Search

We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We focus on two types of parametrized systems -- vertically parametrized and horizontally parametrized systems -- and discuss techniques for computing the tropical data efficiently. We end the paper with several case studies, where we analyze systems arising from chemical reaction networks, coupled oscillators, and rigid graphs., Comment: 34 pages, 6 figures

Let $p\neq{2,3}$ be a prime number and let $\Gamma \subset \mathrm{SL}_{2}(\mathbb{Z})$ be a congruence subgroup with modular curve $X_{\Gamma}/K$ and Jacobian $J(X_{\Gamma})$. In this paper we give an explicit group-theoretic description of the semistable toric rank and component group of $J(X_{\Gamma})$ at the finite places of $K$ lying over $p$. We first produce a suitable deformation retract of the minimal Berkovich skeleton of $X_{\Gamma}$ in terms of Hecke-Iwahori double coset spaces. We call this deformation retract the pruned skeleton of the curve. Our description of this skeleton includes a group-theoretic formula for the edge lengths, allowing us to give the component group of the modular curve as the quotient of a lattice using the monodromy pairing. For $X_{0}(N)$, $X_{1}(N)$, $X_{sp}(N)$ and $X_{sp}^{+}(N)$, we explicitly determine the pruned skeleta using a set of coset schemes over $\mathbb{Z}$. This in particular recovers results by Deligne-Rapoport, Edixhoven, Coleman-McMurdy and Tsushima on the semistable reduction type of $X_{0}(p^{n})$ for $n\leq{4}$. Finally, we determine the geometric Tamagawa number and the prime-to-$2$ structure of the component group of $X_{0}(N)$ over the extension given by Krir's theorem., Comment: 53 pages, 3 figures. Moved a figure to arXiv:2306.03879, added details for the calculations of the edge lengths, slightly changed the title

We study the topology of the moduli space of unramified $\mathbb{Z}/p$-covers of tropical curves of genus $g \geq 2$, where $p$ is a prime number. We use recent techniques by Chan--Galatius--Payne to identify contractible subcomplexes of the moduli space. We then use this contractibility result to show that this moduli space is simply connected. In the case of genus 2, we determine the homotopy type of this moduli space for all primes $p$. This work is motivated by prospective applications to the top-weight cohomology of the space of prime cyclic \'etale covers of smooth algebraic curves., Comment: 39 pages, 11 figures, 5 tables

The fate of thymine upon excitation by ultraviolet radiation has been the subject of intense debate over the past three decades. Today, it is widely believed that its ultrafast excited state decay stems from a radiationless transition from the bright ${\pi}{\pi}^*$ state to a dark $n{\pi}^*$ state. However, conflicting theoretical predictions have made the experimental data difficult to interpret. Here we simulate the ultrafast dynamics in thymine at the highest level of theory to date, performing wavepacket dynamics with a new coupled cluster method. Our simulation confirms an ultrafast ${\pi}{\pi}^*$ to $n{\pi}^*$ transition (${\tau} = 41 \pm 14$ fs). Furthermore, the predicted oxygen-edge X-ray absorption spectra agree quantitatively with the experimental results. Our simulation also predicts an as-yet uncharacterized photochemical pathway: a ${\pi}{\sigma}^*$ channel that leads to hydrogen dissociation at one of the two N-H bonds in thymine. Similar behavior has been identified in other heteroaromatic compounds, including adenine, and several authors have speculated that a similar pathway may exist in thymine. However, this was never confirmed theoretically or experimentally. This prediction calls for renewed efforts to experimentally identify or exclude the presence of this channel., Comment: 42 pages, 23 figures

A dynamical rearrangement in the electronic structure of a molecule can be driven by different phenomena, including nuclear motion, electronic coherence or electron correlation. Recording such electronic dynamics and identifying their fate in aqueous solution has remained a challenge. Here, we reveal the electronic dynamics induced by electronic relaxation through conical intersections in pyrazine molecules using X-ray spectroscopy. We show that the ensuing created dynamics corresponds to a cyclic rearrangement of the electronic structure around the aromatic ring. Furthermore, we find that such electronic dynamics are entirely suppressed when pyrazine is dissolved in water. Our observations confirm that conical intersections can create electronic dynamics that are not directly excited by the pump pulse and that aqueous solvation can dephase them in less than 40 fs. These results have implications for the investigation of electronic dynamics created during light-induced molecular dynamics and shed light on their susceptibility to aqueous solvation., Comment: 62 pages, 34 figures

The X-ray absorption (XA) spectra of aqueous ammonia and ammonium are computed using a combination of coupled cluster singles and doubles (CCSD) with different quantum mechanical and molecular mechanical embedding schemes. Specifically, we compare frozen Hartree--Fock (HF) density embedding, polarizable embedding (PE), and polarizable density embedding (PDE). Integrating CCSD with frozen HF density embedding is possible within the CC-in-HF framework, which circumvents the conventional system-size limitations of standard coupled cluster methods. We reveal similarities between PDE and frozen HF density descriptions, while PE spectra differ significantly. By including approximate triple excitations, we also investigate the effect of improving the electronic structure theory. The spectra computed using this approach show an improved intensity ratio compared to CCSD-in-HF. Charge transfer analysis of the excitations shows the local character of the pre-edge and main-edge, while the post-edge is formed by excitations delocalized over the first solvation shell and beyond., Comment: 35 pages, 12 figures

Humans have created artificial intelligence (AI), not the other way around. This statement is deceptively obvious. In this note, we decided to challenge this statement as a small, lighthearted Gedankenexperiment. We ask a simple question: in a world driven by evolution by natural selection, would neural networks or humans be likely to evolve first? We compare the Solomonoff--Kolmogorov--Chaitin complexity of the two and find neural networks (even LLMs) to be significantly simpler than humans. Further, we claim that it is unnecessary for any complex human-made equipment to exist for there to be neural networks. Neural networks may have evolved as naturally occurring objects before humans did as a form of chemical reaction-based or enzyme-based computation. Now that we know that neural networks can pass the Turing test and suspect that they may be capable of superintelligence, we ask whether the natural evolution of neural networks could lead from pure evolution by natural selection to what we call evolution-bootstrapped simulation. The evolution of neural networks does not involve irreducible complexity; would easily allow irreducible complexity to exist in the evolution-bootstrapped simulation; is a falsifiable scientific hypothesis; and is independent of / orthogonal to the issue of intelligent design., Comment: 6 pages, no figures

In academic literature portfolio risk management and hedging are often versed in the language of stochastic control and Hamilton--Jacobi--Bellman~(HJB) equations in continuous time. In practice the continuous-time framework of stochastic control may be undesirable for various business reasons. In this work we present a straightforward approach for thinking of cross-asset portfolio risk management and hedging, providing some implementation details, while rarely venturing outside the convex optimisation setting of (approximate) quadratic programming~(QP). We pay particular attention to the correspondence between the economic concepts and their mathematical representations; the abstractions enabling us to handle multiple asset classes and risk models at once; the dimensional analysis of the resulting equations; and the assumptions inherent in our derivations. We demonstrate how to solve the resulting QPs with CVXOPT.

Conditional branches pose a challenge for code optimisation, particularly in low latency settings. For better performance, processors leverage dedicated hardware to predict the outcome of a branch and execute the following instructions speculatively, a powerful optimisation. Modern branch predictors employ sophisticated algorithms and heuristics that utilise historical data and patterns to make predictions, and often, are extremely effective at doing so. Consequently, programmers may inadvertently underestimate the cost of misprediction when benchmarking code with synthetic data that is either too short or too predictable. While eliminating branches may not always be feasible, C++20 introduced the [[likely]] and [[unlikely]] attributes that enable the compiler to perform spot optimisations on assembly code associated with likely execution paths. Can we do better than this? This work presents the development of a novel language construct, referred to as a semi-static condition, which enables programmers to dynamically modify the direction of a branch at run-time by modifying the assembly code within the underlying executable. Subsequently, we explore scenarios where the use of semi-static conditions outperforms traditional conditional branching, highlighting their potential applications in real-time machine learning and high-frequency trading. Throughout the development process, key considerations of performance, portability, syntax, and security were taken into account.

We present the first successful application of the coupled cluster approach to simulate the X-ray absorption (XA) spectrum of liquid water. The system size limitations of standard coupled cluster theory are overcome by employing a newly developed coupled cluster method for large molecular systems. This method combines coupled cluster singles, doubles, and perturbative triples in a multilevel framework (MLCC3-in-HF) and is able to describe the delicate nature of intermolecular interactions in liquid water. Using molecular geometries from state-of-the-art path-integral molecular dynamics, we obtain excellent agreement with experimental spectra. Additionally, we show that an accurate description of the electronic structure within the first solvation shell is sufficient to model the XA spectrum of liquid water. Furthermore, we present a rigorous charge transfer analysis with unprecedented reliability, achieved through MLCC3-in-HF. This analysis aligns with previous studies regarding the character of the prominent features of the spectrum., Comment: 32 pages, 4+12 figures

Let $X$ be a normal connected Noetherian scheme. In this paper we give an algorithm to reconstruct the relative poset structure of finite dominant separable morphism $X'\to X$ in terms of topos-theoretic enhancements of the underlying poset of $X$. The different relative poset structures are classified by an object in a $2$-limit of topoi over a graph, which we call the universal poset covering. We interpret the data in the universal covering in terms of glued power series, and we show how to calculate these using iterated symbolic multivariate Newton-Puiseux algorithms. This in particular gives a full local monogenic algorithm to calculate dual intersection graphs of semistable models of curves defined over a discretely valued field. We give a detailed study of this algorithm, with various examples to illustrate the non-trivial gluing phenomena. We also interpret these techniques in terms of analytic spaces, with an eye towards future applications in $p$-adic integration theory., Comment: 42 pages, 7 figures. Rewritten in terms of glued topoi and double coset spaces. Moved a figure from arXiv:2403.09995 to this paper, added a section on rigid analytic interpretations of the algorithm

We use tropical and non-archimedean geometry to study generic root counts of families of polynomial equations. These families are given as morphisms of schemes $X\to{Y}$ that factor through a closed embedding into a relative torus over a parameter space $Y$. We prove a generalization of Bernstein's theorem for these morphisms, showing that the root count of a single well-behaved tropical fiber spreads to an open dense subset of $Y$. We use this to express the generic root count of a wide class of square systems in terms of the matroidal degree of an explicit variety. This in particular gives tropical formulas for the birational intersection indices and volumes of Newton-Okounkov bodies defined by Kaveh and Khovanskii, and the generic root counts of the steady-state equations of chemical reaction networks. An important role in these theorems is played by the notion of tropical flatness, which allows us to infer generic properties of $X\to{Y}$ from a single tropical fiber. We show that the tropical analogue of the generic flatness theorem holds, in the sense that $X\to{{Y}}$ is tropically flat over an open dense subset of the Berkovich analytification of $Y$., Comment: 40 pages, 1 figure. The last section has been reworked into two new sections with new results: we give tropical formulas for the number of steady states in a chemical reaction network and the volumes of Newton-Okounkov bodies

We express the reduction types of Picard curves in terms of tropical invariants associated to binary quintics. We also give a general framework for tropical invariants associated to group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne-Mumford compactification $\overline{M}_{0,n}$., Comment: 25 pages

We present an efficient implementation of the equation of motion oscillator strengths for the closed-shell multilevel coupled cluster singles and doubles with perturbative triples method (MLCC3) in the electronic structure program eT. The orbital space is split into an active part treated with CC3 and an inactive part computed at the coupled cluster singles and doubles (CCSD) level of theory. Asymptotically, the CC3 contribution scales as $O(n_\text{V} n^3_\text{v} n^3_\text{o})$ floating-point operations (FLOP), where $n_V$ is the total number of virtual orbitals while $n_\text{v}$ and $n_\text{o}$ are the number of active virtual and occupied orbitals, respectively. The CC3 contribution, thus, only scales linearly with the full system size and can become negligible compared to the cost of CCSD. We demonstrate the capabilities of our implementation by calculating the UV-VIS spectrum of azobenzene and a core excited state of betaine 30 with more than 1000 molecular orbitals., Comment: 38 page and 10 pages SI and 6 figures

