > 2).

Tracks in the sand: Hooke’s pendulum cum grano salis

The history of science remembers more than just formal facts about scientific discoveries. These side stories are often inspiring. One of them, the story of an unfulfilled death wish of Jacob Bernoulli regarding spirals, inspired us to look around ourselves. And we saw natural spirals around us, which led to the creation of a Hooke’s pendulum, an artificial creator of Bernoulli’s spira mirabilis. Mathematics helped us to investigate that strange curve and to control this little sandy route to making our predecessor’s wish come true.

Strange curves, counting rabbits, and other mathematical explorations

Strange curves, counting rabbits, and other mathematical explorations

Two-strange projective curves

The notion of second order strange projective curve in \( \mathbb{P}^n \), \( n \geqq 3 \), is introduced. A non–degenerate curve C in \( \mathbb{P}^n \), \( n \geqq 3 \), is said to be a second order strange curve if it admits osculating conics and such that all its osculating conics pass through a common point called 2–strange point. An existence–construction result is provided and some examples are furnished.

Osculating subspaces, normal rational curves and generalized strange curves

Osculating subspaces, normal rational curves and generalized strange curves

On the h-vector of a cohen-macaulay domain in positive characteristic

Here we study the Hilbert function of a Cohen-Macaulay homogeneous domain over an algebraically closed field of positive characteristic. The main tool (and an essential part of the main geometrical results) is the study of the Hilbert function of a general hyperplane section X⊂P r of an integral curve C⊂P r+1 , which is pathological in some sense. In §1, we study the case when Cis a strange curve, i.e., all tangent lines to Cat its simple points pass through a fixed point υ∈P r+1 . In §2, we give more refined results under the assumption that the Trisecant Lemma fails for C, i.e., any line spanned by two points of Ccontains one more point of C.

Plane curves whose tangent lines at collinear points are concurrent

We are interested in a particular geometry of plane curves in characteristicp>0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degreeq + 1, whereq is a power ofp.

Strange curves

Strange curves

Nonharmonic Oscillations as Caused by Magnetic Saturation

It is shown that the distorted curve shapes of natural oscillations can be rigorously derived from the saturation characteristic of the oscillatory circuit by means of simple integrations. The natu-ral frequency is not constant as under linear conditions, but varies greatly with the amount of saturation. For forced oscillations, a concise dif-ferential equation is given covering all possible cases by only four parameters. The solution can be evaluated for any initial conditions by a straight-forward step-by-step construction. It shows graphically strange curve shapes of flux, current, and voltages, of no regularity with respect to periodic repetition or sym-metrical behavior during the transient state. In the steady state, a rigorous lineariza-tion of the differential equation allows considering the effect of saturation quan-titatively as a distortion of the impressed voltage. The final effect on magnitude and curve shape can be evaluated by repeated superposition of the residual dis-torting voltage. Intense higher har-monics are produced in this way. Consideration of the resistance voltage as actually present in the state of free oscillations shows that natural oscilla-tions can be sustained in true resonance by an impressed voltage of definite magni-tude and curve shape, requiring in series circuits a highly peaked voltage curve. Hence, the saturated circuit can respond to any constituent harmonic of this shape, leading to the forced development of sub-harmonics to the frequency of the supply voltage within certain ranges of its magni-tude. If the ohmic resistance is small a multitude of such subharmonics and all their higher harmonics may develop.