Integral Version Research Articles

A new integral version of generalized Ostrowski-Grüss type inequality with applications

Our aim is to improve and further generalize the result of integral Ostrowski-Grüss type inequalities involving differentiable functions and then apply these obtained inequalities to probability theory, special means and numerical integration.

Improvements of Slater’s Inequality by Means of 4-Convexity and Its Applications

In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen’s inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater’s inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf–Mandelbrot entropy.

Generalized integral Niezgoda’s inequalities

In this paper, we give generalizations followed by refinements of the integral version of Niezgoda’s inequality in several different ways by using weights, functions with nondecreasing increments and isotonic linear functionals.

On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell

We consider the usual random allocation model of distinguishable particles into distinct cells in the case when there are an even number of particles in each cell. For inhomogeneous allocations, we study the numbers of particles in the first K cells. We prove that, under some conditions, this K-dimensional random vector with centralised and normalised coordinates converges in distribution to the K-dimensional standard Gaussian law. We obtain both local and integral versions of this limit theorem. The above limit theorem implies a χ2 limit theorem which leads to a χ2-test. The parity bit method does not detect even numbers of errors in binary files; therefore, our model can be applied to describe the distribution of errors in those files. For the homogeneous allocation model, we obtain a limit theorem when both the number of particles and the number of cells tend to infinity. In that case, we prove convergence to the finite dimensional distributions of the Brownian bridge. This result also implies a χ2-test. To handle the mathematical problem, we insert our model into the framework of Kolchin’s generalized allocation scheme.

Eigenvalues of singular measures and Connes’ noncommutative integration

In a domain \Omega\subset \mathbb{R}^{\operatorname{N}} we consider compact, Birman–Schwinger type operators of the form \operatorname{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A} with P being a Borel measure in \Omega, containing a singular part, and \mathfrak{A} being an order -\operatorname{N}/2 pseudodifferential operator. Operators are defined by means of quadratic forms. For a class of such operators, we obtain a proper version of H. Weyl's law for eigenvalues, with order not depending on dimensional characteristics of the measure. These results lead to establishing measurability, in the sense of Dixmier–Connes, of such operators and the noncommutative version of integration over Lipschitz surfaces and rectifiable sets.

Bending analysis of nanobeams based on the integral form of nonlocal elasticity using the numerical Rayleigh-Ritz technique

The nonlocal theory is commonly applied for nanomaterials due to its capability in considering size influences. Available studies have shown that the differential version of this theory is not suitable for some problems such as bending of cantilever nanobeams, and the integral version must be used to avoid obtaining inconsistent results. Therefore, an attempt is made in this paper to propose an efficient variational formulation based on the integral nonlocal model for the analysis of nanobeams. The formulation is developed in a general form so that it can be used for arbitrary kernel functions. The nanobeams are modeled using the Bernoulli-Euler beam theory, and their bending behavior is analyzed. Derivation of governing equations is performed according to an energy-based approach. Also, a numerical approach based on the Rayleigh-Ritz method is developed for the solution of problem. Moreover, the results of integral and differential models are compared. It is revealed that by the proposed numerical solution, the paradox in the behavior of nanocantilever is resolved.

Extreme values of the resurgence for homogeneous ideals in polynomial rings

We show that two ostensibly different versions of the asymptotic resurgence introduced by E. Guardo, B. Harbourne and A. Van Tuyl in 2013 are the same. We also show that the resurgence and asymptotic resurgence attain their maximal values simultaneously, if at all, which we apply to a conjecture of E. Grifo. For radical ideals of points, we show that the resurgence and asymptotic resurgence attain their minimal values simultaneously. In addition, we introduce an integral closure version of the resurgence and relate it to the other versions of the resurgence. In closing we provide various examples and raise some related questions, and we finish with some remarks about computing the resurgence.

Spatially discrete Hirota equation: Rational and breather solution, gauge equivalence, and continuous limit

It is well known that the nonlinear Schrödinger equation, the Hirota equation, and their integrable discrete versions are very important integrable equations. These integrable equations not only have deep integrability theory, but also have wide physical applications. In this paper, we focus on an integrable spatially discrete Hirota equation. The new Lax pair, Darboux transformation, rational wave solution, rogue wave solution, breather solutions and gauge equivalent structure of the spatially discrete Hirota equation are investigated. The continuous limit of the rogue wave solution and breather solution of the discrete Hirota equation yields the counterparts of the Hirota equation, which shows that the spatially discrete Hirota equation is a very good model for the numerical analysis for considering the Cauchy problem with a general initial date of the Hirota equation. Besides, we prove that this spatially discrete Hirota equation is gauge equivalent to a discrete integrable generalized Heisenberg spin model under the discrete gauge transformation. In this performance, we see that utilizing the new Lax pair, an essential properties of integrability, of the spatially discrete Hirota equation plays a significant role in the investigation of continuous limit of Lax pair of discrete integrable generalized Heisenberg spin model.

On Hadamard Type Fractional Inequalities for Riemann–Liouville Integrals via a Generalized Convexity

In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,h−m)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions.

New Method of Sensitivity Computation Based on Markov Models with Its Application for Risk Management

Sensitivity analysis is at the core of risk management for financial engineering; to calculate the sensitivity with respect to parameters in models with probability expectation, the most traditional approach applies the finite difference method, whereafter integration by parts formula was developed based on the Brownian environment and applied in sensitivity analysis for better computational efficiency than that of finite difference. Establishing a similar version of integration by parts formula for the Markovian environment is the main focus and contribution of this paper. It is also shown by numerical simulation that our proposed methodology and approach outperform the traditional finite difference method for sensitivity computation. For empirical studies of sensitivity analysis on an NPV (net present value) model, we show the approaches of modeling, especially for parameter estimation of Markov chains given data of company loan states. Applying our newly established integration by parts formula, numerical simulation estimates the variations caused by the capital return rate and multiplier of overdue loan. Furthermore, managemental implications of these results are discussed for the effectiveness of modeling and the investment risk control.

New Generalized Riemann–Liouville Fractional Integral Versions of Hadamard and Fejér–Hadamard Inequalities

In this paper, a new class of functions, namely, exponentially (α, h − m) − p‐convex functions is introduced to unify various classes of functions already defined in the subject of convex analysis. By using this class of functions, generalized versions of well known fractional integral inequalities of Hadamard and Fejér–Hadamard type are obtained. The results of this paper generate fractional integral inequalities of Hadamard and Fejér–Hadamard type for various types of convex and exponentially convex functions simultaneously.

The German Translations of Lanfranc of Milan’s Chirurgia parva: from Latin to Vernacular or from One Vernacular to the Other?*

Abstract Lanfranc of Milan is considered as one of the most influential surgeons of the late Middle Ages and his Latin works – the Chirurgia parva and the Chirurgia magna – have informed surgical practice until the 16th century. The importance of his works (1293/1294 and 1296) is witnessed by the large number of vernacular translations produced after their original compilation. Overall, three High German translations of Lanfranc’s Chirurgia parva have come down to us: 1. the significantly shortened version transmitted in Vatican City, MS Pal. Lat, 1117; 2. the integral version preserved in Kalocsa, Cathedral Library, MS 376 and in Erlangen, University Library, MS B 3; and 3. the early modern printed translation by Otto Brunfels. Even though the Latin source of these translations has never been identified precisely, it has been traditionally taken for granted that they were based on a Latin model. Nevertheless, a series of correspondences between the two manuscript versions and the French translation transmitted in Bern, Burgerbi­bliothek, MS A 85.2 seem to indicate that a vernacular source is also possible. In this study, the two German manuscript translations of Lanfranc’s Chirurgia parva will be taken into consideration with respect to the problems connected with the identification of their source(s), trying to ascertain whether they are really based on a Latin model, or, as suggested by Claude De Tovar, they are rather connected with the French vernacularization of the text witnessed by the Bern manuscript.

Giaccardi Inequality for s‐Convex Functions in the Second Sense for Isotonic Linear Functionals and Associated Results

In this paper, a well‐known inequality called Giaccardi inequality is established for isotonic linear functionals by applying s‐convexity in the second sense, which leads to notable Petrović inequality. As a special case, discrete and integral versions of Giaccardi inequality are derived along with the Petrović inequality as a particular case. In application point of view, newly established inequalities are derived for different time scales.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

DISCO: a database of Deeply Integrated human Single-Cell Omics data.

