relaxation oscillations in a class of predator prey systems abstract we consider a class of three dimensional singularly perturbed predator prey systems having two predators competing exploitatively for the same prey in a constant environment by using dynamical systems techniques and the geometric singular perturbation theory we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators as an application a well known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model

the structure of max plus hyperplanes a max plus hyperplane briefly a hyperplane is the set of all points x x1 xn rmaxn satisfying an equation of the form a1x1 anxn an 1 b1x1 bnxn bn 1 that is max a1 x1 an xn an 1 max b1 x1 bn xn bn 1 with ai bi rmax i 1 n 1 where each side contains at least one term and where ai bi for at least one index i we show that the complements of max plus semispaces at finite points z rn are building blocks for the hyperplanes in rmaxn recall that a semispace at z is a maximal with respect to inclusion max plus convex subset of rmaxn z namely observing that up to a permutation of indices we may write the equation of any hyperplane h in one of the following two forms a1x1 apxp ap 1xp 1 aqxq a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m n 1 are finite or a1x1 apxp ap 1xp 1 aqxq an 1 a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m are finite and an 1 is either finite or we give a formula that expresses a nondegenerate strictly affine hyperplane i e with m n and an 1 as a union of complements of semispaces at a point z rn called the center of h with the boundary of a union of complements of other semispaces at z using this formula we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior we give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center and we characterize the cases in which the boundary bd h of a nondegenerate strictly affine hyperplane h is also a hyperplane next we give the relations between nondegenerate strictly affine hyperplanes h their centers z and their coefficients ai in the converse direction we show that any union of complements of semispaces at a point z rn with the boundary of any union of complements of some other semispaces at that point z is a nondegenerate strictly affine hyperplane we obtain a formula for the total number of strictly affine hyperplanes we give complete lists of all strictly affine hyperplanes for the cases n 1 and n 2 we show that each linear hyperplane h in rmaxn i e with an 1 can be decomposed as the union of four parts where each part is easy to describe in terms of complements of semispaces some of them in a lower dimensional space

remarks on competitive coexistence the asymptotic behavior of two predators competing exploitatively for the same prey in a constant environment is analyzed in this paper assuming that the prey have fast dynamics it is proved through a singular perturbation argument that the two predators can coexist if the prey and predators parameters lie within specified ranges it is also shown that coexistence occurs through a simple periodic behavior a poor season characterized by an almost endemic presence of the prey alternates with a rich season during which prey are abundant and predators regenerate this property follows from the geometric nature of the equilibrium manifold and is therefore generic the results are in agreement with some conjectures supported by computer simulation and with the most recent theoretical findings on competitive coexistence

the entry exit theorem and relaxation oscillations in slow fast planar systems abstract the entry exit theorem for the phenomenon of delay of stability loss for certain types of slow fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations the general existing proofs of this theorem depend on fenichel s geometric singular perturbation theory and blow up techniques in this work we give a short and elementary proof of the entry exit theorem based on a direct study of asymptotic formulas of the underlying solutions we employ this theorem to a broad class of slow fast planar systems to obtain existence global uniqueness and asymptotic orbital stability of relaxation oscillations the results are then applied to a diffusive predator prey model with holling type ii functional response to establish periodic traveling wave solutions furthermore we extend our work to another class of slow fast systems that can have multiple orbits exhibiting relaxation oscillations and subsequently apply the results to a two time scale holling tanner predator prey model with holling type iv functional response it is generally assumed in the literature that the non trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations we do not make this assumption in this paper

stability loss delay in harvesting competing populations abstract when populations are in competition it often happens that one of them disappears harvesting may be used for the control and management of competing species to stabilize the populations at a persistent equilibrium a three dimensional model where the harvesting effort is a dynamic variable is studied in the case where the growth rate of the harvesting effort is very slow the analysis shows that the system can have relaxation oscillations dynamic bifurcation theory is used to determine the maximal and minimal values of harvesting effort along this cycle of oscillations

spectral analysis of time dependent market adjusted return correlation matrix abstract we present an adjusted method for calculating the eigenvalues of a time dependent return correlation matrix in a moving window first we compare the normalized maximum eigenvalue time series of the market adjusted return correlation matrix to that of the logarithmic return correlation matrix on an 18 year dataset of 310 s p 500 listed stocks for small and large window or memory sizes we observe that the resulting new eigenvalue time series is more stationary than the time series obtained without the adjustment second we perform this analysis while sweeping the window size 5 100 500 in order to examine the dependence on the choice of window size this approach demonstrates the multi modality of the eigenvalue distributions we find that the three dimensional distribution of the eigenvalue time series for our market adjusted return is significantly more stationary than that produced by classic method finally our model offers an approximate polarization domain characterized by a smooth l shaped strip the polarization with large amplitude is revealed while there is persistence in agreement of individual stock returns movement with market with small amplitude most of the time

persistent unstable periodic motions i we study the initial value problems of the two dimensional system formula wherei t ii tis a slowly varying parameter f u t i is analytic on all variables and periodic ontwith period 2 andu0 t i are periodic solutions of the system formula whereiis a constant parameter we assume that the variational equations of 0 2 wt fv u0 t i t i w have the corresponding characteristic exponents 1 i 2 i 1 i which move across the imaginary axis from the left half complex plane to the right half complex plane asiincreases pasti we show that under the nonresonant conditions h4 and h5 that 2 n im 1 i forn n in section 4 along with other generic conditions such as h1 h3 below the separations ofu i t fromu0 t1 t i t do not occur at the critical point wherei t i rather the bifurcations are substantially delayed untili t ii treachesiq i which is independent of as 0 in other words u t u0 t1 t i t 0 fort t ii i t iq for someiq iq ii i independent of when 0 an exact formula foriqis given for general situations in near resonance cases a sharp estimate ofiqis also derived

3d extreme value analysis for stock return interest rate and speed of mean reversion it is important to analyze extreme cases of stock return interest rate and speed of mean reversion together while we explore strengths and limitations of heston stochastic volatility model based on behavior of its numerical solutions using milstein method simulations we suggest model improvements in the light of real data applications first we perform high peak and fat tail analysis for the impact of heston model parameters on the simulations of the extreme situations by using the first four standardized moments and extreme value tools such as quantile quantile qq mean excess me and hill plots to examine the fat tailness of the distributions later we illustrate high peak and fat tail analysis for bist 100 index between 02 01 2004 and 17 06 2013 moreover we investigate 3d dynamics of the average logarithmic stock return interest rate and speed of mean reversion variables together furthermore we believe that polarization and the transitions between polarizations and comovements are important part of extreme situation picture we investigate comovement and polarization of interest rates and daily returns of bist 100 index between 2010 and 2013 in order to understand the corresponding behavioral dynamics heston stochastic volatility model predicts that the average logarithmic stock return increases as interest rate rises actually we observe that there are also sufficiently large time intervals where interest rates were decreased and stock prices increased gradually in us stock markets and borsa istanbul unlike the heston stochastic volatility model suggests

the entry exit function and geometric singular perturbation theory this work was partly supported by fwo project g093910 de maesschalck and nsf grant dms 1211707 schecter we thank the referee for a careful reading of the paper which led to a number of clarifications and corrections

decay to uniform states in predator prey chains weakly coupled parabolic systems describing populations undergoing diffusion and in a predator prey chain relationship are discussed the systems are of lotka volterra type with crowding effects included sufficient conditions on the interaction terms are given which imply the existence and global asymptotic stability of a unique equilibrium point with all species coexisting

dynamic hopf bifurcations generated by nonlinear terms abstract we consider the so called delayed loss of stability phenomenon for singularly perturbed systems of differential equations in case that the associated autonomous system with a scalar parameter undergoes the hopf bifurcation at the zero equilibrium point it is assumed that the linearization of the associated system is independent of the parameter and the next terms in the expansion of the right hand parts at zero are positive homogeneous of order 1 simple formulas are presented to estimate the asymptotic delay for the delayed loss of stability phenomenon more precisely we suggest sufficient conditions which ensure that zeros of a simple function defined by the positive homogeneous nonlinear terms are the hopf bifurcation points of the associated system the sign of at other points determines stability of the zero equilibrium and the asymptotic delay equals the distance between the bifurcation point and a zero of some primitive of

a note on unsolvable systems of max min fuzzy equations abstract for unsolvable systems of linear equations of the form a x b over the max min fuzzy algebra r 0 1 max min we propose an efficient method for finding a chebychev best approximation of the matrix a r m n in the set s b d r m n d x b is solvable

application of the heston stochastic volatility model for borsa istanbul using impression matrix norm we study the behavior of solutions for stochastic differential equations such as the heston stochastic volatility model we examine the numerical solutions using euler maruyama milstein and stochastic runge kutta methods to investigate whether there is a role of the methods for different volatility cases or not related to the impact of cumulative errors on this application we perform simulations for different stock market conditions by using the large data set from borsa istanbul 100 bist 100 between 04 01 2007 and 31 12 2012 we use volatilities in terms of extreme values at the overlapping case when we examine initial and long term volatilities for the application of the heston model we also apply unit volatility based on extreme values to approximate volatilities in our analysis we examine the advantages and limitations of the model moreover we introduce 3 dimensional matrix norms furthermore we define market impression matrix norm as an application to the 3 dimensional matrix norms we can benefit from it to quantify market impression approximately by means of the numerical solutions for the stochastic differential equations finally we analyze the simulation results for various parameters

solvability and unique solvability of max min fuzzy equations abstract the problem of solvability and the problem of unique solvability of a fuzzy relation equation in an arbitrary max min algebra are considered and corresponding necessary and sufficient conditions are presented the results allow to solve both problems by an o mnp algorithm where m n p are the dimensions of the corresponding relations in the equation the existence of the greatest and of the least solution is also considered

bifurcation analysis of a prey predator coevolution model we show in this paper how numerical bifurcation analysis can be used to study the evolution of genetically transmitted phenotypic traits for this we consider the standard rosenzweig macarthur prey predator model the american naturalist 97 1963 pp 209 223 and following the so called adaptive dynamics approach we derive from it a second order evolutionary model composed of two odes one for the prey trait and one for the predator trait then we perform a detailed bifurcation analysis of the evolutionary model with respect to various environmental and demographic parameters surprisingly the evolutionary dynamics turn out to be much richer than the population dynamics up to three evolutionary attractors can be present and the bifurcation diagrams contain numerous global bifurcations and codimension 2 bifurcation points interesting biological properties can be extracted from these bifurcation diagrams in particular one can conclude that evolution of the traits can be cyclic and easily prom

population viability in three trophic level food chains the perpetuation of three trophic level ecosystems where the three species exhibit unpredictable time varying survival strategies is described by a specific set the viability kernel gathering all states from which there exists at least one trajectory safeguarding each species over a given density threshold the strategies permitting this property are delineated and called viable strategies all solutions starting outside the viability kernel lead to too low densities or extinction the viability approach highlights the timing of strategy changes necessary for a system to perpetuate itself or alternatively to lead one species to extinction the study of the dependence of the viability kernel on the admissible sets of strategies reveals the minimal flexibilities necessary for the existence of the system the shape of the viability kernel determines whether the exogenous addition or substraction of prey or predator will endanger the system or not thus gathering different experiments with opposite results the comparison of the coexistence kernel with viability kernels for one two or three species points out the importance of repeated strategies not necessarily in a periodic manner thus emphasizing the concept of repetitions in ecosystems instead of cycles as a key feature of coexistence

belyakov homoclinic bifurcations in a tritrophic food chain model complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper first it is shown that the model admits a sequence of pairs of belyakov bifurcations codimension two homoclinic orbits to a critical node then fold and period doubling cycle bifurcation curves associated to each pair of belyakov points are computed and analyzed the overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist the analysis is conducted by combining numerical continuation techniques with theoretical arguments

