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We use the circle method to prove that a density 1 of elements in $\mathbb{F}_q[t]$ are representable as a sum of three cubes of essentially minimal degree from $\mathbb{F}_q[t]$, assuming the Ratios Conjecture and that the characteristic is bigger than 3. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting., Comment: 23 pages

By developing a suitable version of the circle method, we show that the space of degree $e$ rational curves on a smooth hypersurface of degree $d$ has only canonical singularities provided its dimension is sufficiently large with respect to $e$ and $d$., Comment: 13 pages

We study algorithms called rank-revealers that reveal a matrix's rank structure. Such algorithms form a fundamental component in matrix compression, singular value estimation, and column subset selection problems. While column-pivoted QR has been widely adopted due to its practicality, it is not always a rank-revealer. Conversely, Gaussian elimination (GE) with a pivoting strategy known as global maximum volume pivoting is guaranteed to estimate a matrix's singular values but its exponential complexity limits its interest to theory. We show that the concept of local maximum volume pivoting is a crucial and practical pivoting strategy for rank-revealers based on GE and QR. In particular, we prove that it is both necessary and sufficient; highlighting that all local solutions are nearly as good as the global one. This insight elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer, and we implement a version that is observed to be at most $2\times$ more computationally expensive than CPQR. We unify the landscape of rank-revealers by considering GE and QR together and prove that the success of any pivoting strategy can be assessed by benchmarking it against a local maximum volume pivot.

We use a function field version of the circle method to prove that a positive proportion of elements in $\mathbb{F}_q[t]$ are representable as a sum of three cubes of minimal degree from $\mathbb{F}_q[t]$, assuming a suitable form of the Ratios Conjecture and that the characteristic is greater than 3. The analogue of this conjecture for quadratic Dirichlet $L$-functions is known for large fixed $q$, via recent developments in homological stability., Comment: 57 pages

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results are conditional on a conjecture relating the rank of an elliptic curve to its conductor, while they are unconditional in positive characteristic. For quartic or quintic del Pezzo surfaces with a conic bundle structure, we establish even stronger estimates unconditionally as long as the characteristic is not two., Comment: 49 pages, comments welcome!

Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework., Comment: 42 pages, 3 figures, 3 tables

For projection-based linear-subspace model order reduction (MOR), it is well known that the Kolmogorov n-width describes the best-possible error for a reduced order model (ROM) of size n. In this paper, we provide approximation bounds for ROMs on polynomially mapped manifolds. In particular, we show that the approximation bounds depend on the polynomial degree p of the mapping function as well as on the linear Kolmogorov n-width for the underlying problem. This results in a Kolmogorov (n, p)-width, which describes a lower bound for the best-possible error for a ROM on polynomially mapped manifolds of polynomial degree p and reduced size n., Comment: 11 pages, 1 figure

The recent developments and research in distributed ledger technologies and blockchain have contributed to the increasing adoption of distributed systems. To collect relevant insights into systems' behavior, we observe many evaluation frameworks focusing mainly on the system under test throughput. However, these frameworks often need more comprehensiveness and generality, particularly in adopting a distributed applications' cross-layer approach. This work analyses in detail the requirements for distributed systems assessment. We summarize these findings into a structured methodology and experimentation framework called METHODA. Our approach emphasizes setting up and assessing a broader spectrum of distributed systems and addresses a notable research gap. We showcase the effectiveness of the framework by evaluating four distinct systems and their interaction, leveraging a diverse set of eight carefully selected metrics and 12 essential parameters. Through experimentation and analysis we demonstrate the framework's capabilities to provide valuable insights across various use cases. For instance, we identify that a combination of Trusted Execution Environments with threshold signature scheme FROST introduces minimal overhead on the performance with average latency around \SI{40}{\ms}. We showcase an emulation of realistic systems behavior, e.g., Maximal Extractable Value is possible and could be used to further model such dynamics. The METHODA framework enables a deeper understanding of distributed systems and is a powerful tool for researchers and practitioners navigating the complex landscape of modern computing infrastructures.

Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least $23$ over $\mathbb{F}_q(t)$, provided $\text{cha}(\mathbb{F}_q)>3$. Under the same hypotheses, we also verify weak approximation., Comment: 49 pages

This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for representing the high-dimensional system states in a reduced-dimensional coordinate system. While these approximations respect the symplectic nature of Hamiltonian systems, linear basis approximations can suffer from slowly decaying Kolmogorov $N$-width, especially in wave-type problems, which then requires a large basis size. We propose two different model reduction methods based on recently developed quadratic manifolds, each presenting its own advantages and limitations. The addition of quadratic terms to the state approximation, which sits at the heart of the proposed methodologies, enables us to better represent intrinsic low-dimensionality in the problem at hand. Both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than the linear symplectic reduced-order models.