coalescence of particles by differential sedimentation we consider a three dimensional system consisting of a large number of small spherical particles distributed in a range of sizes and heights with uniform distribution in the horizontal direction particles move vertically at a size dependent terminal velocity they are either allowed to merge whenever they cross or there is a size ratio criterion enforced to account for collision efficiency such a system may be described in mean field approximation by the smoluchowski kinetic equation with a differential sedimentation kernel we obtain self similar steady state and time dependent solutions to the kinetic equation using methods borrowed from weak turbulence theory analytical results are compared with direct numerical simulations dns of moving and merging particles and a good agreement is found

gradient based iterative algorithm for solving coupled matrix equations abstract this paper is concerned with iterative methods for solving a class of coupled matrix equations including the well known coupled markovian jump lyapunov matrix equations as special cases the proposed method is developed from an optimization point of view and contains the well known jacobi iteration gauss seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle as special cases we have provided analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm simultaneously the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form the proposed approach requires less computation and is numerically reliable as only matrix manipulation is required some other existing results require either matrix inversion or special matrix products numerical examples show the effectiveness of the proposed algorithm

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

uniqueness of positive solutions of a quasilinear dirichlet problem with exponential nonlinearity uniqueness of positive solutions of the following equation has been obtained where b n is the ball of unit radius and 0 moreover if n 2 then the solutions are nondegenerate

cluster cluster aggregation as an analogue of a turbulent cascade kolmogorov phenomenology scaling laws and the breakdown of self similarity abstract we present a detailed study of the statistical properties of a system of diffusing aggregating particles in the presence of a steady source of monomers we emphasize the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean field description provided by standard smoluchowski kinetic theory the presence of a source of monomers allows the system to reach a statistically stationary state at large times this state is characterized by a constant flux of mass directed from small to large masses it therefore admits a phenomenological description based on the assumption of self similarity and constant mass flux analogous to the kolmogorov s 1941 theory of turbulence unlike turbulence the aggregation problem is analytically tractable using powerful methods of statistical field theory we explain in detail how these methods should be adapted to study the far from equilibrium flux dominated states characteristic of turbulent systems we consider multipoint correlation functions of the mass density by an exact evaluation of the scaling exponents for the one and two point correlation functions we show that the assumption of self similiarity breaks down at large masses for spatial dimensions d 2 we calculate non rigorously the exponents of the higher order correlation functions as an expansion where 2 d we show that the mass distribution exhibits non trivial multiscaling an analogy can be drawn with the case of hydrodynamic turbulence the physical origin of this multiscaling is traced to the presence of strong correlations between particles participating in large mass aggregation events these correlations stem from the recurrence of diffusion processes in d 2 the analytic methods developed here will have more general applicability beyond the study of this specific problem

gradient based iterative algorithms for generalized coupled sylvester conjugate matrix equations abstract by applying the hierarchical identification principle the gradient based iterative algorithm is suggested to solve a class of complex matrix equations with the real representation of a complex matrix as a tool the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices also we solve the problem which is proposed by wu et al 2010 finally some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

weighted sobolev spaces with applications to singular nonlinear boundary value problems abstract variational methods are used in a weighted sobolev space to prove the existence of solutions for a certain class of singular nonlinear ordinary differential equations

uniqueness theorems for negative radial solutions of k hessian equations in a ball abstract in this paper we apply a pohozaev type identity and the monotone separation techniques to prove the uniqueness of negative radial solutions of k hessian equations in a finite ball

statistics of small clusters in system with rare aggregation centers abstract an integral equation describing the statistics of small clusters in a system with rare aggregation centers is considered in this paper this equation is obtained from the probabilistic consideration of transition to cluster cluster aggregation in the small cluster system several different dependences for the characteristic probability are used a new form of dependence for the cluster size distribution is obtained the fluctuation character of used model is discussed in the case of poisson s statistics of growth centers the levy cluster size distribution is obtained

uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations in this article we are concerned with the semilinear elliptic equation where n 2 p 1 and k lxl 0 in r n the correspondence between the initial values of regularly positive radial solutions of the above equation and the associated finite total curvatures will be derived in addition we also conduct the zeros of radial solutions in terms of the initial data under specific conditions on k and p furthermore based on the pohozaev identity and openness for the regions of initial data corresponding to certain types of solutions we obtain the whole structure of radial solutions depending on various situations

infinitely many non radial solutions to a critical equation on annulus abstract in this paper we build infinitely many non radial sign changing solutions to the critical problem p u u 4 n 2 u in u 0 on on the annulus x r n a x b n 3 in particular for any integer k large enough we build a non radial solution which look like the unique positive solution u 0 to p crowned by k negative bubbles arranged on a regular polygon with radius r 0 such that r 0 n 2 2 u 0 r 0 max a r b r n 2 2 u 0 r

positive solutions to singular and non singular second order boundary value problems abstract the existence of positive solutions to second order singular boundary value problems is established our analysis is based on the topological transversality theorem of andrzej granas which relies on the notions of an essential map and a priori estimates on solutions

c i positive solutions of nonlinear singular differential equations for nonmonotonic function terms abstract suppose f t u v is nonnegative continuous on 0 1 r r and nondecreasing on u and nonincreasing on v for each fixed t there exist real numbers a 0 1 b 0 such that for any 0 r 1 f t r u v r a f t u v f t u r v r b f t u v u v r r then a necessary and sufficient condition for the boundary value problem u f t u u 0 0 t 1 u 0 u 0 0 u 1 u 1 0 to have c 0 1 positive solutions is that 0 0 1 g s s f s 1 1 d s where r 0 are nonnegative real numbers 0 and g t s is green s function of the above problem when f x u u 0

multirate sampled data systems all h sub spl infin suboptimal controllers and the minimum entropy controller for a general multirate sampled data sd system the authors characterize explicitly the set of all causal stabilizing controllers that achieve a certain h sub spl infin norm bound moreover they give explicitly a particular controller that further minimizes an entropy function for the sd system the characterization lays the groundwork for synthesizing multirate control systems with multiple mixed control specifications

exact multiplicity and numerical computation of solutions for two classes of non autonomous problems with concave convex nonlinearities abstract we establish the exact multiplicity of positive solutions and the global solution structure for two classes of problems on circular domains the first class involves non autonomous concave convex problems on a ball in r n and the second one deals with concave convex problems on a thin annulus in r n we illustrate our results by numerical computations using a novel algorithm which involves continuation in a global parameter

stationary kolmogorov solutions of the smoluchowski aggregation equation with a source term in this paper we show how the method of zakharov transformations may be used to analyze the stationary solutions of the smoluchowski aggregation equation with a source term for arbitrary homogeneous coagulation kernel the resulting power law mass distributions are of kolmogorov type in the sense that they carry a constant flux of mass from small masses to large they are valid for masses much larger than the characteristic mass of the source we derive a locality criterion expressed in terms of the asymptotic properties of the kernel that must be satisfied in order for the kolmogorov spectrum to be an admissible solution whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the kolmogorov spectrum as an example we compute the exact stationary state for the family of kernels k zeta m 1 m 2 m 1 m 2 zeta 2 which includes both gelling and nongelling cases reproducing the known solution in the case zeta 0 surprisingly the kolmogorov constant is the same for all kernels in this family

a uniqueness result for a semilinear elliptic problem a computer assisted proof abstract starting with the famous article a gidas w m ni l nirenberg symmetry and related properties via the maximum principle comm math phys 68 1979 209 243 many papers have been devoted to the uniqueness question for positive solutions of u u u p in u 0 on where p 1 and ranges between 0 and the first dirichlet eigenvalue 1 of for the case when is a ball uniqueness could be proved mainly by ode techniques but very little is known when is not a ball and then only for 0 in this article we prove uniqueness for all 0 1 in the case 0 1 2 and p 2 this constitutes the first positive answer to the uniqueness question in a domain different from a ball our proof makes heavy use of computer assistance we compute a branch of approximate solutions and prove existence of a true solution branch close to it using fixed point techniques by eigenvalue enclosure methods and an additional analytical argument for close to 1 we deduce the non degeneracy of all solutions along this branch whence uniqueness follows from the known bifurcation structure of the problem

existence results of steady states of semilinear reaction diffusion equations and their applications partially supported by the applied mathematical sciences subprogram of the offlice of energy research u s department of energy under contract w 31 109 eng 38 while a visit ing scientist at argonne national laboratory during the summer of 1987 and m part by the board of regents of the state of louisiana under grant leqsf 86 89 rd a 11 partially supported by the applied mathematical sciences subprogram of the office of energy research u s department of energy under contract w 31 109 eng 38 this author thanks the department of mathematics university of southwestern louisiana for an invitation to spend a semester there in the fall of 1986 during which the research reported in this paper was started

numerical solutions of the isotropic 3 wave kinetic equation we show that the isotropic 3 wave kinetic equation is equivalent to the mean field rate equations for an aggregation fragmentation problem with an unusual fragmentation mechanism this analogy is used to write the theory of 3 wave turbulence almost entirely in terms of a single scaling parameter a new numerical method for solving the kinetic equation over a large range of frequencies is developed by extending lee s method for solving aggregation equations the new algorithm is validated against some analytical calculations of the kolmogorov zakharov kz constant for some families of model interaction coefficients the algorithm is then applied to study some wave turbulence problems in which the finiteness of the dissipation scale is an essential feature firstly it is shown that for finite capacity cascades the dissipation of energy becomes independent of the cut off frequency as this cut off is taken to infinity this is an explicit indication of the presence of a dissipative anomaly secondly a preliminary numerical study is presented of the so called bottleneck effect in a wave turbulence context it is found that the structure of the bottleneck depends non trivially on the interaction coefficient finally some results are presented on the complementary phenomenon of thermalisation in closed wave systems which demonstrates explicitly for the first time the existence of so called mixed solutions of the kinetic equation which exhibit aspects of both kz and equilibrium equipartition spectra

solvability of some nonlinear boundary value problems on etablit l existence des solutions positives de problemes a deux points limites singuliers pour des equations differentielles de la forme y f t y y o o t 1

emergence of fractals in aggregation with stochastic self replication we propose and investigate a simple model which describes the kinetics of aggregation of brownian particles with stochastic self replication an exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings in particular we show analytically that the particle size distribution function exhibits dynamic scaling and we verify it numerically using the idea of data collapse furthermore the conditions under which the resulting system emerges as a fractal are found the fractal dimension of the system is given and the relationship between this fractal dimension and a conserved quantity is pointed out

instantaneous gelation in smoluchowski s coagulation equation revisited we study the solutions of the smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size m we focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization numerical simulations demonstrate that for such kernels with monodisperse initial data the regularized gelation time decreasesas m increases consistent with the expectation that the gelation time is zero in the unregularized system this decrease appears to be a logarithmically slow function of m indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cutoff we also study the case when a source of monomers is introduced in the regularized system in this case a stationary state is reached we present a complete analytic description of this regularized stationary state for the model kernel k m 1 m 2 max m 1 m 2 ensuremath nu which gels instantaneously when m ensuremath rightarrow ensuremath infty if ensuremath nu g1 the stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent ensuremath nu for large cluster sizes the total particle density in the stationary state slowly vanishes as ensuremath nu ensuremath 1 mathrm log m ensuremath 1 2 when m ensuremath rightarrow ensuremath infty the approach to the stationary state is nontrivial oscillations about the stationary state emerge from the interplay between the monomer injection and the cutoff m which decay very slowly when m is large a quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

strictly contractive and positive completions for block matrices abstract this paper deals with the following two problems of completion of block matrices 1 let a ij i j q be given matrices find additional matrices a ij such that the block matrix a a ij n i 1 m j 1 has norm less than one 2 let b ij j i q be given matrices find additional b ij such that the completion b b ij n i j 1 is positive definite the analysis is based on a study of certain linear fractional maps this approach leads to an explicit description of all solutions by linear fractional maps of which the coefficients are given directly in terms of the given data the maximum entropy principle appears as a corollary of a general formula for the determinant of a completion special attention is paid to the toeplitz case

unitary dilation approach to contractive matrix completion in this paper a new method to construct all contractive matrix completion to a block triangular matrix is given this new method uses only elementary matrix operations and has the advantages of better numerical reliability and ease in implementation on computers

a new class of shift varying operators their shift invariant equivalents and multirate digital systems a class of linear shift varying operators that generalize the notion of n periodicity is defined it is shown that shift invariant equivalents for these operators exist and that the equivalence is in a strong sense preserving both algebraic and analytic system properties it is shown that multirate sampled data systems although not generally periodic fall into this class kranc vector switch decomposition and block filter implementations for single input single output multirate systems are connected under the unifying framework of shift invariant equivalents and this framework provides a way to extend them both to multi input multi output systems

nontrivial solutions for the equations of monge ampere type on etudie les solutions strictement convexes des equations de type monge ampere detu xd i xd j u p dans la boule b x 0 p n est un parametre reel on demontre l existence et l unicite

some superlinear problems with nonlocal diffusion coefficient abstract we study a superlinear elliptic problem with a non local diffusion coefficient we show that there exists a drastic change on the structure of the set of positive solutions when the non local coefficient grows fast enough to infinity we combine mainly sub super and bifurcation methods to obtain our results

proof of a conjecture for the perturbed gelfand equation from combustion theory abstract in this paper we prove that when the domain b is a two dimensional ball for all small positive e the solution curve u of the perturbed gelfand equation u eu 1 eu with u b 0 is exactly s shaped this settles a long standing conjecture on the exact number of solutions to this equation

multirate periodic systems and constrained analytic function interpolation problems multirate periodic systems and some related constrained analytic function interpolation problems are studied in this paper after showing how to convert a general multirate periodic system to an equivalent linear time invariant lti system with a structural constraint we formulate some analytic function interpolation problems with such a constraint that can find various applications in the study of multirate and periodic systems both the solvability conditions and characterization of all solutions are presented to these constrained interpolation problems

kang redner small mass anomaly in cluster cluster aggregation the large time small mass asymptotic behavior of the average mass distribution p m t is studied in a d dimensional system of diffusing aggregating particles for 1 or d or 2 by means of both a renormalization group computation as well as a direct resummation of leading terms in the small reaction rate expansion of the average mass distribution it is shown that p m t approximately 1 t d m 1 d sqrt t e kr for m t d 2 where e kr epsilon o epsilon 2 and epsilon 2 d in two dimensions it is shown that p m t approximately ln m ln t t 2 for m t ln t numerical simulations in two dimensions supporting the analytical results are also presented

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar