on the box counting dimension of graphs of harmonic functions on the sierpinski gasket abstract the objective of this paper is to study the box counting dimension of graphs of fractal interpolation functions and harmonic functions on the sierpinski gasket firstly we give construction of a fractal interpolation function on the sierpinski gasket and then with the help of fractal interpolation functions we show the existence of fractal functions in the space dom e consisting of all finite energy functionals on the sierpinski gasket later we provide bounds for the box counting dimension of graphs of some functions belonging to the family of continuous functions which arise as fractal interpolation functions moreover we also obtain bounds for the box counting dimension of graphs of harmonic functions and piecewise harmonic functions also we obtain upper and lower bounds for the box counting dimension of graphs of functions that belong to dom e

boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass we consider a hybrid system consisting of two flexible beams connected by a point mass the constant of rotational inertia is assumed to be nonzero in a previous paper we have proved that in the presence of the point mass the system is well posed in asymmetric spaces in which solutions have one more degree of regularity to one side of the mass we are interested in the problem of controllability when the control acts on the free extreme of one of the beams we prove that when the control time is large enough the system is exactly controllable in an asymmetric space this result is sharp the proofs combine classical techniques from asymptotic analysis and the theory of nonharmonic fourier series

some properties of laplacians on fractals abstract kigami has defined an analog of the laplacian on a class of self similar fractals including the familiar sierpinski gasket we study properties of this operator we show that there is a maximal principle for solutions of certain nonlinear equations of the form u x f x u x we discuss the extension of the laplacian to non compact fractal blow ups and show that it is essentially self adjoint and we prove an analog of liouville s theorem in some cases we also give an explicit algorithm for solving the dirichlet problem for certain domains in the sierpinski gasket and give a characterization of all harmonic functions on those domains

on the hamiltonian structure of ermakov systems a canonical hamiltonian formalism is derived for a class of ermakov systems specified by several different frequency functions this class of systems comprises all known cases of hamiltonian ermakov systems and can always be reduced to quadratures the hamiltonian structure is explored to find exact solutions for the calogero system and for a non central potential with dynamic symmetry some generalizations of these systems possessing exact solutions are also identified and solved

local times for brownian motion on the sierpinski carpet jointly continuous local times are constructed for brownian motion on the sierpinski carpet a consequence is that the brownian motion hits points the method used is to analyze a sequence of eigenvalue problems

on the resistance of the sierpinski carpet let f text n be the nth stage in the construction of the sierpinski carpet let r text n be the electrical resistance of f text n when the left and right sides are each short circuited and a voltage is applied between them we prove that there exists a constant rho such that textstyle frac 1 4 rho n leq r n leq 4 rho n the motivation for this result came from the problem of establishing a the existence and b the value of the spectral dimension of the sierpinski carpet in this and a subsequent paper we settle a and give bounds for b

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

on multi component ermakov systems in a two layer fluid a variational approach by introducing madelung type transformations a two layer fluid model with a circular paraboloidal bottom topography is reduced to coupled nonlinear schrodinger equations incorporating harmonic traps and de broglie bohm quantum potentials a kind of multi component ermakov system is obtained via a variational approach and a multi parameter gaussian ansatz in particular three typical reductions to generalized ermakov systems are discussed notable integrals of motion and additional hamiltonian structures are employed to derive analytical solutions in terms of elliptic integrals for the multi component ermakov systems

null boundary controllability of a one dimensional heat equation with an internal point mass and variable coefficients in this paper we consider a linear hybrid system which is composed by two non homogeneous rods connected by a point mass with dirichlet boundary conditions on the left end and a boundary control acts on the right end we prove that this system is null controllable with dirichlet or neumann boundary controls our approach is mainly based on a detailed spectral analysis together with the moment method in particular we show that the associated spectral gap in both cases dirichlet or neumann boundary controls is positive without further conditions on the coefficients other than the regularities

