Search

We consider solutions to parametrized systems of generalized polynomial equations (with real exponents) in $n$ positive variables, involving $m$ monomials with positive parameters; that is, $x\in\mathbb{R}^n_>$ such that ${A \, (c \circ x^B)=0}$ with coefficient matrix $A\in\mathbb{R}^{l \times m}$, exponent matrix $B\in\mathbb{R}^{n \times m}$, parameter vector $c\in\mathbb{R}^m_>$, and componentwise product $\circ$. As our main result, we characterize the existence of a unique solution (modulo an exponential manifold) for all parameters in terms of the relevant geometric objects of the polynomial system, namely the $\textit{coefficient polytope}$ and the $\textit{monomial dependency subspace}$. We show that unique existence is equivalent to the bijectivity of a certain moment/power map, and we characterize the bijectivity of this map using Hadamard's global inversion theorem. Furthermore, we provide sufficient conditions in terms of sign vectors of the geometric objects, thereby obtaining a multivariate Descartes' rule of signs for exactly one solution.

We present our SageMath package elementary_vectors for computing elementary and sign vectors of real subspaces. In this setting, elementary vectors are support-minimal vectors that can be determined from maximal minors of a real matrix representing a subspace. By applying the sign function, we obtain the cocircuits of the corresponding oriented matroid, which in turn allow the computation of all sign vectors of a real subspace. As an application, we discuss sign vector conditions for existence and uniqueness of complex-balanced equilibria of chemical reaction networks with generalized mass-action kinetics. The conditions are formulated in terms of sign vectors of two subspaces arising from the stoichiometric coefficients and the kinetic orders of the reactions. We discuss how these conditions can be checked algorithmically, and we demonstrate the functionality of our package sign_vector_conditions in several examples., Comment: 10 pages, ICMS 2024

We develop a general toolbox to study $W^{1,p}$ solutions of differential inclusions $\nabla u \in K$ for unbounded sets $K$. A key notion is the concept that a subset $K$ of the space $\mathbb{R}^{d \times m}$ of $d \times m$ matrices can be reduced to another set $K'$. We then use this framework to show that the product rigidity for Sobolev maps fails for $p<2$, and also apply our toolbox to simplify several examples from the literature., Comment: Dedicated to Professor Vladimir \v Sver\'ak on the occasion of his 65th birthday

Let $M$ be a smooth, compact, connected, oriented Riemannian manifold, and let $\imath: M \to \mathbb R^d$ be an isometric embedding. We show that a Sobolev map $f: M \to M$ which has the property that the differential $df(q)$ is close to the set $SO(T_q M, T_{f(q)} M)$ of orientation preserving isometries (in an $L^p$ sense) is already $W^{1,p}$ close to a global isometry of $M$. More precisely we prove for $p \in (1,\infty)$ the optimal linear estimate $$\inf_{\phi \in \mathrm{Isom}_+(M)} \| \imath \circ f - \imath \circ \phi\|_{W^{1,p}}^p \le C E_p(f)$$ where $$ E_p(f) := \int_M {\rm dist}^p(df(q), SO(T_q M, T_{f(q)} M)) \, d{\rm vol}_M$$ and where $\mathrm{Isom}_+(M)$ denotes the group of orientation preserving isometries of $M$. This extends the Euclidean rigidity estimate of Friesecke-James-M\"uller [Comm. Pure Appl. Math. {\bf 55} (2002), 1461--1506] to Riemannian manifolds. It also extends the Riemannian stability result of Kupferman-Maor-Shachar [Arch. Ration. Mech. Anal. {\bf 231} (2019), 367--408] for sequences of maps with $E_p(f_k) \to 0$ to an optimal quantitative estimate. The proof relies on the weak Riemannian Piola identity of Kupferman-Maor-Shachar, a uniform $C^{1,\alpha}$ approximation through the harmonic map heat flow, and a linearization argument which reduces the estimate to the well-known Riemannian version of Korn's inequality.

Chemical reactions can be understood as transformations of multigraphs (molecules) that preserve vertex labels (atoms) and degrees (sums of bonding and non-bonding electrons), thereby implying the atom-atom map of a reaction. The corresponding reaction mechanism is often described by an electron pushing diagram that explains the transformation by consecutive local relocations of invidudal edges (electron pairs). Here, we show that every degree-preserving map between multigraphs, and thus every atom-atom map, can be generated by cyclic electron pushing. Moreover, it is always possible to decompose such an explanation into electron pushing diagrams involving only four electron pairs. This in turn implies that every reaction can be decomposed into a sequence of elementary reactions that involve at most two educt molecules and two product molecules. Hence, the requirement of a mechanistic explantion in terms of electron pushing and small imaginary transition states does not impose a combinatorial constraint on the feasibility of hypothetical chemical reactions.

