Rick Miranda

Algebraic Curves and Riemann Surfaces

Riemann surfaces: Basic definitions Functions and maps More examples of Riemann surfaces Integration on Riemann surfaces Divisors and meromorphic functions Algebraic curves and the Riemann-Roch theorem Applications of Riemann-Roch Abels theorem Sheaves and Cech cohomology Algebraic sheaves Invertible sheaves, line bundles, and

Linear systems of plane curves with base points of equal multiplicity

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PROJECTIVE DEGENERATIONS OF K3 SURFACES, GAUSSIAN MAPS, AND FANO THREEFOLDS

References Index of notation Index of terminology.

Circular nodes in neural networks

In this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m. A conjecture of Harbourne and Hirschowitz implies that when d > 3m, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple (-1)-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all m < 12.

The proto-Lorenz system

SummaryIn this article we exhibit certain projective degenerations of smoothK3 surfaces of degree 2g−2 in ℙg (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of suchK3 surfaces has a corank one Gaussian map, ifg=11 org≥13. We also prove that the general such hyperplane section lies on a uniqueK3 surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which does not rely on the existence of a line.

Exploiting symmetry in boundary element methods

In the usual construction of a neural network, the individual nodes store and transmit real numbers that lie in an interval on the real line; the values are often envisioned as amplitudes. In this article we present a design for a circular node, which is capable of storing and transmitting angular information. We develop the forward and backward propagation formulas for a network containing circular nodes. We show how the use of circular nodes may facilitate the characterization and parameterization of periodic phenomena in general. We describe applications to constructing circular self-maps, periodic compression, and one-dimensional manifold decomposition. We show that a circular node may be used to construct a homeomorphism between a trefoil knot in ℝ3 and a unit circle. We give an application with a network that encodes the dynamic system on the limit cycle of the Kuramoto-Sivashinsky equation. This is achieved by incorporating a circular node in the bottleneck layer of a three-hidden-layer bottleneck network architecture. Exploiting circular nodes systematically offers a neural network alternative to Fourier series decomposition in approximating periodic or almost periodic functions.

Quantum Cohomology of Rational Surfaces

Abstract A quotient of the Lorenz dynamical system is constructed. This “proto-Lorenz” system has the Lorenz system as a double covering, and the double covering explains the two-eared nature of the strange attractor for the Lorenz system. Arbitrary coverings of the proto-Lorenz system are possible, leading to n -eared strange attractors.

The Segre and Harbourne-Hirschowitz Conjectures

Linear operator equations

The method of resultants for computing real solutions of polynomial systems

\mathcal {L}f = g

BIVARIATE HERMITE INTERPOLATION AND LINEAR SYSTEMS OF PLANE CURVES WITH BASE FAT POINTS

are considered in the context of boundary element methods, where the operator