metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

age representation of levy walks partial density waves relaxation and first passage time statistics l vy walks lws define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks alternatively there is a hyperbolic representation of them in terms of partial probability density waves using the latter framework we explore the impact of aging on lws which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution we show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear volterra equations for any initial preparation on this basis relaxation properties i e the convergence towards equilibrium of a generic thermodynamic function dependent on the spatial particle distribution and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics we find that even normal diffusive lws where the long term mean square displacement increases linearly with time may display anomalous relaxation properties such as stretched exponential decay we then discuss the impact of aging on the first passage time statistics of lws by developing the corresponding volterra integral representation as a further natural generalization the concept of lws with wearing is introduced to account for mobility losses

the dancing metric g2 symmetry and projective rolling the dancing metric is a pseudo riemannian metric g of signa ture 2 2 on the space m 4 of non incident point line pairs in the real projec tive plane rp 2 the null curves of m 4 g are given by the dancing condi tion the point is moving towards a point on the line about which the line is turning we establish a dictionary between classical projective geometry incidence projective duality dierential invariants of plane curves and in terms of the osculating conic along a plane curve and its cartan s devel opment the outcome is a correspondence between curves in the projective plane and its dual a projective geometry analog of the familiar rolling without slipping and twisting of a pair of riemannian surfaces

distributions of corank 1 and their characteristic vector fields we prove that any 1 parameter family of corank 1 distributions or pfaff equations on a compact manifold m n is trivializable i e transformable to a constant family by a family of diffeomorphisms if all distributions of the family have the same characteristic line field the characteristic line field is a field of tangent lines which is invariantly assigned to a corank one distribution it is defined on m n if n 2k or on a subset of m n called the martinet hypersurface if n 2k 1 our second main result states that if two corank one distributions have the same characteristic line field and are close to each other then they are equivalent via a diffeomorphism this holds under a weak assumption on the singularities of the distributions the second result implies that the abnormal curves of a distribution determine the equivalence class of the distribution among distributions close to a given one

vessiot structure for manifolds of hyperbolic type darboux integrability and symmetry it is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is darboux integrable then its local cauchy problem may be solved by ordinary differential equations in addition such an equation has infinitely many non trivial conservation laws moreover darboux integrable equations have properties in common with infinite dimensional completely integrable systems in this paper we employ a geometric object intrinsically associated with any hyperbolic partial differential equation its hyperbolic structure to study the darboux integrability of the class e of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables it is shown that the problem of classifying the darboux integrable equations in e contains as a subproblem that of classifying the manifolds of p q hyperbolic type of rank 4 and dimension 2k 3 k 2 p 2 q 2 in turn it is shown that the problem of classifying these manifolds in the two lowest cases p q 2 2 2 3 contains as a subproblem the classification problem for lie groups this generalizes classical results of e vessiot the main result is that if an equation in e is 2 2 or 2 3 darboux integrable on the k jets k 2 then its intrinsic hyperbolic structure admits a lie group of symmetries of dimension 2k 1 or 2k 2 respectively it follows that part of the moduli space for the darboux integrable equations in e is determined by isomorphism classes of lie groups the lie group in question is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic or riemann invariants of the equation the tangential characteristic symmetries the isomorphism class of the tangential characteristic symmetries is a contact invariant of the corresponding darboux integrable partial differential equation

conditional statistical properties of the complex systems having long duration memory national natural science foundation 11305064 11247265 11005041 fujian provincial natural science foundation 2011j01012 program for prominent young talents in fujian province university ja12001 huaqiao university people s republic of china 11bs207

relaxation functions in dipolar materials we compare two simple cartoons of relaxation processes in dipolar materials the first passage relaxation function introduced by k weron 1991 and the average relaxation function expressing the proportion of dipoles which did not change their imposed aligned orientation up to a certain time the latter providing a description closer to what is experimentally measured in some cases the two relaxation functions are proved to coincide

on the invariance of spatially inhomogeneous relaxation processes we consider a general relaxation model based on an ensemble of particles propagating randomly in a general space the particles trajectories are independent and identically distributed copies of an arbitrary random spatial curve and each particle has its own parameter triplet initiation epoch velocity and relaxation coefficient the relaxation rate of each particle is dynamic depending on its spatial location and quantified by an arbitrary spatial landscape function we study the relaxation rate of the entire ensemble and characterize the class of poissonian randomizations of the particles parameters that render the ensemble s relaxation rate invariant with respect to both the particles trajectories and the landscape function the invariant relaxation rates turn out to be governed by power laws and display a statistical behavior analogous to anomalous diffusion applications range from chemical reactions to target search and intelligence sifting

weighted bergman spaces and the integral means spectrum of conformal mappings the classical theory of conformal mappings involves best possible pointwise estimates of the derivative thus supplying a measure of the extremal expansion contraction possible for a conformal mapping it is natural to consider also the integral means of phi t along circles z r where phi is the conformal mapping in question and t is a real parameter 0 1 of the integral means which follows from the classical pointwise estimates is by far too fast better estimates were found by clunie makarov pommerenke bertilsson shimorin and others here we introduce a new method based on area type estimates which discards as little as possible of the information supplied by the area methods the result is a considerable improvement in the estimates of the integral means spectrum known tip to this point

geometric theory for weibull s distribution weibull s distribution is the principal phenomenological law of relaxation in the physical sciences and spans three different relaxation regimes subexponential stretched exponential exponential and superexponential the probabilistic theory of extreme value statistics asserts that the linear scaling limits of minima of ensembles of positive valued random variables which are independent and identically distributed are universally governed by weibull s distribution however this probabilistic theory does not take into account spatial geometry which often plays a key role in the physical sciences in this paper we present a general and versatile model of random reactions in random environments and establish a geometry based theory for the universal emergence of weibull s distribution

on the relationship among three theories of relaxation in disordered systems abstract empirical evidence has accumulated showing the wide occurrence of a stretched exponential relaxation decay law for many diverse condensed matter systems several theories based on different physical mechanisms have been successful in deriving the stretched exponential decay law in a natural way three of these theories the direct transfer model the hierarchically constrained dynamics model and the defect diffusion model are shown here to have an underlying common mathematical structure

corrections to sub riemannian geometry an error in the proof of corollary 6 2 of 1 has been pointed out by gerard ben arous the computation of m x in the case o 0 on p 243 is incorrect because m x 0 when 0 0 and jg x 0 for all k there is also a factor of missing in the formula as stated for 0 0 but this is not significant thus when applying the pontryagin theorem theorem 6 1 it is necessary to consider one additional case o 0 and m x9 0 along the curve the conditions of the theorem in this case lead to the equations

classification of complex and real vacuum spaces of the type n n complex and real vacuum spaces with both self dual and anti self dual parts of the weyl tensor being of the type n are considered such spaces are classified according to two criteria the first one takes into account the properties of the congruences of totally null geodesic 2 dimensional surfaces the null strings the second criterion is the properties of the intersection of these congruences it is proved that there exist six distinct types of the n n spaces new examples of the lorentzian slices of the complex metrics are presented some types of n n spaces which do not possess lorentzian slices are also considered

and the rolling ball 2014 john c baez and john huerta understanding the exceptional lie groups as the symmetry groups of simpler objects is a long standing program in mathematics here we explore one famous realization of the smallest exceptional lie group g2 its lie algebra g2acts locally as the symmetries of a ball rolling on a larger ball but only when the ratio of radii is 1 3 using the split octonions we devise a similar but more global picture of g2 it acts as the symmetries of a spinorial ball rolling on a projective plane again when the ratio of radii is 1 3 we explain this ratio in simple terms use the dot product and cross product of split octonions to describe the g2incidence geometry and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations

submersions hamiltonian systems and optimal solutions to the rolling manifolds problem given a submersion pi q to m with an ehresmann connection mathcal h we describe how to solve hamiltonian systems on m by lifting our problem to q furthermore we show that all solutions of these lifted hamiltonian systems can be described using the original hamiltonian vector field on m along with a generalization of the magnetic force this generalized force is described using the curvature of mathcal h along with a new form of parallel transport of covectors vanishing on mathcal h using the pontryagin maximum principle we apply this theory to optimal control problems m and q to get results on normal and abnormal extremals we give a demonstration of our theory by considering the optimal control problem of one riemannian manifold rolling on another without twisting or slipping along curves of minimal length

