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Spectral analysis of convex processes has led to many results in the analysis of differential inclusions with a convex process. In particular the characterization of eigenvalues with eigenvectors in a given cone has led to results on controllability and stabilizability. However, these characterizations can handle only pointed cones. This paper will generalize all known results characterizing eigenvalues of convex processes with eigenvectors in a given cone. In addition, we reveal the link between the assumptions on our main theorem and classical geometric control theory.

This paper is devoted to an investigation of blow-up phenomena occurring in high-contrast fiber-reinforced composites. When the distance between perfect conductors or between the conductors and the matrix boundary tends to zero, the electric field may appear blow-up. The major objective of this paper is to give a precise description for the singular behavior of such a high concentration in the presence of C 1 , α -inclusions with extreme conductivities. Our results contain the boundary and interior asymptotics of the concentrated field in all dimensions. In particular, the blow-up factor for each dimension is accurately captured.

Discrete-time birth-death processes may or may not have certain properties known as asymptotic aperiodicity and the strong ratio limit property. In all cases known to us a suitably normalized process having one property also possesses the other, suggesting equivalence of the two properties for a normalized process. We show that equivalence may be translated into a property involving Christoffel functions for a type of orthogonal polynomials known as random walk polynomials. The prevalence of this property - and thus the equivalence of asymptotic aperiodicity and the strong ratio limit property for a normalized birth-death process - is proven under mild regularity conditions., 31 pages

We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them., 13 pages. This paper is a substantially reworked version of our paper arXiv:1510.03807, where we now deduce the result of that paper on the global boundedness for Fourier integral operators from a much more general principle for integral operators

We formulate a multi-group and multi-vector epidemic model in which hosts' dynamics is captured by staged-progression $SEIR$ framework and the dynamics of vectors is captured by an $SI$ framework. The proposed model describes the evolution of a class of zoonotic infections where the pathogen is shared by $m$ host species and transmitted by $p$ arthropod vector species. In each host, the infectious period is structured into $n$ stages with a corresponding infectiousness parameter to each vector species. We determine the basic reproduction number $\mathcal{R}_0^2(m,n,p)$ and investigate the dynamics of the systems when this threshold is less or greater than one. We show that the dynamics of the multi-host, multi-stage, and multi-vector system is completely determined by the basic reproduction number and the structure of the host-vector network configuration. Particularly, we prove that the disease-free \mbox{equilibrium} is globally asymptotically stable (GAS) whenever $\mathcal{R}_0^2(m,n,p)1$ and the host-vector configuration is irreducible. That is, either the disease dies out or persists in all hosts and all vector species., 26 pages and one figure

We study asymptotic behavior of global solutions of one-dimensional aerotaxis model proposed in Knosalla and Nadzieja (2015) [9] .

A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary differential equations displays stable and robust oscillations. In this work, we use geometric singular perturbation theory and blow-up method to prove the existence of a strongly attracting limit cycle. This cycle corresponds to a relaxation oscillation of an auxiliary system, whose singular perturbation nature originates from the small Michaelis-Menten constants of the biochemical model. In addition, we give a detailed description of the structure of the limit cycle, and the timescales along it., 38 pages

Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree $n$ and parameters $\alpha$ and $\beta$. From these new results, asymptotic expansions of the zeros are derived and methods are given to obtain the coefficients in the expansions. These approximations can be used as initial values in iterative methods for computing the nodes of Gauss--Jacobi quadrature for large degree and parameters. The performance of the asymptotic approximations for computing the nodes and weights of these Gaussian quadratures is illustrated with numerical examples., Comment: 18 pages, 7 figures

In this paper we consider Lp-regularity estimates for solutions to stochastic evolution equations, which is called stochastic maximal Lp-regularity. Our aim is to find a theory which is analogously to Dore's theory for deterministic evolution equations. He has shown that maximal Lp-regularity is independent of the length of the time interval, implies analyticity and exponential stability of the semigroup, is stable under perturbation and many more properties. We show that the stochastic versions of these results hold.

