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In this paper, we first propose two Chebyshev-type inequalities associated with the general fractional-order (Yang–Abdel–Aty–Cattani) integrals with the Rabotnov fractional-exponential kernel under the condition that μ and ν are synchronous functions. What is more, by the mathematical induction, we prove a new Chebyshev-type inequality in the case that (μ i) i = 1 , ... , n be n positive increasing functions. Finally, we introduce a novel Chebyshev-type inequality via the general fractional-order integrals with the Rabotnov fractional-exponential kernel under the condition that μ and ν are monotonic functions. [ABSTRACT FROM AUTHOR]

High‐precision spectral detection and broadband wavelength sensors have attracted considerable interest for important application in color image sensing. Due to the absorption limitations of silicon, conventional complementary metal–oxide–semiconductors have a relatively limited wavelength region from the ultraviolet to near‐infrared spectrum. Here, a simple‐structured device is developed by assembling two back‐to‐back monolayer graphene (MLG)/Si/MLG heterojunctions for high‐resolution wavelength sensing applications. The single MLG/Si/MLG photodetector exhibits a responsivity of 0.18 A W−1, a specific detectivity of 1.36 × 109 Jones, and an external quantum efficiency of 28.5%. Thanks to the unique device geometry, the distribution of photo‐generation rates in the two photodetectors is completely different, which brings about different photoelectric properties. It is found that the relationship between the photocurrent ratio of both photodetectors and the light wavelength can be fitted by a monotonic function, according to which the wavelength of incident light in the range from 260 to 1050 nm can be accurately estimated. A high resolution of 1 nm is demonstrated across a broadband wavelength from 260 to 1000 nm. This good resolution, along with good repeatability and stability, endow the present device with potential promise in future spectral sensing applications. [ABSTRACT FROM AUTHOR]

In this work, a wavelength sensor that is capable of quantitatively distinguishing the wavelength in the range of ultraviolet (UV) to near-infrared (NIR) light (265–1050 nm) is presented. The wavelength sensor is fabricated by depositing two parallel Au electrodes on the two sides of a $200 ~\mu \text{m}$ Si wafer. It is found that the as-formed two photodetectors display completely different optical properties. And then, the relationship between the photocurrent ratio of two photodetectors and incident wavelength can be described as a numerical function, through which the wavelength from 265 to 1050 nm can be precisely calculated. The unique operation mechanism of the Si wavelength sensor is unveiled by technology computer-aided design (TCAD) simulation. Remarkably, the wavelength sensor easily distinguishes the light with a wavelength difference of 1 nm, which is much better than previously reported devices based on the vertically stacked structures and charge collection narrowing mechanism. [ABSTRACT FROM AUTHOR]

In the present article, topological, metric, and fractal properties of certain sets are investigated. These sets are images of sets whose elements have restrictions on using digits or combinations of digits in own s-adic representations, under the map f, that is a certain distribution function. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of α-fractionally convex functions. We primarily focus on characterizing some of the qualitative properties of convex functions with the assistance of fractional order operators. Also, we discuss the connection of optimality, monotonicity, and convexity of a function in the sense of fractional calculus. Several examples are provided to support the proposed formulation, and one can establish the fractional convexity of a non-convex function. [ABSTRACT FROM AUTHOR]

In this paper, we establish some perturbed versions of the generalized Trapezoid inequality for functions of bounded variation in terms of the cumulative variation function. [ABSTRACT FROM AUTHOR]

In this paper, by applying the generalized Omori-Yau maximum principle for complete spacelike hypersurfaces in warped product spaces, we obtain the sign relationship between the derivative of warping function and support function. Afterwards, by using this result and imposing suitable restrictions on the higher order mean curvatures, we establish uniqueness results for the entire graph in a Riemannian warped product space, which has a strictly monotone warping function. Furthermore, applications to such a space are given. [ABSTRACT FROM AUTHOR]

We show that for any continuous monotonic fixed-point free automor-phism ƒ on a σ-compact subgroup G ⊂ R there exists a binary operation +ƒ such that〈G, +ƒ〉is a topological group topologically isomorphic to 〈G, +〉 and ƒ is a shift with respect to +ƒ. We then show that monotonicity cannot be replaced by a periodic-point free continuous bijections. We explore a few routes leading to generalizations and counterexamples. [ABSTRACT FROM AUTHOR]

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z ∈ [0, 1] is computably random if and only if each nondecreasing computable function [0, 1] → R is differentiable at z. (2) We prove that a real number z ∈ [0, 1] is weakly 2-random if and only if each almost everywhere differentiable computable function [0, 1] → R is differentiable at z. (3) Recasting in classical language results dating from 1975 of the constructivist Demuth, we show that a real z is Martin-L of random if and only if every computable function of bounded variation is differentiable at z, and similarly for absolutely continuous functions. We also use our analytic methods to show that computable randomness of a real is base invariant and to derive other preservation results for randomness notions. [ABSTRACT FROM AUTHOR]

This article presents a range division and contraction algorithm for a class of global optimization problems. In the algorithm, the original problem is first converted into an equivalent monotonic optimization problem whose objective function is just a simple univariate. By exploiting the particularity of this monotonicity, the variable bound in a particular node is then tightened and the bounds on the constraints are calculated to remove the region which doesn't contain optimal solutions. The proposed method can reach an approximate solution within an acceptable error, in which such solution is adequately guaranteed to be feasible and to be close to the actual global optimal solution. Several numerical examples are given to illustrate the feasibility and efficiency of the present algorithm. [ABSTRACT FROM AUTHOR]

In a recent paper the following inequality has been deduced Γ(x+1)<(x2+2)/(x+2),x∈(0, 1). We shall improve this result determining the smallestcfor which the inequality Γ(x+1)<(x2+c)/(x+c),x∈(0, 1) holds. We shall compare this inequality with some known bounds of the Γ function. [ABSTRACT FROM AUTHOR]

This study presents the first attempt of direct continuous‐time model identification using instrumental variable method for Hammerstein–Wiener systems from sampled data. Under the assumption of monotonic function for the Wiener part, the whole non‐linear model is first estimated as an augmented multiple‐input single‐output linear model, from which the model parameters are then extracted by singular value decomposition. A refined instrumental variable method is proposed to consistently identify this non‐linear system acting in a coloured noisy environment. Monte Carlo simulation analysis is presented to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We investigate formal properties, mainly issues connected with propositional logic, of reaction systems introduced by Ehrenfeucht and Rozenberg. We are concerned only with the most simple variant of the systems. Basic properties of propositional formulas are expressed in terms of reaction systems. This leads to NP-completeness (resp. co-NP-completeness) of many problems concerning reaction systems. Among such problems are: (i) deciding whether the function defined by the system is total, (ii) determining the inverse of the function, (iii) deciding whether state sequences always end with a loop. Propositional formulas with monotonic truth-functions yield a particularly simple representation in terms of reaction systems [ABSTRACT FROM AUTHOR]

In the present study, we prove the monotonicity of two functions involving the gamma function Γ(x) and relating to the n-dimensional volume of the unit ball 𝔹 n in ℝ n . [ABSTRACT FROM AUTHOR]

In this paper it is shown that the Hilbert inequality for double series can be improved by introducing a weight function of the form (Multiple line equation(s) cannot be represented in ASCII text), where n ϵ N. A similar result for the Hilbert integral inequality is also given. As applications, some sharp results of Hardy-Littlewood's theorem and Widder's theorem are obtained. [ABSTRACT FROM AUTHOR]

This note describes two conjectures pertaining to repeated partitioning of an arbitrary triangle. The first conjecture turns out to be true, and hence gives rise to a new, more general, conjecture that is also addressed in this article. Both conjectures can be explored in a dynamic geometry environment. The proofs to the conjectures addressed in this article require knowledge of high school Euclidean geometry. [ABSTRACT FROM AUTHOR]

Abstract: We study some properties of the nonlinear Volterra-Stieltjes integral operator of general form which is defined with help of the Stieltjes integral with the kernel depending on two variables. The results obtained in the paper generalize those obtained earlier for the Volterra-Stieltjes integral operators having less general form. We prove also an existence result for nonlinear integral equation of Volterra-Stieltjes type. [Copyright &y& Elsevier]

Abstract: We will study some properties of the integral operator of Volterra-Stieltjes type which is defined with help of the Stieltjes integral with the kernel depending on two real variables. The obtained results are applied to prove an existence result concerning nonlinear integral equation of Volterra-Stieltjes type. [Copyright &y& Elsevier]

This paper presents sufficient conditions on the parameters of the Takagi–Sugeno–Kang (TSK) fuzzy system under which the output of the TSK fuzzy system becomes monotonic with respect to its input. The output of the TSK fuzzy system is first differentiated with respect to its input and the parameter conditions are derived so as to make the derivative nonnegative. The parameter conditions are developed for both single-input and multi-input TSK fuzzy systems where the involved fuzzy membership functions are differentiable everywhere or bar some finite points. The derived monotonicity conditions consist of two parts: the conditions on the consequent parts and the conditions on the fuzzy membership functions. They are further characterized for the most important types of fuzzy membership functions. [Copyright &y& Elsevier]