infinite polytopes in hurwitz stability region we consider hurwitz stability region in the parameter space of monic polynomials firstly a multilinear map from the positive first octant of the cartesian coordinate system to the stability region is defined based on this map a necessary and sufficient condition for stability is obtained starting from stable points polytopes with infinite edge lengths are defined stability of the edges of these polytopes is equivalent to the stability of the convex combinations of third and fourth order factor polynomials if all edges of polytope are stable then by the edge theorem the polytope is robust stable the obtained result is important for understanding the geometry of stability domain and can be used for generation of convex subsets of stable polynomials and for stabilization by fixed order controllers and other related problems

an elementary derivation of the routh hurwitz criterion in most undergraduate texts on control systems the routh hurwitz criterion is usually introduced as a mechanical algorithm for determining the hurwitz stability of a real polynomial unlike many other stability criteria such as the nyquist criterion root locus etc no attempt whatsoever is made to even allude to a proof of the routh hurwitz criterion recent results using the hermite biehler theorem have however succeeded in providing a simple derivation of routh s algorithm for determining the hurwitz stability or otherwise of a given real polynomial however this derivation fails to capture the fact that routh s algorithm can also be used to count the number of open right half plane roots of a given polynomial this paper shows that by using appropriately generalized versions of the hermite biehler theorem it is possible to provide a simple derivation of the routh hurwitz criterion which also captures its unstable root counting capability

removable singularities and non uniqueness of solutions of a singular diffusion equation we prove that the solution u of the equation ut log u u 0 in x0 0 t 2 has removable singularities at x0 0 t if and only if for any 0 0 such that c1 x x0 u x t c2 x x0 holds for all 0 x x0 0 and a t b as a consequence we obtain a sufficient condition for removable singularities at 0 t for solutions of the above equation in 2 0 t and we prove the existence of infinitely many finite mass solutions for the equation in 2 0 t when 0 u0 l1 2 is radially symmetric and u0 lloc1 2

hermite biehler routh hurwitz and total positivity simple proofs of the hermite biehler and routh hurwitz theorems are presented the total nonnegativity of the hurwitz matrix of a stable real polynomial follows as an immediate corollary

evolution equations for nonlinear degenerate parabolic pde abstract we define a convex function on h 1 and characterize its subdifferential by which we introduce an evolution equation as a weak formulation of a class of nonlinear degenerate singular and noncoercive parabolic pdes associated with an arbitrary maximal monotone graph and with the dirichlet boundary condition

classical results on the stability of linear time invariant systems and the schwarz form the paper deals with the classical results of routh and hurwitz biehler and kharitonov concerning the stability of a linear time invariant differential equation it is shown that these results follow directly from a schwarz matrix representation of stable systems

properties of the entire set of hurwitz polynomials and stability analysis of polynomial families it is proved in this paper that all hurwitz polynomials of order not less than n form two simply connected borel cones in the polynomial parameter space based on this result edge theorems for hurwitz stability of general polyhedrons of polynomials and boundary theorems for hurwitz stability of compact sets of polynomials are obtained both cases of families of polynomials with dependent and independent coefficients are considered different from the previous ones our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials moreover our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine

positivity properties of viscosity solutions of a degenerate parabolic equation we consider the problem i u t u u u 2 in r n r u x 0 u 0 x in r n where is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on r n

the routh hurwitz stability criterion revisited the case of multiple poles on imaginary axis this technical note presents a relation between the number of zero rows rows with all zero elements in the routh array and the multiplicity of j axis poles the main result of the technical note is that the existence of more than one zero row in the routh array amounts to instability of the system irrespective of any sign change in the first column

the routh hurwitz array and realization of characteristic polynomials we show that the routh hurwitz array of a given characteristic polynomial provides all the information required to realize the polynomial using an electrical circuit this new interpretation also lends to an intuitive proof of the routh hurwitz stability criterion

unforced polymer translocation compared to the forced case we present results for unforced polymer translocation from simulations using langevin dynamics in two dimensions 2d to four dimensions and stochastic rotation dynamics supporting hydrodynamic modes in three dimensions 3d we compare our results to forced translocation and a simplified model where the polymer escapes from an infinite pore the simple model shows that the scaling behavior of unforced translocation is independent of the dimension of the side to which the polymer is translocating we find that unlike its forced counterpart unforced translocation dynamics is insensitive to pore design hydrodynamics is seen to markedly speed up the unforced translocation process but not to affect the scaling relations average mean squared displacement shows scaling with average transition time in unforced but not in forced translocation the waiting time distribution in unforced translocation follows closely poissonian distribution our measured transfer probabilities align well with those obtained from an equilibrium theory in 3d but somewhat worse in 2d where a polymer s relaxation toward equilibrium with respect to its translocation time is slower consequently in stark contrast to forced translocation unforced translocation is seen to remain close to equilibrium and shows clear universality

a well posed problem in singular fickian diffusion we prove that the problem of solving u t u m 1 u x x text for 1 m leqq 0 with initial conditionu x 0 x and flux conditions at infinity mathop lim limits x to infty u m 1 u x f t mathop lim limits x to infty u m 1 u x g t admits a unique solution u in c infty infty x infty 0 t t for every el1 r 0 0 and every pair of nonnegative flux functionsf g e l loc 0 the maximal existence time is given by t sup left t smallint phi x dx int limits 0 t f s g s ds right this mixed problem is ill posed for anym outside the above specified range

brief paper necessary conditions for hadamard factorizations of hurwitz polynomials a property of hurwitz polynomials is related with the hadamard product garloff and wagner proved that hadamard products of hurwitz polynomials are hurwitz polynomials and garloff and shrinivasan shown that there are hurwitz polynomials of degree 4 which do not have a hadamard factorization into two hurwitz polynomials of the same degree 4 in this paper we give necessary conditions for an even degree hurwitz polynomial to have a hadamard factorization into two even degree hurwitz polynomials such conditions are given in terms of the coefficients of the given polynomial alone furthermore we show that if an odd degree hurwitz polynomial has a hadamard factorization then a system of nonlinear inequalities has at least one solution

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

linked by loops network structure and switch integration in complex dynamical systems abstract simple nonlinear dynamical systems with multiple stable stationary states are often taken as models for switchlike biological systems this paper considers the interaction of multiple such simple multistable systems when they are embedded together into a larger dynamical supersystem attention is focused on the network structure of the resulting set of coupled differential equations and the consequences of this structure on the propensity of the embedded switches to act independently versus cooperatively specifically it is argued that both larger average and larger variance of the node degree distribution lead to increased switch independence given the frequency of empirical observations of high variance degree distributions e g power law in biological networks it is suggested that the results presented here may aid in identifying switch integrating subnetworks as comparatively homogenous low degree substructures potential applications to ecological problems such as the relationship of stability and complexity are also briefly discussed

existence of solutions of the very fast diffusion equation in bounded and unbounded domain we prove the existence of a unique solution of the following neumann problem u t phi m u xx u 0 in a b 0 t u x 0 u0 x 0 in a b phi m u x a t g t and phi m u x b t f t forall 0 0 int a bu 0dx int 0 t f g dt m 0 0 le f g in l loc infty 0 infty 0 le u 0 in l infty a b and the case 1 1 m we also obtain a similar result in higher dimensions as a corollary we will give a new proof of a result of a rodriguez and j l vazquez on the existence of infinitely many finite mass solutions of the above equation in mathbb r times 0 t for any 1 m 0 we also obtain the exact decay rate of the solution at infinity

routh hurwitz test under vanishing leading array elements a slight modification to construct routh s array is presented to cope with the special case of vanishing leading array elements by means of the modified array root locations of a polynomial can readily be determined through counting first column sign changes as is with the normal routh s test without the necessity of adopting the e approach

a note on the routh hurwitz test this note demonstrates how the e method in the original routh hurwitz test can be applied when the routh array contains a row with zero leftmost element together with an all zero row the complete root distribution is determined by applying the criterion only once to the given polynomial d s without either factoring out a common divisor or shifting the imaginary axis since the test may become computationally tedious due to the inclusion of cumbersome e terms the recently reported method of shamash 1 is used to simplify the computational scheme this method being restricted to the case where d s has purely imaginary roots constituting the whole set of roots of the greatest common even divisor is here refined and generalized to accommodate all cases in which all zero rows appear

sufficient algebraic conditions for stability of cones of polynomials in this paper a sufficient condition for a cone of polynomials to be hurwitz is established such condition is a matrix inequality which gives a simple algebraic test for the stability of rays of polynomials as an application to stable open loop systems a cone of gains c such that the function u kctx is a stabilizing control feedback for all k 0 is shown to exist

elementary proof of the routh hurwitz test abstract this note presents an elementary proof of the familiar routh hurwitz test the proof is basically one continuity argument it does not rely on sturm chains cauchy index and the principle of the argument and it is fully self contained in the same style an extended routh hurwitz test is derived which finds the inertia of polynomials

a simple proof of the routh test an elementary proof of the classic routh method for counting the number of left half plane and right half plane zeros of a real coefficient polynomial p sub n s of degree n is given such a proof refers to the polynomials p sub i s of degree i spl les n formed from the entries of the rows of order i and i 1 of the relevant routh array in particular it is based on the consideration of an auxiliary polynomial p sub i s q linearly dependent on a real parameter q which reduces to either polynomial p sub i s or to polynomial p sub i 1 s for particular values of q in this way it is easy to show that i 1 zeroes of p sub i s lie in the same half plane as the zeros of p sub i s and the remaining zero lies in the left or in the right half plane according to the sign of the ratio of the leading coefficients of p sub i s and p sub i 1 s by successively applying this property to all pairs of polynomials in the sequence starting from p sub o s and p sub 1 s the standard rule for determining the zero distribution of p sub n s is immediately derived

well posedness of initial boundary value problem of degenerate parabolic equations abstract we study a nonlinear degenerate parabolic equation associated with an arbitrary maximal monotone graph we impose initial and dirichlet boundary conditions and prove that the problem is well posed

root distribution of a polynomial in subregions of complex plane modified processes are proposed for directly treating singularities in the routh array and procedures are developed for determining the respective orders of simple and or repeated roots lying on the imaginary axis the extended sturm test is developed for determining the root distribution on some specified lines moreover to enhance the routh stability theorem the extended routh theorem for finding the numbers of roots of a polynomial lying within some specified regions is proposed

existence and non existence of a fast diffusion equation inrn in this paper we study the global existence and asymptotic behaviour of solutions tout log ufor the cauchy initial value problem inrn we prove that ifn 3 then every solution satisfies rn up x t dx for any 1 n 2 hence we extend a previous result of vazquez 19 which claims that rn u dx for 0 t tmax

brief paper smooth trivial vector bundle structure of the space of hurwitz polynomials due to the importance of the robustness of stable systems the properties of the set of hurwitz real polynomials are interesting to study among these properties is that the set is an open and not a convex set and recently it was proved that the set of hurwitz polynomials of degree n with positive coefficients h n is contractible now we will prove that the set h n is a smooth trivial vector bundle over h n k of rank k

the linear limit of thin film flows as an obstacle type free boundary problem we study the limit as n to 0 of the nonnegative self similar source type solutions of the thin film equationut unuxxx x 0 we obtain a unique limiting function u which is a solution of an obstacle type free boundary problem with the constraint u geq 0 associated with the linear equationut uxxxx 0 the function u is c1 for t 0 and has finite speed of propagation the positivity set um 0 being bounded by two contact lines x pm a t 1 4 a constant the function u has a dirac mass as initial condition and satisfies the linear equation in the positivity set but not across the free boundaries or contact lines also known as moving boundaries we give an integral representation of u in the positivity set we set up a precise definition of the general non self similar obstacle type free boundary problem which is different from a standard parabolic variational inequality and compare it with the cauchy problem we also consider source type solutions for negative values of n which are so

radial equivalence and study of self similarity for two very fast diffusion equations abstract in this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p laplacian equation begun in a previous work we treat the cases m 0 and p 1 we perform an exhaustive study of self similar solutions for both equations based on a phase plane analysis and the correspondences we discover we also obtain special correspondence relations and self maps for the limit case m 1 p 0 which is particularly important in applications in image processing we also find self similar solutions for the very fast p laplacian equation that have finite mass and in particular some of them that conserve mass while this phenomenon is not true for the very fast diffusion equation

the cauchy problem for ut δu m when 0 m 1 this paper deals with the cauchy problem for the nonlinear diffusion equation u t u u m 1 0 on 0 x rn u 0 u0 when 0 m 1 fast diffusion case we prove that there exists a global time solution for any locally integrable function u0 hence no growth condition at infinity for u0 is required moreover the solution is shown to be unique in that class behavior at infinity of the solution and l loc regularizing effects are also examined when m max n 2 n 0 1

spatially monotone homoclinic orbits in nonlinear parabolic equations of super fast diffusion type this work deals with positive classical solutions of the degenerate parabolic equation u t u p u xx quad quad star when p 2 which via the substitution v u1 p transforms into the super fast diffusion equation v t v m 1 v x x with m frac 1 p 1 in 1 0 it is shown that star possesses some entire positive classical solutions defined for all t in mathbb r and x in mathbb r which connect the trivial equilibrium to itself in the sense that u x t 0 both as t and as t locally uniformly with respect to x in mathbb r moreover these solutions have quite a simple structure in that they are monotone increasing in space the approach is based on the construction of two types of wave like solutions one of them being used for t 0 and the other one for 0 t both types exhibit wave speeds that vary with time and tend to zero as t and t respectively the solutions thereby obtained decay as x uniformly with respect to t in mathbb r but they are unbounded as x it is finally shown that within the class of functions having such a behavior as x there does not exist any bounded homoclinic orbit

inverse routh table construction and stability of delay equations abstract based on an inversion of the routh table construction a unimodular characterization of all hurwitz polynomials is obtained in parameter space the hurwitz polynomials of degree n correspond to the positive 2 n tant the method is then used to construct classes of stable continuous time delay difference equations and delay differential equations of neutral type by a suitable limiting process

a limit case in nonlinear diffusion on considere le probleme suivant u t u 2 u x x si t 0 x r u 0 x u 0 x si x r cette equation est un modele de diffusion rapide en conduction de chaleur non lineaire on suppose que u 0 x est une fonction reelle localement integrable et non negative on determine des conditions sur u 0 qui assurent l existence d une solution u t x 0 dans un intervalle de temps

driven translocation of a polymer fluctuations at work the impact of thermal fluctuations on the translocation dynamics of a polymer chain driven through a narrow pore has been investigated theoretically and by means of extensive molecular dynamics md simulation the theoretical consideration is based on the so called velocity langevin v langevin equation which determines the progress of the translocation in terms of the number of polymer segments s t that have passed through the pore at time t due to a driving force f the formalism is based only on the assumption that due to thermal fluctuations the translocation velocity v s t is a gaussian random process as suggested by our md data with this in mind we have derived the corresponding fokker planck equation fpe which has a nonlinear drift term and diffusion term with a time dependent diffusion coefficient d t our md simulation reveals that the driven translocation process follows a super diffusive law with a running diffusion coefficient d t t where 1 this finding is then used in the numerical solution of the fpe which yields an important result for comparatively small driving forces fluctuations facilitate the translocation dynamics as a consequence the exponent which describes the scaling of the mean translocation time with the length n of the polymer n is found to diminish thus taking thermal fluctuations into account one can explain the systematic discrepancy between theoretically predicted duration of a driven translocation process considered usually as a deterministic event and measurements in computer simulations in the nondriven case f 0 the translocation is slightly subdiffusive and can be treated within the framework of fractional brownian motion fbm

obstructions to existence in fast diffusion equations abstract the study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory thus in the sub field of very fast diffusion it is known that the cauchy problem can be ill posed either because of non uniqueness or because of non existence of solutions with small data the equations we consider take the general form u t d u u x u x x or its several dimension analogue fast diffusion means that d at some values of the arguments typically as u 0 or u x 0 here we describe two different types of non existence phenomena some fast diffusion equations with very singular d do not allow for solutions with sign changes while other equations admit only monotone solutions no oscillations being allowed the examples we give for both types of anomaly are closely related the most typical examples are v t v x v x and u t u xx u x for these equations we investigate what happens to the cauchy problem when we take incompatible initial data and perform a standard regularization it is shown that the limit gives rise to an initial layer where the data become admissible positive or monotone respectively followed by a standard evolution for all t 0 once the obstruction has been removed

a simple algorithm for exact solutions of systems of linear and nonlinear integro differential equations a simple algorithm is used to achieve exact solutions of systems of linear and nonlinear integro differential equations arising in many scientific and engineering applications the algorithm does not need to find the adomain polynomials to overcome the nonlinear terms in adomain decomposition method adm it does not need to create a homotopy with an embedding parameter as in homotopy perturbation method hpm and optimal homotopy asymptotic method oham unlike vim it does not need to find lagrange multiplier in this manuscript no restrictive assumptions are taken for nonlinear terms the applied algorithm consists of a single series in which the unknown constants are determined by the simple means described in the manuscript the outcomes gained by this algorithm are in excellent concurrence with the exact solution and hence proved that this algorithm is effective and easy four systems of linear and nonlinear integro differential equations are solved to prove the above claims and the outcomes are compared with the exact solutions as well as with the outcomes gained by already existing methods

