For a prime p and a polynomial f ∈ Fp[X], we obtain upper bounds on the number of solutions of the congruences f(x) ≡ y (mod p) and f(x) ≡ y (mod p), where (x, y) belongs to an arbitrary square with side length M . Further, we obtain non-trivial upper bounds for the number of hyperelliptic curves Y 2 = X + a2g−1X 2g−1 + . . .+ a1X + a0 over Fp, with coefficients in a 2g-dimensional cube (a0, . ...

Under the excitation of geomagnetic field, normal component of surface scattering magnetic field intensity Hp(y) changes of steel tensile samples with different stress concentration factors α were studied. Constant amplitude sinusoidal tensile load cycles were applied on 18CrNi4A steel plate samples whose α were respectively 1, 3 and 5. Then the correlation of Hp(y) value changes with loading c...

Consider a set of baseline predictors X to predict a binary outcome D and let Y be a novel marker or predictor. This paper is concerned with evaluating the performance of the augmented risk model P(D = 1|Y,X) compared with the baseline model P(D = 1|X). The diagnostic likelihood ratio, DLR(X)(y), quantifies the change in risk obtained with knowledge of Y = y for a subject with baseline risk fac...

For any algebraic variety X defined over a number field K, and height functionHD onX corresponding to an ample divisorD, one can define the counting functionNX,D(B) = #{P ∈ X(K) | HD(P ) ≤ B}. In this paper, we calculate the counting function for hyperelliptic K3 surfaces X which admit a generically two-to-one cover of P1 × P1 branched over a singular curve. In particular, we effectively constr...

Algebraic curves and surfaces play an important and ever increasing role in computer aided geometric design, computer vision, and computer aided manufacturing. Consequently, theoretical results need to be adapted to practical needs. We need efficient algorithms for generating, representing, manipulating, analyzing, rendering algebraic curves and surfaces. In the last years there has been dramat...

In this paper a class of quartic trigonometric Bézier curve, called QT Bézier curve, with two shape parameters is presented. These curves not only inherit most properties of the usual quartic Bézier curves in the polynomial space, but also enjoy some other advantageous properties for shape modelling. The shape of the curve can be adjusted by altering the values of shape parameters while the con...

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K0 be an arbitrary prime field and let r > 1 and s > 1 be integers that are coprime to one another. We show that one can find a finite extension K of K0, a degree-r extension L of K, a degr...

This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of Seiberg-Witten curves in terms of corresponding integrable models.

Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus n ≥ 1 over Q, with a rational Weierstrass point [2, Thm 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus n. This gives us a uniform upper bound of 3 2 on the average ran...

The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shim...