variable elimination in chemical reaction networks with mass action kinetics we consider chemical reaction networks taken with mass action kinetics the steady states of such a system are solutions to a system of polynomial equations even for small systems the task of finding the solutions is daunting we develop an algebraic framework and procedure for linear elimination of variables the procedure reduces the variables in the system to a set of core variables by eliminating variables corresponding to a set of noninteracting species the steady states are parameterized algebraically by the core variables and a graphical condition is given that ensures that a steady state with positive core variables necessarily takes positive values for all variables further we characterize graphically the sets of eliminated variables that are constrained by a conservation law and show that this conservation law takes a specific form

rational conformal field theories and complex multiplication we study the geometric interpretation of two dimensional rational conformal field theories corresponding to sigma models on calabi yau manifolds we perform a detailed study of rcft s corresponding to the t2 target and identify the cardy branes with geometric branes the t2 s leading to rcft s admit complex multiplication which characterizes cardy branes as specific d0 branes we propose a condition for the conformal sigma model to be rcft for arbitrary calabi yau n folds which agrees with the known cases together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories and sometimes appear to be finite in number for calabi yau n folds for n 2 rcft s on k3 may be dense we speculate about the meaning of these special points in the moduli spaces of calabi yau n folds in connection with freezing geometric moduli

quasi steady state reduction for compartmental systems abstract we present a method to determine an asymptotic reduction in the sense of tikhonov and fenichel for singularly perturbed compartmental systems in the presence of slow transport it turns out that the reduction can be derived from the individual interaction terms alone we apply the result to spatially discretized reaction diffusion systems and obtain based on the reduced discretized systems a heuristic to reduce reaction diffusion systems in presence of slow diffusion

new approach to the stability of chemical reaction networks piecewise linear in rates lyapunov functions piecewise linear in rates pwlr lyapunov functions are introduced for a class of chemical reaction networks crns in addition to their simple structure these functions are robust with respect to arbitrary monotone reaction rates of which mass action is a special case the existence of such functions ensures the convergence of trajectories towards equilibria and can be used to establish their asymptotic stability with respect to the corresponding stoichiometric compatibility class we give the definition of these lyapunov functions prove their basic properties and provide algorithms for constructing them examples are provided relationship with consensus dynamics are discussed and future directions are elaborated

polyhedral lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the jacobian is non singular for a vast class of dynamical networks including chemical reaction networks crns with monotonic reaction rates the existence of a polyhedral lyapunov function plf implies structural i e parameter free local stability global structural stability is ensured under the additional assumption that each of the variables chemical species concentrations in crns is subject to a spontaneous infinitesimal dissipation this paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation showing that the existence of a plf structurally ensures global convergence if and only if the system jacobian passes a structural non singularity test it is also shown that if the jacobian is structurally non singular under positivity assumptions for the system partial derivatives the existence of an equilibrium is guaranteed for systems subject to positivity constraints it is shown that if the system admits a plf under structural non singularity assumptions global convergence within the positive orthant is structurally ensured while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise linear in rate lyapunov function

eigenfunctions of the laplacian acting on degree zero bundles over special riemann surfaces we find an infinite set of eigenfunctions for the laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special riemann surfaces of genus greater than one these special surfaces correspond to riemann period matrices satisfying a set of equations which lead to a number theoretical problem it turns out that these surfaces precisely correspond to branched covering of the torus this reflects in a jacobian with a particular kind of complex multiplication

the deficiency zero theorem and global asymptotic stability for a class of chemical reaction networks with arbitrary time delays abstract the deficiency zero theorem dzt is extended to be applied to chemical reaction networks crns with time delays the time evolution of concentrations of species in which networks is described by delay differential equations ddes the extended dzt provides a sufficient condition for a positive equilibrium point of ddes to exist and be locally asymptotically stable furthermore it is proven that the absence of boundary equilibria is a sufficient condition for the positive equilibrium point to be globally asymptotically stable these results are natural analogues of theorems proven for crns without time delays