In this note, by using a discrete analog of a projection formula introduced by A. Seghier in 1978, we calculate the orthogonal polynomials on the unit circle for a rational spectral density having no zeros there, and derive simple matrix representations of themselves, their squared norms, and the Verblunsky coefficients. (C) 2018 Elsevier Inc. All rights reserved.

© 2018 Elsevier Inc. We construct the space of pseudo-quotients that is shown to be isomorphic to the spaces of Beurling tempered ultradistributions.

We use a perturbation approach to prove the occurrence of asymptotically stable periodic solutions in a bilinear oscillator whose one spring has nearly infinite stiffness. This leads to a singularly perturbed problem where the classical theory does not apply.

In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a strict contraction and the case where the functions are matrix polynomials., Comment: 15 pages

The following equation is considered in this paper: x ″ + ( − α + β cos ⁡ ( γ t ) ) x = 0 , where α, β and γ are real parameters and γ > 0 . This equation is referred to as Mathieu's equation when γ = 2 . The parameters determine whether all solutions of this equation are oscillatory or nonoscillatory. Our results provide parametric conditions for oscillation and nonoscillation; there is a feature in which it is very easy to check whether these conditions are satisfied or not. Parametric oscillation and nonoscillation regions are drawn to help understand the obtained results.

Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichm\"uller. Unlike Teichm\"uller's result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations. Finally, we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of R^n \ {0} onto itself., Comment: 25 pages, 2 figures

This paper is concerned with the global dynamics of a PDE viral infection model with cell-to-cell transmission and spatial heterogeneity. The basic reproduction number ℜ 0 , which is a threshold value that predicts whether the infection will go to extinction or not, is defined in a variational characterization. In quite a general setting in which every parameter can be spatially heterogeneous, it is shown that if ℜ 0 ≤ 1 , then the infection-free steady state is globally asymptotically stable, while if ℜ 0 > 1 , then the system is uniformly persistent and the infection steady state is globally asymptotically stable. The proof is based on the construction of the Lyapunov functions and usage of the Green's first identity. Finally, numerical simulation is performed in order to verify the validity of our theoretical results.

We consider the differential geometry of evolutes of singular curves in hyperbolic 2-space and de Sitter 2-space. Firstly, as an application of the basic Legendrian duality theorems, we give the definitions of frontals in hyperbolic 2-space or de Sitter 2-space, respectively. We also give the notions of moving frames along the frontals. By using the moving frames, we define the evolutes of spacelike fronts and timelike fronts, and investigate the geometric properties of these evolutes. As a result, these evolutes can be viewed as wavefronts from the viewpoint of Legendrian singularity theory. At last, we study the relationships among these evolutes.

nkato@se.kanazawa-u.ac.jp, We study abstract linear partial differential equations in Banach spaces and/or Banach lattices related to size-structured population models with spatial diffusion and their dual problems. We introduce mild solutions through semigroup theory and characteristic method and investigate differentiability of mild solutions. Existence of a unique mild solution is shown. Also, a comparison result is obtained as well as the boundedness of mild solutions is investigated in the Banach lattice setting. Furthermore, we consider the dual problems, and then we introduce weak solutions and establish their uniqueness. © 2015 Elsevier Inc., Embargo Period 24 months

The paper considers a scalar conservation law with a discontinuous flux F of the form F ( x , u ) = H ( x ) g ( u ) + ( 1 − H ( x ) ) h ( u ) where H ( x ) is the Heaviside function. Herein, the fluxes g and h are supposed to have one minimum and no maximum and at most one crossing in the interior of the domain of definition. The aim is to verify a weak solution of such a problem in the following way: We are looking for discrete shock profiles for its continuously differentiable perturbation with a parameter e and Godunov's scheme for conservation laws with spatially varying flux functions. The obtained discrete shock profile satisfies a discrete entropy condition of Kruzkhov type and after letting e → 0 , approaches an entropy weak solution of the original equation.

In this paper, we study the global stability of a class of HIV viral infection models with continuous age-structure using the direct Lyapunov method. In each of the cases where the incidence rates are given by nonlinear infection rate F ( T ) G ( V ) , Holling type II functional response and Crowley–Martin functional response, we define the basic reproduction number and prove that it is a sharp threshold determining whether the infection dies out or not. We give a rigorous mathematical analysis on technical materials and necessary arguments, including relative compactness of the orbit and uniform persistence of system, by reformulating the system as a system of Volterra integral equations. We further investigate global behaviors of HIV viral infection models with Holling type II functional response and Crowley–Martin functional response through numerical simulations.

This paper discusses approximation of semigroups of locally Lipschitz operators by discrete semigroups of local quasi-contractions with respect to metric-like functionals. The main approximation result not only extends a product formula for contractive semigroups generated by maximal dissipative operators in real Hilbert spaces but also gives an algorithm for solving a mixed problem for the Kirchhoff equation to which the contractive semigroup theory or Kato's quasilinear theory cannot directly apply.

We prove the Fischer decomposition for the space of spinor-valued polynomials, defined on Euclidean space of four-fold dimension, in terms of irreducible modules for the symplectic group, consisting of so-called osp(4|2)-monogenics. (C) 2017 Elsevier Inc. All rights reserved.

We consider the Cauchy problem for the semilinear Schrodinger equation (NLS){(i∂t+Δ)u=μ|u|p,(t,x)∈[0,Tλ)×Rd,u(0,x)=λf(x),x∈Rd, where u=u(t,x) is a C-valued unknown function, μ∈C\{0}, p>1, λ≥0, f=f(x) is a C-valued given function and Tλ is a maximal existence time of the solution. Our first aim in the present paper is to prove a large data blow-up result for (NLS) in Hs-critical or Hs-subcritical case p≤ps:=1+4/(d−2s), for some s≥0. More precisely, we show that in the case 1 0 such that for any λ>λ0, the following estimates Tλ≤Cλ−κand{limt→Tλ−0⁡‖u(t)‖Hs=∞(if10 are constants independent of λ and (r,ρ) is an admissible pair (see Theorem 2.3). Our second aim is to prove non-existence local weak-solution for (NLS) in the Hs-supercritical case p>ps, which means that if p>ps, then there exists an Hs-function f such that if there exist T>0 and a weak-solution u to (NLS) on [0,T), then λ=0 (see Theorem 2.5).

The lowest eigenvalue of non-commutative harmonic oscillators Q(alpha,beta) (alpha > 0,beta > 0, alpha beta > 1) is studied. It is shown that Q(alpha,beta) can be decomposed into four self-adjoint operators, [GRAPHICS] and all the eigenvalues of each operator Q(sigma p) are simple. We show that the lowest eigenvalue of Q(alpha,beta) is simple whenever alpha not equal beta. Furthermore a Jacobi matrix representation of Q(sigma p) is given and spectrum of Q(sigma p) is considered numerically., Article, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. 415(2):595-609 (2014)

This paper concerns the behaviour of the apsidal angle for orbits of central force system with homogenous potential of degree $-2\leq \alpha\leq 1$ and logarithmic potential. We derive a formula for the apsidal angle as a fixed-end points integral and we study the derivative of the apsidal angle with respect to the angular momentum $\ell$. The monotonicity of the apsidal angle as function of $\ell$ is discussed and it is proved in the logarithmic potential case., Comment: 24 pages, 1 figure

We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply Lp boundedness of them by an extrapolation argument under a sharp condition for the kernels. Also, we prove some weighted Lp inequalities for the operators. © 2012 Elsevier Ltd.

We systematically investigate lower and upper bounds for the modified Bessel function ratio [Formula: see text] by functions of the form [Formula: see text] in case [Formula: see text] is positive for all [Formula: see text], or equivalently, where [Formula: see text] or [Formula: see text] is a negative integer. For [Formula: see text], we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If [Formula: see text], the minimal elements are tangent to [Formula: see text] in exactly one point [Formula: see text], and have [Formula: see text] as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if [Formula: see text] is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively.

