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We introduce the notion of a relative of the Hermitian curve of degree $\sqrt{q}+1$ over $\mathbb{F}_q$, which is a plane curve defined by \[(x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A {}^t \!(x,y,z) =0\] with $A \in GL(3, \mathbb{F}_q)$, and study their basic properties, one of which is that the number of $\mathbb{F}_q$-points of any relative of the Hermitian curve of degree $\sqrt{q}+1$ is congruent to $1$ modulo $\sqrt{q}$. In the latter part of this paper, we classify those curves having two or more rational inflexions., Comment: 18 Pages; The third section is completely different from that of the previous version

We determine the minimal bi-degree(s) of an irreducible filling curve over $\mathbb{F}_q$ for $\mathbb{P}^1\times \mathbb{P}^1$. It is $(q+1, q+1)$ if $q\neq 2$, and they are $(4,3)$ and $(3,4)$ if $q=2$., Comment: 10 Pages

Nano-oscillator devices are gaining more and more attention as a prerequisite for developing novel energy-efficient computing systems based on coupled oscillators. Here, we introduce a highly scalable, frequency-tunable nano-oscillator consisting of one Ovonic threshold switch (OTS) and a field-effect transistor (FET). It is presented that the proposed device shows an oscillating behavior with a natural frequency (f_{nat}) adjustable from 0.5 to 2 MHz depending on the gate voltage applied to the FET. In addition, under a small periodic input, it is observed that the oscillating frequency (f_{osc}) of the device is locked to the frequency (f_{in}) of the input when f_{in} ~ f_{nat}, demonstrating the so-called synchronization phenomenon. It also shows the phase lock of the combined oscillator network using circuit simulation, where the phase relation between the oscillators can be controlled by the coupling strength. These results imply that the proposed device is promising for applications in oscillator-based computing systems., Comment: 18 pages including 5 figures

The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective $3$-space., Comment: 23 pages

A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree $d$ over the finite field of $q$ elements is also given for $d\geq q+1$., Comment: 12 pages Text, 1 figure

A nonsingular surface of degree $d \geq 2$ in $\mathbb{P}^3$ over $\mathbb{F}_q$ has at most $((d-1)q+1)d$ $\mathbb{F}_q$-lines, and this bound is optimal for $d = 2, \sqrt{q}+1, q+1$., Comment: 4 pages. An incorrect argument was fixed

In the previous paper, we established an elementary bound for numbers of points of surfaces in the projective $3$-space over ${\Bbb F}_q$. In this paper, we give the complete list of surfaces that attain the elementary bound. Precisely those surfaces are the hyperbolic surface, the nonsingular Hermitian surface, and the surface of minimum degree containing all ${\Bbb F}_q$-points of the $3$-space., Comment: 8 pages

The nonsingular Hermitian surface of degree $\sqrt{q} +1$ is characterized by its number of $\Bbb{F}_q$-points among the irreducible surfaces over $\Bbb{F}_q$ of degree $\sqrt{q} +1$ in the projective 3-space., Comment: A small gap in the previous version has been filled up

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a generalization of the curve are also presented., Comment: 15pages, presented at AGCT13

We settle the conjecture posed by Sziklai on the number of points of a plane curve over a finite field under the assumption that the curve is nonsingular., Comment: 12 pages. This updated edition contains an addendum, in which we fixed an error in Remark 4.2 of the previous version

Our concern is a nonsingular plane curve defined over a finite field of q elements which includes all the rational points of the projective plane over the field. The possible degree of such a curve is at least q+2. We prove that nonsingular plane curves of degree q+2 having the property actually exist. More precisely, we write down explicitly all of those curves. Actually, Giuseppe Tallini studied such curves in his old paper in 1961. We explain the connection between his work and ours. Moreover we give another proof of his result on the automorphism group of such a curve, from the viewpoint of linear algebra., Comment: 20 pages, A small modification was made