Monotone Continuous Function Research Articles

On the Brownian First-Passage Time Overa One-Sided Stochastic Boundary

Let $B=(B_t)_{t \g 0}$ be standard Brownian motion started at 0 under P, let $S_t=\max_{ 0 \l r \l t} B_r$ be the maximum process associated with B, and let $g: {\bf R}_+ \rightarrow {\bf R}$ be a (strictly) monotone continuous function satisfying $g(s) 0 \vert B_t \l g(S_t) \big \}. $$ Let G be the function defined by $$ G(y) = \exp \bigg(-\int_0^{g^{-1}(y)} {{ds} \over {s-g(s)}} \bigg) $$ for $y \in \bf R$ in the range of g. Then, if g is increasing, we have $$ \lim_{t \rightarrow \infty} \sqrt{t} \bP\{\tau \g t \} = \sqrt{2 \over \pi } \Bigg(-g(0) -\int_{g(0)}^{g(\infty)} G(y)\,dy \Bigg) $$ and this number is finite. Similarly, if g is decreasing, we have $$ \lim_{t \rightarrow \infty} \sqrt{t} \bP\{\tau \g t \} = \sqrt{2 \over \pi } \Bigg(-g(0) + \int_{g(\infty)}^{g(0)} G(y) \,dy \Bigg) $$ and this number may be infinite. These results may be viewed as a {\it stochastic boundary} ext...

A general class of inequalities with mixed means

Suppose (T, Σ, μ) is a space with positive measure,f: ℝ → ℝ is a strictly monotone continuous function, and &(T) is the set of real μ-measurable functions onT. Letx(·) ∈ &(T) andf ○x)(·) ∈L1(T,μ). Comparison theorems are proved for the means\(\mathfrak{M}_{(T,{\mathbf{ }}\mu ,{\mathbf{ }}f)} (x( \cdot ))\) and the mixed means\(\mathfrak{M}_{(T_1 ,{\mathbf{ }}\mu _1 ,{\mathbf{ }}f_1 )} (\mathfrak{M}_{(T_2 ,{\mathbf{ }}\mu _2 ,{\mathbf{ }}f_2 )} (x( \cdot )))\) these inequalities imply analogs and generalizations of some classical inequalities, namely those of Holder, Minkowski, Bellman, Pearson, Godunova and Levin, Steffensen, Marshall and Olkin, and others. These results are a continuation of the author's studies.

The dynamics of continuous maps of finite graphs through inverse limits

Suppose that f: G -* G is a continuous piecewise monotone function on a finite graph G. Then the following are equivalent: (i) f has positive topological entropy; (ii) there are disjoint intervals I1 and I2 and a positive integer n with I1 U I2 C fn(JI) n fn(I2); (iii) the inverse limit space constructed by using f on G as a single bonding map contains an indecomposable subcontinuum. This result generalizes known results for the interval and circle.

An Intelligent Vehicle Highway Information Management System

Abstract: An IVHS (Intelligent Vehicle Highway System) information management system obtains information from road sensors, city maps and event schedules, and generates information to drivers, pafie controllers and researchers. We extend the relational database to model traffic information in a relational database by abstract data types and triggers. Abstract data types are needed for eficient modeling of spatial and temporal information, which create inefficiencies in traditional databases. We use monotonic continuous functions to map the object locations to disk addresses to save disk space and computation time. Model of spatial data is created to efficiently process moving objects. We provide schema for IVHS databases with the relevant abstract data types. We also have a large sample of the relations needed to model IVHS data. Several interesting queries are presented to show the power of the model. Triggers are defined, using rule‐definition mechanisms to represent incident detection and warning systems. An eficient physical model with MoBiLe access method is provided.

On the commutant of a multiplication operator and conditions of invertibility of its elements

We obtain a description of isomorphisms acting in the space of square summable functions and commuting with the operator of multiplication by a continuous piecewise monotone function. Preliminary corrections are made of some mistakes found in the statements of the paper [3].