integral estimates of conformal derivatives and spectral properties of the neumann laplacian abstract in this paper we study integral estimates of derivatives of conformal mappings d of the unit disc d c onto bounded domains that satisfy the ahlfors condition these integral estimates lead to estimates of constants in sobolev poincare inequalities and by the rayleigh quotient we obtain spectral estimates of the neumann laplace operator in non lipschitz domains quasidiscs in terms of the quasi conformal geometry of the domains specifically the lower estimates of the first non trivial eigenvalues of the neumann laplace operator in some fractal type domains snowflakes were obtained

universal self similarity of propagating populations this paper explores the universal self similarity of propagating populations the following general propagation model is considered particles are randomly emitted from the origin of a d dimensional euclidean space and propagate randomly and independently of each other in space all particles share a statistically common yet arbitrary motion pattern each particle has its own random propagation parameters emission epoch motion frequency and motion amplitude the universally self similar statistics of the particles displacements and first passage times fpts are analyzed statistics which are invariant with respect to the details of the displacement and fpt measurements and with respect to the particles underlying motion pattern analysis concludes that the universally self similar statistics are governed by poisson processes with power law intensities and by the fr echet and weibull extreme value laws

versal normal form for nonsemisimple singularities abstract the theory of versal normal form has been playing a role in normal form since the introduction of the concept by v i arnol d in 1 2 but there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields that is the linear part should be done completely first before one computes the nonlinear terms in this paper we address this issue by giving a complete description of a first order calculation in the case of the two and three dimensional irreducible nilpotent cases which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics l 4 problem

a relationship between asymmetric levy stable distributions and the dielectric susceptibility this paper as a complement to the work of montroll and bendler is concerned with the levy stable distributions and their connection to the dielectric response of dipolar materials in the frequency domain the necessary and sufficient condition for this connection is found the presented probabilistic analysis is based on the mathematically correct representation of the meaning of the relaxation function of a system of dipoles and shows why the same form of a distribution of relaxation rates namely the completely asymmetric levy stable distribution should apply in all different relaxing systems this is in contrast to the traditional definition of the relaxation function expressed as a weighted average of exponential relaxation functions which does not explain the universality of the dielectric relaxation law it also follows from the present considerations that not only is the imaginary part of the dielectric susceptibility directly related to the levy stable distribution as was found by montroll and bendler but so is the real part as a consequence the relation cot n 2 for p and 0 n 1 implied by experimental results is obtained

mass action law versus local contagion dynamics a mean field statistical approach with application to the theory of epidemics a mean field approach for epidemic processes with high migration is suggested by analogy with non equilibrium statistical mechanics for large systems a limit of the thermodynamic type is introduced for which both the total size of the system and the total number of individuals tend to infinity but the population density remains constant in the thermodynamic limit the infection rate is proportional to the product of the proportion of individuals susceptible to infection and the average probability of infection the limit form of the average probability of infection is insensitive to the detailed behaviour of the fluctuations of the number of infectious individuals and may belong to two universality classes 1 if the fluctuation of the number of infectives is non intermittent it increases with the increase of the partial density of infectives and approaches exponentially the asymptotic value one for large densities 2 for intermittent fluctuations obeying a power law scaling the average probability of infection also displays a saturation effect for large densities of infectives but the asymptotic value one is approached according to a power law rather than exponentially for low densities of infectives both expressions for the average probability of infection are linear functions of the proportion of infectives and the infection rate is given by the mass action law

numerical methods for computing nonlinear eigenpairs part i iso homogeneous cases with a rayleigh quotient formulation a local minimax method is developed to solve a class of iso homogeneous nonlinear eigenpair problems for multiple solutions in banach spaces following their instability order the algorithm is implemented to compute weighted eigenpairs of the p laplacian numerical eigenfunctions are illustrated by their graphics several interesting phenomena have been observed and are open for further investigation mathematical analysis related to convergence and instability order of computed eigenfunctions etc is also presented

normal forms and versal deformations of linear hamiltonian systems abstract equivalence classes of time independent linear real hamiltonian systems can be identified up to canonical transformations with the orbits of the adjoint action of the real symplectic group on its lie algebra a new set of representatives also called normal forms for these orbits is given versal deformations of systems in normal form are constructed applications of versal deformations to the study of bifurcations of linear systems with small codimension are indicated

