Brand Awareness via Online Media: An Evidence Using Instagram Medium with Statistical Analysis

Online marketing refers to the practices of promoting a company’s brand to its potential customers. It helps the companies to find new venues and trade worldwide. Numerous online media such as Facebook, YouTube, Twitter, and Instagram are available for marketing to promote and sell a company’s product. However, in this study, we use Instagram as a marketing medium to see its impact on sales. To carry out the computational process, the approach of linear regression modeling is adopted. Certain statistical tests are implemented to check the significance of Instagram as a marketing tool. Furthermore, a new statistical model, namely a new generalized inverse Weibull distribution, is introduced. This model is obtained using the inverse Weibull model with the new generalized family approach. Certain mathematical properties of the new generalized inverse Weibull model such as moments, order statistics, and incomplete moments are derived. A complete mathematical treatment of the heavy-tailed characteristics of the new generalized inverse Weibull distribution is also provided. Different estimation methods are discussed to obtain the estimators of the new model. Finally, the applicability of the new generalized inverse Weibull model is established via analyzing Instagram advertising data. The comparison of the new distribution is made with two other models. Based on seven analytical tools, it is observed that the new distribution is a better model to deal with data in the business, finance, and management sectors.

Review: Horizontal, directionally drilled and radial collector wells

Horizontal wells play an often overlooked role in hydrogeology and aquifer remediation but can be an interesting option for many applications. This study reviews the constructional and hydraulic aspects that distinguish them from vertical wells. Flow patterns towards them are much more complicated than those for vertical wells, which makes their mathematical treatment more demanding. However, at some distance, the drawdown fields of both well types become practically identical, allowing simplified models to be used. Due to lower drawdowns, the yield of a horizontal well is usually higher than that of a vertical well, especially in thin aquifers of lower permeability, where they can replace several of the latter. The lower drawdown, which results in lower energy demand and slower ageing, and the centralized construction of horizontal wells can lead to lower operational costs, which can make them an economically feasible option.

A Critical Review on the Complex Potentials in Linear Elastic Fracture Mechanics

Introducing a crack in an elastic plate is challenging from the mathematical point of view and relevant within an engineering context of evaluating strength and reliability of structures. Accordingly, a multitude of associated works is available to date, emanating from both applied mathematics and mechanics communities. Although considering the same problem, the given complex potentials prove to be different, revealing various inconsistencies in terms of resulting stresses and displacements. Essential information on crack near-tip fields and crack opening displacements is nonetheless available, while intuitive adaption is required to obtain the full-field solutions. Investigating the cause of prevailing deficiencies inevitably leads to a critical review of classical works by Muskhelishvili or Westergaard. Complex potentials of the mixed-mode loaded Griffith crack, sparing restrictive assumptions or limitations of validity, are finally provided, allowing for rigorous mathematical treatment. The entity of stresses and displacements in the whole plate is finally illustrated and the distributions in the crack plane are given explicitly.

Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion

<abstract><p>It is the objective to provide a mathematical treatment of a model to predict the behaviour of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and uniqueness of solutions together with a comparison principle. Once the regularity principles are shown, the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW profile follows a potential law in the stable connection that converges to the stationary solution. Such potential law suggests that the pressure induced by the invasive over the hostile area increases over time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent invasion.</p></abstract>

The critical values of coupling strength of the non-self-dual Ising lattices under decimation transformation

In this work, the values of critical coupling strengths of the Ising lattices which are changing their lattice structure (or non-self-dual) under decimation transformations, such as the honeycomb, the triangular and the body centered cubic Ising lattices, are obtained by a modified real space renormalization group approach (RSRG). This modification is necessary to obtain a proper relation between the coupling strengths of the original and the decimated lattices. Indeed, we have achieved to obtain a proper renormalized coupling strength relation for honeycomb and triangular lattices readily, since the decimation transformation of the honeycomb lattice produces the triangular lattice or vice versa. Here, the problem of having physically untractable interactions between degrees of freedom in the renormalized Hamiltonian, which leads eventually to inevitable approximations in the treatment, except for the 1D Ising chain, has been solved with a proper approximation. Especially for the 3D Ising lattices, the physically untractable interactions appearing in the renormalized Hamiltonian make the mathematical treatment too cumbersome. As a result, there is not enough research dealing with the 3D Ising lattices using RG theory. Our approximation is based on using the simple relation [Formula: see text], which is, of course, a very relevant first-order approximation, if [Formula: see text]. With the help of this approximation, decimation transformation process produces only pairwise interactions in the renormalized Hamiltonian instead of having four spins, six spins, or even eight spin interactions which appear naturally if all the terms are kept in the renormalized Hamiltonians of the Ising lattices in 2D and higher dimensions. Without this approximation, one cannot apply analytic RG treatment feasibly to even simple cubic lattice, let alone applying it to the body centered cubic lattice. Using this modified RG approach, the values of critical coupling strengths of the honeycomb, the triangular and the body centered cubic Ising lattices are obtained analytically as [Formula: see text], [Formula: see text] and [Formula: see text] respectively. Apparently, these estimations are really close to the results obtained from cumbersome exact treatments which are [Formula: see text], [Formula: see text] and [Formula: see text] for the honeycomb, the triangular and the body centered cubic lattices.

Super-resolved 3D mapping of molecular orientation with vibrational techniques

When a sample has an anisotropic structure, it is possible to obtain different information, when changing polarization of incident light. Using polarized light of a single vibrational band to determine the in-plane orientation and internal ordering of a sample is a typical practice in materials science. Acquiring mapping data at four different polarizations with a stationary sample than just at two polarizations offers much more insight into the sample structure with proper mathematical treatment. A concurrent analysis of two vibrational bands with perpendicular transition moment orientations allows the understanding of the orientational ordering in three dimensions. We show here, to the best of our knowledge, the first application of concurrent analysis to IR spectromicroscopy data and obtain orientation angles of a model spherulite polycaprolactone sample. Moreover, we show that this method can be easily applied to high resolution, diffraction limited FT-IR and Raman imaging and even to sub-diffraction limit O-PTIR imaging. Due to the non-tomographic experimental approach, no image distortion is visible and nanometer scale orientation domains can be observed. 3D bond orientation maps will enable in-depth characterization of sample structure in a quantitative manner enabling more precise control of their physicochemical properties and function.

Super-resolved 3D mapping of molecular orientation with vibrational techniques

When a sample has an anisotropic structure, it is possible to obtain different information, when changing polarization of incident light. Using polarized light of a single vibrational band to determine the in-plane orientation and internal ordering of a sample is a typical practice in materials science. Acquiring mapping data at four different polarizations with a stationary sample than just at two polarizations offers much more insight into the sample structure with proper mathematical treatment. A concurrent analysis of two vibrational bands with perpendicular transition moment orientations allows the understanding of the orientational ordering in three dimensions. We show here, to the best of our knowledge, the first application of concurrent analysis to IR spectromicroscopy data and obtain orientation angles of a model spherulite polycaprolactone sample. Moreover, we show that this method can be easily applied to high resolution, diffraction limited FT-IR and Raman imaging and even to sub-diffraction limit O-PTIR imaging. Due to the non-tomographic experimental approach, no image distortion is visible and nanometer scale orientation domains can be observed. 3D bond orientation maps will enable in-depth characterization of sample structure in a quantitative manner enabling more precise control of their physicochemical properties and function.

Automated classification technique for edge-on galaxies based on mathematical treatment of brightness data

Classification of edge-on galaxies is important to astronomical studies due to our Milky Way galaxy being an edge-on galaxy. Edge-on galaxies pose a problem to classification due to their less overall brightness levels and smaller numbers of pixels. In the current work, a novel technique for the classification of edge-on galaxies has been developed. This technique is based on the mathematical treatment of galaxy brightness data from their images. A special treatment for galaxies’ brightness data is developed to enhance faint galaxies and eliminate adverse effects of high brightness backgrounds as well as adverse effects of background bright stars. A novel slimness weighting factor is developed to classify edge-on galaxies based on their slimness. The technique has the capacity to be optimized for different catalogs with different brightness levels. In the current work, the developed technique is optimized for the EFIGI catalog and is trained using a set of 1800 galaxies from this catalog. Upon classification of the full set of 4458 galaxies from the EFIGI catalog, an accuracy of 97.5% has been achieved, with an average processing time of about 0.26 seconds per galaxy on an average laptop.

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

Effect of Compliance on Residual Stresses in Manufacturing with Moving Heat Sources

Manufacturing processes involving moving heat sources include additive manufacturing, welding, laser processing (cladding and heat treatment), machining, and grinding. These processes involve high local thermal stresses that induce plasticity and result in permanent residual stress and distortion. The residual stresses are typically calculated numerically at great computational expense despite the fact that the inelastic fraction of the domain is very small. Efforts to decouple the small plastic part from the large elastic part have led to the development of the tendon force concept. The tendon force can be predicted analytically for the case of infinitely rigid components; however, this limitation has prevented the broader use of the concept in practical applications. This work presents a rigorous mathematical treatment using dimensional analysis, asymptotics, and blending which demonstrates that the effect of geometric compliance depends on a single dimensionless group, the Okerblom number. Closed-form expressions are derived to predict the effect of compliance without the need for empirical ad-hoc fitting or calibration. The proposed expressions require input of only material properties and tabulated process parameters, and are thus ideally suited for use in metamodels and design calculations, as well as incorporation into engineering codes and standards.