We extend a theorem of Gehring and Osgood from 1979–relating to the distortion of the quasihyperbolic metric by a quasiconformal mapping between Euclidean domains–to the setting of metric measure spaces of Q -bounded geometry. When the underlying target space is bounded, we require that the boundary of the image has at least two points. We show that even in the manifold setting, this additional assumption is necessary.

In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L 2 consisting of square integrable random vectors. We show that for the solution X to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x for this inclusion that is a ‖ ⋅ ‖ L 2 -continuous selection of X . This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.

We introduce a definition for the $p$-Laplace operator on positive and finite Borel measures that satisfy an Adams-type embedding condition., Comment: 17 pages

A new deterministic model for the spread of a communicable disease that is controllable using mass quarantine is designed. Unlike in the case of the vast majority of prior quarantine models in the literature, the new model includes a quarantine-adjusted incidence function for the infection rate and the quarantine of susceptible individuals suspected of being exposed to the disease (thereby making it more realistic epidemiologically). The earlier quarantine models tend to only explicitly consider individuals who are already infected, but show no clinical symptoms of the disease (i.e., those latently-infected), in the quarantine class (while ignoring the quarantine of susceptible individuals). In reality, however, the vast majority of people in quarantine (during a disease outbreak) are susceptible. Rigorous analysis of the model shows that the assumed imperfect nature of quarantine (in preventing the infection of quarantined susceptible individuals) induces the phenomenon of backward bifurcation when the associated reproduction threshold is less than unity (thereby making effective disease control difficult). For the case when the efficacy of quarantine to prevent infection during quarantine is perfect, the disease-free equilibrium is globally-asymptotically stable when the reproduction threshold is less than unity. Furthermore, the model has a unique endemic equilibrium when the reproduction threshold exceeds unity (and the disease persists in the population in this case).

We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has a six dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution to the free wave equation. It is assumed that the charge density satisfies Wiener condition which is a version of Fermi Golden Rule, and that the momenta of the charge distribution vanish up to the fourth order. The proof is based on a development of the general strategy introduced by Buslaev and Perelman: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component., Comment: 31 pages

Recently, we discussed optimality conditions for quasiconvex programming by introducing ‘Q-subdifferential’, which is a notion of differential of quasiconvex functions. In this paper, we investigate basic and fundamental properties of the Q-subdifferential. Especially, we show results of a chain rule for composition with non-decreasing functions, monotonicity of the Q-subdifferential, mean-value theorem, a sufficient condition for a global minimizer for quasiconvex programming, and the calculus of the Q-subdifferential of the supremum of quasiconvex functions.

We prove the existence of non-radially symmetric solutions for semilinear degenerate elliptic equations with radially symmetric coefficients in the plane. We adapt the viscosity solution for the weak solution. The key arguments consist of the analysis of the structure of 2 π -periodic solutions for the associated Laplace–Beltrami operator and construction of super- and sub-solutions which have the prescribed asymptotic structures.

In [M. Kato, L. Maligranda, On James and Jordan–von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl. 258 (2001) 457–465], the James constant of the 2-dimensional Lorentz sequence space d ( 2 ) ( ω , q ) is computed in the case where 2 ⩽ q ∞ . It is an open problem to compute it in the case where 1 ⩽ q 2 . In this paper, we completely determine the James constant of d ( 2 ) ( ω , q ) in the case where 1 ⩽ q 2 .

In this article we prove new sufficient conditions under which the feedback operator associated with the Linear Quadratic Regulator control design for distributed parameter systems is nuclear or Hilbert-Schmidt. Examples illustrating the main results are also given. (c) 2006 Elsevier Inc. All rights reserved.

We show that the Alexander transform of a β-spirallike function is univalent when cos β ⩽ 1 / 2 , which settles the problem posed by Robertson. We also solved a problem considered by Y.J. Kim and Merkes.

Optimal control problems are studied for the equation of membrane with strong viscosity. The Gâteaux differentiability of solution mapping on control variables is proved and the various types of necessary optimality conditions corresponding to the distributive and terminal values observations are established.

Let -Domega((.), z)D + q be a differential operator in L-2(0, infinity) whose leading coefficient contains the eigenvalue parameter z. For the case that omega((.), z) has the particular formomega(t, z) = p(t) + c(t)(2)/(z - r (t)), z is an element of C \ R,and the coefficient functions satisfy certain local integrability conditions, it is shown that there is an analog for the usual limit-point/limit-circle classification. In the limit-point case mild sufficient conditions are given so that all but one of the Titchmarsh-Weyl coefficients belong to the so-called Kac subclass of Nevanlinna functions. An interpretation of the Titchmarsh-Weyl coefficients is given also in terms of an associated system of differential equations where the eigenvalue parameter appears linearly. (C) 2004 Elsevier Inc. All rights reserved.

人間社会研究域学校教育系, We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces Hwp (0 < P ≤ 1) and on the weighted Lp spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means. © 2003 Elsevier Science (USA). All rights reserved.

Stein's method provides a way of finding approximations to the distribution, ρ say, of a random variable, which at the same time gives estimates of the approximation error involved. In essence the method is based on a defining equation, or equivalently an operator, of the distribution ρ and a related Stein equation. Up to now it was not clear which equation to take. One could think of a lot of equations. We show how for a broad class of distributions, there is one convenient equation. We give a systematic treatment, including the Stein equation and its solution. A key tool in Stein's theory is the generator method developed by Barbour and we suggest employing the generator of a Markov process for the operator of the Stein equation. For a given distribution there may be various Markov processes which fit in Barbour's method and, up to now, it was still not clear which Markov process to take to obtain good results. We show how for a broad class of distributions there is a special Markov process, a birth and death process, or a diffusion, which takes a leading role in the analysis. Furthermore, a key role is played by the classical orthogonal polynomials. It turns out that the defining operator is based on a hypergeometric difference or differential equation, which lies at the heart of the classical orthogonal polynomials. Furthermore, the spectral representation of the transition probabilities of the Markov process involved are in terms of orthogonal polynomials closely related to the distribution to be approximated. This systematic treatment together with the introduction of orthogonal polynomials in the analysis seems to be new. Furthermore some earlier uncovered examples like the beta, the Student's t, and the hypergeometric distribution are now worked out.

a b s t r a c t We study the global behavior of the price dynamics in a commodity market governed by a balance between demand and supply. While the dependence of demand on price is considered instantaneous, the supply term contains a delay, leading to a delay–differential equation. A discrete model is naturally defined as a limit case of this equation. We provide a thorough study of the discrete case, and use these results to get new sufficient conditions for the global convergence of the solutions to the positive equilibrium in the continuous case. For when the equilibrium is unstable, we provide some bounds for the amplitude of the oscillations that are quite sharp when the delay is large.

a b s t r a c t We use Stein’s method to prove a generalization of the Lindeberg–Feller CLT providing an upper and a lower bound for the superior limit of the Kolmogorov distance between a normally distributed random variable and the rowwise sums of a rowwise independent triangular array of random variables which is asymptotically negligible in the sense of Feller. A natural example shows that the upper bound is of optimal order. The lower bound improves a result by Andrew Barbour and Peter Hall.

Time-independent swirling flows in rotationally symmetric pipes of constant and varying diameter are constructed using variational techniques. In Part I, by E. van Groesen, B. W. van de Fliert, and E. Fledderus (J. Math. Anal. Appl.,192(1995), 764–788.) critical flows in pipes of uniform cross-section were found by extremizing the cross-sectional energy at constrained value of the cross-sectional helicity and the axial flow rate. In this paper we add the cross-sectional angular momentum to the constraints and find an unfolding of the family of flows obtained in Part I. The flows are steady Beltrami-type flows, independent of the axial coordinate, and show the phenomenon of flow-reversal at the central axis of the pipe. Closely related to the unfolding is the introduction of a complexity parameter; instead of taking only discrete values for the flows studied in Part I, now the complexity can change in a continuous way. An approximation for swirling flows in a pipe with varying circular cross-section is constructed in a quasi-homogeneous way as a succession in the axial direction of critical flows in uniform pipes. The evolution of the constraint values in the axial direction is determined by the conservation laws for the energy, angular momentum, and the axial flux. A clear presentation of the mechanism behind the evolution of parameters in the base-flow is so obtained. The consistency conditions that follow from the conservation laws are shown to be related to the solvability conditions of the linearized equation.

We find all sequences of polynomials (pn)n⩾0with persistent roots (i.e.,pn(x)=cn(x−r1)(x−r2)···(x−rn)) that are of binomial type in Viskov's generalization of Rota's umbral calculus to generalized Appell polynomials. We show that such sequences only exist in the classical umbral calculus, the divided difference umbral calculus, and the new “hyperbolic” umbral calculus generated by[formula]. In each of these three umbral calculi, we also find all Sheffer sequences with persistent roots.

Time-independent swirling flows in rotationally symmetric pipes of constant and varying diameter are constructed using variational techniques. Critical flows in pipes of uniform cross section are found by extremizing the cross-sectional energy at constrained value of the cross-sectional helicity and the axial how rate. Discrete classes of hows are found explicitly, parameterized by the radius and the value of the cross-sectional helicity. These flows are steady Beltrami-type hows and are independent of the axial co-ordinate. Solution branches can show the phenomenon of flow-reversal at the central axis of the pipe: the axial velocity becomes negative with increasing radius. An approximation for swirling flows in a pipe with varying circular cross section is constructed in a quasi-homogeneous way as a succession in the axial direction of critical flows in uniform pipes. With the changing value of the radius, the value of the cross-sectional helicity is determined by the requirement that the approximation satisfies the correct energy and flux conservation. It will be shown that these requirements are equivalent to satisfying the necessary solvability conditions that arise from a perturbation analysis around the quasi-homogeneous approximation. The approximate flows have the property that in lowest order approximation the local cross-sectional energy density increases linearly with R, while the local cross-sectional helicity density is constant.

Starting from the Riemann-Liouville and the Weyl calculus, compositions of fractional integral and fractional differential operators are studied in this paper. These composite operators and their inverses admit descriptions as integral transformations with Gegenbauer functions in their kernel. Rodrigues-type formulas for Gegenbauer functions and new relations for fractional differential and integral operators are derived. Thus classical results on integral equations of Mellin convolution type are extended and unified.

We establish integral representation results for suitably pointwise continuous and comonotonic additive functionals of bounded variation de…ned on Stone lattices. As an application, we prove a comonotonic version of the Daniell-Stone Theorem. 2000 Mathematics Subject Classi…cation: Primary 28A12, 28A25, 46G12; Secondary 91B06.

Discrete Fourier analysis is used to obtain simple proofs of certain inequalities about finite number sequences determined by Fan, Taussky, and Todd [Monatsh. Math. 59 (1955), 73-90] and their converses determined by Milovanovic and Milovanovic [J. Math., Anal. Appl. 88 (1992), 378-387]. Using the same techniques, the inequality (2 sin pi/2(n + 1))4 SIGMA(k = 1)n b(k)2 less-than-or-equal-to SIGMA(k = 0)n - 1 (b(k) - 2b(k + 1) + b(k + 2)2 less-than-or-equal-to (2 + 2 cos pi/n + 1)2 SIGMA(k = 1)n b(k)2 is proved for all real numbers 0 = b0, b1, ..., b(n), b(n+1) = 0, which answers a question raised by Alzer [J. Math. AnaL Appl. 161 (1991), 142-147]. Second, the method is used to obtain the eigenvalues and eigenvectors of matrices (a(ij)) that are rotation-invariant, i.e., that obey (a(ij)) = (a(i + 1)(j + 1)). (C) 1994 Academic Press, Inc.