on the existence of three solutions for the dirichlet problem on the sierpinski gasket we apply a recently obtained three critical point theorem of b ricceri to prove the existence of at least three solutions of certain two parameter dirichlet problems defined on the sierpinski gasket we also show the existence of at least three nonzero solutions of certain perturbed two parameter dirichlet problems on the sierpinski gasket using both the mountain pass theorem of ambrosetti and rabinowitz and that of pucci and serrin

quantum mechanics as a multidimensional ermakov theory i time independent systems presents a multidimensional ermakov theory applicable to both separable and nonseparable time independent quantum mechanical systems the separability of the system being determined by the separability of the wavefunction in configuration space a consequence of the theory is the existence of a new exact invariant for multidimensional quantum systems in one dimension the author shows that this new invariant reduces to the well known ermakov lewis invariant two applications of the theory are given in the first case a new exact invariant is obtained for planar optical and particle channelling systems in the second case he obtains the ermakov invariants and electron ray path equations for the hydrogen atom

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

control of a network of euler bernoulli beams abstract the aim is to study the boundary controllability of a system modeling the vibrations of a network of n euler bernoulli beams connected by n vibrating point masses using the classical hilbert uniqueness method the control problem is reduced to the obtention of an observability inequality the solution is then expressed in terms of fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value this property called spectral gap holds as soon as the roots of a function denoted by f and giving the asymptotic behaviour of the eigenvalues are all simple for a network of n 2 different beams this assumption on the multiplicity of the roots of f denoted by a is proved to be satisfied and controllability follows for higher values of n a numerical approach allows one to prove a in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open

on the simplicity of eigenvalues of two nonhomogeneous euler bernoulli beams connected by a point mass abstract in this paper we consider a linear system modeling the vibrations of two nonhomogeneous euler bernoulli beams connected by a point mass this system is generated by the following equations x y t t t x x y x x t x x x q x y x t x x 0 t 0 x 1 0 0 1 m y t t t 0 t y t x x 0 t y t x x 0 t 0 with hinged boundary conditions at both ends where m 0 x 0 x 0 q x 0 and t y x y x x x q x y x for x 1 0 0 1 we prove that all the associated eigenvalues n n 1 are algebraically simple furthermore the corresponding eigenfunctions n n 1 satisfy n 1 t n 1 0 and n 1 t n 1 0 for all n 1 these results are essential for the proof of the exact boundary controllability problem related to this system by means of the non harmonic fourier series method

lie symmetries nonlinear equations of motion and new ermakov systems it is shown that a lagrangian of the form l 1 1 rho 2 omega 2 t rho 2 g t f k t rho said to be in factored form yields an equation of motion that is equivalent to the most general equation derivable via noether s theorem from the unfactored lagrangian l 1 2 rho 2 p rho t in view of this equivalence the theory of extended lie groups is applied to the factored nonlinear equation of motion p omega 2 t rho g t f k t rho to obtain its lie symmetries the latter are obtained when g t and k t initially arbitrary are determined in terms of a function x t which satisfies the auxiliary equation x omega 2 t x k x3 it is then possible with the auxiliary equation and the equation of motion to form a coupled pair of nonlinear equations an ermakov system whose first integral is not invariant under the action of the symmetry group in contrast to previous ermakov systems

ermakov systems of arbitrary order and dimension structure and linearization ermakov systems of arbitrary order and dimension are constructed these inherit an underlying linear structure based on that recently established for the classical ermakov system as an application alignment of a 2 1 dimensional ermakov and integrable ernst system is shown to produce a novel integrable hybrid of a 2 1 dimensional sinh gordon system and of a conventional ermakov system

riesz basis approach to the stabilization of a flexible beam with a tip mass using an abstract condition of riesz basis generation of discrete operators in the hilbert spaces we show in this paper that a sequence of generalized eigenfunctions of an euler bernoulli beam equation with a tip mass under boundary linear feedback control forms a riesz basis for the state hilbert space in the meanwhile an asymptotic expression of eigenvalues and the exponential stability are readily obtained the main results of siam j control optim 36 1998 pp 1962 1986 are concluded as a special case and the additional conditions imposed there are removed

semilinear equations on fractal blowups abstract the paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal where imbedding of the energy space into l p is not compact in absence of invariant transformations involved in conventional concentration compactness argument the paper develops convergence reasoning based on the fractal s self similarity

nonlinear superposition higher order nonlinear equations and classical linear invariants we find a class of nonlinear ordinary differential equations in of any order m 2 whose solutions are given by the nonlinear superposition law t x t r d t dt where dmr d m f r r and x t satisfies a special self adjoint linear equation of order m the coefficients of the self adjoint equation in x which are identical to those of the nonlinear equations in can be deduced from the well known invariants of the classical theory of linear differential equations

ermakov ray reid systems with additive noise abstract using the methods developed by us in cervantes lopez et al 2014 for multiplicative noises we present results on the effects of the additive noise on the ermakov lewis invariant this case can be implemented in the euler maruyama numerical method if the additive noise is considered as the forcing term of the parametric oscillator and presented as a particular case of the ermakov ray reid systems the results are obtained for the same particular examples as for the multiplicative noise and show a tendency to less robustness of the ermakov lewis invariant to the additive noise as compared to the multiplicative noise

oscillation properties for the equation of vibrating beam with irregular boundary conditions abstract in this paper we study the oscillatory properties for the eigenfunctions of some fourth order eigenvalue problems where the boundary conditions are irregular in the sense of the classification of s janczewski oscillation theorems for the differential boundary value problems of the fourth order ann of math 29 1928 521 542 in this case we show that these oscillatory properties are different from those of the sturm liouville problem

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

can one hear the dimension of a fractal we consider the spectrum of the laplacian in a bounded open domain of n with a rough boundary i e with possibly non integer dimension and we discuss a conjecture by m v berry generalizing weyl s conjecture then using ideas mark kac developed in his famous study of the drum we give upper and lower bounds for the second term of the expansion of the partition function the main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on minkowski dimensions and on minkowski measures rather than on haussdorff ones

ermakov systems velocity dependent potentials and nonlinear superposition we derive new ermakov systems with velocity dependent potentials the extended ermakov system presented contains all known one dimensional cases and many new systems these ermakov systems lead to a nonlinear superposition law for the solutions

resistance and spectral dimension of sierpinski carpets the authors announce results which prove the existence of the spectral dimension ds for sierpinski carpets in two dimensions their method employs the einstein relation zeta d omega df using this numerical calculations of the resistance of approximations to the sierpinski carpet yield an accurate estimate for ds

harmonization and homogenization on fractals this paper suggests a direct approach to define the laplacian the spectral dimension of nested fractals and the pre sierpinski carpet conductivity we find a geometric construction of the harmonic functions on the gasket and therefore can describe effectively the dense set of functions having finite energy the paper is mostly aimed at the homogenization on the pre sierpinski gasket whose horizontal and nonhorizontal bonds have different conductivities a andb respectively we prove the convergence of the rescaled energies on the pre sierpinski gasket to a b e where e is the standard energy on the gasket with uniform conductivities we also find an explicit expression for the effective conductivity a b and deduce that its set of singularities turns out to be the julia set of a certain rational function a special section is devoted to the problem of the pre sierpinski carpet conductivity asymptotic behavior for this problem a new proof of barlow bass inequalities with sharper constants is given

semilinear pdes on self similar fractals a laplacian may be defined on self similar fractal domains in terms of a suitable self similar dirichlet form enabling discussion of elliptic pdes on such domains in this context it is shown that that semilinear equations such as u u p 0 with zero dirichlet boundary conditions have non trivial non negative solutions if 0 1 or if 2 and 1 p 2 2 where is the intrinsic dimension or spectral dimension of the system thus the intrinsic dimension takes the r o le of the euclidean dimension in the classical case in determining critical exponents of semilinear problems

harmonic calculus on fractals a measure geometric approach ii differentiation of functions w r t finite atomless measures with compact support on the real line is introduced the related harmonic calculus is similar to that of the classical lebesgue case as an application we obtain the weyl exponent for the spectral asymptotics of the laplacians w r t linear cantor type measures with arbitrary weights

non linear elliptical equations on the sierpinski gasket this paper investigates properties of certain nonlinear pdes on fractal sets with an appropriately defined laplacian we obtain a number of results on the existence of non trivial solutions of the semilinear elliptic equation u a x u f x u with zero dirichlet boundary conditions where u is defined on the sierpinski gasket we use the mountain pass theorem and the saddle point theorem to study such equations for different classes of a and f a strong sobolev type inequality leads to properties that contrast with those for classical domains

nonlinear wave equations on a class of bounded fractal sets the nonlinear wave equation u tt u f u with given initial data and zero boundary conditions on a class of bounded self similar fractal sets is investigated the sobolev type inequality is the starting point of this work which holds for a class of fractals including the well known sierpinski gasket we obtain the global existence of strong solutions for suitable f if the spectral dimension ds of the fractal satisfies ds 2 the key is to construct the wave propagator and hilbert spaces of functions on the fractal the main difficulty in obtaining the global existence of a weak solution is establishing a priori estimates depending on a regularity property for f t he regularity property of a weak solution is obtained through a fine analysis in which the sobolev type inequality plays a crucial role 2002 elsevier science usa all rights reserved

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

harmonic calculus on p c f self similar sets the main object of this paper is the laplace operator on a class of fractals first we establish the concept of the renormalization of difference operators on post critically finite p c f for short self similar sets which are large enough to include finitely ramified self similar sets and extend the results for sierpinski gasket given in 10 to this class under each invariant operator for renormalization the laplace operator green function dirichlet form and neumann derivatives are explicitly constructed as the natural limits of those on finite pre self similar sets which approximate the p c f self similar sets also harmonic functions are shown to be finite dimensional and they are characterized by the solution of an infinite system of finite difference equations

one parameter dirichlet problems on the sierpinski gasket abstract using a three critical point theorem due to arcoya and carmona we show that under suitable conditions and for sufficiently large real parameters the following nonlinear elliptic equation defined on the sierpinski gasket and with zero dirichlet boundary conditions u x b x u x f x u x has at least two nonzero weak solutions

on 2 1 dimensional ermakov systems ermakov type systems in 2 1 dimensions are introduced multiwave solutions of a 2 1 dimensional pinney equation and a modulated 2 1 dimensional sine gordon equation are thereby constructed

rational ermakov systems of fuchsian type studies a subclass of a class of systems of coupled nonlinear oscillators that have well behaved singularities which can be studied using an exact linearization this allows the author to classify such systems and to isolate those with special properties

on integrable ermakov painleve iv systems novel hybrid ermakov painleve iv systems are introduced and an associated ermakov invariant is used in establishing their integrability backlund transformations are then employed to generate classes of exact solutions via the linked canonical painleve iv equation

ermakov ray reid systems in nonlinear optics a hydrodynamics type system incorporating a madelung bohm type quantum potential as derived by wagner et al via maxwell s equations and the paraxial approximation in nonlinear optics is reduced to a nonlinear schrodinger canonical form a two parameter nonlinear ermakov ray reid system that arises from this model and which governs the evolution of beam radii in an elliptically polarised medium is shown to be reducible to a classical poschl teller equation a class of exact solutions to the ermakov type system is constructed in terms of elliptic dn functions it is established that integrable two component ermakov ray reid subsystems likewise arise in a coupled 2 1 dimensional nonlinear optics model descriptive of the two pulse interaction in a kerr medium the hamiltonian structure of these subsystems allows their complete integration

the classical korteweg capillarity system geometry and invariant transformations a class of invariant transformations is presented for the classical korteweg capillarity system the invariance is an extension of a kind originally introduced in an anisentropic gasdynamics context in a particular instance application of the invariant transformation leads to a deformed one parameter class of karman tsien type capillarity laws associated with a deformation of an integrable nonlinear schrodinger type equation which incorporates a de broglie bohm potential the latter and another integrable case associated with the classical boussinesq equation may be linked to the motion of curves in euclidean and projective space so that both the invariant transformation and the galilean invariance of the capillarity system may be interpreted in a geometric and soliton theoretic manner the work is set in the broader context of other connections of invariant transformations in gasdynamics with soliton theory

the relations among a special type of solutions in some d 1 dimensional nonlinear equations among a special type of solutions of some kinds of nonlinear equations in d 1 dimensions such as the cubic nonlinear klein gordon sine gordon double sine gordon ginzburg landau equations etc some mapping relations exist that allow one to find a solution of one system from a solution of another furthermore one can find some new solutions from a known one within the same system some new n kinklike solutions are also presented

nonlinear problems on the sierpinski gasket abstract this paper concerns with a class of elliptic equations on fractal domains depending on a real parameter our approach is based on variational methods more precisely the existence of at least two non trivial weak strong solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive banach spaces a special case of the main result improves a classical application of the mountain pass theorem in the fractal setting given by falconer and hu in 14

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold

properties of natural frequencies and harmonic bending vibrations of a rod at one end of which is concentrated inertial load abstract in this paper we consider a spectral problem that describes the bending vibrations of a homogeneous rod in cross sections of which the longitudinal force acts the left end of which is fixed and on the right end an inertial mass is concentrated we give a general characteristic of the location of the eigenvalues on the real axis we study the structure of root spaces and oscillation properties of eigenfunctions we investigate the basic properties in the space l p 1 p of the system of eigenfunctions of this problem

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

higher order special self adjoint equations and particle dynamics we exhibit a remarkable connection between a hierarchy of higher order special self adjoint ordinary differential equations and the description of motion of a cluster of particles in classical mechanics the cluster is assumed to consist of equal mass particles all moving in one dimension in a perturbation schema based on the first order equation of motion of the center of mass point the time evolution of the moments of order m 1 is governed by the solution of a special self adjoint equation of order m a similar connection exists for the moments of a wave packet in quantum mechanics

classification scheme for two dimensional ermakov type systems and generalizations we describe a classification scheme for certain pairs of coupled nonlinear second order ordinary differential equations having a first integral which can be obtained by an ermakov type elimination procedure the classification produces a hierarchy of systems having different levels of generality different cases within the same level are distinguished by the integrability of a certain one form which is related to a nonlinear superposition law between the two equations the study of these one forms and the classification are made systematic by using the general reduction theorem of pfaffian forms the paper contains illustrations of the way previous examples fit into the classification scheme and a discussion of its generalization to higher dimensions

exact controllability for string with attached masses we consider the problem of boundary control for a vibrating string with n interior point masses we assume the control is at the left end and the string is fixed at the right end singularities in waves are smoothed out to one order as they cross a point mass we characterize the reachable set for a l 2 control the control problem is reduced to a moment problem which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results

regularity and exact controllability for a beam with piezoelectric actuator we consider an initial and boundary value problem modelling the vibrations of a bernoulli euler beam with an attached piezoelectric actuator we show that the sobolev regularity of the solution is by frac 1 2 epsilon higher than that one obtains by simply using the sobolev regularity of the control term the main results concern the dependence of the space of exactly controllable initial data on the location of the actuator our approach is based on the hilbert uniqueness method introduced by lions em contr olabilit e exacte des syst emes distribu es masson paris 1988 combined with some results from the theory of diophantine approximation