symmetries of the rolling model in the present paper we study the infinitesimal symmetries of the model of two riemannian manifolds m g and hat m hat g rolling without twisting or slipping we show that under certain genericity hypotheses the natural bundle projection from the state space q of the rolling model onto m is a principal bundle if and only if hat m has constant sectional curvature additionally we prove that when m and hat m have different constant sectional curvatures and dimension n ge 3 the rolling distribution is never flat contrary to the two dimensional situation of rolling two spheres of radii in the proportion 1 3 which is a well known system satisfying e cartan s flatness condition

response theory for random channel kinetics in complex systems application to lifetime distributions of active intermediates we present a response theory for random channel kinetics based on the method of characteristic functionals we assume that the uctuations of the rate coecients are generated by the contributions of a large number of reaction channels and can be described in terms of a set of grand canonical probability density functionals we consider linear as well as nonlinear responses for systems with static or dynamic disorder for all cases considered we derive analytical expressions for the ensemble average of the response of the system to an external perturbation the theory is applied for developing a new approach to the analysis of the kinetics in complex chemical systems based on a random channel approach we assume that the total transition rate from one state of the system to another is made up of the contributions of a large number of random reaction pathways which can be described in terms of a stochastic point process we study the response of such a system to an external perturbation and show that it can be expressed in terms of a set of lifetime distributions of reaction intermediates we examine the possibilities of applying the technique to the study of large chemical systems by means of tracer experiments and suggest methods for computing rate distributions from measured lifetime distributions we generalize the method of spectral kinetic analysis for the interpretation of measured lifetime distributions we investigate the relationships between spectral kinetics analysis and random channel statistics c 2000 published by elsevier science b v all rights reserved

classification of engel knots let m mathcal d be an engel 4 manifold we show that the scanning map from the space of engel knots to the space of formal engel knots is a weak homotopy equivalence when restricted to the complement of the closed ker mathcal d orbits this is a relative parametric and c 0 close h principle

nonequilibrium fluctuation dissipation relations for independent random rate processes with dynamical disorder a class of rate processes with dynamical disorder is investigated based on the two following assumptions a the system is composed of a random number of particles or quasiparticles which decay according to a first order kinetic law b the rate coefficient of the process is a random function of time with known stochastic properties the formalism of characteristic functionals is used for the direct computation of the dynamical averages the suggested approach is more general than the other approaches used in the literature it is not limited to a particular type of stochastic process and can be applied to any type of random evolution of the rate coefficient we derive an infinity of exact fluctuation dissipation relations which establish connections among the moments of the survival function and the moments of the number of surviving particles the analysis of these fluctuation dissipation relations leads to the unexpected result that in the thermodynamic limit the fluctuations of the number of partic

stretched exponential decay laws of general defect diffusion models we calculate a correlation function of a dipole which flips upon contact with one of the defects making generally non gaussian diffusions other than the memory effect in the fractal random walk model the non gaussian property can be an origin of the stretched exponential law of the correlation function

twistor space for rolling bodies on a natural circle bundle mathbb t m over a 4 dimensional manifold m equipped with a split signature metric g whose fibers are real totally null selfdual 2 planes we consider a tautological rank 2 distribution mathcal d obtained by lifting each totally null plane horizontally to its point in the fiber over the open set where g is not antiselfdual the distribution mathcal d is 2 3 5 in mathbb t m we show that if m is a cartesian product of two riemann surfaces 1 g1 and 2 g2 and if g g 1 oplus g 2 then the circle bundle mathbb t sigma 1 times sigma 2 is just the configuration space for the physical system of two surfaces 1 and 2 rolling on each other the condition for the two surfaces to roll on each other without slipping or twisting identifies the restricted velocity space for such a system with the tautological distribution mathcal d on mathbb t sigma 1 times sigma 2 we call mathbb t sigma 1 times sigma 2 the twistor space and mathcal d the twistor distribution for the rolling surfaces among others we address the following question for which pairs of surfaces does the restricted velocity distribution which we identify with the twistor distribution mathcal d have the simple lie group g2 as the group of its symmetries apart from the well known situation when the surfaces 1 and 2 have constant curvatures whose ratio is 1 9 we unexpectedly find three different types of surfaces that when rolling without slipping or twisting on a plane have mathcal d with the symmetry group g2 although we have found the differential equations for the curvatures of 1 and 2 that gives mathcal d with g2 symmetry we are unable to solve them in full generality so far

