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We consider the approximation of the solutions of the integral equations of the form $$\lambda x(s) = \int\limits_a^b {K(s,t,x(t))dt + y(s)} ,a \leqq s \leqq b,$$ or of the form $$x(s) = \int\limits_a^b {K(s,t)f(t,x(t))dt + y(s)} ,a \leqq s \leqq b$$ by using the techniques of monotone operators.

Integral absorption in a spectrum band is represented as a series. Monotone growth of the coefficients in the series and their limits are established. The results are used to analyze known absorption formulas and to compare approximate formulas.

Improved sufficient conditions are derived for the exponential stability of a nonlinear time varying feedback system having a time invariant blockG in the forward path and a nonlinear time varying gain ϕ(.)k(t) in the feedback path. φ(.) being an odd monotone nondecreasing function. The resulting bound on is less restrictive than earlier criteria.

We shall deal with the inequality (2) and we prove among others the following results. Every solution of the inequality (2) is monotone increasing. Every solution of the inequality (2) is differentiable ifn≥3. A functionf satisfies the inequality (3) if and only if its saltus part and absolutely continuous part satisfy also the inequality (3). We give the general solution of (3) in the field of saltus function. Finally by the help of [Muszely] we give the general solution of (3). This results are generalisations of one ofJ. Aczel andJ. Pfanzagl.

We consider in this paper cones of generalized absolutely monotone functions with respect to an infinite sequence ofC ∞ [a, b] functions {u i } i = 0/∞ such that for alln, n=0,1,2,...{u 0, u1,...,un} constitutes an Extended Tchebycheff System on [a,b]. We find a necessary and sufficient condition for the functionsu i, i=0,1,2,... to generate all the extreme rays in this cone. We conclude by constructing a cone of generalized absolutely monotone functions where theu i′s do not generate all of its extreme rays.

The numbers in each column of ann ×m matrix of multivariate data are interpreted as giving the measured values of alln of the objects studied on one ofm different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show (a) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but (b) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.

In this paper we establish the stability of the Ritz procedure for the nonlinear equationAu+F(u)?f=0 whereA is a linear positive definite operator in a real Hilbert spaceH andF is a monotone nonlinear operator.

Inherent in the proof of Renyi's Theorem [9] (as observed by Kallenberg [4] and M6nch [8]) is the fact that the distribution of an orderly point process (a process without multiple points) is determined by the zero probability function 4)(B)--P{N(B)=O} (N(B)=number of points in B). It is natural to ask what set functions ~b can arise as zero probability functions of point processes. In this paper we characterize these set functions in terms of a property that can naturally be called complete monotonicity. In his classical paper on capacities [1], Choquet studies a class of set functions which he calls alternating capacities of order infinity. Based on the integral representation of these functions in terms of extremal functions, Choquet observes that a class of these functions can be interpreted as giving the probability that sets intersect a random compact set. The extremal functions are of the form

Here f = (fi 1)~ in a continuous function from ~ x A into IR M and fl = (fli)~ in a maximal monotone (multivalued) operator in the euclidean space ~M. In § 2 we develop abstract results which are applied to Eqs. (1.1) and (i,2) in § 3. The results and techniques used here are somewhat analogous to those in Brezis [2], Brezis et al. [3], Browder [-4], Da Prato [5], Hess [7, 8], Martin [10], Ton [ i2] , as well as several references cited in these papers.

The refractory periods of an aggregate of simple “counter” neurons are assumed distributed according to some probability frequency. The output of the aggregate is computed for rectangular and triangular distributions. In particular, it is shown that the maximum output of an aggregate with any triangular distribution cannot exceed the maximum output of its average neuron by a factor greater than 2 ln 2. This puts an upper bound on the amount of departure from the behavior of the average neuron which an aggregate characterized by a certain type of distribution can show. Next, the aggregate is supposed to be subjected to regularly spaced stimuli. Under these conditions, a single neuron will give a discontinuous output curve. If, however, the refractory periods are distributed according to some frequency, the output curve may be “smoothed out.” A general condition on the distribution is derived which makes the output monotone increasing with the input. The condition is applied to some special cases.

Since closed cones C in finite dimensional real vector spaces V play the role of “Lie algebras” for finite dimensional compact abelian semigroups, we discuss the structure of congruence relations on such cones. We show in particular that a monotone closed congruence on such a cone C is entirely determined by a countable collection of closed ideals of C and a countable collection of linear subspaces of V (see theorem 3.3).

We state three theorems concerning iterative methods for solving equations containing monotone, but not necessarily uniformly monotone operators in Hilbert space, which may be not Lipschitz bounded, thus extending results obtained by Sibony [9, 10]. We apply the method to a nonlinear elliptic Dirichlet problem of laminar flow theory of non Newtonian fluids, thus improving the rather experimental treatment given in Ames [1] and removing the condition of nonzero difference quotients assumed by Cryer [3].

It is proved that the monotone σ-closure of the self-adjoint part of anyC*-algebraA is the self-adjoint part of aC*-algebra ℬ. IfA is of type I it is proved that ℬ is weakly σ-closed, i.e. ℬ is aΣ*-algebra. The physical importance ofΣ*-algebras was explained in [1] and [7].

in a matched r • r contingency table. The matching essentially consists in the fact that each of the n = E Z nij individuals is grouped simultaneously with respect to the appropriate row and column classifications which are the same. Thus e.g. if the effect of an educational campaign on cigarette smoking amongst high-school students is being reviewed, the same person will be classified according to his admitted smoking habits (Non Smoker, Light, Moderate, Heavy) prior to and subsequent to a series of school lectures plus a film on lung cancer. Let P = [pij] stand for the corresponding r • r matrix of multinominal probabilities with the usual marginal values Pi. = EP~i,P. j = E Pij such that E ]2 Pi~ = 1. j i i j Particular interest then attaches to the hypothesis of marginal symmetry :Pi. =P.~ for i = 1 . . . . . r. STUART [1955] has suggested one test for/-/~, and FINNEY [1959] extended STUART'S method to the case where some monotone scale is associated with the row and column classifications. Earlier BOWKER [1948] proposed a test for the hypothesis of symmetry of the elements of P, or

LetL be a linear uniformly elliptic second order operator. The boundary value problem is solved for the nonlinear elliptic equationLu=f(x, u) wheref(x, u) is a monotone increasing function ofu for each pointx in the domain. A descent technique based on Newton's method is shown to yield a sequence of iterates which converges uniformly and quadratically to the solution. The convergence is independent of the choice for the initial iterate. Numerical results in two dimensions are presented.

This part deals with a subclass of T [T as defined in 2.1 of Part I of this paper]. We designate the direct and inverse transformations of this class by U and. U-', and prove that these transformations, and others defined by their products are commutative. We further shew that transformations corres¬ponding to differences of any real order form a subclass of the group defined by U, U-1-, and their products. In (16) we shew that some important theorems of Anderson (A. 1)* are, either deducible from, or particular cases of, theorems of Parts I and II of this paper. In (17) we discuss the generaliza¬tion of Knopp's results on " Mehrfach monotone folgen " (K. 2).

For the problem of minimizing a suitable type of integral functionalJ[y] in the class\(\mathfrak{M}\)k of real, monotone nonincreasing functionsy which are Lipschitzian on a compact interval [a, b] with Lipschitz constantk, there is presented an existence theorem and a characterization of minimizing functions as solutions, in the sense of Filippov, of associated differential equations whose members involve discontinuities. For the problem of minimizingJ[y] in the class\(\mathfrak{M}\) of all real, monotone nonincreasing functions for whichJ[y] exists, there is established an existence theorem and proof that, under suitable hypotheses, a solution of this second problem is the limit of solutions of the aforementioned problem ask → ∞. For the particular case in whichJ[y] is the integral of 1/2[y −h(t)]2, whereh(t) is measurable and bounded on [a, b], it is shown that the minimizing function forJ[y] in the class\(\mathfrak{M}\) is the derivative almost everywhere of the least concave majorant of the functionH(t)=∫0th(s)ds, t e [a,b].

