Elementary vectors and conformal sums in polyhedral geometry and their relevance for metabolic pathway analysis

A fundamental result in metabolic pathway analysis states that every flux mode can be decomposed into a sum of elementary modes. However, only a decomposition without cancelations is biochemically meaningful, since a reversible reaction cannot have different directions in the contributing elementary modes. This essential requirement has been largely overlooked by the metabolic pathway community. Indeed, every flux mode can be decomposed into elementary modes without cancelations. The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace is a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors). In this work, we extend the theorem, first to "subspace cones" and then to general polyhedral cones and polyhedra. Thereby, we refine Minkowski's and Carath\'eodory's theorems, two fundamental results in polyhedral geometry. We note that, in general, elementary vectors need not be support-minimal, in fact, they are conformally non-decomposable and form a unique minimal set of conformal generators. Our treatment is mathematically rigorous, but suitable for systems biologists, since we give self-contained proofs for our results and use concepts motivated by metabolic pathway analysis. In particular, we study cones defined by linear subspaces and nonnegativity conditions - like the flux cone - and use them to analyze general polyhedral cones and polyhedra. Finally, we review applications of elementary vectors and conformal sums in metabolic pathway analysis

Scattering of gap solitons by PT-symmetric defects

The resonant scattering of gap solitons (GS) of the periodic nonlinear Schr\"odinger equation with a localized defect which is symmetric under the parity and the time-reversal (PT) symmetry, is investigated. It is shown that for suitable amplitudes ratios of the real and imaginary parts of the defect potential the resonant transmission of the GS through the defect becomes possible. The resonances occur for potential parameters which allow the existence of localized defect modes with the same energy and norm of the incoming GS. Scattering properties of GSs of different band-gaps with effective masses of opposite sign are investigated. The possibility of unidirectional transmission and blockage of GSs by PT defect, as well as, amplification and destruction induced by multiple reflections from two PT defects, are also discussed

Parametrized systems of polynomial inequalitites with real exponents via linear algebra and convex geometry

We consider parametrized systems of generalized polynomial inequalities (with real exponents) in $n$ positive variables $x$ and involving $m$ monomial terms $c \circ x^B$ (with exponent matrix $B$, positive parameters $c$, and component-wise product $\circ$). More generally, we study solutions $x \in \mathbb{R}^n_>$ to $(c \circ x^B) \in C$ for a "coefficient cone" $C \subseteq \mathbb{R}^m_>$. Our framework encompasses systems of generalized polynomial equations, as studied in real fewnomial and reaction network theory. We identify the relevant geometric objects of the problem, namely a bounded set $P$ arising from the coefficient cone and two subspaces representing monomial differences and dependencies. The dimension of the latter subspace (the monomial dependency $d$) is crucial. As our main result, we rewrite the problem in terms of $d$ binomial conditions on the "coefficient set" $P$, involving $d$ monomials in the parameters. In particular, we obtain a classification of polynomial inequality systems. If $d=0$, solutions exist (for all $c$) and can be parametrized explicitly, thereby generalizing monomial parametrizations (of the solutions). Even if $d>0$, solutions on the coefficient set can often be determined more easily than solutions of the original system. Our framework allows a unified treatment of multivariate polynomial inequalities, based on linear algebra and convex geometry. We demonstrate its novelty and relevance in three examples from real fewnomial and reaction network theory. For two mass-action systems, we parametrize the set of equilibria and the region for multistationarity, respectively, and for univariate trinomials, we provide a "solution formula" involving discriminants and "roots".Comment: arXiv admin note: substantial text overlap with arXiv:2304.05273. author note: we now separated material between this manuscript and arXiv:2304.05273, in particular, there is no overlap anymore regarding examples. there is still overlap in the outline of the underlying geometric objects objects, but this is required for a self-contained presentatio