electric double layer on fractal electrodes the electric double layer around fractal electrodes is studied with direct numerical simulations of the nernst planck equation coupled with the poisson equation the capacitance and the relaxation time obey a power law as a function of the system size the temperature and the concentration the time evolution of the charging process exhibits a stretched exponential law and the exponent beta is numerically evaluated

dimension gap under conformal mappings abstract we give an estimate for the hausdorff gauge dimension of the boundary of a simply connected planar domain under p integrability of the hyperbolic metric p 1 this estimate does not degenerate when p tends to one for p 1 the boundary can even have positive area the same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric we also give an example which shows that our estimates are essentially sharp

genericity results for singular curves let m be a smooth manifold and cal d m m geq 2 be the set of rank m distributions on m endowed with the whitney c infty topology we show the existence of an open set o m dense in cal d m so that every nontrivial singular curve of a distribution d of o m is of minimal order and of corank one in particular for m geq 3 every distribution of o m does not admit nontrivial rigid curves as a consequence for generic sub riemannian structures of rank greater than or equal to three there does not exist nontrivial minimizing singular curves

non minimality of corners in subriemannian geometry we give a short solution to one of the main open problems in subriemannian geometry namely we prove that length minimizers do not have corner type singularities with this result we solve problem ii of agrachev s list and provide the first general result toward the 30 year old open problem of regularity of subriemannian geodesics

versal normal form at the lagrange equilibrium l4 a new normal form called versal for the linearized hamiltonian vector field of the planar restricted three body problem at the lagrange equilibrium point l4 depending smoothly on the mass ratio for all values close to the critical routh s ratio is described then a canonical transformation also depending smoothly on the mass ratio which brings the linear hamiltonian vector field into this versal normal form is explicitly calculated

existence of engel structures we develop a construction of engel structures on 4 manifolds based on decompositions of manifolds into round handles this allows us to show that all parallelizable 4 manifolds admit an engel structure we also show that given two engel manifolds m1 m2 satisfying a certain condition on the characteristic foliations there is an engel structure on m1 m2 s2 s2 which is closely related to the original engel structures

parameter calibration between models and simulations connecting linear and non linear descriptions of anomalous diffusion abstract anomalous diffusion is an ubiquitous phenomenon which have been studied by several approaches including simulation analytical methods and experiments partial differential equations either fractional linear ones as non linear are able to describe the phenomenon in particular the heavy tails observed in the probability distribution functions besides one can observe some difficulties to tame the anomalous diffusion parameters in many simulations studies consequently the relationship between simulations results to the corresponding model becomes somewhat impaired in this work we present a systematic calibration between simulations and models exploring the relationship among the diffusion coefficients and diffusion exponents with the order of fractional derivatives q gaussian parameter and model diffusion constant by means of a statistical fitting procedure of the simulations data we establish a connection between linear and non linear approaches simulations considered a ctrw with a criterion to control mean waiting time and step length variance and a full range of well controlled cases ranging from subdiffusion to superdiffusion regimes were generated theoretical models were expressed by means of generalised diffusion equations with fractional derivatives in space and time and by the non linear porous medium equation to assess the diffusion constant the order of fractional derivatives and the q gaussian parameter of simulations data in each case of anomalous diffusion we compare the accuracy of two methods 1 by analysis of the dispersion of the variance over time and 2 by the optimisation of solutions of the theoretical models to the histogram of positions the relative accuracies of the models were also analysed for each regimen of anomalous diffusion we highlight relations between simulations parameters and model parameters and discuss methods to link them among those tsallis buckman scaling law was verified

extremals for eigenvalues of laplacians under conformal mapping g polya and g szeg showed in 1951 that for simply connected plane domains the first eigenvalue of the laplacian with dirichlet boundary conditions is maximal for a disk under a conformal mapping normalization that is iff z is a conformal map of a diskdonto a bounded simply connected plane domain normalized by f 0 1 then 1 1 d later polya and m schiffer showed that actually j 1n1 j j 1n1 j d for eachn 1 2 3 this paper shows that for every convex increasing function j 1n 1 j j 1n 1 j d for eachn 1 2 3 in particular taking a asfor a fixeds 1 gives that the zeta function of the eigenvalues of the laplacian is minimal for the disk under dirichlet boundary conditions the bulk of the paper addresses similar questions for simply and doubly connected domains on cones and cylinders and on surfaces of variable curvature extending the work of c bandle t gasser and j hersch also letmgbe ann dim riemannian manifold with boundary and for each smooth functionwon the closure ofm write j w for thejth eigenvalue of the operatorw 1 g under dirichlet boundary conditions note that in two dimensions w 1 gis the laplace beltrami operator of the metricwg it is proved here that the zeta type functional nj 1 j w 1 is convex with respect to the mass densityw

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

generating functional approach to multichannel parallel relaxation with application to the problem of direct energy transfer in fractal systems with dynamic disorder a model for multichannel parallel relaxation is suggested based on the following assumptions a an individual channel is characterized by a set of continuous state variables the corresponding relaxation rate is a function of the state variables as well as of the time interval for which the channel is open b the number of channels is a random variable described by a correlated point process defined in the space of state parameters of an individual channel analytical expressions for the generating functional of the overall relaxation rate and for the average survival function are derived in terms of the generating functional of the point process the general formalism is applied to the problem of direct energy transfer from excited donors to acceptors in fractal systems with dynamic disorder it is assumed that the number of acceptors obeys a poissonian distribution law with a constant average density in a df dimensional fractal structure embedded in a ds dimensional euclidean space ds 1 2 3 and th

universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder dynamical generalizations of stretched exponential the asymptotic behavior of multichannel parallel relaxation processes for systems with dynamical disorder is investigated in the limit of a very large number of channels an individual channel is characterized by a state vector x which due to dynamical disorder is a random function of time a limit of the thermodynamic type in the x space is introduced for which both the volume available and the average number of channels tend to infinity but the average volume density of channels remains constant scaling arguments combined with a stochastic renormalization group approach lead to the identification of two different types of universal behavior of the relaxation function corresponding to nonintermittent and intermittent fluctuations respectively for nonintermittent fluctuations a dynamical generalization of the static huber s relaxation equation is derived which depends only on the average functional density of channels w t d w t the channels being classified according to their different relaxation rates w w t which are random functions of time for intermittent fluctuations a more complicated relaxation equation is derived which in addition to the average density of channels w t d w t depends also on a positive fractal exponent h which characterizes the fluctuations of the density of channels the general theory is applied for constructing dynamical analogs of the stretched exponential relaxation function for nonintermittent fluctuations the type of relaxation is determined by the regression dynamics of the fluctuations of the relaxation rate if the regression process is fast and described by an exponential attenuation function then after an initial stretched exponential behavior the relaxation process slows down and it is not fully completed even in the limit of very large times for self similar regression obeying a negative power law the relaxation process is less sensitive to the influence of dynamical disorder both for small and large times the relaxation process is described by stretched exponentials with the same fractal exponent as for systems with static disorder for large times the efficiency of the relaxation process is also slowed down by fluctuations similar patterns are found for intermittent fluctuations with the difference that for very large times and a slow regression process a crossover from a stretched exponential to a self similar algebraic relaxation function occurs some implications of the results for the study of relaxation processes in

transformation to versal deformations of matrices abstract in the paper versal deformations of matrices are considered the versal deformation is a matrix family inducing an arbitrary multi parameter deformation of a given matrix by an appropriate smooth change of parameters and basis given a deformation of a matrix it is suggested to find transformation functions the change of parameters and the change of basis dependent on parameters in the form of taylor series the general method of construction of recurrent procedures for calculation of coefficients in the taylor expansions is developed and used for the cases of real and complex matrices elements of classical lie and jordan algebras and infinitesimally reversible matrices several examples are given and studied in detail applications of the suggested approach to problems of stability singularity and perturbation theories are discussed

two forms of self similarity as a fundamental feature of the power law dielectric response a new mathematical representation of the cluster model for dielectric relaxation in a bound dipole case is established by employing the extremal value theory two distinct probabilistic mechanisms which drive the dielectric response function to acquire the power law form are presented consequently two forms of self similarity one of which dominates the response at short times and the other at long times leading to a general relaxation equation are identified finally the conditions under which the derived response function takes the well known empirical forms williams watts cole cole cole davidson broadened debye and flat loss responses are recognized