The most important problem in the theory of entire functions is exploring the relation between the growth of an entire function and the distribution of its apoints. The classical investigation dealt with the connection between the rate of growth of M(r,f) and the function N(r,a). The growth scale was further refined by introducing as comparison functions, monotone functions that grow more slowly than any of the iterated log arithms.

This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matricesS, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements ofS. A comparable and more easily proven theorem for the vector model is also introduced. At mostn-2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.

The idea of norms monotone in the nonnegative orthant is extended to partially ordered finite dimensional real spaces. Relations between the monotonicity of a norm and of the dual and the subordinate matrix norm are investigated.

The present paper is intended to give an axiomatic approach to rounded computations. A rounding is defined as a monotone mapping of an ordered set into a subset, which in general is called a lower respectively an upper screen. The first chapter deals with roundings in ordered sets. In the second chapter further properties of roundings in linearly ordered sets are studied. The third chapter deals with the two most important applications, the approximation of the real arithmetic on a finite screen and the approximation of the real interval arithmetic on an upper screen. Beyond these examples various further applications are possible.

The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.

Let X(t) be a continuous n×n symmetric matrix function of t for 0≤t≤1, monotone in the sense that X(t)−X(s) is non-negative definite for 1≥t≥s≥0. Denote by [X(t)−X(s)]1/2 the unique non-negative definite square root of X(t)−X(s) for t≥s. Take 0≤t1≤t2≤…≤tN=1 to be a sub-division of [0, 1] and consider the sum $$S_N = \sum\limits_{i = 1}^{N - 1} {\left[ {X(t_{i + 1} ) - X(t_i )} \right]} ^{1/2} F(t_i )\left[ {X(t_{i + 1} ) - X(t_i )} \right]^{1/2} $$ , where F(t) is a given continuous matrix function of t in [0, 1].

In this paper, some properties of monotone mappings and quasi-compact mappings have been studied.

Any one parameter exponential family of distributions has monotone likelihood ratios. As the product probabilities of n identical distributions of an exponential family form again an exponential family, it has monotone likelihood ratios for arbitrary n. Furthermore, the members of an exponential family are mutually absolutely continuous. In Part 1, we show that these properties uniquely characterize the exponential family. The application of this result to the theory of testing hypotheses (Part 2) shows that if a family of mutually absolutely continuous distributions has uniformly most powerful tests for arbitrary levels of significance, and arbitrary sample sizes, then it is necessarily an exponential family.

The parametric linear complementarity problem under study here is given by the conditionsq + αp + Mz ≥ 0,α ≥ 0,z ≥ 0,z T (q + αp + Mz) = 0 whereq is nonnegative. This paper answers three questions including the following one raised by G. Maier: What are the necessary and sufficient conditions onM guaranteeing that for every nonnegative starting pointq and every directionp the components of the solution to the parametric linear complementarity problem are nondecreasing functions of the parameterα?

Evolution of the orbital elements of a two-body system with slowly decreasing mass according to Jeans' mode is described by a non-linear, non-autonomous system of differential equations. In general the system contains one stationary solution (e=1,f=π), for which an instability criterion is derived. For example the stationary solution is unstable for all Jeans-Eddington functionsm n (t) with 1≦n≦3 which characterize the loss of mass. Furthermore, it is possible to describe the quantitative behaviour ofE+ω,e anda for arbitrarym(t) in a large number of cases. In the case of the Jeans-Eddington functions we find that the amplitude of the oscillations ine is monotone decreasing with time ifn>3 and it is monotone increasing with time ifn

We shall deal with the inequality (2) and we prove among others the following results. Every solution of the inequality (2) is monotone increasing. Every solution of the inequality (2) is differentiable ifn≥3. A functionf satisfies the inequality (3) if and only if its saltus part and absolutely continuous part satisfy also the inequality (3). We give the general solution of (3) in the field of saltus function. Finally by the help of [Muszely] we give the general solution of (3). This results are generalisations of one ofJ. Aczel andJ. Pfanzagl.

Let (a n ) be a strictly monotone and convergent sequence of real numbers. Necessary and sufficient conditions are given that the sequence (b n ) defined by $$b_n = \frac{{a_{n + 1} - k_n a_n }}{{1 - k_n }}$$ is strictly monotone either in the same or in the opposite direction as (a n ), provided (k n ) is monotone in the same direction as (a n ). Two numerical examples are presented.

