transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

poisson lie group of pseudodifferential symbols we introduce a lie bialgebra structure on the central extension of the lie algebra of differential operators with scalar or matrix coefficients on the line and on the circle this defines a poisson lie structure on the dual group of pseudodifferential symbols of an arbitrary real or complex order we show that the usual second benney gl n kdv or gl n adler gelfand dickey and kp poisson structures are naturally realized as restrictions of this poisson structure to submanifolds of this universal poisson lie group moreover the reduced sl n versions of these manifolds orw n algebras in physical terminology can be viewed as certain subspaces of the quotient of this poisson lie group by the dressing action of the group of functions on the circle or as a result of a poisson reduction finally we define an infinite set of commuting functions on the poisson lie group that give the standard families of hamiltonians when restricted to the submanifolds mentioned above the poisson structure and hamiltonians on the whole group interpolate between the poisson structures and hamiltonians of benney kp and kdv flows we also discuss the geometrical meaning ofw as a limit of poisson algebraswe as e 0

maximum rank array codes and their application to crisscross error correction a mu n n k array code c over a field f is a k dimensional linear space of n n matrices over f such that every nonzero matrix in c has rank or mu it is first shown that the dimension of such array codes must satisfy the singleton like bound k

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

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

on two extremal matrix problems abstract let w be a linear subspace of symmetric n n matrices whose rank is at most t it is shown that if the underlying field has more than n elements then let l be an affine space of n n matrices all having rank at least k it is shown that if the underlying field is algebraically closed then dim l n 2 k 1 2 the last result is applied to a problem of valiant concerning permanents and determinants

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

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

equivalence of two approaches to integrable hierarchies of kdv type the equivalence between the approaches of drinfeld sokolov and feigin frenkel to the mkdv and kdv hierarchies is established a new derivation of the mkdv equations in the zero curvature form is given connection with the baker akhiezer function and the tau function is also discussed

two constructions of affine lie algebra representations and boson fermion correspondence in quantum field theory we establish an isomorphism between the vertex and spinor representations of affine lie algebras for types dl 1 and dl 1 2 we also study decomposition of spinor representations using the infinite family of casimir operators and prove that they are either irreducible or have two irreducible components we show that the vertex and spinor constructions of the representations can be reformulated in the language of two dimensional quantum field theory in this physical context the two constructions yield the generalized sine gordon and thirring models respectively already in renormalized form the isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson fermion correspondence

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

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

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

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

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

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

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

spaces of hankel matrices over finite fields abstract let f be a finite field for each 1 n we construct a 2 n dimensional linear space h n of n n hankel matrices over f such that rank a for all 0 a h n it follows that if w is a linear space of hankel matrices over f such that rank a for all a w then dim w

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

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

sine gordon equation and representations of affine kac moody algebra sl2 abstract we give a representation theoretic interpretation of the sine gordon equation using the action of affine kac moody algebra sl 2 on the weyl algebra in this setup functions become functions in non commuting variables the skew casimir operators that we introduce here give rise to a hierarchy of partial differential equations that includes the sine gordon equation and two copies of the korteweg de vries equation

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