The ability to study cellular heterogeneity at single cell resolution is making single-cell sequencing increasingly popular. However, there is no publicly available resource that offers an integrated cell atlas with harmonized metadata that users can integrate new data with. Here, we present DISCO (https://www.immunesinglecell.org/), a database of Deeply Integrated Single-Cell Omics data. The current release of DISCO integrates more than 18 million cells from 4593 samples, covering 107 tissues/cell lines/organoids, 158 diseases, and 20 platforms. We standardized the associated metadata with a controlled vocabulary and ontology system. To allow large scale integration of single-cell data, we developed FastIntegration, a fast and high-capacity version of Seurat Integration. We also developed CELLiD, an atlas guided automatic cell type identification tool. Employing these two tools on the assembled data, we constructed one global atlas and 27 sub-atlases for different tissues, diseases, and cell types. DISCO provides three online tools, namely Online FastIntegration, Online CELLiD, and CellMapper, for users to integrate, annotate, and project uploaded single-cell RNA-seq data onto a selected atlas. Collectively, DISCO is a versatile platform for users to explore published single-cell data and efficiently perform integrated analysis with their own data.

Denjoy-type Integrals in Locally Convex Topological Vector Space

In this paper, we introduce AC* and ACG*-type properties and then, using theseconditions along with other concepts, define two Denjoy-type integrals of a function with values in a locally convex topological vector space (LCTVS). We show, among others, that these newly defined integrals are included in the SH integral, a stronger version of the Henstock integral for LCTVS-valued functions.

ON THE SL-INTEGRAL OF LCTVS-VALUED FUNCTIONS

A function F:[a,b]→X is said to be an SL function if it satisfies the Strong Lusin (SL) condition given as follows: for every θ-nbd U and a set E⊂[a,b] of measure zero, there exists a gauge δ such that for every δ-fine partial partition D={([xi-1,xi],ti):1≤i≤n} of [a,b] with ti∈E, there exist θ-nbds U1,U2,…,Un such that ∑i=1nUi⊆V and F(xi)-F(xi-1)∈Ui for each i=1,2,…,n. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.

GENERALIZATION AND REFINEMENTS OF JENSEN INEQUALITY

We give generalizations and refinements of Jensen and Jensen− Mercer inequalities by using weights which satisfy the conditions of Jensen and Jensen− Steffensen inequalities. We also give some refinements for discrete and integral version of generalized Jensen−Mercer inequality and shown to be an improvement of the upper bound for the Jensen’s difference given in [32]. Applications of our work include new bounds for some important inequalities used in information theory, and generalizing the relations among means.

ISS and integral-ISS of switched systems with nonlinear supply functions

The problem of input-to-state stability (ISS) and its integral version (iISS) are considered for switched nonlinear systems with inputs, resets and possibly unstable subsystems. For the dissipation inequalities associated with the Lyapunov function of each subsystem, it is assumed that the supply functions, which characterize the decay rate and ISS/iISS gains of the subsystems, are nonlinear. The change in the value of Lyapunov functions at switching instants is described by a sum of growth and gain functions, which are also nonlinear. Using the notion of average dwell-time (ADT) to limit the number of switching instants on an interval, and the notion of average activation time (AAT) to limit the activation time for unstable systems, a formula relating ADT and AAT is derived to guarantee ISS/iISS of the switched system. Case studies of switched systems with saturating dynamics and switched bilinear systems are included for illustration of the results.

Bounds for the Jensen Gap in terms of Power Means with Applications

Jensen’s and its related inequalities have attracted the attention of several mathematicians due to the fact that Jensen’s inequality has numerous applications in almost all disciplines of mathematics and in other fields of science. In this article, we propose new bounds for the difference of two sides of Jensen’s inequality in terms of power means. An example has been presented for the importance and support of the main results. Related results have been given in quantum calculus. As consequences, improvements of quantum integral version of Hermite-Hadamard inequality have been derived. The obtained inequalities have been applied for some well-known inequalities such as Hermite-Hadamrd, Hölder, and power mean inequalities. Finally, some applications are given in information theory. The tools performed for obtaining the main results may be applied to obtain more results for other inequalities.