mean first passage time of surface mediated diffusion in spherical domains we present an exact calculation of the mean first passage time to a target on the surface of a 2d or 3d spherical domain for a molecule alternating phases of surface diffusion on the domain boundary and phases of bulk diffusion the presented approach is based on an integral equation which can be solved analytically numerically validated approximation schemes which provide more tractable expressions of the mean first passage time are also proposed in the framework of this minimal model of surface mediated reactions we show analytically that the mean reaction time can be minimized as a function of the desorption rate from the surface

an l2 quotient algorithm for finitely presented groups abstract the paper develops algorithmic methods to enumerate all normal subgroups of a finitely presented group such that the factor groups are either isomorphic to psl 2 p n or to pgl 2 p n the case of two generators is treated in detail a range of examples starting with the free group on two generators and ending with groups having only finitely many normal subgroups of this type is discussed

a hybrid asymptotic numerical method for low reynolds number flows past a cylindrical body the classical problem of slow steady two dimensional flow of a viscous incompressible fluid around an infinitely long straight cylinder is considered the cylinder cross section is symmetric about the direction of the oncoming stream but otherwise is arbitrary for low reynolds number the well known singular perturbation analysis for this problem shows that the asymptotic expansions of the drag coefficient and of the flow field start with infinite logarithmic series we show that the entire infinite logarithmic expansions of the flow field and of the drag coefficient are contained in the solution to a certain related problem that does not involve the cross sectional shape of the cylinder the solution to this related problem is computed numerically using a straightforward finite difference scheme the drag coefficient for a cylinder of a specific cross sectional shape which is asymptotically correct to within all logarithmic terms is given in terms of a single shape dependent constant that is determi

on shift dynamics for cyclically presented groups abstract a group defined by a finite presentation with cyclic symmetry admits a shift automorphism that is periodic and word length preserving it is shown that if the presentation is combinatorially aspherical and orientable in the sense that no relator is a cyclic permutation of the inverse of any of its shifts then the shift acts freely on the non identity elements of the group presented for cyclic presentations defined by positive words of length at most three the shift defines a free action if and only if the presentation is combinatorially aspherical and the shift itself is fixed point free if and only if the group presented is infinite

knot theory for self indexed graphs we introduce and study so called self indexed graphs these are oriented finite graphs endowed with a map from the set of edges to the set of vertices such graphs naturally arise from classical knot and link diagrams in fact the graphs resulting from link diagrams have an additional structure an integral flow we call a self indexed graph with integral flow a comte the analogy with links allows us to define transformations of comtes generalizing the reidemeister moves on link diagrams we show that many invariants of links can be generalized to comtes most notably the linking number the alexander polynomials the link group etc we also discuss finite type invariants and quandle cocycle invariants of comtes

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

orbit decidability and the conjugacy problem for some extensions of groups given a short exact sequence of groups with certain conditions 1 f g h 1 we prove that g has solvable conjugacy problem if and only if the corresponding action subgroup a 6 aut f is orbit decidable from this we deduce that the conjugacy problem is solvable among others for all groups of the form z2 fm f2 fm fn z and zn a fm with virtually solvable action group a 6 gln z also we give an easy way of constructing groups of the form z4 fn and f3 fn with unsolvable conjugacy problem on the way we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem as an application an alternative solution to the conjugacy problem in aut f2 is given

small cancellation theory and automatic groups part ii the main results obtained in part i can be rephrased in the language of root systems and euclidean planar tessellations as saying that finite a1 x a1 and a 2 piecewise euclidean abbreviated pe complexes of non positive curvature have automatic fundamental groups a 2 dimensional complex is a a 1 a1 respectively a2 pe complex if every 2 cell can be identified with a unit square resp equilateral triangle in the euclidean plane in such a way that the induced metrics agree on intersections such a complex has non positive curvature if the link of each vertex contains no circuits without backtracking of length less than 2 n here we prove using similar techniques that finite 2 complexes modelled on the other two root systems b 2 and g2 corresponding to tessellations by 45 right angle triangles and by 30 60 right angle triangles also have automatic fundamental groups theorems 2 3 and 3 2 to be precise

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

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

analysis of the poisson nernst planck equation in a ball for modeling the voltage current relation in neurobiological microdomains abstract the electro diffusion of ions is often described by the poisson nernst planck pnp equations which couple nonlinearly the charge concentration and the electric potential this model is used among others to describe the motion of ions in neuronal micro compartments it remains at this time an open question how to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a domain bounded by an impermeable dielectric membrane the purpose of this paper is to construct an asymptotic approximation to the solution of the stationary pnp equations in a d dimensional ball d 1 2 3 in the limit of large total charge in this geometry the pnp system reduces to the liouville gelfand bratu lgb equation with the difference that the boundary condition is neumann not dirichlet and there is a minus sign in the exponent of the exponential term the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of poisson s equation these differences replace attraction by repulsion in the lgb equation thus completely changing the solution we find that the voltage is maximal in the center and decreases toward the boundary we also find that the potential drop between the center and the surface increases logarithmically in the total number of charges and not linearly as in classical capacitance theory this logarithmic singularity is obtained for d 3 from an asymptotic argument and cannot be derived from the analysis of the phase portrait these results are used to derive the relation between the outward current and the voltage in a dendritic spine which is idealized as a dielectric sphere connected smoothly to the nerve axon by a narrow neck this is a fundamental microdomain involved in neuronal communication we compute the escape rate of an ion from the steady density in a ball which models a neuronal spine head to a small absorbing window in the sphere we predict that the current is defined by the narrow neck that is connected to the sphere by a small absorbing window as suggested by the narrow escape theory while voltage is controlled by the pnp equations independently of the neck

on some questions about a family of cyclically presented groups abstract we study various questions about the generalised fibonacci groups a family of cyclically presented groups which includes as special cases the fibonacci sieradski and gilbert howie groups

coherence subgroup separability and metacyclic structures for a class of cyclically presented groups abstract we study a class m of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations this class includes many finite metacyclic generalized fibonacci groups that have been previously identified in the literature by analyzing their shift extensions we show that the groups in the class m are coherent subgroup separable satisfy the tits alternative possess finite index subgroups of geometric dimension at most two and that their finite subgroups are all metacyclic many of the groups in m are virtually free some are free products of metacyclic groups and free groups and some have geometric dimension two we classify the finite groups that occur in m giving extensive details about the metacyclic structures that occur and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three we show that any finite group in the class m that has fixed point free shift automorphism must be cyclic

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

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

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

asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture abstract let x t be a positive recurrent diffusion process corresponding to an operator l on a domain d r d with oblique reflection at d if d r d for each x d we define a volume preserving norm that depends on the diffusion matrix a x we calculate the asymptotic behavior as e 0 of the expected hitting time of the e ball centered at x and of the principal eigenvalue for l in the exterior domain formed by deleting the ball with the oblique derivative boundary condition at d and the dirichlet boundary condition on the boundary of the ball this operator is non self adjoint in general the behavior is described in terms of the invariant probability density at x and det a x in the case of normally reflected brownian motion the results become isoperimetric type equalities

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

efficient finite groups arising in the study of relative asphericity we study a class of two generator two relator groups denoted j n m k that arise in the study of relative asphericity as groups satisfying a transitional curvature condition particular instances of these groups occur in the literature as finite groups of intriguing orders here we find infinite families of non elementary virtually free groups and of finite metabelian non nilpotent groups for which we determine the orders all mersenne primes arise as factors of the orders of the non metacyclic groups in the class as do all primes from other conjecturally infinite families of primes we classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups f a b c with trivalent cayley graphs that was introduced by c m campbell h s m coxeter and e f robertson the theory of cyclically presented groups informs our methods and we extend part of this theory namely on connections with polynomial resultants to bicyclically presented groups that arise naturally in our analysis as a corollary to our main results we obtain new infinite families of finite metacyclic generalized fibonacci groups

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

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

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

on a certain class of cyclically presented groups abstract we continue the study of the cyclically presented groups g g n w where w x 0 x i x j x i x j we prove that under certain conditions on the parameters n i j the group g is infinite these results support our conjecture which states for n 4 exactly when g is trivial and that if g is non trivial then g is infinite

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

tadpole labelled oriented graph groups and cyclically presented groups abstract we study a class of labelled oriented graph log group where the underlying graph is a tadpole graph we show that such a group is the natural hnn extension of a cyclically presented group and investigate the relationship between the log group and the cyclically presented group we relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the log group in the case where the label on the tail of the log spells a positive word in the vertices in the circuit we show that the logs and groups coincide with those considered by szczepanski and vesnin we obtain new presentations for these cyclically presented groups and show that the groups of fibonacci type introduced by johnson and mawdesley are of this form these groups generalise the fibonacci groups and the sieradski groups and have been studied by various authors we continue these investigations using small cancellation and curvature methods to obtain results on hyperbolicity automaticity sq universality and solvability of decision problems

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

algorithms determining finite simple images of finitely presented groups we address the question for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection we prove both positive and negative results for a collection of finite simple groups that contains infinitely many alternating groups or contains classical groups of unbounded dimensions we prove that there is no such algorithm on the other hand for families of simple groups of lie type of bounded rank we obtain positive results for example for any fixed untwisted lie type x there is an algorithm that determines whether or not any given finitely presented group has simple images of the form x q for infinitely many q and if there are finitely many the algorithm determines them

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

exact mean exit time for surface mediated diffusion we present an exact expression for the mean exit time through the cap of a confining sphere for particles alternating phases of surface and of bulk diffusion the present approach is based on an integral equation which can be solved analytically in contrast to the statement of berezhkovskii and barzykin j chem phys 136 54115 2012 we show that the mean exit time can be optimized with respect to the desorption rate under analytically determined criteria

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

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

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

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

link indices of hurwitz maps abstract let m be a regular map of type m n with automorphism group a smooth quotient of the ordinary 2 m n triangle group then by recent work of melekoglu and singerman we can associate with m some positive integers called the link indices of m the number of link indices is always one two or three depending on the parity of m and n in the case where m and n are both odd which happens when m is a hurwitz map of type 3 7 there is a unique link index in this paper we prove that every positive integer divisible by 2 or 3 is the link index of some hurwitz map we also find the link indices of all hurwitz maps of genus up to 10000 and of all the hurwitz maps with automorphism group psl 2 q for some q 700 and we show that every integer between 2 and 360 is the link index of some hurwitz map

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

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

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

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

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

c groups of psl 2 q and pgl 2 q abstract we classify the c group representations of the groups psl 2 q and pgl 2 q we obtain that the c rank of psl 2 q and pgl 2 q is 3 except when q 7 9 11 19 31 for psl 2 q and when q 5 for pgl 2 q in which case it is 4 we provide all representations of rank four

nonlinearity saturation as a singular perturbation of the nonlinear schroźdinger equation saturation of a kerr type nonlinearity in the nonlinear schro dinger equation nlse can be regarded as a singular perturbation which regularizes the well known blow up phenomenon in the cubic nlse an asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions in one dimension this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum in two dimensions we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast scale operator this leads to an outer solution with a free logarithmic singularity whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves however for a certain value of wave power we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field we test our results by numerical simulation of both the full equation and free boundary reductions

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

an l2 quotient algorithm for finitely presented groups on arbitrarily many generators abstract we generalize the plesken fabianska l 2 quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators the main difficulty lies in a constructive description of the invariant ring of gl 2 k on m copies of sl 2 k by simultaneous conjugation by giving this description we generalize and simplify some of the known results in invariant theory an implementation of the algorithm is available in the computer algebra system magma

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

representations of generalized equipped posets and posets with automorphisms over galois field extensions abstract generalized equipped posets and their representations over a galois field extension k l are introduced and studied an equipment of a poset p is a pair s consisting of a finite group and a collection s x y x y of its non empty subsets such that x y y z x z for any chain x y z in p equipped posets in the old sense of our former papers correspond to the case 2 with some additional restriction since the category of representations of a strictly equipped poset p s is isomorphic to the category of representations of its evolvent q which is a poset with automorphisms we study in fact representations of posets with automorphisms over the pair k l in the case where gal l k we define the complexification functor c rep k l q rep l q and the realification functor r rep l q rep k l q and show that they induce reciprocal bijections between the isoclasses of indecomposables of the category rep k l q and the orbits of isoclasses of indecomposables of the category rep l q in this way the problem on classification of indecomposables of equipped posets and posets with automorphisms over k l actually is reduced to that one for representations of ordinary posets over l

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

families of group presentations related to topology we study some algebraic properties of a class of group presentations depending on a finite number of integer parameters this class contains many well known groups which are interesting from a topological point of view we find arithmetic conditions on the parameters under which the considered groups cannot be fundamental groups of hyperbolic 3 manifolds of finite volume then we investigate the asphericity for many presentations contained in our family

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

representations commutative algebra and hurwitz groups abstract characteristic zero representations of the 2 3 7 triangle group in degrees up to seven are constructed by using janet s algorithm for solving polynomial equations these are used to find families of hurwitz groups i e finite epimorphic images for some varieties of representations it is investigated whether additional relations can be uniformly imposed and still result in subvarieties of representations the methods are of more general interest some remarks about the interaction of positive characteristics and characteristic zero are made