Monotone Functions on Linear Lattices

1963 ◽  
Vol 15 ◽  
pp. 226-236 ◽  
Author(s):  
H. W. Ellis ◽  
Hidegorô Nakano
Keyword(s):  
Monotone Function ◽  
Continuous Linear ◽  
Monotone Functions ◽  
Linear Lattice ◽  
Image Position ◽  
Linear Lattices

If R is a sequentially continuous linear lattice, a function f(x), denned on R+ = {x:0 ≤ x ∊ R} with 0 ≤ f(x) ≤ + ∞, will be called a monotone function if it satisfies

Kolmogorov, Linear and Pseudo-Dimensional Widths of Classes of s-Monotone Functions in 𝕃p, 0 < p < 1

2008 ◽  
Vol 51 (2) ◽  
pp. 236-248
Author(s):  
Victor N. Konovalov ◽  
Kirill A. Kopotun
Keyword(s):  
Unit Ball ◽  
Finite Interval ◽  
Divided Differences ◽  
Monotone Functions ◽  
Image Position

AbstractLet Bp be the unit ball in 𝕃p, 0 < p < 1, and let , s ∈ ℕ, be the set of all s-monotone functions on a finite interval I, i.e., consists of all functions x : I ⟼ ℝ such that the divided differences [x; t0, … , ts] of order s are nonnegative for all choices of (s + 1) distinct points t0, … , ts ∈ I. For the classes Bp := ∩ Bp, we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces , 0 < q < p < 1:

Generalized Resolvent Equations and Unsymmetric Dirichlet Spaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1111-1141
Author(s):  
Joanne Elliott
Keyword(s):  
Linear Operators ◽  
Dirichlet Spaces ◽  
Resolvent Equation ◽  
Continuous Linear ◽  
Generalized Resolvent ◽  
Resolvent Equations ◽  
Measure Spaces ◽  
Image Position ◽  
Generalized Resolvent Equations ◽  
Continuous Linear Operators

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {Uλ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ Uλϕ ≧ 0) continuous linear operators from L2(X, dμ′) to L2(X,dμ) with the following additional properties: if ϕ ≧ 0 then Uλϕ is non-decreasing as λ increases, while λ−1Uλϕ is nonincreasing.A family {Mλ:λ > 0} of continuous linear operators from L2(X, dμ) to L2(X, dμ′) satisfies the “generalized resolvent equation” relative to {Uλ} if(0.1)for positive λ and v. If Uλ = λI, then (0.1) is just the well-known resolvent equation. The family {Mλ} is called submarkov if Mλ is a positive operator and(0.2)it is conservative if(0.3)

scholarly journals Factorization of hyponormal operators

1972 ◽  
Vol 13 (3) ◽  
pp. 323-326
Author(s):  
Mary R. Embry
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyponormal Operators ◽  
Image Position ◽  
Continuous Linear Operators

In [1]R. G. Douglas proved that if A and B are continuous linear operators on a Hilbert space X, the following three conditions are equivalent:

Continuous and Köthe–Toeplitz duals of certain sequence spaces

1969 ◽  
Vol 65 (2) ◽  
pp. 431-435 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Sequence Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Paranormed Space ◽  
Complex Sequences ◽  
Image Position

If (X, g) is a paranormed space, with paranorm g (see (2)), then we denote by X* the continuous dual of X, i.e. the set of all continuous linear functionals on X. If E is a set of complex sequences x = (xk) then E† will denote the generalized Köthe–Toeplitz dual of E

scholarly journals A nonlinear complementarity problem for monotone functions

1979 ◽  
Vol 20 (2) ◽  
pp. 227-231 ◽  
Author(s):  
Sribatsa Nanda ◽  
Ujagar Patel
Keyword(s):  
Unique Solution ◽  
Complementarity Problem ◽  
Monotone Function ◽  
Nonlinear Complementarity Problem ◽  
Monotone Functions ◽  
Nonlinear Complementarity

In this note we prove that for a monotone function that fixes the origin, the complementarity problem for Cn always admits a solution. If, moreover, the function is strictly monotone, then zero is the unique solution. These results are stronger than known results in this direction for two reasons: firstly, there is no condition on the nature of the cone and secondly, no feasibility assumptions are made.

scholarly journals On Extensions of Monotone Functions from Linear SubLattices

1965 ◽  
Vol 8 (3) ◽  
pp. 329-343
Author(s):  
H. W. Ellis
Keyword(s):  
Existence And Uniqueness ◽  
Order Conditions ◽  
Point Of View ◽  
Monotone Functions ◽  
Linear Lattice

In this note real valued functions, defined on a linear sublattice S of a linear lattice R and satisfying the two order conditions (M1) and (M2), are studied from the point of view of the existence and uniqueness of extensions to R. The paper is partly expository and supplements and extends §3 of [4] where S was assumed to be an ℓ-ideal.

scholarly journals Curves with zero derivative in F-spaces

1981 ◽  
Vol 22 (1) ◽  
pp. 19-29 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Characteristic Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Map ◽  
Left And Right ◽  
The Mean ◽  
Image Position ◽  
Locally Convex

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

scholarly journals Idempotent and compact matrices on linear lattices: a survey of some lattice results and related solutions of finite relational equations

1993 ◽  
Vol 16 (2) ◽  
pp. 301-309 ◽  
Author(s):  
Fortunata Liguori ◽  
Giulia Martini ◽  
Salvatore Sessa
Keyword(s):  
Linear Lattice ◽  
Linear Lattices

After a survey of some known lattice results, we determine the greatest idempotent (resp. compact) solution, when it exists, of a finite square rational equation assigned over a linear lattice. Similar considerations are presented for composite relational equations.

scholarly journals METHOD OF STEPWISE SMOOTHING EXPERIMENTAL DEPENDENCES FOR PROBLEMS OF SHORT-TERM FORECASTING

2021 ◽  
Vol 2021 (3) ◽  
pp. 126-133
Author(s):  
Anatoly Anatolevich Ermakov ◽  
Tatyana Klimentyevna Kirillova
Keyword(s):  
Monotone Function ◽  
Operating Time ◽  
Smoothing Method ◽  
Monotone Functions ◽  
Short Term ◽  
Distribution Law ◽  
True Value ◽  
Complex Technical Systems ◽  
Short Term Forecasting ◽  
True Values

The article considers the correspondence of the step-by-step smoothing method as one of the possible algorithms for short-term forecasting of statistics of equal-current measurements of monotone functions, which represent the values of the determining parameters that evaluate the dynamics of the states of complex technical systems based on the operating time. The true value of the monitored parameter is considered unknown, and the processed measurement values are distributed normally. The measurements are processed by step-by-step smoothing. As a result of processing, a new statistic is formed, which is a forecast statistic, each value of which is a half-sum of the measurement itself and the so-called private forecast. First, the forecasts obtained in this way prove to have the same distribution law as the distribution law of a sample of equally accurate measurements. Second, the forecast trend should be the same as the measurement trend and correspond to the theoretical trend, that is, the true values of the monotone function. Third, the variance of the obtained statistics should not exceed the variance of the original sample. It is inferred that the method of step-by-step smoothing method can be proposed for short-term forecasting

Properties (V) and (w V) on C(Ω X)

1995 ◽  
Vol 117 (3) ◽  
pp. 469-477 ◽  
Author(s):  
Elizabeth M. Bator ◽  
Paul W. Lewis
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Linear Functional ◽  
Formal Series ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Weakly Compact ◽  
Image Position ◽  
Continuous Function Space ◽  
Fundamental Paper

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* iffor each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

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