Showing papers on "Bounded function published in 1978"

A generalization of the Kermack-McKendrick deterministic epidemic model☆

Abstract: In this paper the Kermack-McKendrick deterministic model is generalized, introducing an interaction term in which the dependence upon the number of infectives occurs via a nonlinear bounded function which may take into account saturation phenomena for large numbers of infectives. An extension of the well-known threshold theorem is obtained, after a stability analysis of the equilibrium points of the system. A numerical example is carried out in detail.

Reversal-Bounded Multicounter Machines and Their Decision Problems

Abstract: Decidable and undecldable properties of various classes of two-way multlcounter machines (deterministic, nondetermmlstlc, multttape, pushdown store augmented) with reversal-bounded input and/or counters are investigated In particular It IS shown that the emptiness, infiniteness, dlsjointness, containment, universe, and equivalence problems are decidable for the class of deterministic two-way multlcounter machines whose input and counters are reversal-bounded

Complexity-based induction systems: Comparisons and convergence theorems

Abstract: In 1964 the author proposed as an explication of {\em a priori} probability the probability measure induced on output strings by a universal Turing machine with unidirectional output tape and a randomly coded unidirectional input tape. Levin has shown that if tilde{P}'_{M}(x) is an unnormalized form of this measure, and P(x) is any computable probability measure on strings, x , then \tilde{P}'_{M}\geqCP(x) where C is a constant independent of x . The corresponding result for the normalized form of this measure, P'_{M} , is directly derivable from Willis' probability measures on nonuniversal machines. If the conditional probabilities of P'_{M} are used to approximate those of P , then the expected value of the total squared error in these conditional probabilities is bounded by -(1/2) \ln C . With this error criterion, and when used as the basis of a universal gambling scheme, P'_{M} is superior to Cover's measure b\ast . When H\ast\equiv -\log_{2} P'_{M} is used to define the entropy of a rmite sequence, the equation H\ast(x,y)= H\ast(x)+H^{\ast}_{x}(y) holds exactly, in contrast to Chaitin's entropy definition, which has a nonvanishing error term in this equation.

Large time behavior of solutions of systems of nonlinear reaction-diffusion equations*

Abstract: We discuss the asymptotic behavior of solutions of weakly coupled parabolic equations describing systems undergoing diffusion, convection and nonlinear interaction in a bounded spatial domain, $\Omega $. If the system admits a bounded invariant region $\Sigma $ of phase space then we isolate a parameter $\sigma $ which depends upon the size of $\Omega $, the lower bound for the diffusion matrix, the magnitude of convection and a measure of the strength or sensitivity of the reaction. For $\sigma > 0$ we show that every solution with initial values in $\Sigma $ and subject to homogeneous Neumann boundary conditions decays exponentially to a spatially homogeneous function of time. This limiting function is a solution of an ordinary differential equation whose $\omega $-limit sets are determined by the reaction mechanism alone. This result may be interpreted as giving a sufficient condition for the validity of the “lumped parameter” approximation of distributed systems by solutions of ordinary differential e...

The Boltzmann equation. I. Uniqueness and local existence

Abstract: An abstract form of the spatially non-homogeneous Boltzmann equation is derived which includes the usual, more concrete form for any kind of potential, hard or soft, with finite cutoff. It is assumed that the corresponding “gas” is confined to a bounded domain by some sort of reflection law. The problem then considered is the corresponding initial-boundary value problem, locally in time.

Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brownian Motions

Abstract: The origins of this paper lie in a question posed to us by Frank Spitzer who, in fact, ended up solving most of his problem on his own. His problem is the following. Consider an infinite system of independent (^-dimensional branching Brownian motions which at their branching times disappear or double with equal probabilities, and assume that the initial distribution of the system is a Poisson point process. Denote by 5?,(-O, *^0 and r^ oo. The answer is yes if d>3 and no if d — \ or 2 (cf. [2], [4], [6] and for a related situation [3]). The question in which Spitzer was interested is what happens if (when d>3) one appropriately rescales the limiting random variable 5700. To be precise, given tf^>0, define for bounded

NC machining of arbitrarily shaped pockets

Abstract: Machining mechanical parts requires a great deal of area cleaning, i.e. cutting away all material within given boundaries. This paper describes a fast algorithm, implemented in the SAAB-Adapt processor, to compute the necessary cutter paths for machining arbitrarily shaped areas (pockets) bounded by straight lines and arcs.

An Analysis of the Greedy Heuristic for Independence Systems

Abstract: The worst case behaviour of the greedy heuristic for independence systems is analyzed by deriving lower bounds for the ratio of the greedy solution value to the optimal value. For two special independence systems, this ratio can be bounded by 1/2, for two other independence systems, it converges with increasing problem size to zero. The main theorem states that for every independence system ( E , F ) the ratio is bounded by I/k, k such that ( E , F ) can be represented as the intersection of k matroids.

A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations

Abstract: Let L be a real linear operator with a positive definite symmetric part M. In certain applications a number of problems of the form $Mv = g$ can be solved with less human or computational effort than the original equation $Lu = f$. An iterative Lanczos method, which requires no a priori information on the spectrum of the operators, is derived for such problems. The convergence of the method is established assuming only that $M^{ - 1} L$ is bounded. If $M^{ - 1} L$ differs from the identity mapping by a compact operator the convergence is shown to be superlinear. The method is particularly well suited for large sparse systems arising from elliptic problems. Results from a series of numerical experiments are presented. They indicate that the method is numerically stable and that the number of iterations can be accurately predicted by our error estimate.

Criteria for Recurrence and Existence of Invariant Measures for Multidimensional Diffusions

Abstract: Let $L = \frac{1}{2} \sum^k_{i,j=1} a_{ij}(x)(\partial^2/\partial x_i \partial x_j) + \sum^k_{i=1} b_i(x)(\partial/\partial x_i)$ be an elliptic operator such that $a_{ij}(\bullet)$ are continuous and $b_i(\bullet)$ are measurable and bounded on compacts. Criteria for transience, null recurrence, and positive recurrence of diffusions on $R^k$ governed by $L$ are derived in terms of the coefficients of $L$.

Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids

Abstract: In a bounded domain of the space R n (n=2 or 3) we consider the initialand boundary-value problem regarding the determination of the velocity vector of the fluid, the pressure, and the density from the system of Navier-Stokes equation and the continuity equations, as well as from the initial conditions for the velocity and from the adherence boundary conditions. It is proved that the three-dimensional problem is uniquely solvable on some finite time interval and, in the case of a small initial velocity vector and a small volume force, also on an infinite interval; however, the two-dimensional problem is uniquely solvable for all t⩾0 without any smallness restrictions.

A bound on solutions of linear integer equalities and inequalities

Abstract: Consider a system of linear equalities and inequalities with integer coefficients. We describe the set of rational solutions by a finite generating set of solution vectors. The entries of these vectors can be bounded by the absolute value of a certain subdeterminant. The smallest integer solution of the system has coefficients not larger than this subde- terminant times the number of indeterminates. Up to the latter factor, the bound is sharp. Let A, B, C, D be m x zz-, m x \-,p x n-,p x 1-matrices respectively with integer entries. The rank of A is r, and s is the rank of the (m + p) X n- matrix (c). Let M be an upper bound on the absolute values of those (s — 1) X (s — 1)- or s X i-subdeterminants of the (m + p) X (n + l)-matrix (c d)> which are formed with at least r rows from (A, B). Theorem. If Ax = B and Cx > D have a common integer solution, then they have one with coefficients bounded by (n + \)M. Let Mx, M2, and M3 be upper bounds on the absolute values of the r X r-subdeterminants, the subdeterminants, and the entries of (A, B) respectively. Taking the zz X zz-identity matrix for C and D = 0, we have the following

Almost periodic functions and partial differential operators

Abstract: ContentsIntroduction § 1. The Bohr compactum of a locally compact Abelian group and almost periodic functions § 2. Approximations of almost periodic functions by trigonometric polynomials and periodic functions § 3. Spaces of almost periodic functions and almost periodic operators § 4. Exact theorems on boundedness and regularity, essential self-adjointness, and a theorem on equality of spectra § 5. The behaviour of bounded solutions at the points of the Bohr compactum, the existence of bounded solutions, and the structure of the inverse operator and of the Green's functionReferences

Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral

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On the Convergence of a New Conjugate Gradient Algorithm

Abstract: This paper studies the convergence of a conjugate gradient algorithm proposed in a recent paper by Shanno. It is shown that under loose step length criteria similar to but slightly different from those of Lenard, the method converges to the minimizes of a convex function with a strictly bounded Hessian. Further, it is shown that for general functions that are bounded from below with bounded level sets and bounded second partial derivatives, false convergence in the sense that the sequence of approximations to the minimum converges to a point at which the gradient is bounded away from zero is impossible.

