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The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.

We investigate a known problem whether a Sobolev homeomorphism between domains in Rn can change sign of the Jacobian. The only case that remains open is when f∈W1,[n/2], n≥4. We prove that if n≥4, and a sense-preserving homeomorphism f satisfies f∈W1,[n/2], f-1∈W1,n-[n/2]-1 and either f is Hölder continuous on almost all spheres of dimension [n / 2], or f-1 is Hölder continuous on almost all spheres of dimensions n-[n/2]-1, then the Jacobian of f is non-negative, Jf≥0, almost everywhere. This result is a consequence of a more general result proved in the paper. Here [x] stands for the greatest integer less than or equal to x.

It is well known that every smooth cubic threefold is the zero locus of the Pfaffian of a 6×6 skew-symmetric matrix of linear forms in P4 . To compactify the space of such Pfaffian representations of a given cubic and to study the construction in families for singular or reducible cubics as well as, it is thus natural to consider the incidence correspondence of Pfaffian representations inside the product of the space of semistable skew-symmetric 6×6 matrices of linear forms in P4 and the space of cubics. Here we describe concretely the irreducible component of this incidence correspondence dominating the space of skew matrices.

We present a conjecture for the Chow ring of the universal moduli stack of bundles over hyperelliptic curves and prove it for rank and genus two. Consequently, we obtain explicit generators and relations to conclude that the Chow ring is tautological. In addition, we compute the Chow rings of products of universal Jacobians over genus two curves.

This work presents a new formulation to holistically control four cooperative multi-rotor drones controlled in two pairs. This approach uses a modular relative Jacobian with components consisting of the Jacobians of each individual drone. This type of controller relies mainly on the relative motion between the drones, consequently releasing unnecessary constraints inherent to the control of drones in absolute motion. We present the derivations of all the necessary equations of the modular relative Jacobian to control the four multi-rotor drones. We also present the derivations of the Jacobian for each drone. We implement our proposed method in the Gazebo RotorS simulation using four hexa-rotor drones modeled from Ascending Technologies called firefly drones. We present the simulation results and analyze them to show the effectiveness of our proposed approach.

We give a somewhat informal introduction to the integrable systems approach to the Schottky problem, explaining how the theta functions of Jacobians can be used to provide solutions of the KP equation, and culminating with the exposition of Krichever's proof of Welters' trisecant conjecture in the most degenerate (flex line) case.

We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties \\(A_f\\) attached by Shimura to normalized newforms \\(f \\in S_2( \\Gamma_0(N))\\). We present all the curves corresponding to principally polarized surfaces \\(A_f\\) for \\(N\\leq500\\).

Softmax Self-Attention (SSA) is a key component of Transformer architectures. However, when utilised within skipless architectures, which aim to improve representation learning, recent work has highlighted the inherent instability of SSA due to inducing rank collapse and poorly-conditioned Jacobians. In this work, we design a novel attention mechanism: Orthogonal Self-Attention (OSA), which aims to bypass these issues with SSA, in order to allow for (non-causal) Transformers without skip connections and normalisation layers to be more easily trained. In particular, OSA parametrises the attention matrix to be orthogonal via mapping a skew-symmetric matrix, formed from query-key values, through the matrix exponential. We show that this can be practically implemented, by exploiting the low-rank structure of our query-key values, resulting in the computational complexity and memory cost of OSA scaling linearly with sequence length. Furthermore, we derive an initialisation scheme for which we prove ensures that the Jacobian of OSA is well-conditioned.

We extend Moonen's definition of Ekedahl-Oort types of smooth curves in terms of Hasse-Witt triples to all stable curves and show that it matches Ekedahl and van der Geer's definition of Ekedahl-Oort types of their generalized Jacobians as semi-abelian varieties. Using this intrinsic insight, we can compute the dimensions of certain Ekedahl-Oort loci of curves and generalize some previously known results about the dimensions of the \\(p\\)-rank and \\(a\\)-number loci of curves.

We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves \\(C_{p^2}: y^2=x^{p^2}-1\\), where \\(p\\) is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper.