degree dispersion increases the rate of rare events in population networks there is great interest in predicting rare and extreme events in complex systems and in particular understanding the role of network topology in facilitating such events in this letter we show that degree dispersion the fact that the number of local connections in networks varies broadly increases the probability of large rare fluctuations in population networks generically we perform explicit calculations for two canonical and distinct classes of rare events network extinction and switching when the distance to threshold is held constant and hence stochastic effects are fairly compared among networks we show that there is a universal exponential increase in the rate of rare events proportional to the variance of a network s degree distribution over its mean squared

nonlinear elliptic equations related to weighted sum operators abstract we study a new kind of degenerate operators related to some weighted sum operators although the operators are severely degenerate we show the smoothness of the weak solutions

perturbed polynomials with zeros in the left sector abstract we investigate the directional perturbation of a real nominal polynomial we propose an algorithm for computing the largest allowable perturbation bound which will simultaneously maintain the zeros of the perturbed polynomial within the left sector in the complex plane

the stationary arrival process of independent diffusers from a continuum to an absorbing boundary is poissonian we consider the arrival process of infinitely many identical independent diffusion processes from an infinite bath to an absorbing boundary previous results on this problem were confined to independent brownian particles arriving at an absorbing sphere the present paper extends these results to general diffusion processes without any symmetries and without resorting to explicit expressions for solutions to the relevant equations it is shown that for general absorbing boundaries and force fields the steady stream of arrivals is poissonian with rate equal to the total flux on the absorbing boundary as calculated from the continuum theory of diffusion with transport the considered arrival problem arises in the theory of langevin simulations of ions in electrolytic solutions in a langevin simulation ions enter and exit the simulation region and it is necessary to compute the probability laws for their entrance times into the simulation while the simulated ions inside the small simulation region inte

varying boundary conditions in a general class of elliptic problems of mixed type abstract in this paper we use the linear theory developed in cano casanova and lopez gomez j differential equations 178 2002 123 to show the continuous dependence of the positive solutions of a general class of sublinear elliptic boundary value problems of mixed type with respect to the boundary operator besides working with mixed non classical boundary conditions this paper considers problems where bifurcation from infinity occurs treating this kind of problems is far from easy since a complete knowledge of the set of values of the parameters where they exhibit a positive solution is imperative our results are far from immediate as a consequence of the vanishing weight functions involved in our general setting now a days the analysis of these general problems where the coefficients are allowed to vary and eventually vanishing or changing sign is focusing a great deal of attention

uniform asymptotic approximation of diffusion to a small target generalized reaction models the diffusion of a reactant to a binding target plays a key role in many biological processes the reaction radius at which the reactant and target may interact is often a small parameter relative to the diameter of the domain in which the reactant diffuses we develop uniform in time asymptotic expansions in the reaction radius of the full solution to the corresponding diffusion equations for two separate reactant target interaction mechanisms the doi or volume reactivity model and the smoluchowski collins kimball partial absorption surface reactivity model in the former the reactant and target react with a fixed probability per unit time when within a specified separation in the latter upon reaching a fixed separation they probabilistically react or the reactant reflects away from the target expansions of the solution to each model are constructed by projecting out the contribution of the first eigenvalue and eigenfunction to the solution of the diffusion equation and then developing matched asymptotic expansions in laplace transform space our approach offers an equivalent but alternative method to the pseudopotential approach we previously employed isaacson and newby phys rev e 88 012820 2013 pleee81539 375510 1103 physreve 88 012820 for the simpler smoluchowski pure absorption reaction mechanism we find that the resulting asymptotic expansions of the diffusion equation solutions are identical with the exception of one parameter the diffusion limited reaction rates of the doi and partial absorption models this demonstrates that for biological systems in which the reaction radius is a small parameter properly calibrated doi and partial absorption models may be functionally equivalent

closed form of the generalized green s function for the helmholtz operator on the two dimensional unit sphere the closed representation of the generalized known also as reduced or modified green s function for the helmholtz partial differential operator on the surface of the two dimensional unit sphere is derived in its compact form the derived formula contains a legendre polynomial and a derivative of the legendre function of the first kind with respect to its index an explicit expression for that derivative is found and used to obtain an expanded and potentially more suitable in applications form of the generalized green s function for the operator in question the related problem of constructing the closed form of the generalized green s function for the legendre ordinary differential operator on the segment 1 x 1 with the boundary conditions of finiteness at x 1 is also solved

properties of the principal eigenvalues of a general class of non classical mixed boundary value problems in this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions our theory is a substantial extension of the classical theory by p hess and t kato 1980 comm partial differential equations5 999 1030 in obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself these auxiliary results complement and in some sense complete the theory of d daners and e n dancer 1997 j differential equations138 86 132 the main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to h amann and j lopez gomez 1998 j differential equations146 336 374

perturbation of eigenvalues due to gaps in two dimensional boundaries motivated by problems involving diffusion through small gaps we revisit two dimensional eigenvalue problems with localized perturbations to neumann boundary conditions we recover the known result that the gravest eigenvalue is o ln 1 where is the ratio of the size of the hole to the length scale of the domain and provide a simple and constructive approach for summing the inverse logarithm terms and obtaining further corrections comparisons with numerical solutions obtained for special geometries both for the dirichlet patch problem where the perturbation to the boundary consists of a different boundary condition and for the gap problem confirm that this approach is a simple way of obtaining an accurate value for the gravest eigenvalue and hence the long term outcome of the underlying diffusion problem

the mean first rotation time of a planar polymer we estimate the mean first time called the mean rotation time mrt for a planar random polymer to wind around a point this polymer is modeled as a collection of n rods each of them being parameterized by a brownian angle we are led to study the sum of i i d imaginary exponentials with one dimensional brownian motions as arguments we find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function finally we obtain an asymptotic formula for the mrt whose leading order term depends on the square root of n and interestingly depends weakly on the mean initial configuration our analytical results are confirmed by brownian simulations our analytical results are confirmed by brownian simulations

regularity for a class of strongly degenerate quasilinear operators abstract we prove a harnack inequality and regularity for solutions of a quasilinear strongly degenerate elliptic equation we assume the coefficients of the structure conditions to belong to suitable stummel kato classes

harnack inequality and regularity for degenerate quasilinear elliptic equations we prove harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions the degeneracy is given by a suitable power of a strong a weight regularity results are achieved under minimal assumptions on the coefficients and as an application we prove c1 local estimates for solutions of a degenerate equation in non divergence form

a new fast algorithm for robust stability analysis of linear control systems with linearly dependent parametric uncertainties abstract in this paper a fast algorithm is proposed for the computation of stability margins in parameter space for linear control systems subject to structured uncertainties the case in which plant transfer function coefficients are affine in a set of physical uncertain parameters is considered the paper shows also how the proposed algorithm can be used to solve another interesting problem in robust control analysis i e the determination of the region of pole location of a closed loop control system including an entire family of uncertain plants

partially reflected diffusion the radiation reactive or robin boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries the underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary however the relation between the reaction constant in the robin boundary condition and the reflection probability is not well defined in this paper we define the partially reflected process as a limit of the markovian jump process generated by the euler scheme for the underlying ito dynamics with partial boundary reflection trajectories that cross the boundary are terminated with probability p sqrt delta t and otherwise are reflected in a normal or oblique direction we use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the fokker planck equation in a half space wi

domain perturbation for elliptic equations subject to robin boundary conditions abstract we prove domain perturbation theorems for linear and nonlinear elliptic equations under robin boundary conditions the theory allows very singular perturbation of domains in particular it includes cutting holes parts degenerating to a set of measure zero such as the dumbbell problem or wildly oscillating boundaries in the last case we show that the limiting problem is the dirichlet problem

multiscale models and stochastic simulation methods for computing rare but key binding events in cell biology abstract the main difficulty in simulating diffusion processes at a molecular level in cell microdomains is due to the multiple scales involving nano to micrometers few to many particles have to be simulated and simultaneously tracked while there are exploring a large portion of the space for binding small targets such as buffers or active sites bridging the small and large spatial scales is achieved by rare events representing brownian particles finding small targets and characterized by long time distribution these rare events are the bottleneck of numerical simulations a naive stochastic simulation requires running many brownian particles together which is computationally greedy and inefficient solving the associated partial differential equations is also difficult due to the time dependent boundary conditions narrow passages and mixed boundary conditions at small windows we present here two reduced modeling approaches for a fast computation of diffusing fluxes in microdomains the first approach is based on a markov mass action law equations coupled to a markov chain the second is a gillespie s method based on the narrow escape theory for coarse graining the geometry of the domain into poissonian rates the main application concerns diffusion in cellular biology where we compute as an example the distribution of arrival times of calcium ions to small hidden targets to trigger vesicular release

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

narrow escape problem with a mixed trap and the effect of orientation we consider the mean first passage time mfpt of a two dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections high order asymptotic formulas for the mfpt and the fundamental eigenvalue of the laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the mean time to capture we obtain a simple geometric condition which gives the optimal trap alignment in terms of the gradient of the regular part of a regular part of a green s function and a certain alignment vector we find that subdividing the absorbing portions of the trap reduces the mean first passage time of the diffusing particle in the scenario where the trap undergoes prescribed motion in the domain the mfpt is seen to be particularly sensitive to the orientation of the trap

strong localized perturbations of eigenvalue problems this paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded o

conditions on the boundary of the zero set and application to stabilization of systems with uncertainty abstract we describe an analytic method for finding the location of the zero set of a vector valued function which depends on m real variables and n complex parameters we apply the method to robust stabilization of multivariable linear feedback systems we find exact measures of the extent of permissible perturbations in the plant and or the compensator that maintain feedback stability

analytical solutions for an escape problem in a disc with an arbitrary distribution of exit holes along its boundary we analytically construct solutions for the mean first passage time and splitting probabilities for the escape problem of a particle moving with continuous brownian motion in a confining planar disc with an arbitrary distribution i e of any number size and spacing of exit holes absorbing sections along its boundary the governing equations for these quantities are poisson s equation with a non zero constant forcing term and laplace s equation respectively and both are subject to a mixture of homogeneous neumann and dirichlet boundary conditions our solutions are expressed as explicit closed formulae written in terms of a parameterising variable via a conformal map using special transcendental functions that are defined in terms of an associated schottky group they are derived by exploiting recent results for a related problem of fluid mechanics that describes a unidirectional flow over no slip no shear surfaces as well as results from potential theory all of which were themselves derived using the same theory of schottky groups they are exact up to the determination of a finite set of mapping parameters which is performed numerically their evaluation also requires the numerical inversion of the parameterising conformal map computations for a series of illustrative examples are also presented

addendum to on the stability properties of polynomials with perturbed coefficients this note serves to point out that the method presented by soh et al 1 can be used to obtain the largest hypersphere which contains the coefficients of polynomials which have zeros only on or within any simply connected region d in the complex plane which is symmetric with respect to the real axis it is then extended to obtain the largest right circular hypercone which contains the coefficients of polynomials which have zeros only on or within any simply connected region d

a doubly degenerate diffusion equation in multiplicative noise removal models abstract in this paper we consider the neumann problem of a doubly degenerate diffusion equation which is proposed to deal with multiplicative noise we prove the existence and the extremum principle of weak solutions for the problem

holder continuity for solutions of linear degenerate elliptic equations under minimal assumptions abstract in this paper we study the local regularity of solutions to linear degenerate elliptic equations of the form i j 1 n a ij u x i d j u x j i 1 n b i u x i cu f i 1 n f i x i we prove harnack s inequality for positive solutions and the holder continuity of the solutions of assuming on the coefficients of the lower order terms very general hypotheses involving appropriate degenerate morrey spaces

maximal perturbation bound for hurwitz property of analytic functions the robust stability problem for a family of analytic functions subjected to structured perturbations is studied in particular the maximal allowable perturbation bound on the system parameters is characterized

summing logarithmic expansions for singularly perturbed eigenvalue problems strong localized perturbations of linear and nonlinear eigenvalue problems in a bounded two dimensional domain d are considered the effects on an eigenvalue lambda 0 of the lapla cian and on the fold point lambda c0 of a nonlinear eigenvalue problem of removing a small subdomain d epsilon of radius epsilon from d and imposing a condition on the boundary of the resulting hole are determined using the method of matched asymptotic expansions it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start with infinite series in powers of 1 log epsilon d kappa here d kappa is a constant that depends on the shape of d epsilon and on the precise form of the boundary condition on the hole in each case it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole this related problem is not stiff and can be solved numerically

complex resonances of a hard rectangular cavity coupled to a half space through a narrow aperture electromagnetic coupling through a small slot into a two dimensional rectangular cavity is computed via a simple ritz galerkin expansion of the aperture field for the case of transverse electric te polarization one term expansions for each of the symmetric and antisymmetric dynamic aperture fields use the exact solutions for harmonic potentials that solve the related neumann problem of a slot in a hard plane source free solutions exist at discrete complex values of the frequency variable these eigenfunctions of the negative laplacian are improper modes and several of the low order complex resonances are found from zeros of the variational expression that relates the aperture field to the excitation the complex resonances are perturbations of the real closed cavity resonances and their presence is unmistakable in the net stored energy in the cavity under plane wave excitation from the exterior half space

a dual result to kharitonov s theorem a dual problem to kharitonov s problem involving a diamond instead of a rectangle is considered the results show that a family of polynomials with coefficients varying in the diamond is strictly hurwitz if and only if eight one dimensional exposed edges of the diamond are strictly hurwitz

an asymptotic analysis of intracellular signaling gradients arising from multiple small compartments intracellular signaling gradients naturally arise through a local activation of a dif fusible signaling molecule e g by a localized kinase and a subsequent deactivation at a distant cellular site e g by a cytosolic phosphatase here we consider a spherical cell containing a finite number of small spherical compartments where a signaling molecule becomes activated by a localized enzyme for the activation rate two cases are considered a saturated enzyme with a constant rate and an unsaturated enzyme with a linear rate using the method of matched asymptotic expansions we derive approximate solutions of the steady state diffusion equation with a linear deactivation rate to obtain the three dimensional concentration profile of activated signaling molecules inside the cell depending on the diffusion length of the signaling molecule the profiles decay either exponentially or algebraically where the mode of decay is described by an associated green s function our analysis provides simple expressions for the local concentration profile in the neighborhood of a signaling compartment which can be used to estimate the amplitude and the extent of the gradients in the respective regimes the global concentration profiles also depend on the cell size and the particular spatial arrangement of the compartments relative to each other and to the cell boundary

narrow escape part iii non smooth domains and riemann surfaces we consider the narrow escape problem in two dimensional riemannian manifolds with a metric g with corners and cusps in an annulus and on a sphere specifically we calculate the mean time it takes a brownian particle diffusing in a domain to reach an absorbing window when the ratio varepsilon frac partial omega a g partial omega g between the absorbing window and the otherwise reflecting boundary is small if the boundary is smooth as in the cases of the annulus and the sphere the leading term in the expansion is the same as that given in part i of the present series of papers however when it is not smooth the leading order term is different if the absorbing window is located at a corner of angle then e tau frac omega g alpha d log frac 1 varepsilon o 1 if near a cusp then e tau grows algebraically rather than logarithmically thus in the domain bounded between two tangent circles the expected lifetime is e tau frac omega d 1 1 d frac 1 varepsilon o 1 where d 1 is the ratio of the radii for the smooth boundary case we calculate the next term of the expansion for the annulus and the sphere it can also be evaluated for domains that can be mapped conformally onto an annulus this term is needed in real life applications such as trafficking of receptors on neuronal spines because log frac 1 varepsilon is not necessarily large even when varepsilon frac partial omega a g partial omega g is small in these two problems there are additional parameters that can be small such as the ratio of the radii of the annulus the contributions of these parameters to the expansion of the mean escape time are also logarithmic in the case of the annulus the mean escape time is e tau frac omega g pi d log frac 1 varepsilon frac 12 log frac 1 delta o 1

a note on robust pole placement pole placement algorithm in single input single output siso systems is discussed with respect to the corresponding real stability radius of the resulting closed loop polynomial

diffusion escape through a cluster of small absorbing windows we study the first eigenvalue of the laplace equation in a bounded domain in r d d 2 3 with mixed neumann dirichlet zaremba boundary conditions the neumann condition is imposed on most of the boundary and the dirichlet boundary consists of a cluster of small windows when the windows are well separated the first eigenvalue is asymptotically the sum of eigenvalues of mixed problems with a single dirichlet window however when two or more dirichlet windows cluster tightly together they interact nonlinearly we compare our asymptotic approximation of the eigenvalue to the escape rate of simulated brownian particles through the small windows

regularity for quasilinear degenerate elliptic equations equations like 1 1 have been studied by many authors in the case x 1 see e g 2 and the references therein or an a2 muckenhoupt weight 6 and 17 here 3 is a strong a weight and 31 p n 1 p n the novelty here is the degeneracy condition given by choice of the weight to be a power of a strong a weight moreover we assume very mild integrability conditions on the lower order terms and known term in 1 1 these conditions are sharp and at least in some instances are necessary and sufficient see e g 5 or 12 the non degenerate case was studied in 16 and 15 our main result is contained in theorem 4 3 and it states the harnack inequality for nonnegative weak solutions of equation 1 1 as a direct consequence smoothness for weak solutions is obtained in particular we have continuity result under stummel kato type assumptions and holder continuity result under morrey type assumptions see theorems 5 1 and 5 2 we will prove our results using as main tool a weighted fefferman poincare embedding see theorem 3 2 mixed with the moser iteration technique this investigation was suggested by b franchi and we thank him for enlightening discussions about strong a weights we also thank the anonymous referee for very good suggestions to improve the presentation and l saloff coste for useful discussions

narrow escape part i a brownian particle with diffusion coefficient d is confined to a bounded domain by a reflecting boundary except for a small absorbing window partial omega a the mean time to absorption diverges as the window shrinks thus rendering the calculation of the mean escape time a singular perturbation problem in the three dimensional case we construct an asymptotic approximation when the window is an ellipse assuming the large semi axis a is much smaller than omega 1 3 omega is the volume and show that the mean escape time is e tau sim frac omega 2 pi da k e where e is the eccentricity and k cdot is the complete elliptic integral of the first kind in the special case of a circular hole the result reduces to lord rayleigh s formula e tau sim frac omega 4ad which was derived by heuristic considerations for the special case of a spherical domain we obtain the asymptotic expansion e tau frac omega 4ad 1 frac a r log frac r a o frac a r this result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function if is a two dimensional bounded riemannian manifold with metric g and varepsilon partial omega a g omega g ll1 we show that e tau frac omega g d pi log frac 1 varepsilon o 1 this result is applicable to diffusion in membrane surfaces

diffusion on a curved surface coupled to diffusion in the volume an algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume a problem often arising in cell biology it applies to pixilated surfaces obtained from experimental images and performs at low computational cost in the method the laplace beltrami operator is approximated locally by the laplacian on the tangential plane and then a finite volume discretization scheme based on a voronoi decomposition is applied convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2 the method is applied to a cell biological problem where a signaling molecule g protein rac cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior simulations on realistic cell geometry are performed to validate and determine the accuracy of a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching the method is implemented within the virtual cell computational framework freely accessible at www vcell org

maximal polytopic stability domains in parameter space for uncertain systems abstract some results on the stability and pole location of families of uncertain systems with characteristic polynomial coefficients affected by structured perturbations are presented the case in which the coefficients are affine in a set of system physical parameters is considered assuming that the parameters are subject to polytopic perturbations of a given structure for these cases which often arise in system stability robustness analysis as for example in the robustification of controllers for multivariable closed loop systems a procedure is given to estimate maximal stability polytopic regions in the space of parameters this result is also used to derive simple necessary and sufficient conditions for the discrete time stability of convex combinations of two polynomials

stability of solitary waves in finite ablowitz ladik lattices we discuss the effects of different types of boundary conditions b c on the linear stability of a solitary wave in a finite length dynamical lattice described by the ablowitz ladik al model types of b c considered are fixed dirichlet no flux neumann and free as well as periodic b c the behaviour of eigenvalues around a stationary nonlinear wave consistent with several types of b c is studied analytically and numerically the translational eigenvalues are found to move away from the origin it is shown that for b c of the fixed type these eigenvalues bifurcate into a neutrally stable oscillatory kind while the no flux and free b c lead to an exponentially weak instability the rotational eigenmodes those related to the gauge invariance of the lattice strictly remain at the origin as gauge invariance is not broken by the linear homogeneous b c these numerical results are compared to the analytical and semi analytical predictions obtained within the approximation of the images generated by the solitary wave beyond the lattice edges a potential effect of continuous spectrum band edge bifurcations on stability is also evaluated with the conclusion that that these bifurcations cannot destabilize the soliton for periodic b c the translational invariance is preserved and so is the lax pair structure sustaining the integrable nature of the problem the al model was chosen so that the effects of boundary conditions are monitored in the absence of non integrable discreteness finally the possibility of the generation of multipulse configurations due to the interplay of finiteness effects with the exponential tail tail interaction of the pulses is examined

diffusion of protein receptors on a cylindrical dendritic membrane with partially absorbing traps we present a model of protein receptor trafficking within the membrane of a cylin drical dendrite containing small protrusions called spines spines are the locus of most excitatory synapses in the central nervous system and act as localized traps for receptors diffusing within the dendritic membrane we treat the transverse intersection of a spine and dendrite as a spatially extended partially absorbing boundary and use singular perturbation theory to analyze the steady state distribution of receptors we compare the singular perturbation solutions with numerical solutions of the full model and with solutions of a reduced one dimensional model and find good agreement between them all we also derive a system of fokker planck equations from our model and use it to exactly solve a mean first passage time mfpt problem for a single receptor traveling a fixed axial distance along the dendrite this is then used to calculate an effective diffusion coefficient for receptors when spines are uniformly distributed along the length of the cable and to show how a nonuniform distribution of spines gives rise to anomalous subdiffusion

activation through a narrow opening the escape of brownian motion through a narrow absorbing window in an otherwise reflecting boundary of a domain is a rare event in the presence of a deep potential well there are two long time scales the mean escape time from the well and the mean time to reach the absorbing window we derive a generalized kramers formula for the mean escape time through the narrow window

vibrations of thin plates with small clamped patches eigenvalues of fourth order elliptic operators feature prominently in stability analysis of elastic structures this paper considers out of plane modes of vibration of a thin elastic plate perforat

finding elliptic fekete points sets two numerical solution approaches abstract the objective of this work is to provide a methodology for approximating globally optimal fekete point configurations this problem is of interest in numerical mathematics and scientific modeling following a brief discussion of the analytical background lipschitz global optimization lgo is applied to determine i e to numerically approximate fekete point configurations next to this optimization approach an alternative strategy by formulating a set of differential algebraic equations daes of index 2 will be considered the steady states of the daes coincide with the optima of the function to be minimized illustrative numerical results with configurations of up to 150 fekete points are presented to show the viability of both approaches

stabilization of linear systems with fixed order controllers abstract this paper deals with the problem of stabilizing a linear time invariant plant with a feedback compensator of fixed dynamic order this is an important unsolved problem in control theory because existing solutions to the regulator problem can only generate controllers that are of high enough order that arbitrary pole placement becomes possible in general the dynamic order of controllers can be much lower if only stabilization is desired in the transfer function domain a new necessary condition for stabilizability by a fixed order controller is derived this result which can be used sequentially to derive a lower bound on the order of a stabilizing controller is based on a classical result 1873 of linear programming known as gordan s theorem of the alternative the stabilization algorithm successively selects hurwitz vectors to make the largest stability hypersphere intersect a linear subspace determined by the plant parameters and the order of the controller occurrence of such an intersection is a sufficient condition for stabilizability and the intersection also determines a stabilizing compensator the algorithm is illustrated by practical examples which demonstrate that a large reduction in controller order is achievable by this method

matched asymptotic analysis to solve the narrow escape problem in a domain with a long neck in this study we mainly consider the narrow escape problem nep in a two dimensional domain with a long neck which is the two dimensional analogue of a dendritic spine geometry the nep requires the computation of the mean escape time of a brownian particle starting from the head until it exits from the end of the neck where the particle is absorbed we divide the domain into the neck part n and the head part h with the common boundary the escape time in h can be considered to be the time from the head to the end of the neck while the escape time in n can be considered to be the time from the neck to the end of the neck we compute the two exit times separately and match them by considering some boundary value problem with an impedance boundary condition on which we refer to as the neumann robin boundary model

motion of three point vortices on a sphere abstract we solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius r the system is more general than the corresponding one in the plane 1 3 5 19 28 33 34 37 38 and reduces to it in the limit r as long as the three vortices remain sufficiently close to each other during the course of their motion instead of using spherical coordinates we use cartesian coordinates in which the vector i points from the center of the sphere to the vortex with strength i an important conserved quantity is the center of vorticity vector c i i i which with no loss of generality we align with the z axis based on the size of this vector relative to the radius of the sphere we classify the motion into one of five types super radial radial sub radial degenerate or a limiting super radial case this categorization allows us to draw several conclusions about the qualitative motion of the vortices we then fully characterize all fixed and relative equilibria on the sphere for fixed equilibria the vortices must lie on great circles geodesics if the strengths are equal they form an equilateral triangle otherwise the triangle shape is specified once the strength of the vortices is given the relative equilibria are classified as either degenerate c 0 or non degenerate c 0 for each type the shape of the vortex triangle is described and the frequency of rotation is computed as in the planar problem it is possible to introduce trilinear coordinates and study the motion in a phase plane which allows us to locate all the equilibria as well as to characterize more complex relative dynamics we then describe self similar vortex collapse on the sphere stating necessary and sufficient conditions for collapse to occur computing the collapse times and vortex trajectories on the route towards collapse comparisons are made with the collapse formulas in the plane derived in 1 30

diffusion on a sphere with localized traps mean first passage time eigenvalue asymptotics and fekete points a common scenario in cellular signal transduction is that a diffusing surface bound molecule must arrive at a localized signaling region on the cell membrane before the signaling cascade can be completed the question then arises of how quickly such signaling molecules can arrive at newly formed signaling regions here we attack this problem by calculating asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere in the presence of n partially absorbing traps of small radii the rate at which the small diffusing molecule becomes captured by one of the traps is determined by asymptotically calculating the principal eigenvalue for the laplace operator on the sphere with small localized traps the asymptotic analysis relies on the method of matched asymptotic expansions together with detailed properties of the green s function for the laplacian and the helmholtz operators on the surface of the unit sphere the asymptotic results compare favorably with

hybrid asymptotic numerical approach for estimating first passage time densities of the two dimensional narrow capture problem a hybrid asymptotic numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two dimensional domain with a reflecting boundary as motivation for this study we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean this implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined to solve the underlying diffusion equation the method of laplace transforms is used to obtain an elliptic problem of modified helmholtz type in the limit of vanishing trap sizes each trap is represented as a dirac point source that permits the solution of the transform equation to be represented as a superposition of helmholtz green s functions using this solution we construct asymptotic short time solutions of the first passage time density which captures peaks associated with rapid capture by the absorbing traps when numerical evaluation of the helmholtz green s function is employed followed by numerical inversion of the laplace transform the method reproduces the density for larger times we demonstrate the accuracy of our solution technique with a comparison to statistics obtained from a time dependent solution of the diffusion equation and discrete particle simulations in particular we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged the hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes

a geometric approach to stabilization by output feedback in this paper we present a geometric approach to the stabilizability of linear systems by output feedback the geometric approach is completely analysed for both discrete time and continuous time single input single output systems two examples are worked out in detail to illustrate the theoretical development we also show how our approach can be extended to dyadic systems the simplicity of the geometric approach in the special case when the number of inputs or outputs is one less than the order of the system is illustrated with the help of examples present address bell laboratories south plainfield new jersey 07080 u s a

noncommuting limits in electromagnetic scattering asymptotic analysis for an array of highly conducting inclusions we consider formulations for the helmholtz operator for periodic media containing high contrast inclusions in the limit when the wavelength outside the inclusions tends to infinity applications are to problems of electromagnetism the main focus is on the analysis of the effect of noncommuting limits an effect which indicates that linear boundary value problems of electromagnetism give formally different results for the long wavelength limits in cases where highly conducting inclusions have refractive indices of different orders of magnitude specifically the effective moduli of the homogenized material will depend on the path used to approach the origin in the coordinate space wave number normalized refractive index of the inclusions 1 this mathematical observation gives a physical subtlety which is studied in this paper the dispersion relation for the lowest frequency or acoustic mode is investigated as are the conditions for existence of an acoustic mode cases of both nondispersive and di

potential and field singularity at a surface point charge the behavior of the magnetic potential near a point charge fluxon located at a curved regular boundary surface is shown to be essentially different from that of a volume point charge in addition to the usual inverse distance singularity two singular terms are generally present the first of them a logarithmic one is axially symmetric with respect to the boundary normal at the charge location and proportional to the sum of the two principal curvatures of the boundary surface at this point that is to the local mean curvature the second term is asymmetric and proportional to the difference of the two principal curvatures in question it is also bounded at the charge location both terms vanish apparently if the charge is at a planar point of the boundary and only in this case the field in the charge vicinity behaves accordingly featuring generally two singular terms proportional to the inverse distance in addition to the main inverse distance squared singularity this result is significant in particular for studying the interaction of magnetic vortices in type ii superconductors

the transition to a point constraint in a mixed biharmonic eigenvalue problem the mixed order eigenvalue problem delta delta 2 u delta u lambda u 0 with delta 0 modeling small amplitude vibrations of a thin plate is analyzed in a bounded two dimensional domain omega that contains a single small hole of radius varepsilon centered at some x 0 in omega clamped conditions are imposed on the boundary of omega and on the boundary of the small hole in the limit varepsilon to 0 and for delta mathcal o 1 the limiting problem for u must satisfy the additional point constraint u x 0 0 to determine how the eigenvalues of the laplacian in a domain with a small hole are perturbed by adding the small fourth order term delta delta 2 u together with an additional boundary condition on partial omega and on the hole boundary the asymptotic behavior of the eigenvalues of the mixed order eigenvalue problem are studied in the dual limit varepsilon to 0 and delta to 0 leading order behaviors of eigenvalues are determined for three ranges

layer potential techniques in spectral analysis part i complete asymptotic expansions for eigenvalues of the laplacian in domains with small inclusions we provide a rigorous derivation of new complete asymptotic expansions for eigenvalues of the laplacian in domains with small inclusions the inclusions somewhat apart from or nearly touching the boundary are of arbitrary shape and arbitrary conductivity contrast vis a vis the background domain with the limiting perfectly conducting inclusion by integral equations we reduce this problem to the study of the characteristic values of integral operators in the complex plane powerful techniques from the theory of meromorphic operator valued functions and careful asymptotic analysis of integral kernels are combined for deriving complete asymptotic expansions for eigenvalues our asymptotic formulae in this paper may be expected to lead to efficient algorithms not only for solving shape optimization problems for laplacian eigenvalues but also for determining specific internal features of an object based on scattering data measurements

harnack inequality and continuity of weak solutions for doubly degenerate elliptic equations we prove harnack inequality and regularity for solutions of a quasilinear doubly degenerate elliptic equation generated by grushin vector fields we assume the coefficients of the structure conditions to belong to suitable stummel kato classes

analytical and numerical results for an escape problem a particle moves with brownian motion in a unit disc with reflection from the boundaries except for a portion called a window or gate in which it is absorbed the main problems are to determine the first hitting time and spatial distribution a closed formula for the mean first hitting time is discovered and proven for a gate of any size also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed numerical simulations of the stochastic process with finite step size and a sufficient number of sample paths are compared with the exact solution to the brownian motion the limit of zero step size providing an empirical formula for the difference histograms of first hitting times are also generated

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

narrow escape and leakage of brownian particles questions of flux regulation in biological cells raise a renewed interest in the narrow escape problem the determination of a higher order asymptotic expansion of the narrow escape time depends on determining the singularity behavior of the neumann green s function for the laplacian in a three dimensional 3d domain with a dirac mass on the boundary in addition to the usual 3d coulomb singularity this green s function also has an additional weaker logarithmic singularity by calculating the coefficient of this logarithmic singularity we calculate the second term in the asymptotic expansion of the narrow escape time and in the expansion of the principal eigenvalue of the laplace equation with mixed dirichlet neumann boundary conditions with small dirichlet and large neumann parts we also determine the leakage flux of brownian particles that diffuse from a source to an absorbing target on a reflecting boundary of a domain if a small perforation is made in the reflecting boundary

robustness of pole location in perturbed systems abstract in this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients a geometric approach in coefficient space is exploited to derive maximal deviations in a given class of admissible perturbations of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane it is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane this last result can be considered an extension of the result of the well known theorem of kharitonov it is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations

criticality in reactors under domain or external temperature perturbations the conditions for the onset of thermal runaway in reactors with small non uniformities is investigated the reaction is modelled by an arrhenius heat generation term with a finite activation energy and the dimensionless temperature u o is taken to satisfy a nonlinear equation of the form u 0 0f u 0 0 x d vu 0 bu 0 0 x d we investigate three classes of perturbations of this problem first we treat a small temperature variation maintained on the boundary of the domain secondly we consider a small distortion of the boundary of a circular cylindrical domain and thirdly we analyse the effect of a small hole in the domain in each case we derive asymptotic expansions for the critical frank kamenetskii parameter c where is a measure of the size of the perturbation a numerical scheme is then used to determine numerical values for the coefficients in the asymptotic expansion of c finally some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem

the robustness properties of univariate and multivariate reciprocal polynomials the author investigates the robustness properties of univariate and multivariate reciprocal polynomials that are nonzero on the unit circle and the unit polycircle respectively he shows that any polytope of univariate reciprocal polynomials are nonzero on the unit circle if and only if a set of real valued rationals corresponding to its vertices are entirely either positive or negative on the unit circle ensuring that these vertex rationals are entirely either positive or negative on the unit circle can be carried out by the tests described by lakshmanan 1992 when these existing tests are combined with the results contained in this paper it provides a complete procedure for testing the nonzeroness of polytopes of univariate reciprocal polynomials over the unit circle he shows that this result generalizes to the case of multivariate polynomials for any polytope of multivariate polynomials to be nonzero on the unit polycircle it is necessary and sufficient that a set of real valued multivariate rationals corresponding to its vertices are entirely either positive or negative on the unit polycircle again by using the test the positivity or the negativity of the vertex rationals can be ensured as well thereby resulting in a complete procedure for testing the nonzeroness of an entire polytope of multivariate reciprocal polynomials over the unit polycircle although he develops the results for polytopic families he then extends those results to the case of non polytopic reciprocal polynomial families

sensitivities of stability constraints and their applications let the real polynomial a s a 0 a 1 s a n s n with the coefficients being known differentiable functions a k x be given and let the constraints g i x 0 determine the strictly hurwitz property of the polynomial a s a simple and efficient method to calculate the derivatives partial g i x partial x j is proposed then the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed also a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given

state space measures of robustness of pole locations for structured and unstructured perturbations abstract measures of pole location robustness for linear feedback systems are derived from a state space model of the system the robustness tests ensure that the eigenvalues of the perturbed systems matrix a e remain in a desired region d of the complex plane containing the eigenvalues of the nominal system matrix a the region d may be any open set of the complex plane whatsoever the results are expressed in terms of induced matrix norms and apply to structured perturbations of the form e b c where b and c define the structure of e and may be nonsquare matrices rank one perturbations e of minimal norm and with the given structure that will cause a e to have an eigenvalue outside of d are constructed for the cases when the matrix norm is induced by the vector 1 norm or the vector norm the advantages of having robustness measures for several matrix norms that can be computed are illustrated with a simple example that demonstrates how the conservatism of single tests can be reduced using several tests i e several matrix norms a method for computing numerically the robustness measures for particular norms is presented it can be used to compute with a guaranteed degree of accuracy the maximum of the norm of the frequency response of a system

mean first passage times in confined media from markovian to non markovian processes we review recent theoretical works that enable the accurate evaluation of the mean first passage time mfpt of a random walker to a target in confinement for markovian memory less and non markovian walkers for the markovian problem we present a general theory which allows one to accurately evaluate the mfpt and its extensions to related first passage observables such as splitting probabilities and occupation times we show that this analytical approach provides a universal scaling dependence of the mfpt on both the volume of the confining domain and the source target distance in the case of general scale invariant processes this analysis is applicable to a broad range of stochastic processes characterized by length scale invariant properties and reveals the key role that can be played by the starting position of the random walker we then present an extension to non markovian walks by taking the specific example of a tagged monomer of a polymer chain looking for a target in confinement we show that the mfpt can be calculated accurately by computing the distribution of the positions of all the monomers in the chain at the instant of reaction such a theory can be used to derive asymptotic relations that generalize the scaling dependence with the volume and the initial distance to the target derived for markovian walks finally we present an application of this theory to the problem of the first contact time between the two ends of a polymer chain and review the various theoretical approaches of this nonmarkovian problem

some results on robust stability of discrete time systems the theorem of kharitonov on the hurwitz property of interval families of polynomials cannot be extended in genera to obtain sufficient conditions for the stability of families of characteristic polynomials of discrete time systems necessary and sufficient conditions for this stability problem are given such conditions naturally give rise to a computationally efficient stability test which requires the solution of a one parameter optimization problem and which can be considered as a counterpart to the kharitonov test for continuous time systems at the same time the method used to derive the stability conditions provides a procedure for solving another stability robustness problem i e the estimation of the largest domain of stability with a rectangular box shape around given nominal values of the polynomial coefficients

complete characterization of the spherical spectral set the problem of computing the spectral set of any given spherical family of polynomials is considered an algorithm based on the decomposition of the spectral set in three disjoint subsets with relate points having the same dimension of the back image in the parameter space is devised moreover the structure of the polynomial function which characterizes the main subset is analyzed such an analysis leads to significant simplifications for boundary generation and bounds computation of the spectral set

green functions for weighted subelliptic p laplace operator constructed by hormander s vector fields abstract this paper deals with the weighted subelliptic p laplace operator constructed by hormander s vector fields l p u d i v x a x x u x x u x p 2 2 a x x u x where u w 1 p u w 1 p q and a x is a bounded measurable and m m symmetric matrix satisfying 1 w x 2 p 2 a x w x 2 p 2 r m w x a p we first prove existence of the modified green function for l p by virtue of minty browder theorem and then existence of the green function for l p by checking the convergence of sequence of modified green functions next we derive upper bounds of the modified green function for l p by establishing the interpolation inequality in the weighted weak l p spaces finally the bounds of the green function for l p are also obtained by virtue of results for the modified green function and the compact weighted embedding theorem

the narrow escape problem the narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain the problem is equivalent to solving the mixed dirichlet neumann boundary value problem for the poisson equation with small dirichlet and large neumann parts the mixed boundary value problem which goes back to lord rayleigh originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the neumann problem in domains with bottlenecks we review here recent developments in the non standard asymptotics of the problem which are based on several ingredients a better resolution of the singularity of neumann s function resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks and the breakup of composite domains into simpler components the new methodology applies to two and higher dimensional problems selected applications are r

first passage quantities of brownian motion in a bounded domain with multiple targets a unified approach in this paper we introduce a general computation method to systematically determine the mean first passage time mfpt the global mean first passage time gmfpt and splitting probabilities for a continuous brownian motion in a confined 2d or 3d domain with multiple absorbing targets in the bulk or on the boundary this method is applied to spherically symmetric domains in the limit of small sized targets and asymptotic expansions of the mpft gmfpt and splitting probabilities are obtained in four distinct cases 3d domains with targets in the bulk 3d domains with targets on the boundary 2d domains with targets on the bulk and 2d domains with targets on the boundary this approach gives a unified description of existing exact results which were obtained using specific technics and also yields new results in particular for n targets splitting probabilities

computation of capacity this article introduces a method for computing upper and lower bounds for the logarithmic capacity of a compact plane set if the set has the holder continuity property then these bounds converge to the value of the capacity a number of examples are discussed in detail including the cantor middle third set for which we estimate c e 0 220949102189507

narrow escape problem for the unit sphere homogenization limit optimal arrangements of large numbers of traps and the n 2 conjecture a narrow escape problem is considered to calculate the mean first passage time mfpt needed for a brownian particle to leave a unit sphere through one of its n small boundary windows traps a procedure is established to calculate optimal arrangements of n 1 equal small boundary traps that minimize the asymptotic mfpt based on observed characteristics of such arrangements a remarkable property is discovered that is the sum of squared pairwise distances between optimally arranged n traps on a unit sphere is integer equal to n 2 it is observed for 2 n 1004 with high precision it is conjectured that this is the case for such optimal arrangements for all n a dilute trap limit of homogenization theory when n can be used to replace the strongly heterogeneous dirichlet neumann mfpt problem with a spherically symmetric robin problem for which an exact solution is readily found parameters of the robin homogenization problem are computed that capture the first four terms of the asymptotic mfpt close agreement of asymptotic and homogenization mfpt values is demonstrated the homogenization approach provides a radically faster way to estimate the mfpt since it is given by a simple formula and does not involve expensive global optimization to determine locations of n 1 boundary traps

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

harnack inequality and smoothness for quasilinear degenerate elliptic equations we prove local and global regularity for the positive solutions of a quasilinear variational degenerate equation assuming minimal hypothesis on the coefficients of the lower order terms as an application we obtain holder continuity for the gradient of solutions to nonvariational quasilinear equations

one extending hypersphere method to handle dominant pole assignment it is pointed out that the stability hypersphere approach presented by c b soh et al see ibid vol ac 32 p 239 244 mar 1987 can be extended to handle the case where the region is not simply connected the extension is useful in application such as dominant pole assignment and filter design where the separation of zeros is required

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

pole assignment with robust stability this paper uses convex analysis for the pole assignment design of discrete time siso systems incorporating robust stability against norm bounded parametric perturbations in the plant transfer function the method involves designing an overparameterized pole assignment controller for the nominal plant with the overparameterization chosen to reduce the size of changes in the closed loop characteristic equation which result from plant perturbations sufficiency bounds on the l sub p norm of the perturbation guaranteeing stability are obtained

universal formula for extreme first passage statistics of diffusion the timescales of many physical chemical and biological processes are determined by first passage times fpts of diffusion the overwhelming majority of fpt research studies the time it takes a single diffusive searcher to find a target however the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers this fastest fpt depends on extremely rare events and has a drastically faster timescale than the fpt of a given single searcher in this work we prove a simple explicit formula for every moment of the fastest fpt the formula is remarkably universal as it holds for d dimensional diffusion processes i with general space dependent diffusivities and force fields ii on riemannian manifolds iii in the presence of reflecting obstacles and iv with partially absorbing targets our results rigorously confirm generalize correct and unify various conjectures and heuristics about the fastest fpt

narrow escape part ii the circular disk we consider brownian motion in a circular disk whose boundary partial omega is reflecting except for a small arc partial omega a which is absorbing as varepsilon partial omega a partial omega decreases to zero the mean time to absorption in partial omega a denoted e tau becomes infinite the narrow escape problem is to find an asymptotic expansion of e tau for varepsilon ll1 we find the first two terms in the expansion and an estimate of the error the results are extended in a straightforward manner to planar domains and two dimensional riemannian manifolds that can be mapped conformally onto the disk our results improve the previously derived expansion for a general domain e tau frac omega d pi big log frac 1 varepsilon o 1 big d is the diffusion coefficient in the case of a circular disk we find that the mean first passage time from the center of the disk is e tau boldmath x 0 bf 0 frac r 2 d big log frac 1 varepsilon log 2 frac 1 4 o varepsilon big the second term in the expansion is needed in real life applications such as trafficking of receptors on neuronal spines because log frac 1 varepsilon is not necessarily large even when e is small we also find the singular behavior of the probability flux profile into partial omega a at the endpoints of partial omega a and find the value of the flux near the center of the window