existence of solutions of the very fast diffusion equation abstract let m u um m if 1 m u log u if m 0 for any m 1 r t 0 0 u 0 l loc 1 r with 0 u 0 d x 0 u 0 d x and 0 f bv 0 such that 0 f d s for some constant 0 we prove the existence of solution of the equation ut m u xx in r 0 t 0 u x u0 in r for some constant t0 t such that 0 t u m 1 u x 0 s d s 0 u y t d y m 1 t t f d s 0 0 which satisfies a b u m 1 u x x s d s a b f d s as x for all 0 log u x as x uniformly on a b if m 0 for all 0 0 u x t d x and 0 u x t d x for all 0 log u x f t as x uniformly on a b if m 0 for all 0 0 u 0 l loc 1 r with 0 u 0 d x 0 u 0 d x we prove the existence of infinitely many finite mass solutions of the above equation with prescribed mass and line integral at time t and show that the solution will satisfy some uniform decay rate as x

a phase plane discussion of convergence to travelling fronts for nonlinear diffusion the paper is concerned with the asymptotic behavior as t of solutions u x t of the equation u t u xx f left u right 0 x in left infty infty right in the case f 0 f 1 0 with f u non positive for u 0 sufficiently close to zero and f u non negative for u 1 sufficiently close to 1 this guarantees the uniqueness but not the existence of a travelling front solution u u x ct u 0 u and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front if it exists and to a stacked combination of travelling fronts if it does not the approach is to use the monotonicity to take u and t as independent variables and p ux as the dependent variable and to apply ideas of sub and super solutions to the diffusion equation for p

critical evaluation of the computational methods used in the forced polymer translocation in forced polymer translocation the average translocation time tau scales with respect to pore force f and polymer length n as tau approximately f 1 n beta we demonstrate that an artifact in the metropolis monte carlo method resulting in breakage of the force scaling with large f may be responsible for some of the controversies between different computationally obtained results and also between computational and experimental results using langevin dynamics simulations we show that the scaling exponent beta or 1 nu is not universal but depends on f moreover we show that forced translocation can be described by a relatively simple force balance argument and beta to arise solely from the initial polymer configuration

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

extended table for eliminating the singularities in routh s array a simple procedure for eliminating the classical epsilon method introduced by the routh hurwitz test is given the procedure extends the construction of the routh table in the special case of vanishing leading array elements without introducing complementary multiplications it is shown that it is always possible to associate with any polynomial g an extended routh table the number of sign changes in the first column of the table equals the number of zeros of g with positive real part

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

nonlinear degenerate parabolic equations with neumann boundary condition abstract we study neumann problem for a class of nonlinear degenerate parabolic pde a typical nonlinearity we have in mind is for instance u 1 u u 0 we establish a necessary and sufficient condition on given data for existence of solution

thermal waves in absorbing media abstract we discuss the existence of travelling wave solutions with interfaces for the nonlinear heat equation with absorption u t a u m xx bu n with a b 0 and m n r several situations occur depending on the relative strength of the diffusion and absorption terms reflected by their exponents m and n we characterize the existence of finite travelling waves in terms of m and n show their uniqueness up to translations in space and time and derive their velocity from the wave profile near the interface or front

nearest stable system using successive convex approximations stability is a crucial property in the study of dynamical systems we focus on the problem of enforcing the stability of a system a posteriori the system can be a matrix or a polynomial either in continuous time or in discrete time we present an algorithm that constructs a sequence of successive stable iterates that tend to a nearby stable approximation x of a given system a the stable iterates are obtained by projecting a onto the convex approximations of the set of stable systems some possible applications for this method are correcting the error arising from some noise in system identification and a possible solver for bilinear matrix inequalities based on convex approximations in the case of polynomials a fair complexity is achieved by finding a closed form solution to first order optimality conditions

on the fisher kpp equation with fast nonlinear diffusion in this paper we examine the cauchy problem for a nonlinear fisher kolmogorov petrovski piskunov fisher kpp reaction diffusion equation with fast diffusion namely u t u n u u 1 u for x in r n 0 0 we show in particular that the problem does not have permanent form travelling wave solutions and we derive the large time asymptotic form of the solution

on polytopes in hurwitz region abstract we consider the stability region of monic polynomials in the coefficient space starting from specially chosen stable points we define polytopes whose edges are either stable or belong to the stability boundary therefore by the edge theorem the interiors are stable stable polytopes in the reverse directions which are directed to infinity have been studied in the work dzhafarov et al 2019 some applications of the obtained results to the synthesis of fixed order stabilizing controllers are given