Sufficient conditions for generalized absolutely monotone functions to possess a Taylor-type expansion in terms of the corresponding Extended Tchebycheff systems were found by Karlin and Ziegler. The question of necessary conditions, however, was left open. In this paper we solve this question by finding necessary and sufficient conditions for the validity of the expansion. The structure of the cone of generalized absolutely monotone functions and its extreme rays are also discussed.

It is considered a one dimensional heat diffusion probelm in a medium bounded by two parallel planes, x=0 and x=s(t), with a temperature specification on the moving face and a nonlinear relationship between the heat flux and the temperature on the other one. Existence, uniquenesses and computability of the solution are investigated. Continuous and monotone dependence on the data and related properties are also studied. The case s(0)=0 is then widely discussed. The class of boundary problems for which these results hold seems to be very wide.

In the theory of lattice ordered groups there are considered several types of convergence. In this work it is shown that for nets (r)-convergence is essentially stronger than (o)-convergence, while for sequences these notions are not comparable (as is known, in K-lineals, (r)-convergence for sequences as well as for nets is stronger than (o)-convergence); in Kσ-groups (r)-convergence of sequences is stronger than (o)-convergence. (A sequence is considered (o)-convergent if it is compressed by monotone sequences to a common limit.)

It is shown that the set of points for which a monotone mappingT:X→X * from a separable Banach space into its dual is not single-valued has no interior; if dimX

Recently B. Simon proved a remarkable theorem to the effect that the Schrodinger operatorT=−Δ+q(x) is essentially selfadjoint onC 0 ∞ (R m if 0≦q ∈L 2(R m). Here we extend the theorem to a more general case,T=−Σ =1/ (∂/∂x j −ib j(x))2 +q 1(x) +q 2(x), whereb j, q1,q 2 are real-valued,b j ∈C(R m),q 1 ∈L loc 2 (R m),q 1(x)≧−q*(|x|) withq*(r) monotone nondecreasing inr ando(r 2) asr → ∞, andq 2 satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq 1 is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.

Existence of a phase-transition is proved for an infinite linear chain of spins μj=±1, with an interaction energy $$H = - \sum J(i - j)\mu _i \mu _j ,$$ whereJ(n) is positive and monotone decreasing, and the sums ΣJ(n) and Σ (log logn) [n3J(n)]−1 both converge. In particular, as conjectured byKac andThompson, a transition exists forJ(n)=n−α when 1 < α < 2. A possible extension of these results to Heisenberg ferromagnets is discussed.

Such a set G is said to be maximal monotone if it is maximal among monotone sets in the sense of inclusion, and a mapping T is said to be maximal monotone if its graph G(T) is a maximal monotone set. If D(T) is convex and T is hemicontinuous from D(T) to X* (i.e. T is continuous from each line segment of D(T) to the weak* topology of X*), then T is maximal monotone if and only if it is maximal in the sense of operator inclusion in the family of monotone mappings from X to X*. For operators T with D(T) = X , the basic result obtained independently by BROWNER [5] and MINTY [31] states that for a reflexive space X, if T is a hemicontinuous monotone operator from D(T) = X to X* and if in addition, T is coercive, i.e.

A real-valued discrete time Markov Chain {X n} is defined to be stochastically monotone when its one-step transition probability function pr {X n+1≦y¦ X n=x} is non-increasing in x for every fixed y. This class of Markov Chains arises in a natural way when it is sought to “bound” (stochastically speaking) the process {X n} by means of a “smaller” or “larger” process with the same transition probabilities; the class includes many simple models of applied probability theory. Further, a given stochastically monotone Markov Chain can readily be “bounded” by another chain {Y n}, with possibly different transition probabilities and not necessarily stochastically monotone, and this is of particular value when the latter process leads to simpler algebraic manipulations. A stationary stochastically monotone Markov Chain {X n} has cov(f(X 0), f(X n)) ≧ cov(f(X 0), f(X n+1))≧0 (n =1, 2,...) for any monotonic function f(·). The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.