The PSI Positron Production (P$^3$ or P-cubed) experiment is a demonstrator for a e+ source and capture system with potential to improve the state-of-the-art e+ yield by an order of magnitude. The experiment is driven by the FCC-ee injector study and will be hosted in the SwissFEL facility at the Paul Scherrer Institute in Switzerland. This paper is an overview of the P$^3$ design at an advanced stage, with a particular emphasis on a novel e+ capture system and its associated beam dynamics. Additionally, a concept for the experiment diagnostics is presented, as well as the key points of the ongoing installation works.

We provide fundamental results on positive solutions to parametrized systems of generalized polynomial $\textit{inequalities}$ (with real exponents and positive parameters), including generalized polynomial $\textit{equations}$. In doing so, we also offer a new perspective on fewnomials and (generalized) mass-action systems. We find that geometric objects, rather than matrices, determine generalized polynomial systems: a bounded set/"polytope" $P$ (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency $d$, is crucial. As our main result, we rewrite $\textit{polynomial inequalities}$ in terms of $d$ $\textit{binomial equations}$ on $P$, involving $d$ monomials in the parameters. In particular, we establish an explicit bijection between the original solution set and the solution set on $P$ via exponentiation. (i) Our results apply to any generalized polynomial system. (ii) The dependency $d$ and the dimension of $P$ indicate the complexity of a system. (iii) Our results are based on methods from linear algebra and convex/polyhedral geometry, and the solution set on $P$ can be further studied using methods from analysis such as sign-characteristic functions (introduced in this work). We illustrate our results (in particular, the relevant geometric objects) through 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 even for univariate trinomials, we offer new insights: We provide a "solution formula" involving discriminants and "roots"., Comment: arXiv admin note: substantial text overlap with arXiv:2304.05273. author note: this manuscript and arXiv:2304.05273 are parallel works; there is no overlap regarding examples. there is overlap in the outline of the underlying theory for a self-contained presentation

We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric objects, rather than matrices: a polytope $P$ (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency $d$, is crucial. Indeed, we rewrite $\textit{polynomial}$ equations in terms of $d$ $\textit{binomial}$ equations on the coefficient polytope $P$, involving $d$ monomials in the parameters. We further study the solution set on $P$ using methods from analysis such as sign-characteristic functions and Wronskians. In this work, we present first applications to fewnomial systems through five (classes of) examples. In particular, we study (i) $n$ trinomials involving ${n+2}$ monomials in $n$ variables, having dependency $d=1$, and (ii) one trinomial and one $t$-nomial (with $t\ge3$) in two variables, having $d=t-1\ge2$. For (i), we bound the number of positive solutions using the number of roots of a univariate polynomial of degree at most $n$. We also show that this number is always less than or equal to the number of sign changes in an optimal Descartes' rule given in Bihan et al. (2021). For (ii), we improve upper bounds given in Li et al. (2003) and Koiran et al. (2015). Further, for two trinomials ($t=3$), we refine the known upper bound of five in terms of the exponents, and we find an example with five positive solutions that is even simpler than the smallest "Haas system"., Comment: author note: this manuscript and arXiv:2306.13916 are parallel works; there is overlap in the outline of the underlying theory, required for a self-contained presentation. more specifically, in this manuscript, we apply results from arXiv:2306.13916

Generalized mass-action systems are power-law dynamical systems arising from chemical reaction networks. Essentially, every nonnegative ODE model used in chemistry and biology (for example, in ecology and epidemiology) and even in economics and engineering can be written in this form. Previous results have focused on existence and uniqueness of special steady states (complex-balanced equilibria) for all rate constants, thereby ruling out multiple (special) steady states. Recently, necessary conditions for linear stability have been obtained. In this work, we provide sufficient conditions for the linear stability of complex-balanced equilibria for all rate constants (and also for the non-existence of other steady states). In particular, via sign-vector conditions (on the stoichiometric coefficients and kinetic orders), we guarantee that the Jacobian matrix is a $P$-matrix. Technically, we use a new decomposition of the graph Laplacian which allows to consider orders of (generalized) monomials. Alternatively, we use cycle decomposition which allows a linear parametrization of all Jacobian matrices. In any case, we guarantee stability without explicit computation of steady states. We illustrate our results in examples from chemistry and biology: generalized Lotka-Volterra systems and SIR models, a two-component signaling system, and an enzymatic futile cycle.

The journal impact factor (JIF) is often equated with journal quality and the quality of the peer review of the papers submitted to the journal. We examined the association between the content of peer review and JIF by analysing 10,000 peer review reports submitted to 1,644 medical and life sciences journals. Two researchers hand-coded a random sample of 2,000 sentences. We then trained machine learning models to classify all 187,240 sentences as contributing or not contributing to content categories. We examined the association between ten groups of journals defined by JIF deciles and the content of peer reviews using linear mixed-effects models, adjusting for the length of the review. The JIF ranged from 0.21 to 74.70. The length of peer reviews increased from the lowest (median number of words 185) to the JIF group (387 words). The proportion of sentences allocated to different content categories varied widely, even within JIF groups. For thoroughness, sentences on 'Materials and Methods' were more common in the highest JIF journals than in the lowest JIF group (difference of 7.8 percentage points; 95% CI 4.9 to 10.7%). The trend for 'Presentation and Reporting' went in the opposite direction, with the highest JIF journals giving less emphasis to such content (difference -8.9%; 95% CI -11.3 to -6.5%). For helpfulness, reviews for higher JIF journals devoted less attention to 'Suggestion and Solution' and provided fewer Examples than lower impact factor journals. No, or only small differences were evident for other content categories. In conclusion, peer review in journals with higher JIF tends to be more thorough in discussing the methods used but less helpful in terms of suggesting solutions and providing examples. Differences were modest and variability high, indicating that the JIF is a bad predictor for the quality of peer review of an individual manuscript., Comment: 44 pages

We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible $\textit{core matrix}$, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states (complex-balanced equilibria). Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given $\textit{monomial evaluation orders}$ (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain $\textit{binomial differential inclusions}$. In particular, this holds for complex-balanced mass-action systems, and hence we also obtain a polyhedral-geometry proof of the classical result.

Reaction networks (RNs) comprise a set $X$ of species and a set $\mathscr{R}$ of reactions $Y\to Y'$, each converting a multiset of educts $Y\subseteq X$ into a multiset $Y'\subseteq X$ of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix $\mathbf{S} \in \mathbb{R}^{X\times\mathscr{R}}$ have been discussed extensively in the literature. Here, we provide sufficient conditions for $\mathbf{S}$ that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively. Chemically plausible RNs allow neither a perpetuum mobile, i.e., a "futile cycle" of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of $\mathbf{S}$. The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and - in a stricter sense - futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. Furthermore, it is sufficient for the existence of a realization in terms of sum formulas, obeying conservation of "atoms". In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless $\mathbf{S}$ implies that they are "obligatory isomers". In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved.

Motivated by experiments and formal asymptotic expansions in the physics literature, Maor and Shachar (J. Elasticity 134 (2019), 149-173) studied the behaviour of a model elastic energy of maps between manifolds with incompatible metrics. For thin objects they analysed the scaling of the minimal elastic energy as a function of the thickness. In particular they showed that for maps from a ball of radius h in an oriented Riemannian manifold to Euclidean space, the infimum of a model elastic energy per unit volume scales like the fourth power of h and after rescaling one gets convergence to a quadratic expression in the curvature tensor R(p), where p denotes the centre of the ball. In this paper we show the same result for general compact oriented Riemannian targets with R(p) replaced by a suitable difference of the curvature tensors in the target and the domain, thus answering Open Question 1 in the paper by Maor and Shachar. The result extends noncompact targets provided they satisfy a uniform regularity condition. A key idea in the proof is to use Lipschitz approximations to define a suitable notion of convergence., Comment: Typos corrected, exposition slightly expanded, reference [2] added

Let $\Omega \subset \mathbb{R}^3$ be an open and bounded set with Lipschitz boundary and outward unit normal $\nu$. For $11 \quad \text{if $p = \frac32$.}$$ Specifically, there exists a constant $c=c(p,\Omega,r)>0$ such that the inequality \[ \|P \|_{L^p}\leq c\,\left(\|\operatorname{sym} P \|_{L^p} + \|\operatorname{dev}\operatorname{sym} \operatorname{Curl} P \|_{L^{r}}\right) \] holds for all tensor fields $P\in W^{1,\,p, \, r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl})$. Here, $\operatorname{dev} X := X -\frac13 \operatorname{tr}(X)\,\mathbb{1}$ denotes the deviatoric (trace-free) part of a $3 \times 3$ matrix $X$ and the boundary condition is understood in a suitable weak sense., Comment: Ref.numbers are now visible