the slow passage through a hopf bifurcation delay memory effects and resonance this paper explores analytically and numerically in the context of the fitzhugh nagumo model of nerve membrane excitability an interesting phenomenon that has been described as a delay or memory effect it can occur when a parameter passes slowly through a hopf bifurcation point and the system s response changes from a slowly varying steady state to slowly varying oscillations on quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects the dynamic aspect of the parameter variation this delay and its dependence on the speed of the parameter variation are described the model involves several parameters and particular singular limits are investigated one in particular is the slow passage through a low frequency hopf bifurcation where the system s response changes from a slowly varying steady state to slowly varying relaxation oscillations we find in this case the onset o

analytical study of a triple hopf bifurcation in a tritrophic food chain model we provide an analytical proof of the existence of a stable periodic orbit contained in the region of coexistence of the three species of a tritrophic chain the method used consists in analyzing a triple hopf bifurcation for some values of the parameters three limit cycles born via this bifurcation one is contained in the plane where the top predator is absent another one is not contained in the domain of interest where all variables are positive the third one is contained where the three species coexist the techniques for proving these results have been introduced in previous articles by the second author and are based on the averaging theory of second order existence of this triple hopf bifurcation has been previously discovered numerically by kooij et al

uncertainty transformation via hopf bifurcation in fast slow systems propagation of uncertainty in dynamical systems is a significant challenge here we focus on random multiscale ordinary differential equation models in particular we study hopf bifurcation in the fast subsystem for random initial conditions we show that a random initial condition distribution can be transformed during the passage near a delayed dynamic hopf bifurcation i to certain classes of symmetric copies ii to an almost deterministic output iii to a mixture distribution with differing moments and iv to a very restricted class of general distributions we prove under which conditions the cases i iv occur in certain classes vector fields

on bifurcation delay an alternative approach using geometric singular perturbation theory abstract to explain the phenomenon of bifurcation delay which occurs in planar systems of the form x f x z z g x z z where f x 0 0 0 and g x 0 0 changes sign at least once on the x axis we use the exchange lemma in geometric singular perturbation theory to track the limiting behavior of the solutions using the trick of extending dimension to overcome the degeneracy at the turning point we show that the limiting attracting and repulsion points are given by the well known entry exit function and the minimum of z on the trajectory is of order exp 1 also we prove smoothness of the return map up to arbitrary finite order in

low and high frequency oscillations in three dimensional food chain systems the dynamic behavior of a three trophic level food chain prey predator superpredator is analyzed in this paper prey is logistic while predator and superpredator have functional response of the holling type moreover the trophic levels are assumed to be characterized by increasing and quite diversified time responses using a singular perturbation approach explicit conditions for the persistence of the three populations are derived and the structure of the corresponding attractors is noted as well as the nature of transients the analysis shows that the system can have low frequency cycles due to the interactions between predator and superpredator nevertheless the interactions between prey and predator can give rise to high frequency oscillations which can arise during the transients toward the attractor or during the low frequency cycle this periodic burst of high frequency oscillations develops in particular when the two predators are fairly efficient these results note in a very concise w

period doubling cascades in a predator prey model with a scavenger the dynamics of the classic planar two species lotka volterra predator prey model are well understood we introduce a scavenger species that scavenges the predator and is also a predator of the common prey for this model we analytically prove that all trajectories are bounded in forward time and numerically demonstrate persistent bounded paired cascades of period doubling orbits over a wide range of parameter values standard analytical and numerical techniques are used in the analysis of this model making it an ideal pedagogical tool we include exercises and an open ended project to promote mastery of these techniques

delayed oscillation phenomena in the fitzhugh nagumo equation we study the problem of the slow passage through a hopf bifurcation point for the fitzhugh nagumo equation fhn vt dvxx v w i0 t 0 1a wt bv b w 0 1b where has some properties so that the system has a hopf bifurcation at i i when 0 and i i0 t is regarded as a parameter independent of t the experimental results of jakobsson and guttman showed that large amplitude oscillations occurred only after i reaches a value well above i when is positive and small the paper of s m baer t erneux and j rinzel siam j appl math 49 1989 55 71 studied these phenomena numerically and produced a prediction of the ignition jumping time for the system in this work we provide a rigorous proof of the results conjectured by baer erneux and rinzel referenced above we show that if we start the solution of 0 1 at any point near the frame solution which is the zero of the right hand side of 0 1 at any ii i then the solution stays near the frame solution until i reaches some iq i furthermore for those cases in which ii is close to i we show that the solution moves from the frame solution to become a large amplitude solution after i moves above some iq i

imperfect bifurcation with a slowly varying control parameter we consider a general class of imperfect bifurcation problems described by the following first order nonlinear differential equation y i ky p lambda t y delta where k 1 or 1 and p 2 or 3 are fixed quantities the solution depends on the values of the imperfection parameter delta 0 delta ll 1 and the time dependent control parameter lambda t lambda 0 varepsilon t and 0 varepsilon ll 1 if delta varepsilon 0 this equation admits at lambda 0 a bifurcation from the basic state y 0 to nonzero steady states in the first part of the paper we analyze the perturbation of the bifurcation solutions produced both by the small imperfection delta ne 0 and the slow variation of lambda varepsilon ne 0 we show that lambda 0 does not correspond to the transition between the two branches of slowly varying steady states this transition appears at a larger value of lambda lambda 1 provided that delta is sufficiently

contributions to max min convex geometry i segments abstract we give some contributions to the theory of max min convex geometry that is convex geometry in the semimodule r max min n over the max min semiring r max min r we introduce elementary segments that generalize from n 2 the horizontal vertical or oblique segments contained in the main bisector of r max min 2 we show that every segment in r max min n is a concatenation of a finite number of elementary subsegments at most 2 n 1 respectively at most 2 n 2 in the case of comparable respectively incomparable endpoints x y in this first part we study max min segments and in the subsequent second part submitted we study max min semispaces and some of their relations to max min convex sets

eigenvalues and eigen functionals of diagonally dominant endomorphisms in min max analysis the so called min analysis may be viewed as an extension to the continuous case and to functional spaces of shortest path algebras in graphs we investigate here min max analysis which extends in some similar way minimum spanning tree problems and maximum capacity path problems in graphs an endomorphisms a of the functional min max semi module acts on any functional to produce a where a x miny max a x y y we present here a complete characterization of eigenvalues and eigen functionals of diagonally dominant endomorphisms i e such that x y a x x 0a a x y 0a it is shown in particular that any real value 0a is an eigenvalue and that the associated eigen semi module has a unique minimal generator

residuation in fuzzy algebra and some applications abstract the system 0 1 max min has a number of well known applications many of which involve the inversion of linear relations we show the advantage of residuation theory for this process and apply it to the analysis of the problem of chebychev approximation

contributions to max min convex geometry ii semispaces and convex sets abstract in this article continuing v nitica i singer contributions to max min convex geometry i segments linear algebra appl 2007 doi 10 1016 j laa 2007 09 032 we give some further contributions to the theory of max min geometry the max min semifield is the set r r endowed with the operations max min in r a subset c of r n is said to be max min convex if the relations x y c r the neutral element of imply x y c where is understood componentwise and x x 1 x n for r x x 1 x n r n in analogy with the definition of semispaces for usual linear spaces see e g p c hammer maximal convex sets duke math j 22 1955 103 106 a max min semispace at a point z r n is a maximal with respect to inclusion max min convex subset of r n z in contrast to the case of linear spaces where there exist an infinity of semispaces at each point we show that in r n there exist at most n 1 max min semispaces at each point and exactly n 1 at each point whose all coordinates are finite we determine these max min semispaces and give some consequences for separation of max min convex sets from outside points we show that max min convexity restricted to the finite part r n of r n is a multi order convexity

an interval version of separation by semispaces in max min convexity abstract in this paper we study separation of a closed box from a max min convex set by max min semispaces this can be regarded as an interval extension of the known separation results we give a constructive proof of the separation in the case when the box satisfies a certain condition and we show that the separation is never possible when the condition is not satisfied we also study the separation of two max min convex sets by a box and by a box and a semispace

resolving infeasibility in extremal algebras abstract two extremal algebras b based on a linearly ordered set b are considered in the maxmin algebra max min and in the maxgroup algebra max and is a group operation if a system a x b of linear equations over an extremal algebra is insolvable then any subset of equations such that its omitting leads to a solvable subsystem is called a relieving set we show that the problem of finding the minimum cardinality relieving set is np complete in the maxmin algebra already for bivalent systems while it is polynomially solvable for bivalent systems in maxgroup algebra and also np complete for trivalent systems

minimax algebra and applications publisher summary this chapter discusses the applications of minimax algebra the chapter discusses a discrete event system des in which the individual components move from event to event rather than varying continuously through time a characteristic of many such dess which is focussed in the chapter is that any given component must wait before proceeding to its next event until certain others have completed their current events the chapter presents a hypothetical des containing four machines called the model system max algebra depends on a crucial change of notation in place of the operator max use of the symbol reminiscent of an addition symbol instead of use reminiscent of a multiplication symbol an and gate is a device used in signal processing it is characterized by a single output and a number of inputs each of which may be active or quiescent a delay is characterized by a single input a number of outputs and a given fixed time interval the chapter discusses several processes of max algebra scheduling and approximation chebyshev approximation and others

an application of the separation principle for detecting slow fast limit cycles in a three dimensional system a geometric method called the separation principle is used in this paper for detecting limit cycles in a slow fast dynamical system describing a food chain with trophic levels with diversified time responses the analysis is very simple and based on singular perturbation arguments the limit cycle resulting from the method is composed of a concatenation of catastrophic transitions occuring at different speeds

the structure of max min hyperplanes abstract in this article continuing 12 13 further contributions to the theory of max min convex geometry are given the max min semiring is the set r r endowed with the operations max min in r a max min hyperplane briefly a hyperplane is the set of all points x x 1 x n r n satisfying an equation of the form a 1 x 1 a n x n a n 1 b 1 x 1 b n x n b n 1 with a i b i r i 1 n 1 where each side contains at least one term and where a i b i for at least one index i the main result is a description of a hyperplane in terms of simple polyhedral blocks as an application one shows that the separation of max min closed convex sets by max min hyperplanes is not possible in general

a dynamical system with hopf bifurcations and catastrophes a well known second order positive dynamical system describing the dynamics of a renewable resource together with its consumers prey predator model is first considered such a dynamical system is recognized to be a subtle hopf bifurcation without catastrophes then the model is slightly modified by including a control parameter which represents the indirect exploitation of the resource this new system is shown to have a much richer and interesting dynamics in particular catastrophes of the fold type hysteresis may arise moreover when the control value corresponding to the hopf bifurcation is greater than the lower threshold of the hysteresis one of the two transitions characterizing the catastrophe is a transition between a stableequilibrium and a limit cycle the results are illustrated in the control phase portrait

decay to uniform states in food webs weakly coupled parabolic systems describing populations undergoing diffusion and interaction dynamics are discussed the systems are of lotka volterra type and model complex food webs in which there can be several trophic levels involving predator prey competition commensal and amensal interactions sufficient conditions on the interaction terms are given which imply the existence and global asymptotic stability of a unique equilibrium point with all species coexisting the theory of contracting rectangles is the main tool used in proving the stability result

on delayed oscillation in nonspatially uniform fitzhugh nagumo equation abstract we study the problem of the slow passage through a hopf bifurcation point for the fitzhugh nagumo equation fhn v t dv xx f v w x i i t 0 1a w t bv b w 0 1b where f has some properties so that the system has a hopf bifurcation at i i when 0 and i ii t is regarded as a parameter independent of t the experimental results of e jackobsson and r guttman 1981 in biophysical approach to excitable systems plenum new york showed that large amplitude oscillations occurred only after i reached a value well above i when is positive and small s m baer t erneux and j rinzel 1989 siamappl math 49 55 71 studied these phenomena numerically and produced a prediction of the ignition jumping time for the system j su 1993 j differential equations105 180 215 1990 delayed oscillation phenomena in fitzhugh naguma equation ph d thesis proved the delayed oscillation phenomena when x 1 in this work we show that delayed oscillations occur when is small enough for any x 0

on the dimension of max min convex sets we introduce a notion of dimension of max min convex sets following the approach of tropical convexity we introduce a max min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope we describe the relation between this rank and the notion of strong regularity in max min algebra which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property