On Finitely Additive Priors, Coherence, and Extended Admissibility

Abstract: A decision maker is seen to be coherent in the sense of de Finetti if, and only if, his probabilities are computed in accordance with some finitely additive prior. If a bounded loss function is specified, then a decision rule is extended admissible (i.e., not uniformly dominated) if and only if it is Bayes for some finitely additive prior. However, if an improper countably additive prior is used, then decisions need not cohere and decision rules need not be extended admissible. Invariant, finitely additive priors are found and their posteriors calculated for a class of problems including translation parameter problems.

A global representation of multidimensional piecewise-linear functions with linear partitions

Abstract: An analytical representation is introduced for m -dimensional piecewise-linear functions which are affine over convex polyhedral regions bounded by linear partitions. Explicit formulas are presented to compute the coefficients associated with this representation along with an example.

On the bounded solutions of a nonlinear convolution equation

Abstract: This investigation is concerned with the nonlinear convolution equation u(x) − (gou) * k(x) = 0 on the real line IR. The kernel k is nonnegative and integrable on IR , with ∫ IR k(x)dx = 1; the function g is real-valued and continuous on IR, g(0) = 0 , and there exists a p > 0 such that g(x) > x for x ∈ (0,p) and g(p) = p. Sufficient conditions are given for the non-existence of bounded nontrivial solutions. Implications for the solution of the inhomogeneous equation u(x) − (g o u) * k(x) = f(x), x ∈ IR , are discussed. Finally, uniqueness (modulo translation) is shown to hold. The results are applied to a problem of mathematical epidemiology.

On the reflection principle in several complex variables

Abstract: The edge-of-the-wedge theorem is used to extend a biholomorphic map across a nondegenerate real analytic boundary in C' under some differentiability assumption at the boundary. The classical theorem of H. A. Schwarz says that if a function is holomorphic in a plane region and extends continuously to an analytic arc in the boundary of the region, then it extends holomorphically across the arc. An analogue in several complex variables is C. Fefferman's theorem [1] that a biholomorphic map between two bounded strongly pseudoconvex domains with real analytic boundaries extends biholomorphically across the boundary. Recently, H. Lewy [2] has given an elementary proof of a local variant of this result for a holomorphic map which is continuously differentiable to the boundary. See also Pincuk [4]. The purpose of this note is to give a similar elementary proof of extension of a biholomorphic map, assuming some regularity to the boundary. The main idea is to use the edge-of-the-wedge theorem. The distributional version of this theorem permits the reduction of the differentiability assumptions somewhat. We also introduct a natural reflection associated to a nondegenerate analytic real hypersurface. Let M and M' be analytic real hypersurfaces having nondegenerate Levi forms and bounding on domains D and D', respectively, in C', n > 2. Let f: D D' be a biholomorphic map which extends continuously to M and maps M into M'. We first define a reflection mapping, i.e. a local anti-holomorphic involution. Let Cn x Pn -I be the (2n 1)-dimensional complex manifold of tangent (n 1)-planes in Cn. We take as local coordinates za, Zn Pa, 1 < a ? n 1. M is given locally by a real analytic equation r(z, z-) = O, dr # 0. Varying z and zindependently, we define local complex hypersurfaces Qw = {z: r(z, we) = 0) for points w near M. The.reality condition on r implies that z is in Qw if and only if w is in Qz. Let Tz Qw denote the tangent plane to Qw at z. It exists for z and w near a point of M, since ar(z, we) =# 0. For Received by the editors February 10, 1978. AMS (MOS) subject classifications (1970). Primary 32H99.

Numerical solution of an exterior Neumann problem using a double layer potential

Abstract: Introduction. Solving boundary value problems for partial differential operators by integral equation methods is not a new idea. However, the classical way to do it consists in representing the unknown solution as a potential of the type that will lead to an integral equation of the second kind. Then, Fredholm's theorems can be used. Thus, the Dirichlet problem is usually solved with the help of a double layer potential, and the Neumann problem with the use of a single layer potential. We shall have a different point of view. Our aim will be to obtain a variational formulation of the problem in order to obtain the existence and unicity of a solution and error estimates. This philosophy leads to opposite choices for the representation of the solution. Thus, J. C. Nedelec and J. Planchard, for the three-dimensional case, and M. N. Leroux for the two-dimensional case, have solved the Dirichlet problem by using a single layer potential. We propose here the solution of a Neumann problem by using a double layer potential. Let 2 be a bounded open set of R3. Let r be the boundary of 2 and Qc denote the complementary set of Q. We assume that r is sufficiently smooth, and we put the coordinates' origin in 2. We shall write nt, for the exterior normal to r, r, for the distance to the origin, [V] = vlint vie,xt, for the jump through r, of the function v defined in Ri3.

The Distance in BMO to L

Abstract: (9dx T I -III XI~ is the mean of cp over I. Every bounded (measurable) function is in BMO and IIpII* 0 and x0 =x(s) such that (1.2) sup, 1 | {x G I: I 9(x) -' l > A} I < e-/'

Presburger arithmetic with bounded quantifier alternation

Abstract: This paper concerns both the complexity aspects of PA and the pragmatics of improving algorithms for dealing with restricted subcases of PA for uses such as program verification. We improve the Cooper-Presburger decision procedure and show that the improved version permits us to obtain time and space upper bounds for PA classes restricted to a bounded number of alternations of quantifiers. The improvement is one exponent less than the upper bounds for the decision problem for full PA. The pragmatists not interested in complexity bounds can read the results as substantiation of the intuitive feeling that the improvement to the Cooper-Presburger algorithm is a real, rather than ineffectual, improvement. (It can be easily shown that the bounds obtained here are not achievable using the Cooper-Presburger procedure).

A new type of convergence

Abstract: In his important paper (1), Lorentz defined the space f of almost convergent sequences, using the idea of Banach limits. If x ∈ l∞(R), the space of bounded real sequences, andwhere the inf is taken over all sets n(1), n(2), …, n(r) of natural numbers, then a Banach limit L may be defined as a linear functional on l∞(R) which satisfies

Perturbation of translation invariant positivity preserving semigroups on

Abstract: The theory of singular local perturbations of translation invariant positivity preserving semigroups on L2(R", d"x) is developed A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations Estimates on the local singularities of the kernels of the semigroups, e~'H, are given Eigenfunction expansions are derived The local singularities of the eigenfunction and generalized eigenfunctions are discussed Results are illustrated with examples involving singular perturbations of —A I Introduction The sum of an operator, //0, which generates a positivity preserving translation invariant semigroup on L2(RN, dNx) and a potential V is the subject of the present work In §11 the class of such H0's is discussed more fully Here we only remark that the operators corresponding to nonrel- ativistic and relativistic energy in quantum mechanics are included The potentials considered are, in general, too singular to be operators and are given as forms, so that H0 + V must be defined as a form sum A detailed description of the potentials is given in §111 and IV The success of the perturbation program for the investigation of operator sums is impressive (20) For form sums such an analysis is more difficult because functions of forms are generally undefined One technique for analyzing functions,/, of H0 + V is to: (1) approximate Kby bounded functions V"; (2) show f(H0 + V") approximates/(//0 + V); and (3) analyze f(H0 + V) by (2) and a direct analysis of f(H0 + V") Such a procedure for (1) and (2) was developed by Kato (20) and Faris (10), using/(x) = (x + X)-1 It employs monotone convergence arguments and so is applicable only when the potential V can be written as the sum of a rather general nonnegative function V+ and a nonpositive function V_ which is a small form perturbation of H0 One truncates V+ and K_ to obtain functions V+ n and V_ " which are absolutely bounded by the integer n Then, for all