On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

We have studied the Opial-type inequalities for superquadratic functions proved for arbitrary kernels. These are estimated by applying mean value theorems. Furthermore, by analyzing specific functions, the fractional integral and fractional derivative inequalities are obtained.

Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

Multiscale Tomographic Wave-Matter Interaction Modeling to Enable Artifact-Free Material Defect Reconstruction

Technologies for material defect detection/metrology are often based on measuring the interactions between defects and waves. These interactions frequently create artifacts that skew the quantitative character of the relevant measurements. Since defects can have a significant impact on the functional behavior of the materials and structures they are embedded in, accurate knowledge of their geometric shape and size is necessary. Responding to this need, the present work introduces preliminary efforts towards a multiscale modeling and simulation framework for capturing the interactions of waves with materials bearing defect ensembles. It is first shown that conventional approaches such as ray tracing result in excessive geometric errors. Instead, a more robust method employing solutions to the wave equation (calculated using the Finite Element Method) is developed. Although the use of solutions to the general wave equation permits application of the method to many wave-based defect detection technologies, this work focuses exclusively on the application to X-ray computed tomography (XCT). A general parameterization of defect geometries based on superquadratic functions is also introduced, and the interactions of defects modeled in this fashion with X-rays are investigated. A synthetic two-dimensional demonstration problem is presented. It is shown that the combination of parameterization and modeling techniques allows the recovery of an accurate, artifact-free defect geometry utilizing classical inverse methods. The path forward to a more complete realization of this technology, including extensions to other wave-based technologies, three-dimensional problem domains, and data derived from physical experiments is outlined.

Inequalities of Hardy Type via Superquadratic Functions with General Kernels and Measures for Several Variables on Time Scales

We use the properties of superquadratic functions to produce various improvements and popularizations on time scales of the Hardy form inequalities and their converses. Also, we include various examples and interpretations of the disparities in the literature that exist. In particular, our findings can be seen as refinements of some recent results closely linked to the time-scale inequalities of the classical Hardy, Pólya-Knopp, and Hardy-Hilbert. Some continuous inequalities are derived from the main results as special cases. The essential results will be proved by making use of some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales.

Multiscale Tomographic Wave-Matter Interaction Modeling to Enable Artifact-Free Material Defect Reconstruction

The fabrication and subsequent in-service excitation of materials/structures invariably generates significant defects including cracks, voids and inclusions, spanning many length scales. The various technologies available for detecting and qualitatively describing such defects suffer from the introduction of various degrees inaccuracy when it comes to the quantification of defect geometry. This can have an adverse impact on modeling and understanding how the material/part/structure performance is effected by these defects. The main cause of this shortcoming is that aspects of the physical processes used to interrogate the material system, using monochromatic or polychromatic waves such as X-ray, mm-wave, or ultrasound, are not taken into account. These waves interact with the multiscale defect ensemble in a complex fashion that inevitably produces spurious “artifacts” in the resulting data, which cannot be removed via conventional data post-processing. These artifacts then introduce unacceptable levels of error when reconstructing defect geometry and computing the remaining lifespan of defect-bearing materials/structures. The present work introduces preliminary efforts towards a multiscale modeling and simulation framework for capturing the interactions of waves (such as X-rays) with materials bearing defect ensembles. It is shown that conventional approaches such as ray tracing are not adequate, and a more robust solution to the relevant wave equations utilizing the Finite Element discretization is employed. A general parameterization of defect geometries based on superquadratic functions is also introduced, and the interactions of defects modeled in this fashion with X-rays are investigated. It is also shown that this combination of parameterization and modeling techniques allows the recovery of true, artifact-free defect geometry utilizing classical inverse methods. The methodology is demonstrated using synthetic tomographic data, and the path forward to a more complete realization of this technology is outlined.

Klein's Trace Inequality and Superquadratic Trace Functions

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left| {\frac{{A - B}}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular improvement of Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.

External Jensen-type operator inequalities via superquadraticity

In this paper we establish several Jensen-type operator inequalities for a class of superquadratic functions and self-adjoint operators. Our results are given in the so-called external form. As an application, we give improvements of the H?lder?McCarthy inequality and the classical discrete and integral Jensen inequality in the corresponding external forms. In addition, the established Jensen-type inequalities are compared with the previously known results and we show that our results provide more accurate estimates in some general settings.