linear relations among holomorphic quadratic differentials and induced siegel s metric on mg we find the explicit form of the volume form on the moduli space of non hyperelliptic riemann surfaces induced by the siegel metric a long standing question in string theory this question is related to the explicit form of the g 2 g 3 2 linearly independent relations among the 2 fold products of holomorphic abelian differentials that are provided in the case of canonical curves of genus g 4 such relations can be completely expressed in terms of determinants of the standard normalized holomorphic abelian differentials remarkably it turns out that the induced volume form is the kodaira spencer map of the square of the bergman reproducing kernel

systematic measures of biological networks ii degeneracy complexity and robustness this paper is part ii of a two part series devoted to the study of systematic measures in a complex bionetwork modeled by a system of ordinary differential equations in this part we quantify several systematic measures of a biological network including degeneracy complexity and robustness we will apply the theory of stochastic differential equations to define degeneracy and complexity for a bionetwork robustness of the network will be defined according to the strength of attractions to the global attractor based on the study of stationary probability measures and entropy made in part i of this series we will investigate some fundamental properties of these systematic measures in particular the connections between degeneracy complexity and robustness 2016 wiley periodicals inc

a reaction network approach to the theory of acoustic wave turbulence abstract we propose a new approach to study the long time dynamics of the wave kinetic equation in the statistical description of acoustic turbulence the approach is based on rewriting the discrete version of the wave kinetic equation in the form of a chemical reaction network then employing techniques used to study the global attractor conjecture to investigate the long time dynamics of the newly obtained chemical system we show that the solution of the chemical system converges to an equilibrium exponentially in time in addition a resonance broadening modification of the acoustic wave kinetic equation is also studied with the same technique for the near resonance equation if the resonance broadening frequency is larger than a threshold the solution of the system goes to infinity as time evolves

sensitivity and network topology in chemical reaction systems in living cells biochemical reactions are catalyzed by specific enzymes and connect to one another by sharing substrates and products forming complex networks in our previous studies we established a framework determining the responses to enzyme perturbations only from network topology and then proved a theorem called the law of localization explaining response patterns in terms of network topology in this paper we generalize these results to reaction networks with conserved concentrations which allows us to study any reaction system we also propose network characteristics quantifying robustness we compare e coli metabolic network with randomly rewired networks and find that the robustness of the e coli network is significantly higher than that of the random networks

input to state stability of rate controlled biochemical networks in this paper the study of the class of biochemical systems known as zero deficiency networks is extended to the case of time varying kinetic parameters we show that the resulting class of nonlinear systems with inputs satisfies a notion of input to state stability uniformly over a set of parameters in particular the input to state stability estimates allow us to characterize the robustness of zero deficiency networks with respect to perturbations in the parameters as well as study their stability when the reaction rates are controlled by an independent process

piecewise linear lyapunov functions for structural stability of biochemical networks we consider the problem of assessing structural stability of biochemical reaction networks with monotone reaction rates namely of establishing if all the networks with a certain structure are stable regardless of specific parameter values we investigate stability by absorbing the network equations in a linear differential inclusion and seeking for a polyhedral lyapunov function proper to the considered network structure a numerical recursive procedure is devised to test stability for a wide class of mono and bimolecular reaction networks which we name unitary the procedure is shown to be very efficient since due to the particular structure of the problem it requires iterations in the space of integer valued matrices we also consider a similar less conservative procedure that allows us to test even when the lyapunov function cannot be found whether the system evolution is structurally bounded in this case we absorb the equations in a positive linear differential inclusion to show the effectiveness of the proposed procedure we report the outcomes of both a stability and a boundedness test for many non trivial biochemical reaction networks and we analyze well established models in the literature

preclusion of switch behavior in networks with mass action kinetics we study networks taken with mass action kinetics and provide a jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants we are concerned with the characterization of injective networks that is networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class we show that a network is injective if and only if the determinant of the jacobian of a certain function does not vanish the function consists of components of the species formation rate function and a maximal set of independent conservation laws the determinant of the function is a polynomial in the species concentrations and the rate constants linear in the latter and its coefficients are fully determined the criterion also precludes the existence of degenerate steady states further we relate injectivity of a network to that of the network obtained by adding outflow or degradation reactions for all species

