monotone iterative technique for partial differential equations of first order in view of the fact that first order partial differential equations arise naturally in modelling growth population of cells which constanctly change their properties a study of stable and chaotic solutions of such equations was initiated in 3 in this paper we develop monotone iterative technique for first order partial differential equations and for this purpose we need to discuss existence uniqueness and comparison results we also indicate how the monotone sequences obtained may be employed as candidates for lyapunov functions in stability theory

bonnesenʼs inequality for john domains in rn abstract we prove sharp quantitative isoperimetric inequalities for john domains in r n we show that the bonnesen style inequalities hold true in r n under the john domain assumption which rules out cusps our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit we use the sharp quantitative inequalities proved in fusco et al 2008 7 and fuglede 1989 4 to reduce our problem to symmetric domains

steiner symmetric extremals in polya szego type inequalities abstract the cases of equality are analyzed in steiner symmetrization inequalities for dirichlet type integrals in particular minimal assumptions are determined under which functions attaining equality are necessarily steiner symmetric

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

transitions and kinematics of reaction convection fronts in a cell population model abstract here we consider cell population dynamics in which there is a simultaneous proliferation and maturation the mathematical model of this process is formulated as a nonlinear first order partial differential equation for the cell density u t x in which there is retardation delay in the temporal t variable thus we consider a transient reaction convection equation in which the cell density is convected with maturation velocity r for localizedd initial perturbations the equation has positive and negative traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution and negative fronts correspond to the reversed case three classes of fronts are found according to the strength of the convection velocity i for strong convection r 1 the fronts are simple translations of the initial data regardless of delay strength ii for weak convection r 1 two types of fronts exist a reaction convection fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state and b convection fronts arise if the localized initial perturbation acts at zero maturation on a finite amplitude state for weak convection cases ii a and ii b a further classification arises according to whether the magnitude of the temporal delay is larger or smaller than a critical value b it is found that the critical delay b corresponds to the hopf bifircation of the reaction equation that is obtained in the absence of convection r 0 for delays larger than b the convective and reaction convection fronts are oscillatory in addition all the reaction convection fronts reverse their direction of motion undergo a positive to negative transition and display non uniform kinematics simulation results are validated and interpreted using solutions for the unretarded equations and by local stability analysis

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

stability of polya szegő inequality for log concave functions abstract a quantitative version of polya szeg inequality is proven for log concave functions in the case of steiner and schwarz rearrangements

symmetry of constrained minimizers of the cahn hilliard energy on the torus abstract we establish sufficient conditions for a function on the torus to be equal to its steiner symmetrization and apply the result to so called volume constrained minimizers of the cahn hilliard energy the resulting connectedness of superlevel sets is used in two dimensions together with the bonnesen inequality to quantitatively estimate the sphericity of minimizers we also show how two point rearrangements can be used to give an alternate proof of symmetry for the constrained minimizers of the cahn hilliard model

affine orlicz polya szego principle for log concave functions abstract the affine l p polya szego principle significantly strengthens the classical polya szego principle it is an open problem whether there exists an affine orlicz polya szego principle which includes the affine l p polya szego principle as special case in this paper an affine orlicz polya szego principle for log concave functions is established by using functional steiner symmetrizations

the sharp quantitative sobolev inequality for functions of bounded variation abstract the classical sobolev embedding theorem of the space of functions of bounded variation bv r n into l n r n is proved in a sharp quantitative form

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

ensemble and trajectory statistics in a nonlinear partial differential equation we have examined the influence of parametric noise on the solution behavioru t x of a nonlinear initial value problem arising in cell kinetics in terms of ensemble statistics the eventual limiting solution mean mathop xi limits u and variance mathop sigma u 2 limits are well characterized functions of the noise statistics and mathop xi limits u and mathop sigma u 2 limits depend on when noise is continuously present along the trajectory mathop xi limits u and mathop sigma u 2 limits are independent of the noise statistics and however in their evolution toward mathop xi limits u and mathop sigma u 2 limits both u t x and u2 t x depend on the noise and

capacite et inegalite de faber krahn dans rn abstract in this paper we define a new capacity which allows us to control the behaviour of the dirichlet spectrum of a compact riemannian manifold with boundary with small subsets which may intersect the boundary removed this result generalises a classical result of rauch and taylor the crushed ice theorem in the second part we show that the dirichlet spectrum of a sequence of bounded euclidean domains converges to the spectrum of a ball with the same volume if the first eigenvalue of these domains converges to the first eigenvalue of a ball

computational issues in solid boundary evaluation abstract boundary representation is an attractive formalism for describing the geometry of three dimensional solid objects providing a complete and explicit enumeration of the contiguous surface elements constituting their bounding surfaces the computational procedure whereby boundary representations are derived from informationally complete but implicit input specifications for solids e g a sequence of boolean operations acting on elementary solid primitives is known as boundary evaluation recent research in boundary evaluation has emphasized the topological adjacency relationships defined by the surface elements of a boundary file which furnish an organized data structure for storing and accessing boundary information however the issues of the computational procedures by which these surface elements are actually derived and their formal mode of representation have been largely neglected consequently reliable boundary evaluation has not made significant progress beyond polyhedral and quadric surface objects the basic computational problem in boundary evaluation is the reliable determination of surface intersections the surface elements participating in a boundary file are known as trimmed surfaces since one can imagine them as being cut out of unbounded analytic parent surfaces along intersection curve segments the issues of surface intersection computation and trimmed surface representation are thus intimately coupled and their consistent treatment is the key to reliable boundary evaluation in this paper we survey the available procedures for computing surface intersections representing trimmed surfaces and processing solid boundary representations and briefly describe a novel trimmed surface formulation which adopts a unified systematic approach to the boundary evaluation problem the focus is on the reliable computation and representation of individual trimmed surface elements rather than data structures for expressing their topological adjacencies or the processing and applications of boundary files per se since boundary evaluation is usually not an end in itself however we describe elementary interrogation functions which form the basis for real applications

a quantitative sobolev inequality in bv a quantitative version of the standard sobolev inequality with sharp constant for functions u in w1 1 rn or bv rn is established in terms of a distance of u from the manifold of all multiples of characteristic functions of balls inequalities involving non euclidean norms of the gradient are discussed as well

on the uniqueness and monotonicity of energy minimisers in the homotopy classes of incompressible mappings and related problems the goal of this paper is to prove the existence and uniqueness of the so called energy minimisers in homotopy classes for the variational energy integral f u x z x f x 2 u 2 u 2 2 dx with f c 0 of class c 2 and satisfying suitable conditions and u lying in the sobolev space of weakly differentiable incompressible mappings of a finite open symmetric plane annulus x onto itself specifically lying in a x u w 1 2 x r 2 det u 1 a e in x and u x on x it is well known that the space a x admits a countably infinite homotopy class decomposition a x s ak with k z we prove that the energy integral f has a unique minimiser in each of these homotopy classes ak furthermore we show that each minimiser is a homeomorphic monotone radially symmetric twist mapping of class c 3 x x or as smooth as f allows thereafter whilst also being a local minimiser of f over a x with respect to the l 1 metric to our best knowledge this is the first uniqueness result for minimisers in homotopy classes in the context of incompressible mappings

on the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems it is the purpose of this paper to discuss the equality sign in d and in related inequalities in particular we shall give an answer to the following question does the equality in d imply that u u this question has been raised by g polya g szego in their book on isoperimetric inequalities l l p 186 they state that their method of proving d unfortunately leaves little hope for discussing the sign of equality another comment on this question can be found in 8 where e lieu conjectures that probably a strict version of d is true as it turns out the doubts of g polya g szeg were indeed justified because there are functions u for which equality holds in d while u does not coincide with u in section 1 of this paper i give an example of such a function on the other hand the conjecture of e lieb is also reasonable since generically i can give a positive answer to the question above there is a dense subset of nonnegative functions in wol 2 1 1 for which the equality in d implies

new polya szego type inequalities and an alternative approach to comparison results for pde s abstract we prove some polya szego type inequalities which involve couples of functions and their rearrangements our inequalities reduce to the classical polya szego principle when the two functions coincide as an application we give a different proof of a comparison result for solutions to dirichlet boundary value problems for laplacian equations proved in 1

dimension of measures invariant with respect to the wazewska partial differential equation abstract we study asymptotic properties of a nonlinear first order partial differential equation which describes the reproduction of blood cells this equation under conditions proposed by wazewska generates a semigroup of transformations with highly chaotic behaviour of trajectories we show that this semigroup has invariant measures with arbitrary large dimension

on the stability of the polygonal isoperimetric inequality abstract we obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths the variance of its radii and its deviation from being convex our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices

second order γ limit for the cahn hilliard functional the goal of this paper is to solve a long standing open problem namely the asymptotic development of order 2 by convergence of the mass constrained cahn hilliard functional

bifurcations of banach space operators on etudie des familles a un parametre d operateurs differentiables de frechet t c u r c u r sur un espace de fonctions c u r si cet espace a un cone positif generateur il existe des conditions sous lesquelles la famille a une suite de regimes periodiques qui apparaissent quand le parametre varie on definit le concept de famille isomodale

a polya szego inequality in a fermi metric abstract let n k n with n 2 and 1 k n given a positive function c r n k we form the riemannian metric g on r n associated to the differential expression d s 2 d x 2 x 2 d y 2 where we write r n x x y with x r n k and y r k let be a log convex measure on r k with smooth density and the product measure l n k on r n where c r n k is a positive function we obtain a polya szego inequality of the form r n f u j g u d r n f u s j g u s d for sobolev functions u where the operation s refers to the k n steiner symmetrisation with respect to the gradient operator g is associated to the metric g and the mapping j may be seen as interpolating between the tangent space at x and r n the nonnegative integrand f is continuous and convex in the gradient variable and satisfies some additional hypotheses as an application we derive a polya szego inequality in the hyperbolic plane that takes the above form

generic properties of measures invariant with respect to the wazewska partial differential equation it is shown that measure with the concentration dimension equal to zero is typical in family of measures invariant with respect to the wazewska equation

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

strong mixing gaussian measures for chaotic semigroups abstract in this paper we will be concerned with the problem of the existence of an invariant mixing measure considering its connection with the chaotic behavior of linear semigroups on separable banach spaces we first prove an identity characterizing invariant gaussian measure involving its covariance operator and the infinitesimal generator of the semigroup this gives an answer to a question raised by rudnicki in his inspiring review paper 32 under suitable conditions we use the proved identity to give an invariant mixing gaussian measure as distribution of a wiener integral

a nonlinear age and maturity structured model of population dynamics ii chaos instability hypercyclicity and chaos are investigated in a nonlinear model of age and maturity structured population dynamics it is demonstrated that the behavior of solutions depends on the viewpoint of the observer viewed in the direction of age structure the population stabilizes to a regular distribution viewed in the direction of maturity structure there is the possibility of chaotic behavior

chaos and hypercyclicity for solution semigroups to some partial differential equations abstract we investigate the solution semigroup to some partial differential equations and give some sufficient conditions for the solution semigroup to be hypercyclic or chaotic

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

a variational principle for fractal dimensions abstract a new definition of the dimension of probability measures is introduced it is related with the fractal dimension of sets by a variational principle this principle is applied in the theory of iterated function systems

symmetry of solutions to the optimal exit time control problem in this paper we study the solutions to the optimal exit time control problem such a problem tries to find the state feedback control law with a fixed cost that can keep the state of a randomly perturbed system inside a subset of the state space called the safe set for as long as possible on average by formulating the problem as an optimization problem with pde constraints and using symmetrization techniques we show that when the safe set is a ball the optimal feedback control if it exists must be radially symmetric furthermore we show that among all safe sets with a fixed volume the ball is the best in that it yields the most efficient optimal exit time control the proofs make essential use of the general isoperimetric inequality

faber krahn inequalities for the robin laplacian a free discontinuity approach isoperimetric inequalities for the principal eigenvalues of the robin laplacian are interpreted as free discontinuity problems of unusual type we prove a full range of faber krahn inequalities in a nonlinear setting and for non smooth domains including the open case of the torsional rigidity the key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation as a byproduct we obtain the best constants for a class of poincare inequalities with trace terms in mathbb r n

chaotic c0 semigroups induced by semiflows in lebesgue and sobolev spaces abstract we give characterizations of chaos for c 0 semigroups induced by semiflows on l p for open r similar to the characterizations of hypercyclicity and mixing of such c 0 semigroups proved in 18 moreover we characterize hypercyclicity mixing and chaos for these classes of c 0 semigroups on w 1 p i for a bounded interval i r and prove that these c 0 semigroups are never hypercyclic on w 1 p i we apply our results to concrete first order partial differential equations such as the von foerster lasota equation

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

second order derivatives and rearrangements the classical polya szego principle states that the lebesgue norm of the gradient of a weakly differentiable function on r decaying to 0 at infinity decreases under spherically symmetric rearrangement we present extensions of this principle to second order derivatives in two different directions on one hand when dimension n 1 we show that the space of those functions whose second order distributional derivative is a measure with finite total variation is closed under the operation of decreasing rearrangement moreover a polya szego type principle is proved for such a variation on the other hand in any dimension an inequality between the decreasing rearrangement of the second order derivatives of a function and the first order derivative of its decreasing rearrangement is established as a consequence a new unified approach to second order sobolev type inequalities in rearrangement invariant spaces via 1 dimensional hardy type inequalities is derived dipartimento di matematica e applicazioni per l architettura universita di firenze piazza ghiberti 27 50122 firenze italy e mail cianchi cesit1 unifi it

pointwise and renyi dimensions of an invariant measure of random dynamical systems with jumps we estimate the lower pointwise dimension and the generalized renyi dimension of an invariant measure of random dynamical systems with jumps it is worthwhile to note that the dimensions are a useful tool in studying the hausdorff dimension of measures and sets our model generalizes markov processes corresponding to iterated function systems and poisson driven stochastic differential equations it can be used as a description of many physical and biological phenomena

a quantitative isoperimetric inequality for fractional perimeters abstract recently frank and seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from sobolev seminorms of fractional order this isoperimetric inequality is improved here in a quantitative form

continuous random dynamical systems abstract we study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by poisson noise the main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system since the dynamical system is defined on an arbitrary banach space possibly infinite dimensional to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by lasota and yorke and extended recently to infinite dimensional spaces by szarek finally it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study hence we obtain their stability as well

localization for random walks among random obstacles in a single euclidean ball place an obstacle with probability 1 p independently at each vertex of mathbb z d and run a simple random walk before hitting one of the obstacles for d ge 2 and p strictly above the critical threshold for site percolation we condition on the environment where the origin is contained in an infinite connected component free of obstacles and we show that for environments with probability tending to one as n rightarrow infty there exists a unique discrete euclidean ball of volume d log 1 p n asymptotically such that the following holds conditioned on survival up to time n we have that at any time t in epsilon n n n for some epsilon n rightarrow n rightarrow infty 0 with probability tending to one the simple random walk is in this ball this work relies on and substantially improves a previous result of the authors on localization in a region of volume poly logarithmic in n for the same problem

the hausdorff dimension of invariant measures for random dynamical systems abstract we consider random dynamical systems with jumps the hausdorff dimension of invariant measures for such systems is estimated

convex symmetrization and polya szego inequality abstract we prove a polya szego inequality involving a convex symmetrization of functions and we investigate the equality case