Nonstandard and standard compactifications of ordered topological spaces

We construct the Nachbin ordered compactification and the ordered realcompactification, a notion defined in the paper, of a given ordered topological space as nonstandard ordered hulls. The maximal ideals in the algebras of the differences of monotone continuous functions are completely described. We give also a characterization of the class of completely regular ordered spaces which are closed subspaces of products of copies of the ordered real line, answering a question of T.H. Choe and Y.H. Hong. The methods used are topological (standard) and nonstandard.

A Complete Classification of the Piecewise Monotone Functions on the Interval

We define two functions $f$ and $g$ on the unit interval $[0,1]$ to be strongly conjugate iff there is an order-preserving homeomorphism $h$ of $[0,1]$ such that $g = {h^{ - 1}}fh$ (a minor variation of the more common term "conjugate", in which $h$ need not be order-preserving). We provide a complete set of invariants for each continuous (strictly) piecewise monotone function such that two such functions have the same invariants if and only if they are strongly conjugate, thus providing a complete classification of all such strong conjugacy classes. In addition, we provide a criterion which decides whether or not a potential invariant is actually realized by some piecewise monotone continuous function.

A complete classification of the piecewise monotone functions on the interval

We define two functions f f and g g on the unit interval [ 0 , 1 ] [0,1] to be strongly conjugate iff there is an order-preserving homeomorphism h h of [ 0 , 1 ] [0,1] such that g = h − 1 f h g = {h^{ - 1}}fh (a minor variation of the more common term "conjugate", in which h h need not be order-preserving). We provide a complete set of invariants for each continuous (strictly) piecewise monotone function such that two such functions have the same invariants if and only if they are strongly conjugate, thus providing a complete classification of all such strong conjugacy classes. In addition, we provide a criterion which decides whether or not a potential invariant is actually realized by some piecewise monotone continuous function.

Monotone and comonotone polynomial approximation revisited

Using a suitable Peetre functional that weighs differently the behaviour of the function in the middle of the interval and near the endpoints, we obtain estimates of the Jackson type on the rate of monotone polynomial approximation to a monotone continuous function. These estimates involve the second modulus of smoothness related to the Peetre functional. Then we apply these estimates to get estimates on the degree of comonotone polynomial approximation of a piecewise monotone function and on the degree of copositive polynomial approximation of a continuous function that changes sign finitely often in the interval.

Characterization of a Marshall-Olkin type class of distributions

A class of bivariate distributions that generalize Marshall-Olkin's one is characterized through a functional equation which involves two associative operations. The obtained distributions concentrate positive mass on the linex=y when the two associative operations coincide; otherwise a positive mass is concentrated on a continuous monotone function.

QUASIHYPONORMAL θ-CLASS AND BN-CLASS COMPOSITION OPERATORS

Let H be a complex Hilbert space, and T be a linear bounded operator on H. T is called a quasihyponormal operator if there exists a strict monotone continuous function φ(t) on [0, + ∞) with φ(0)= 0 (standard function) such that φ(T~*T)-φ(TT~*) D_φ ≥ 0. T is called a complete hyponormal operator if D_≥ 0 is true for every standard function. T is called a θ-class operator if [T~* + T, T~*T] = 0. T is called a BN-class operator if [T~*T,TT~*] = 0. T is called a strong complete hyponormal operator if

Typical monotone continuous functions

Typical monotone continuous functions

Ein elementarer beweis des satzes von malmquist-yosida-steinmetz

In this paper we give an elementary proof of the famous theorem of J. Malmquist from 1913 and its refinement given by N. Steinmetz 1978. This result says, that if a differential equation of type (1.1) has a transcendental meromorphic solution, then it can be transformed into one of the six normalforms given in theorem 3.2 or a power of it. In this note we make no use of Nevanlinna's theory or the theory of central index used in former proofs. We only need the theorem of Liouville and the wellknown inequalities for meromorphic functions g with a finite number of poles, the Euclidean algorithm and a lemma of E Borel about monotone continuous functions.

INTEGRAL MODULI OF SMOOTHNESS AND THE FOURIER COEFFICIENTS OF THE COMPOSITION OF FUNCTIONS

Using the integral modulus of smoothness, estimates for the Fourier coefficients of a composition of functions are obtained in this paper. It is proved, for example, that for any function and any positive sequence with there exists a monotone continuous function (, ) such that where and are the Fourier coefficients of the function . Bibliography: 4 titles.

The curvature of most convex surfaces vanishes almost everywhere

At least since 1904 we know about the existence of singular monotone functions, i.e. continuous monotone functions of one real variable with vanishing derivative a.e. In that year Lebesgue [7] and Minkowski [8] gave their famous examples. In 1910 Faber [4], in 1916 Sierpinski [10] and then many others greatly enriched the variety of known examples. By integrating anyone of these functions we get a differentiable convex function with vanishing second derivative a.e. Thus we obtain smooth convex curves with vanishing curvature a.e. A very beautiful and simple example is due to de R h a m [3J: Take a convex polygon, consider on each side the two points dividing it into three equal parts and take the convex polygon with all these division points as vertices. Repeat the procedure. As a limit we get a smooth, strictly convex curve with a.e. vanishing curvature! The purpose of this paper is to show that, in the sense of Baire categories, most convex curves have the above property. More generally, most convex surfaces in IR" have a.e. a vanishing sectional curvature in every tangent direction. We start with some definitions. By a convex surface we shall always understand a closed convex surface in Busemann's sense (see [11, p. 3). A convex curve is a one-dimensional convex surface. A point x of a convex surface S is called smooth if S is differentiable at x. S is smooth if each of its points is smooth. Any half-line in a supporting plane (this is a hyperplane, see [1], p. 4) of a convex surface S, originating at a point x ~ S will be called supporting direction at x. A supporting direction at a smooth point is called tangent direction. The union of a circular with a square 2-cell, such that the radius and centre of the circle coincide with the side-length and a vertex of the square, is called corner-disk. The two points lying on both the circle and the square are called touching points of the corner-disk. A 2-cell having the union of half a circle with a segment as boundary is called semidisk. The points of the boundary of a semidisk which are not smooth are called corners of the semidisk. The radius of a corner-disk or a semidisk is the radius of the circle appearing in their definitions.

Internal reflection from an amplifying layer

The reflectance of a homogeneous amplifying layer between two transparent regions is determined theoretically with the help of the Fresnel formulas. The transparent region containing the incident beam has a higher refractive index, corresponding to an internal reflection configuration. In the limit of infinite layer thickness, the reflectance is a continuous monotonic function of incident angle, greater than unity for all angles. For certain finite values of layer thickness, the reflectance has a singular point. The results explain the reported observation of a reflectance of 1000 from an excited laser dye in contact with a quartz prism.

Invertible Composition Operators on L 2 (λ)

Let C<, be a composition operator on L?(X), where X is a a-finite measure defined on the Borel subsets of a standard Borel space. In this paper a necessary and sufficient condition for the invertibility of C<, is given in terms of invertibility of 0. Also all invertible composition operators on L2(R) induced by monotone continuous functions are characterised. Introduction. Let (X, S, X) be a a-finite measure space, and let ( be a measurable transformation from X into itself. Let L3(X) denote the Hilbert space of all square-integrable functions on X. Define a composition transformation C. on L3(X) as C-f = f o p for every f E L3(X). In case C, is a bounded operator with the range in L3(A), we call it a composition operator induced by 4. The Banach space of all bounded linear operators will be denoted by S. The purpose of this paper is to study the invertible composition operators. The invertible composition operators on HP (of the unit disk) are studied by Schwartz [5], where he proves that the invertibility of the inducing function on the unit disk into itself is a necessary and sufficient condition for the invertibility of the corresponding composition operator. This is true because the inducing functions are analytic functions and hence they are nicely behaved. In case of composition operators on L2(X) the above statement is not completely true as is shown later by an example. However, a little more addition to the hypothesis makes it go through both ways in case of L2(X). 2. An invertibility theorem. DEFINITION. Let (X,S,X) be a measure space. Let 0 be a measurable transformation on X into itself. Then 0 is said to be one-to-one if there exists a measurable transformation 4' on X into itself such that (4p o +)(x) = x a.e. 0 is said to be onto if there exists a measurable transformation X such that (po w) (x) = x a.e. 0 is said to be invertible if there is a measurable transformation 4, such that (0 o A)(x) = (4o +)(x) = x a.e. Such 4 is called the inverse of 0 and is denoted as 0DEFINITION. A standard Borel space X is a Borel subset of a complete separable metric space T. The class S will consist of all the sets of the form X n B, where B is a Borel subset of T. Received by the editors March 21, 1975. AMS (MOS) subject classifications (1970). Primary 47B99; Secondary 47B99.

Invertible composition operators on 𝐿²(𝜆)

Let C ϕ {C_\phi } be a composition operator on L 2 ( λ ) {L^2}(\lambda ) , where λ \lambda is a σ \sigma -finite measure defined on the Borel subsets of a standard Borel space. In this paper a necessary and sufficient condition for the invertibility of C ϕ {C_\phi } is given in terms of invertibility of ϕ \phi . Also all invertible composition operators on L 2 ( R ) {L^2}({\mathbf {R}}) induced by monotone continuous functions are characterised.

Graphical Solution of Imbibition Equations Used To Predict Oil Recovery by Water Influx in Naturally Fractured Reservoirs

Introduction This article presents graphical solutions to some imbibition equations that describe the recovery of oil from naturally fractured reservoirs with water influx. The equations, presented originally by Aronofsky et al., can be written as Equation(1) for the case of a single-model element and Equation(2) for the case of a total fractured reservoir where the velocity of advance of the water table is uniform. In Eqs. 1 and 2, r is recovery at any time t, R is the limit toward which the recovery converges, and X is a constant giving the rate of convergence. By abstract reasoning, these equations have proven to be useful in practical situations. Working the solution by hand, practical situations. Working the solution by hand, however, is difficult because it requires a trial-and-error effort to determine the constants X and R. The basic assumptions describing the abstract model are (1) the oil production from a small volume, dv, of porous matrix saturated with oil at time zero is a porous matrix saturated with oil at time zero is a continuous monotonic function of time, and it converges to a finite limit, and (2) none of the properties that determine the rate of convergence change sufficiently during the process to affect this rate or the limit. This implies, for example, that variations in wettability and residual oil saturation are not accounted for. The charts presented in this article were generated to simplify the solution. They are based on work presented by Kern to analyze fouling problems in heat presented by Kern to analyze fouling problems in heat exchangers. (Full-scale graphs are available to the interested reader upon request.) Notice that the graphical solutions are limited to cases where the continuous monotonic function is of the assumed exponential form. For other cases, a quantitative analysis is not possible. JPT P. 1526

Quantized fluctuations in the Josephson oscillations of a shunted superconducting point contact

By detecting the oscillation of a tuned circuit resonant at frequency v0 coupled to a shunted superconducting point contact we have observed oscillations not only at the Josephson voltage V=φ0v0, where φ0 is the flux quantum, but also at voltages Vn=gn(T)φ0v0, where gn(T) is a continuous monotonic function of temperature. For most temperatures gn(T)=N(T)/n, where N and n are integers, however there are ranges of temperature over which g1(T) varies rapidly through nonintegral values and the voltages Vn are not harmonically related. The level of oscillation of the tuned circuit was used to measure the power spectrum of the voltage waveform at the point contact and the linewidths were found to be oscillatory functions of temperature with minima when V1=Nφ0v0 and maxima when V1=(N+1/2)φ0v0. The Josephson oscillation of the shunted point contact consists of pulses of N flux quanta (Nφ0) crossing the point contact at a repetition frequency v0. The temperature dependencies are interpreted in terms of fluctuations in N. The relationship between these results and some temperature dependent features observed on the I-V curves of point contacts and some implications for noise thermometry are discussed.