a network dynamics approach to chemical reaction networks abstracta treatment of a chemical reaction network theory is given from the perspective of nonlinear network dynamics in particular of consensus dynamics by starting from the complex balanced assumption the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in the chemical reaction network theory and which directly relates to the thermodynamics and port hamiltonian formulation of the system central in this formulation is the definition of a balanced laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality this immediately leads to the characterisation of the set of equilibria and their stability furthermore the assumption of complex balancedness is revisited from the point of view of kirchhoff s matrix tree theorem both the form of the dynamics and the deduced behaviour are very similar to consensus dynamics and provide additional perspectives to the latte

nonlinear dynamics of networks the groupoid formalism a formal theory of symmetries of networks of coupled dynamical systems stated in terms of the group of permutations of the nodes that preserve the network topology has existed for some time global network symmetries impose strong constraints on the corresponding dynamical systems which affect equilibria periodic states heteroclinic cycles and even chaotic states in particular the symmetries of the network can lead to synchrony phase relations resonances and synchronous or cycling chaos symmetry is a rather restrictive assumption and a general theory of networks should be more flexible a recent generalization of the group theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network the input sets now the symmetry group becomes a groupoid which is an algebraic structure that resembles a group except that the product of two elements may not be defined the groupoid formalism makes it possible to extend group theoretic methods to more general networks and in particular it leads to a complete classification of robust patterns of synchrony in terms of the combinatorial structure of the network many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology a network of dynamical systems is not just a dynamical system with a high dimensional phase space it is also equipped with a canonical set of observables the states of the individual nodes of the network moreover the form of the underlying ode is constrained by the network topology which variables occur in which component equations and how those equations relate to each other the result is a rich and new range of phenomena only a few of which are yet properly understood

multiple equilibria in complex chemical reaction networks ii the species reaction graph for mass action kinetics the capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the structure of the underlying network of chemical reactions we suggest a new graph theoretical method for discriminating between complex reaction networks that can admit multiple equilibria and those that cannot in particular we associate with each network a species reaction graph which is similar to reaction network representations drawn by biochemists and we show that if the graph satisfies certain weak conditions the differential equations corresponding to the network cannot admit multiple equilibria em no matter what values the rate constants take because these conditions are very mild they amount to powerful and quite delicate necessary conditions that a network must satisfy if it is to have the capacity to engender multiple equilibria broad qualitative results of this kind are especially apt for individual reaction rate constants are rare

weak dynamical nonemptiability and persistence of chemical kinetics systems a frequently desirable characteristic of chemical kinetics systems is that of persistence the property that if all the species are initially present then none of them may tend toward extinction it is known that solutions of deterministically modeled mass action systems may approach only portions of the boundary of the positive orthant which correspond to semilocking sets alternatively called siphons consequently most recent work on persistence of these systems has been focused on these sets in this paper we focus on a result which states that for a conservative mass action system persistence holds if every critical semilocking set is dynamically nonemptiable and the system contains no nested locking sets we will generalize this result by introducing the notion of a weakly dynamically nonemptiable semilocking set and making novel use of the well known farkas lemma we will also connect this result to known results regarding complex balanced systems and systems with facets

nontangency based lyapunov tests for convergence and stability in systems having a continuum of equilibria this paper focuses on the stability analysis of systems having a continuum of equilibria two notions that are of particular relevance to such systems are convergence and semistability convergence is the property whereby every solution converges to a limit point that may depend on the initial condition semistability is the additional requirement that all solutions converge to limit points that are lyapunov stable we give new lyapunov function based results for convergence and semistability of nonlinear systems these results do not make assumptions of sign definiteness on the lyapunov function instead our results use a novel condition based on nontangency between the vector field and invariant or negatively invariant subsets of the level or sublevel sets of the lyapunov function or its derivative and represent extensions of previously known stability results involving semidefinite lyapunov functions to illustrate our results we deduce convergence and semistability of the kinetics of the michaelis menten chemical reaction and the closed loop dynamics of a scalar system under a universal adaptive stabilizing feedback controller

correction to structure and stability of certain chemical networks and applications to the kinetic proofreading model of t cell receptor signal transduction deals with the theory of structure stability robustness and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks the results given here extend but are also heavily based upon certain previous work by feinberg 1977 1979 1987 1995 and horn and jackson 1972 of which a self contained and streamlined exposition is included the theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by mckeithan 1995 for t cell receptor signal transduction

analysis of structures causing instabilities we present a simple new method to systematically identify all topological structures e g positive feedback loops potentially leading to locally unstable steady states icsa the instability causing structure analysis systems without any instability causing structure i e not fulfilling the necessary topological condition for instabilities cannot have unstable steady states it follows that common bistability or multistability and hopf bifurcations are excluded and sustained oscillations and deterministic chaos are most unlikely the icsa leads to new insights into the topological organization of chemical and biochemical systems such as metabolic gene regulatory and signal transduction networks

exponential time decay of solutions to reaction cross diffusion systems of maxwell stefan type the large time asymptotics of weak solutions to maxwell stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated the diffusion matrix of the system is generally neither symmetric nor positive definite but the equations admit a formal gradient flow structure which provides entropy free energy estimates the main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite dimensional inequality the key elements of the proof are the existence of a unique detailed balance equilibrium and the derivation of an inequality relating the entropy and the entropy production the main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations the idea is to enlarge the space of n partial concentrations by adding the total concentration viewed as an independent variable thus working with n 1 variables further results concern the existence of global bounded weak solutions to the parabolic system and an extension of the results to complex balance systems

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces

instantaneous limits of reversible chemical reactions in presence of macroscopic convection abstract chemically reacting systems frequently involve fast reversible reactions additional slow reactions as well as mass transport due to macroscopic convection in this situation the passage to infinite reaction speed is a means to reduce the complexity of the reaction kinetics and to avoid the need for explicit values of the rate constants thereby the large stiffness of the original system of differential equations is also removed in the present paper this instantaneous reaction limit is studied for systems with independent fast reversible reactions where the rate functions are given by mass action kinetics under realistic assumptions the limiting dynamical system is derived and convergence of the solutions is obtained as the rate constants tend to infinity the proof is based on lyapunov functions techniques and exploits the structure of rate functions that results from mass action kinetics this approach is complementary to the quasi steady state approximation which is often applied in chemical engineering the differences are illustrated by means of a classical enzyme substrate reaction scheme

analysis of autocatalytic networks in biology autocatalytic networks in particular the glycolytic pathway constitute an important part of the cell metabolism changes in the concentration of metabolites and catalyzing enzymes during the lifetime of the cell can lead to perturbations from its nominal operating condition we investigate the effects of such perturbations on stability properties e g the extent of regions of attraction of a particular family of autocatalytic network models numerical experiments demonstrate that systems that are robust with respect to perturbations in the parameter space have an easily verifiable in terms of proof complexity region of attraction properties motivated by the computational complexity of optimization based formulations we take a compositional approach and exploit a natural decomposition of the system induced by the underlying biological structure into a feedback interconnection of two input output subsystems a small subsystem with complicating nonlinearities and a large subsystem with simple dynamics this decomposition simplifies the analysis of large pathways by assembling region of attraction certificates based on the input output properties of the subsystems it enables numerical as well as analytical construction of block diagonal lyapunov functions for a large family of autocatalytic pathways

constitutive equations for micropolar hyper elastic materials in this paper the concept of hyper elasticity in the micropolar continuum theory is investigated the restrictions on the fourth order elasticity tensors are investigated using the representation theorems a general form of constitutive equations for micropolar hyper elastic isotropic materials is presented as some special cases generalizations of the neo hookean and mooney rivlin type materials to the micropolar continuum theory are presented the generalized constitutive equations reduce to those of the microplar linear elasticity theory when the deformations are infinitesimal also updated lagrangian finite element formulations for the micropolar hyper elastic materials are presented considering two planar examples it is shown that an increase in the micropolar parameter results in the reduction of the deformation of the bodies also it is shown that for a specimen with very small dimensions e g in the micron level the micropolar effects are more sensible furthermore it is shown that the influence of the micropolar parameters is dependent not only on the size of the body but also to its geometry and loading conditions for the problems in which the deformation is very close to a homogeneous state the micropolar effects are negligible