droplet minimizers of an isoperimetric problem with long range interactions we give a detailed description of the geometry of single droplet patterns in a non local isoperimetric problem in particular we focus on the sharp interface limit of the ohta kawasaki free energy for diblock copolymers regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long range interactions exploiting fine properties of the regularity theory for minimal surfaces we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns where u is the order parameter of a two phases system confined in r n and and m are two nonnegative numerical parameters the two terms in the energy mimic attractive short range and repulsive long range energies between the phases more precisely the first term is local it favors minimal interface area and drives the system toward a partition into few pure phases while the second term involving a coulomb like kernel g is non local and favor a fine mixing of the phases a detailed description of the energy is given in 1 the competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns whose geometry strongly depends on the choice of the parameters and m e g spherical spots cylinders gyroids and lamellae 0 1 the ohta kawasaki functional for diblock copolymers the model we consider arises as a simplification of a ginzburg landau functional proposed by ohta and kawasaki in their pioneering paper 41 as a possible description of a diblock copolymers dbc system even though it is questionable whether such an energy actually describes dbcs see choksi and ren 15 muratov 39 and niethammer and oshita 40 nevertheless it is a first and mathematically non trivial attempt to capture some of the main features of these systems for such a reason it deserved over the last twenty years great attention from both the mathematical and the physical

on starshaped rearrangement and applications a radial symmetrization technique is investigated and new properties are proven the method transforms functions u into new functions u with starshaped level sets and is therefore called starshaped rearrangement this rearrangement is in general not equimeasurable a circumstance with some surprising consequences the method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems in particular the lipschitz continuity of free boundaries can now be easily obtained in a new fashion introduction in 26 g szego introduced the concept of radial symmetrization this is a nonequimeasurable rearrangement which transforms functions u q r q c r2 with starshaped level sets qc x e flu x c into functions u with symmetric starshaped level sets later 2 3 15 16 22 his approach was generalized to transform functions with level sets containing the origin into starshaped and not necessarily symmetric level sets and to higher dimensions we call a set d c rn starshaped with respect to xo if x e d implies tx 1 t xo e d for every t e 0 1 we call a set d starshaped if it is starshaped with respect to the origin therefore a more appropriate name for this rearrangement might be starshaped rearrangement in particular since the notion of radial symmetrization can be easily confused with circular or spherical symmetrization or spherically symmetric rearrangement one of the standard results in the theory of rearrangements is that the capacity of a condenser decreases under rearrangement i e j vu x lpdx 1 vu x ipdx for p 1 where u 1 on ql u 0 on 9q and u o c q c q e 0 0 j ivu p x ipdx for p 1 provided q c rn is a bounded starshaped domain containing ue 0 and u 0 1 u e c0 1 q u 1 in u 0 and u 0 on iq another convenient feature of other rearrangements is their equimeasurability a rearrangement u u is called equimeasurable if the n dimensional lebesgue received by the editors march 26 1984 and in revised form september 24 1985 1980 mathematics subject classification primary 35j20 35r35 35b05 26d20 lthis research was supported by the deutsche forschungsgemeinschaft dfg and by e reid 1986 american mathematical society 0002 9947 86 1 00 25 per page

on population resilience to external perturbations we study a spatially explicit harvesting model in periodic or bounded environments the model is governed by a parabolic equation with a spatially dependent nonlinearity of kolmogorov petrovsky piskunov type and a negative external forcing term delta using sub and supersolution methods and the characterization of the first eigenvalue of some linear elliptic operators we obtain existence and nonexistence results as well as results on the number of stationary solutions we also characterize the asymptotic behavior of the evolution equation as a function of the forcing term amplitude in particular we define two critical values delta and delta 2 such that if delta is smaller than delta the population density converges to a significant state which is everywhere above a certain small threshold whereas if delta is larger than delta 2 the population density converges to a remnant state everywhere below this small threshold our results are shown to be useful for studying

a nonlinear age and maturity structured model of population dynamics i basic theory a nonlinear model of age and maturity structured population dynamics is analyzed the population is structured by age and maturity of the individuals and the nonlinearity in the equations corresponds to density dependent limitation of population growth the existence and asymptotic behavior of solutions are studied

pointwise symmetrization inequalities for sobolev functions and applications we develop a technique to obtain new symmetrization inequalities that provide a unified framework to study sobolev inequalities concentration inequalities and sharp integrability of solutions of elliptic equations

compactness via symmetrization consider two types of translation invariant functionals i and j on rm and a sequence of functions fn whose corresponding symmetric rearrangements f n are convergent we show that fn themselves converge up to translations if either limn i fn limn i fn or limn j fn limn j fn these compactness results lead to applications in variational problems and stability problems in stellar dynamics

chaoticity and invariant measures for a cell population model abstract we present a structured model of a cell reproduction system given by a partial differential equations with a nonlocal division term this equation generates semiflows acting on some subspaces of locally integrable functions we show that these semiflows possess invariant mixing measures positive on open sets from this it follows that the system is chaotic i e it has dense trajectories and each trajectory is unstable we also show the chaoticity of this system in the sense of devaney

verifiable implementation of geometric algorithms using finite precision arithmetic abstract two methods are proposed for correct and verifiable geometric reasoning using finite precision arithmetic the first method data normalization transforms the geometric structure into a configuration for which all finite precision calculations yield correct answers the second method called the hidden variable method constructs configurations that belong to objects in an infinite precision domain without actually representing these infinite precision objects data normalization is applied to the problem of modeling polygonal regions in the plane and the hidden variable method is used to calculate arrangements of lines

on a comparison theorem via symmetrisation talenti s theorem allows us to estimate the norms of the solution of an elliptic boundary value problem posed on a bounded open set in terms of those of the symmetrised problem posed on the ball centred at 0 and such that we are interested in the limiting case when equality is achieved between the norms of the two solutions we show then that and that the solution is radially symmetric and decreasing the novelty of this exposition is that the method used is very elementary compared to the usual arguments

functions of bounded variation and rearrangements the action of the symmetric rearrangement of functions of bounded variation is investigated in particular pointwise regularizing effects are established polya szeg type inequalities with a characterization of equality cases are also proved

confinement of brownian motion among poissonian obstacles in ℝ d d 3 consider brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of d d 1 at the points of a poisson cloud of constant intensity v 0 assume that brownian motion is absorbed instantaneously upon entering the obstacle set in szn conf sznitman has shown that in d 2 conditionally on the event that the process does not enter the obstacle set up to time t the probability that brownian motion remains within distance t1 4 from its starting point is going to 1 as t goes to infinity we show that the same result holds true for d 3 with t1 4 replaced by t1 d 2 the proof is based on sznitmans refined method of enlargement of obstacles 10 as well as on a quantitative isoperimetric inequality due to hall 4

a quantitative polya szego principle andrea cianchidipartimento di matematica e applicazioni per l architetturapiazza ghiberti 27 50122 firenze italye mail cianchi uni itluca espositodipartimento di ingegneria dell informazione e matematica applicatavia ponte don melillo 84084 fisciano sa italye mail esposito diima unisa itnicola fuscodipartimento di matematica e applicazionivia cintia 80126 napoli italye mail n fusco unina itcristina trombettidipartimento di matematica e applicazionivia cintia 80126 napoli italye mail cristina unina itjune 27 2006

cylindrical algebraic decomposition ii an adjacency algorithm for the plane in part i of the present paper we defined cylindrical algebraic decompositions cad s and described an algorithm for cad construction in part ii we give an algorithm that provides information about the topological structure of a cad of the plane informally two disjoint cells in e r r 1 are adjacent if they touch each other formally they are adjacent if their union is connected in a picture of a cad eg fig 2 of part i it is obvious to the eye which pairs of cells are adjacent however the cad algorithm of part i does not actually produce this information nor did the original version of that algorithm in collins 1975

on stability of some general random dynamical system we consider a new random dynamical system which generalizes markov processes corresponding to iterated function systems and poisson driven stochastic differential equations it can be used as a description of many physical and biological phenomena under the suitable assumption will be proved its stability

problem of stability and chaotic properties of multidimensional lasota equation in the context of matuszewska orlicz indices abstract we study asymptotic properties of the dynamical system generated by the multidimensional lasota equation we give the conditions of its stability and chaos in the sense of devaney in orlicz spaces we apply matuszewska orlicz indices to a description of asymptotic behavior of the considered semigroup we analyze also the conditions under which asymptotic behavior for more general form of the equation is provided an example illustrates the criteria when the semigroups generated by the equations have not asymptotic behavior

a selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

on the hausdorff dimension of invariant measures of weakly contracting on average measurable ifs we consider measures which are invariant under a measurable iterated function system with positive place dependent probabilities in a separable metric space we provide an upper bound of the hausdorff dimension of such a measure if it is ergodic we also prove that it is ergodic iff the related skew product is

fluctuations of principal eigenvalues and random scales we investigate the principal dirichlet eigenvalue of the laplacian with soft poissonian obstacles in large boxes of d 2 with the help of our recent version of the method of enlargement of obstacles 18 we derive quantitative confidence intervals for these eigenvalues we also provide less quantitative estimates which however point out the correct size of fluctuations and indicate a stiffness in their behavior in the two dimensional case we derive geometric controls which relate these eigenvalues to certain empty circular droplets our results also have natural applications to the study of the location of minima of certain intermittent random variational problems motivated by 13 17

a cell population model described by impulsive pdes existence and numerical approximation this paper considers a cell population model described by impulsive pdes existence and numerical approximation of solutions are investigated by showing that the related operator a x ddx on c0 generates a strongly conitunous semigroups under general assumption on a x we prove the existence of the solution of the nonlinear problem with impulses and given approximation of the solution of the homogenuous equation

strong ergodic properties of a first order partial differential equation on etudie l equation differentielle d ordre 1 u t c x u x f x u pour t x 0 0 1 avec la condition initiale u 0 x v x pour x 0 1 on demontre l existence d une mesure invariante exacte ayant certaines proprietes supplementaires

stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ for convex bodies d in rn the deviation d from spherical shape is estimated from above in terms of the dimensionless isoperimetric deficiency a of d as follows d f a for a sufficiently small here f is an explicit elementary function vanishing continuously at 0 the estimate is sharp as regards the order of magnitude of f the dimensions n 2 and 3 present anomalies as to the form of f in the planar case n 2 the result is contained in an inequality due to t bonnesen a qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies d in rn in the sense that as d varies d p 0 for a a 0 the proof of the estimate d f a is based on a related estimate in the case of domains not necessarily convex that are supposed a priori to be nearly spherical in a certain sense

the central limit theorem for random dynamical systems we consider random dynamical systems with randomly chosen jumps the choice of deterministic dynamical system and jumps depends on a position the central limit theorem for random dynamical systems is established

periodic solutions of hyperbolic partial differential equations abstract constructive methods by obtaining multiple solutions of first and second order nonlinear hyperbolic periodic boundary value problems are given appropriate partial differential inequalities are developed and are then employed to derive the iterative schemes which yield the desired solutions the advantage of these methods is that the solutions of the given nonlinear problems are constructed as uniform limits of monotone iterates these iterates being solutions of linear problems are explicitly computable

propagation of population pulses and fronts in a cell replication problem non locality and dependence on the initial function abstract here we study the solution types in a simplified equation arising from a cell population dynamics model formulated as a non linear first order partial differential equation for the cell density u t x in the presence of non local maturational x interactions this can be considered as a reaction convection equation in which the cell density is convected with maturation velocity r the non local action affects only the reaction birth rate an initial function representing a pulse of small amplitude on an almost extinguished population may be dissipated may produce multiple pulses or may remain intact as it travels driven by reaction and convection depending on the magnitude of the non local action alternately for localized initial perturbations the equation also has positive traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution for weak convection r 1 which is the only case treated here reaction driven fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state a classification arises according to whether the magnitude of the maturation delay is larger or smaller than a critical value and scaling arguments are used to determine the critical delay for delays larger than critical the reaction driven fronts are oscillatory and correspond to the case for which the correlation length for non local action is of the same order as the characteristic length of maturation population gradients that yield oscillatory birth rates simulation results are validated and interpreted using scaling arguments and solutions for the local equations