Subtropical Real Root Finding

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients. Then we employ linear programming to heuristically find roots. There is a specialized variant for roots with exclusively positive coordinates, which is of considerable interest for applications in chemistry and systems biology. An implementation of our method combining the computer algebra system Reduce with the linear programming solver Gurobi has been successfully applied to input data originating from established mathematical models used in these areas. We have solved several hundred problems with up to more than 800000 monomials in up to 10 variables with degrees up to 12. Our method has failed due to its incompleteness in less than 8 percent of the cases

Testing Binomiality of Chemical Reaction Networks Using Comprehensive Gröbner Systems

We consider the problem of binomiality of the steady state ideals of biochemical reaction networks. We are interested in finding polynomial conditions on the parameters such that the steady state ideal of a chemical reaction network is binomial under every specialisation of the parameters if the conditions on the parameters hold. We approach the binomiality problem using Comprehensive Gr\"obner systems. Considering rate constants as parameters, we compute comprehensive Gr\"obner systems for various reactions. In particular, we make automatic computations on n-site phosphorylations and biomodels from the Biomodels repository using the grobcov library of the computer algebra system Singular

First-Order Tests for Toricity

Motivated by problems arising with the symbolic analysis of steady state ideals in Chemical Reaction Network Theory, we consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a coset of a multiplicative group. That property corresponds to Shifted Toricity, a recent generalization of toricity of the corresponding polynomial ideal. The key idea is to take a geometric view on varieties rather than an algebraic view on ideals. Recently, corresponding coset tests have been proposed for complex and for real varieties. The former combine numerous techniques from commutative algorithmic algebra with Gr\"obner bases as the central algorithmic tool. The latter are based on interpreted first-order logic in real closed fields with real quantifier elimination techniques on the algorithmic side. Here we take a new logic approach to both theories, complex and real, and beyond. Besides alternative algorithms, our approach provides a unified view on theories of fields and helps to understand the relevance and interconnection of the rich existing literature in the area, which has been focusing on complex numbers, while from a scientific point of view the (positive) real numbers are clearly the relevant domain in chemical reaction network theory. We apply prototypical implementations of our new approach to a set of 129 models from the BioModels repository

Nanometer-scale Tomographic Reconstruction of 3D Electrostatic Potentials in GaAs/AlGaAs Core-Shell Nanowires

We report on the development of Electron Holographic Tomography towards a versatile potential measurement technique, overcoming several limitations, such as a limited tilt range, previously hampering a reproducible and accurate electrostatic potential reconstruction in three dimensions. Most notably, tomographic reconstruction is performed on optimally sampled polar grids taking into account symmetry and other spatial constraints of the nanostructure. Furthermore, holographic tilt series acquisition and alignment have been automated and adapted to three dimensions. We demonstrate 6 nm spatial and 0.2 V signal resolution by reconstructing various, previously hidden, potential details of a GaAs/AlGaAs core-shell nanowire. The improved tomographic reconstruction opens pathways towards the detection of minute potentials in nanostructures and an increase in speed and accuracy in related techniques such as X-ray tomography

Linear Integer Arithmetic Revisited

We consider feasibility of linear integer programs in the context of verification systems such as SMT solvers or theorem provers. Although satisfiability of linear integer programs is decidable, many state-of-the-art solvers neglect termination in favor of efficiency. It is challenging to design a solver that is both terminating and practically efficient. Recent work by Jovanovic and de Moura constitutes an important step into this direction. Their algorithm CUTSAT is sound, but does not terminate, in general. In this paper we extend their CUTSAT algorithm by refined inference rules, a new type of conflicting core, and a dedicated rule application strategy. This leads to our algorithm CUTSAT++, which guarantees termination

TimbreCLIP: Connecting Timbre to Text and Images

We present work in progress on TimbreCLIP, an audio-text cross modal embedding trained on single instrument notes. We evaluate the models with a cross-modal retrieval task on synth patches. Finally, we demonstrate the application of TimbreCLIP on two tasks: text-driven audio equalization and timbre to image generation.Comment: Submitted to AAAI workshop on creative AI across modalitie