Enhanced Toolpath Planning for Fused Filament Fabrication

The slicing software process the 3D geometry into 2D slices and toolpaths for additive manufacturing processes. Most slicing software allows users to select from an array of infill patterns and to specify the overall infill volume fraction globally. However, the ability to locally control the volume fraction, and mechanical properties, is often limited. In this paper, we propose a novel toolpath enhancing algorithm to enable the local control on the volume fraction of various stock and custom infill patterns. In particular, the algorithm widens the infill pattern by directly modifying their toolpath with connected Fermat curves. By preserving the topology of the original toolpath, the connected Fermat curve not only produces predictable boosts in part performance but also minimized the printing time by eliminating extruder traversals without material deposition. The field that controls local volume fraction can be designed either manually or through optimization. The effectiveness of the proposed approach in toolpath generation is demonstrated through volume fraction fields designed by both approaches.

Rational points on Fermat curves over finite fields

Let [Formula: see text] be the [Formula: see text]-rational point on the Fermat curve [Formula: see text] with [Formula: see text]. It has recently been proved that if [Formula: see text] then each [Formula: see text] is a cube in [Formula: see text]. It is natural to wonder whether there is a generalization to [Formula: see text]. In this paper, we show that the result cannot be extended to [Formula: see text] in general and conjecture that each [Formula: see text] is a cube in [Formula: see text] if and only if [Formula: see text].

Homology of the Fermat Tower and Universal Measures for Jacobi Sums

We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson’s adelic beta functions, in a manner similar to Ihara’s definition of ℓ-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

Let denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ, k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

The maximum likelihood degree of Fermat hypersurfaces

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.

RATIONAL POINTS ON SOME FERMAT CURVES AND SURFACES OVER FINITE FIELDS

We give an explicit description of the 𝔽qi-rational points on the Fermat curve uq-1 + vq-1 + wq-1 = 0, for i ∈{1, 2, 3}. As a consequence, we observe that for any such point (u, v, w), the product uvw is a cube in 𝔽qi. We also describe the 𝔽q2-rational points on the Fermat surface uq-1 + vq-1 + wq-1 + xq-1 = 0, and show that the product of the coordinates of any such point is a square.