Related papers: A tropical geometry for bounded biochemical state …
Binding and unbinding of ligands to specific sites of a macromolecule are one of the most elementary molecular interactions inside the cell that embody the computational processes of biological regulations. The interaction between…
Topological materials are renowned for their ability to harbor states localized at their peripheries, such as surfaces, edges, and corners. Accompanying these states, fractional charges appear on peripheral unit cells. Recently,…
Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted…
We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is…
We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…
The numerical analysis of elastic wave propagation in unbounded media may be difficult to handle due to spurious waves reflected at the model artificial boundaries. Several sophisticated techniques such as nonreflecting boundary conditions,…
Preparing and certifying bound entangled states in the laboratory is an intrinsically hard task, due to both the fact that they typically form narrow regions in the state space, and that a certificate requires a tomographic reconstruction…
Analogously as in classical algebraic geometry, linear pencils of tropical plane curves are parameterized by tropical lines in a coefficient space. A special example of such a linear pencil is the set of tropical plane curves with an…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
Sufficiently accurate finite state models, also called symbolic models or discrete abstractions, allow one to apply fully automated methods, originally developed for purely discrete systems, to formally reason about continuous and hybrid…
Possible forms of obstructed atomic limits in quasi-one-dimensional systems are studied using line group symmetry. This is accomplished by revisiting the standard theory with an emphasis on its group-theoretical background, synthesizing the…
The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an…
Molecules have seemed like a natural fit to deep learning's tendency to handle a complex structure through representation learning, given enough data. However, this often continuous representation is not natural for understanding chemical…
Nonequilibrium quantum field theory is often used to derive an approximation for the evolution of number densities and asymmetries in astroparticle models when a more precise treatment of quantum thermal effects is required. This work…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
Understanding the behavior of biomolecules such as proteins requires understanding the critical influence of the surrounding fluid (solvent) environment--water with mobile salt ions such as sodium. Unfortunately, for many studies, fully…
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and…
Motivated by problems from Chemical Reaction Network Theory, we investigate whether steady state ideals of reversible reaction networks are generated by binomials. We take an algebraic approach considering, besides concentrations of…
We investigate topological properties of density matrices motivated by the question to what extent phenomena like topological insulators and superconductors can be generalized to mixed states in the framework of open quantum systems. The…
We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach…