joint similarities and parameterizations for naimark complementary frames abstract it is known that the naimark complementary frames for a given frame are not necessarily unique up to the similarity in this paper we introduce the concept of joint complementary frame pairs for a given dual frame pair and prove that they are unique up to the joint similarity as an application we give a necessary and sufficient condition under which two naimark complementary frames are similar for different pairs of dual frames we present an operator parameterization for their joint complementary frame pairs

an introduction to mathematical models of coagulation fragmentation processes a discrete deterministic mean field approach abstract we summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity an integer multiple of some basic unit size in particular we discuss smoluchowski s equation for aggregation the becker doring model of simultaneous aggregation and fragmentation and more general models involving coagulation and fragmentation

orthonormal dilations of parseval wavelets we prove that any parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the baumslag solitar group bs 1 2 langle u t utu 1 t 2 rangle we give a precise description of this representation in some special cases and show that for wavelet sets it is related to symbolic dynamics theorem 3 14 we prove that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics theorem 3 24 we give concrete examples of parseval wavelets for which we compute the orthonormal dilations in detail we construct parseval wavelet sets which have infinitely many non isomorphic orthonormal dilations

on solution of the periodic riccati equations results presented by mori in the article named above ibid vol ac 30 pp 162 164 feb 1985 which give some bounds for the trace and the determinant of the positive definite solution to the algebraic riccati matrix equation are shown to be erroneous and suggestions to remedy these errors are made

periodic solutions of periodic riccati equations for periodically time varying matrix riccati equations controllability and observability in the usual sense are shown to be sufficient for the existence of a unique positive definite periodic solution

a formula for the general solution of riccati type matrix differential equations abstract in this paper a closed formula for the general solution of a class of non symmetric riccati matrix differential equations in given in terms of exponentials of matrices related to the data

boundary problems and periodic riccati equations we consider the resolution problem of a periodic riccati equation from the point of view of a generalized boundary problem that lets us reduce the differential problem to an algebraic equation of riccati type sufficient conditions for the existence and calculus of special types of solutions are given

some analytical and numerical solutions for colloidal aggregation with fragmentation we study the kinetic equation which describes the evolution of cluster size distribution of the aggregation fragmentation problem the so called generalized smoluchowski equation for several dynamics of cluster growth by using the method of the generating function we find analytical solutions for some cases besides numerical solutions are found for the time evolution of clusters where the aggregation and fragmentation kernels do not allow a complete analytical solution

solutions and critical times for the monodisperse coagulation equation when aij a b i j cij the author shows that if aij a b i j cij the solutions of smoluchowski s coagulation equation monodisperse case can be expressed as equilibrium distributions conditioned on a deterministically changing and known parameter a partial converse is also given because the equilibrium distributions are already known this effectively solves the equation for the given aij if c not 0 then there is a finite critical time t tc when the moments cease to converge the author finds tc for all the given aij

exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition the coagulation fragmentation equation describes the concentration f i t of particles of size i in nn 0 at time t geq 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break up we show that when the rate of fragmentation is sufficiently stronger than that of coalescence f i t i in nn 0 tends to an unique equilibrium as t tends to infinity although we suppose that the initial datum is sufficiently small we do not assume a detailed balance or reversibility condition the rate of convergence we obtain is furthermore exponential

some results on the coagulation equation eq 1 represents the evolution of particles c t of size 0 at time t 0 undergoing a change in size governed by reaction kernel k a physical interpretation of the terms in 1 can be found in melzak 5 or the review article by drake 2 global existence and uniqueness of solutions to 1 has been proven by aizenman and bak 1 for constant kernels with suitable conditions imposed on the initial particle distribution c0 for the product kernel k mcleod 4 has shown again for a suitably restricted initial particle distribution that a mass conserving solution exists for some nite time that solutions in this case need not conserve mass for all time was demonstrated by stewart 6 the phenomenon whereby conservation of mass breaks down in nite time is known as gelation and is physically interpreted as being caused by the appearance of an in nite gel or superparticle 3

a formula for the post gelation mass of a coagulation equation with a separable bilinear kernel abstract it is well known that solutions of the coagulation equation do not conserve mass if the coagulation kernel grows too rapidly the phenomenon whereby conservation of mass breaks down in finite time is known as gelation and is physically interpreted as being caused by the appearance of an infinite gel or superparticle in this paper we derive a formula for the post gelation mass for the case of a separable bilinear kernel k and investigate the asymptotic behaviour of the solution

gelation time in the discrete coagulation fragmentation equations with a bilinear coagulation kernel the discrete coagulation fragmentation equations with a bilinear coagulation kernel describing the evolution of clusters of particles undergoing binary reactions are studied we compute the gelation time and post gelation mass for the pure coagulation equation a method of characteristics is developed to solve numerically the partial differential equation satisfied by a moment generating function for a product coagulation kernel and a constant fragmentation rate the accuracy of the numerical method is verified by comparison of the numerical results with an exact solution for the number density of monomers for a given coagulation rate a critical value of the fragmentation rate values greater than which lead to mass conservation is identified

periodic solutions of riccati equations applied to multirate sampling from some basic relations in estimation theory a comprehensive set of formulae are obtained for the periodic solution of riccati equations to obtain numerically stable algorithms square root formulations are introduced both the discrete time and continuous time cases are considered closed loop properties for periodic systems are also investigated the results are applied to multirate measurement sampling the control of a continuous stirred tank reactor serves as an illustrative example the temperature measurements are then complemented by concentration analysis available at a slower rate than the basic sampling and control rate the results give some guidelines about the benefits of multirate sampling

solving algebraic and differential riccati operator equations abstract by application of the riesz dunford functional calculus a method for solving generalized algebraic riccati operator equations in hilbert space is given results are applied to the problem of finding explicit solutions for cauchy problems and two point boundary value problems for generalized riccati operator differential equations

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions