Global attractivity for a nonautonomous Nicholson’s equation with mixed monotonicities

We consider a Nicholson’s equation with multiple pairs of time-varying delays and nonlinear terms given by mixed monotone functions. Sufficient conditions for the permanence, local stability and global attractivity of its positive equilibrium K are established. The main novelty here is the construction of a suitable auxiliary difference equation x n+1 = h(x n ) with h having negative Schwarzian derivative, and its application to derive the attractivity of K for a model with one or more pairs of time-dependent delays. Our criteria depend on the size of some delays, improve results in recent literature and provide answers to open problems.

The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.

On sums of monotone functions over smooth numbers

In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers ( A i ) i = 1 ∞ ⊆ d m n ( f ) \left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right) , which requirements are sufficient for the asymptotic ∑ n ∈ A N P ( n ) ≤ N θ f ( n ) ∼ ρ ( 1 / θ ) ∑ n ∈ A N f ( n ) \sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.

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

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

One Concave-Convex Inequality and Its Consequences

Our starting point is an integral inequality that involves convex, concave and monotonically increasing functions. We provide some interpretations of the inequality, in terms of both probability and terms of linear functionals, from which we further generate completely monotone functions and means. The latter application is seen from the perspective of monotonicity and convexity.

On the q-Monotonicity Preservation of Durrmeyer-Type Operators

We prove that various Durrmeyer-type operators preserve q-monotonicity in [0, 1] or $$[0,\infty )$$ [ 0 , ∞ ) as the case may be. Recall that a 1-monotone function is nondecreasing, a 2-monotone one is convex, and for $$q>2$$ q > 2 , a q-monotone function possesses a convex $$(q-2)$$ ( q - 2 ) nd derivative in the interior of the interval. The operators are the Durrmeyer versions of Bernstein (including genuine Bernstein–Durrmeyer), Szász and Baskakov operators. As a byproduct we have a new type of characterization of continuous q-monotone functions by the behavior of the integrals of the function with respect to measures that are related to the fundamental polynomials of the operators.

Learning of monotone functions with single error correction

Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM (n) of learning of monotone Boolean functions equals C n ⌊ n / 2 ⌋ $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor} \end{array}$ + C n ⌊ n / 2 ⌋ + 1 $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor+1} \end{array}$ (φM (n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.

Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

For a continuous and positive function w (λ), λ> 0 and µ a positive measure on [0, ∞) we consider the following 𝒟-logarithmic integral transform 𝒟 ℒ o g ( w , μ ) ( T ) : = ∫ 0 ∞ w ( λ ) 1 n ( λ + T λ ) d μ ( λ ) , \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among others that, if A, B > 0 with BA + AB ≥ 0, then 𝒟 ℒ o g ( w , μ ) ( A ) + 𝒟 ℒ o g ( w , μ ) ( B ) ≥ 𝒟 ℒ o g ( w , μ ) ( A + B ) . \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right). In particular we have 1 6 π 2 + di log ( A + B ) ≥ di log ( A ) + di log ( B ) , {1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right), where the dilogarithmic function dilog : [0, ∞) → ℝ is defined by di log ( t ) : = ∫ 1 t 1 n s 1 - s d s , t ≥ 0. {\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0. Some examples for integral transform 𝒟Log (·, ·) related to the operator monotone functions are also provided.

An Extensive Operational Law for Monotone Functions of LR Fuzzy Intervals with Applications to Fuzzy Optimization

The operational law put forward by Zhou et al. on strictly monotone functions with regard to regular LR fuzzy numbers makes a valuable push to the development of fuzzy set theory. However, its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which restricts its application in practice to a certain degree. In this paper, we propose an extensive operational law that generalizes the one proposed by Zhou et al. to apply to monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs), of which regular fuzzy numbers can be regarded as particular cases. By means of the extensive operational law, the inverse credibility distributions (ICDs) of monotone functions regarding regular LR-FIs can be calculated efficiently and effectively. Moreover, the extensive operational law has a wider range of applications, which can deal with the situations hard to be handled by the original operational law. Subsequently, based on the extensive operational law, the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs are presented. Furthermore, the proposed operational law is also applied to dispose fuzzy optimization problems with regular LR-FIs, for which a solution strategy is provided, where the fuzzy programming is converted to a deterministic equivalent first and then a newly-devised solution algorithm is utilized. Finally, the proposed solution strategy is applied to a purchasing planning problem, whose performances are evaluated by comparing with the traditional fuzzy simulation-based genetic algorithm. Experimental results indicate that our method is much more efficient, yielding high-quality solutions within a short time.