on the equivalence of the parallel channel and the correlated cluster relaxation models the question of the origins of nonexponential relaxation is addressed in terms of the probabilistic approach to relaxation the interconnection between two differently rooted probabilistic models i e between the parallel channel and the correlated cluster models is presented we show that clearly different probabilistic origins yield in both approaches a well defined class of universally valid two power law responses with the stretched exponential and exponential decay laws as special cases the equivalence of both models indicates that variations in the local environment of the relaxing configurational units parallel channel relaxation can provide a basis for self similar relaxation dynamics without the need for hierarchically constrained dynamics correlated clusters relaxation

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

topography of chance we present a model of multiplicative langevin dynamics that is based on two foundations the langevin equation and the notion of multiplicative evolution the model is a nonlinear mechanism transforming a white noise input to a dynamic equilibrium output using a single control an underlying convex u shaped potential function the output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories the model generates each admissible combination of the output s shape and randomness in a universal and robust fashion moreover practically all the probability distributions that are supported on the positive half line and that are commonly encountered and applied across the sciences can be reverse engineered by this model hence this model is a universal equilibrium mechanism in the context of multiplicative dynamics for the robust generation of chance the model s output in turn the properties of the produced chance the output s shape and randomness are determined with mathematical precision by the control s landscape its topography thus a topographic map of chance is established as a particular application probability distributions with power law tails are shown to be universally and robustly generated by controls on the edge of convexity convex u shaped potential functions with asymptotically linear wings

wait and switch relaxation model relationship between nonexponential relaxation patterns and random local properties of a complex system the wait and switch stochastic model of relaxation is presented using the random variable formalism of limit theorems of probability theory we explain the universality of the short and long time fractional power laws in relaxation responses of complex systems we show that the time evolution of the nonequilibrium state of a macroscopic system depends on two stochastic mechanisms one which determines the local statistical properties of the relaxing entities and the other one which determines the number random or deterministic of the microscopic and mesoscopic relaxation contributions within the proposed framework we derive the havriliak negami and kohlrausch williams watts functions we also discuss the influence of the random walk characteristics of migrating defects on the homogeneous and heterogeneous relaxation scenarios and show the origins of the stretched exponential integral kernel in the integral representation of the ensemble averaged relaxation function

on the first eigenvalues of free vibrating membranes in conformal regular domains in 1961 g polya published a paper about the eigenvalues of vibrating membranes the free vibrating membrane corresponds to the neumann laplace operator in bounded plane domains in this paper we obtain estimates for the first non trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains this class includes convex domains john domains etc on the basis of our estimates we conjecture that the eigenvalues of the neumann laplace operator depend on the hyperbolic metrics of plane domains we propose a new method for the estimates which is based on weighted poincare sobolev inequalities obtained by the authors recently

langevin unification of fractional motions we present a physical langevin based theory explaining the emergence and pervasiveness of the following fractional motions brownian motion l vy motion fractional brownian motion and fractional l vy motion a general form of micro level langevin dynamics with infinitely many degrees of freedom is considered transcending from the micro level to the macro level the infinitely many degrees of freedom collapse to two characteristic exponents and the aforementioned fractional motions emerge universally the exponents categorize the fractional motions and determine their statistical and topological properties the theory establishes a unified langevin bedrock to fractional motions which are widely applied prototypical models of random transport in the sciences

randomized central limit theorems a unified theory the central limit theorems clts characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables the clts assert that the universal probability laws governing ensembles aggregate statistics are either gaussian or levy and that the universal probability laws governing ensembles extreme statistics are frechet weibull or gumbel the scaling schemes underlying the clts are deterministic scaling all ensemble components by a common deterministic scale however there are random environment settings in which the underlying scaling schemes are stochastic scaling the ensemble components by different random scales examples of such settings include holtsmark s law for gravitational fields and the stretched exponential law for relaxation times in this paper we establish a unified theory of randomized central limit theorems rclts in which the deterministic clt scaling schemes are replaced with stochastic scaling schemes and present randomized counterparts to the classic clts the rclt scaling schemes are shown to be governed by poisson processes with power law statistics and the rclts are shown to universally yield the levy frechet and weibull probability laws