A time-dependent equation of motion coupled cluster singles and doubles (TD-EOM-CCSD) method is implemented, which uses a reduced basis calculated with the asymmetric band Lanczos algorithm. The approach is used to study weak-field processes in small molecules induced by ultrashort valence pump and core probe pulses. We assess the reliability of the procedure by comparing TD-EOM-CCSD absorption spectra to spectra obtained from the time-dependent coupled-cluster singles and doubles (TDCCSD) method and observe that spectral features can be reproduced for several molecules, at much lower computational times. We discuss how multiphoton absorption and symmetry can be handled in the method and general features of the core-valence separation (CVS) projection technique. We also model the transient absorption of an attosecond X-ray probe pulse by the glycine molecule., Comment: 15 pages and 5 figures

In this paper we generalize the $j$-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves $X$ given by $y^{n}=f(x)$. We first define a set of tropical invariants for $f(x)$ using symmetrized Pl\"{u}cker coordinates and we show that these invariants determine the tree associated to $f(x)$. We then prove that this tree completely determines the reduction type of $X$ for $n$ that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials $f(x)$ of degree $d\leq{5}$., Comment: 35 pages, 9 figures

We present the first investigation of excited state dynamics by resonant Auger-Meitner spectroscopy (also known as resonant Auger spectroscopy) using the nucleobase thymine as an example. Thymine is photoexcited in the UV and probed with X-ray photon energies at and below the oxygen K-edge. After initial photoexcitation to a {\pi}{\pi}* excited state, thymine is known to undergo internal conversion to an n{\pi}* excited state with a strong resonance at the oxygen K-edge, red-shifted from the ground state {\pi}* resonances of thymine (see our previous study Wolf et al., Nat. Commun., 2017, 8, 29). We resolve and compare the Auger-Meitner electron spectra associated both with the excited state and ground state resonances, and distinguish participator and spectator decay contributions. Furthermore, we observe simultaneously with the decay of the n{\pi}* state signatures the appearance of additional resonant Auger-Meitner contributions at photon energies between the n{\pi}* state and the ground state resonances. We assign these contributions to population transfer from the n{\pi}* state to a {\pi}{\pi}* triplet state via intersystem crossing on the picosecond timescale based on simulations of the X-ray absorption spectra in the vibrationally hot triplet state. Moreover, we identify signatures from the initially excited {\pi}{\pi}* singlet state which we have not observed in our previous study.

In this paper we give an algorithm that calculates the skeleton of a tame covering of curves over a complete discretely valued field. The algorithm relies on the {{tame simultaneous semistable reduction theorem}}, for which we give a short proof. To use this theorem in practice, we show that we can find extensions of chains of prime ideals in normalizations using compatible power series. This allows us to reconstruct the skeleton of the covering. In studying the connections between power series and extensions of prime ideals, we obtain generalizations of classical theorems from number theory such as the Kummer-Dedekind theorem and Dedekind's theorem for cycles in Galois groups., Comment: 45 pages, 7 figures

We present a new and efficient implementation of the closed shell coupled cluster singles and doubles with perturbative triples method (CC3) in the electronic structure program $e^T$. Asymptotically, a ground state calculation has an iterative cost of $4n_{\text{V}}^{4}n_{\text{O}}^{3}$ floating point operations (FLOP), where $n_{\text{V}}$ and $n_{\text{O}}$ are the number of virtual and occupied orbitals respectively. The Jacobian and transpose Jacobian transformations, required to iteratively solve for excitation energies and transition moments, both require $8n_{\text{V}}^{4}n_{\text{O}}^{3}$ FLOP. We have also implemented equation of motion (EOM) transition moments for CC3. The EOM transition densities require recalculation of triples amplitudes, as $n_{\text{V}}^{3}n_{\text{O}}^{3}$ tensors are not stored in memory. This results in a noniterative computational cost of $10n_{\text{V}}^{4}n_{\text{O}}^{3}$ FLOP for the ground state density and $26n_{\text{V}}^{4}n_{\text{O}}^{3}$ FLOP per state for the transition densities. The code is compared to the CC3 implementations in CFOUR, Dalton and Psi4. We demonstrate the capabilities of our implementation by calculating valence and core excited states of L-proline., Comment: 33 pages, 2 figures and TOC

The eT program is an open source electronic structure package with emphasis on coupled cluster and multilevel methods. It includes efficient spin adapted implementations of ground and excited singlet states, as well as equation of motion oscillator strengths, for CCS, CC2, CCSD, and CC3. Furthermore, eT provides unique capabilities such as multilevel Hartree-Fock and multilevel CC2, real-time propagation for CCS and CCSD, and efficient CC3 oscillator strengths. With a coupled cluster code based on an efficient Cholesky decomposition algorithm for the electronic repulsion integrals, eT has similar advantages as codes using density fitting, but with strict error control. Here we present the main features of the program and demonstrate its performance through example calculations. Because of its availability, performance, and unique capabilities, we expect eT to become a valuable resource to the electronic structure community., Comment: 31 pages and 10 figures - supplementary information included in the uploaded files

We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle of length $-v(j(E))$. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves $(E,P)$, where $E$ has multiplicative reduction and $P$ is an inflection point that reduces to the singular point on the reduction of $E$. We then follow the strategy as in \cite[Theorem 6.2]{BPR11} and construct an embedding such that its tropicalization contains a cycle of length $-v(j(E))$. We call this a numerically faithful tropicalization. A key difference between this approach and the approach in \cite{BPR11} is that we do not require any of the analytic theory on Berkovich spaces such as the {\it{Poincar\'{e}-Lelong formula}} or \cite[Theorem 5.25]{BPR11} to establish the numerical faithfulness of this tropicalization., Comment: 24 pages, 12 figures. This is a pre-print of an article published in "Arnold Mathematical Journal". The final authenticated version is available online at: https://doi.org/10.1007/s40598-019-00121-y

In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first generalize a result by Liu and Lorenzini by proving a simultaneous semistable reduction theorem for residually tame coverings. We then use this to construct a functor from the category of residually tame coverings of a marked curve $(X,D)$ to the category of tame coverings of a metrized complex $\Sigma$ associated to $(X,D)$. We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. Using this equivalence, we then define filtrations of the fundamental group of the marked curve, giving for instance the absolute decomposition and inertia groups of the metrized complex. We then use the analytic slope formula to prove that the extensions that arise from the abelianizations of the decomposition and inertia quotients coincide with the extensions that arise from the toric and connected parts of the analytic Jacobian of the curve., Comment: 21 pages, 1 figure

In this paper we show that if $\phi_{i}:A_{i}\rightarrow{A}$ is a semisimple pointed $K$-rational $\ell$-isogeny graph of order $n$ for a prime $\ell$, then the group of $\ell$-torsion points $A[\ell](\overline{K})$ contains a subspace of dimension $n$ generated by $K$-rational points. We also show that the same result is true for elliptic curves without the semisimplicity condition. Furthermore, we give an explicit counterexample for abelian varieties of higher dimension to show that the semisimplicity condition is indeed necessary.

In this paper we show that for any prime number $p$ not equal to $11$ or $19$, the Tribonacci number $T_{p-1}$ is divisible by $p$ if and only if $p$ is of the form $x^2+11y^2$. We first use class field theory on the Galois closure of the number field corresponding to the polynomial $x^3-x^2-x-1$ to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for $T_{p-1}$. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight $2$ and level $11$., Comment: Added a connection with the Fourier coefficients of the unique newform of weight $2$ and level $11$

In this paper, we study the problem of tropicalizing tame degree three coverings of the projective line. Given any degree three covering $C\longrightarrow{\mathbb{P}^{1}}$, we give an algorithm that produces the Berkovich skeleton of $C$. In particular, this gives an algorithm for finding the Berkovich skeleton of a genus $3$ curve. The algorithm uses a continuity statement for inertia groups of semistable Galois coverings, which we prove first. After that we give a formula for the decomposition group of an irreducible component $\Gamma\subset{\mathcal{C}_{s}}$ for a semistable Galois covering $\mathcal{C}\longrightarrow{\mathcal{D}}$. We conclude the paper with a simple application of these $S_{3}$-coverings to elliptic curves, giving another proof of the familiar semistability criterion for elliptic curves using a natural degree three morphism to $\mathbb{P}^{1}$ instead of the usual degree two morphism.