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A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order α, 0 < α ≤ 1, called F^α-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called F^α-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The F^α-derivative is local unlike the classical fractional derivative. The F^α-calculus retains much of the simplicity of ordinary calculus. Several results including analogues of fundamental theorems of calculus are proved.

The integral staircase function, which is a generalization of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension.

Spaces of F^α-differentiable and F^α-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and F^α-differentiability is generalized using Sobolev-like construction.

F^α-differential equations are equations involving F^α-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviors are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one-dimensional motion of a particle undergoing friction in a fractal medium.

Calculus on fractals, or F^α-calculus, developed in a previous paper, is a calculus based fractals F ⊂ R, and involves F^α-integral and F^α-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The F^α-integral is suitable for integrating functions with fractal support of dimension α, while the F^α-derivative enables us to differentiate functions like the Cantor staircase. Several results in F^α-calculus are analogous to corresponding results in ordinary calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of F^α-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour.

In this paper we show that, as operators, the F^α-integral and F^α-derivative are conjugate to the Riemann integral and ordinary derivative respectively. This is accomplished by constructing a map ψ which takes F^α-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes F^α-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in ordinary calculus and F^α-calculus.

This conjugacy is useful, among other things, to find solutions to F^α-differential equations: they can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

Over the last five years, Industry 4.0 (I4.0) has gained a lot of attention from industry leaders, policymakers, and government officials worldwide. In an era where new concepts and techniques are introduced continuously, there is a lack of systematic literature review (SLR) on identifying main dimensions, levels, methods to assess the maturity and readiness level toward I4.0. To address this gap, we have chosen our primary objective to provide a critical review of existing literature on dimensions, methods, levels, and current trends to evaluate the I4.0 maturity and readiness models. A Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) methodology was adopted to make sure that there is no replication and to maintain complete transparency in the review process. A total of 53 papers were deemed relevant for thematic analysis. From the literature, we have found and proposed 10 main dimensions — Strategy and Organization, Manufacturing and Operations, Supply Chain, Business Model, IT, People, Customers, Product, Services, and Culture, to assess the I4.0 maturity and readiness levels of an organization. Further, a conceptual framework was proposed for the same. This study contributes theoretically to the development of I4.0 maturity and readiness models. So far, this is the first review paper on dimensions of I4.0 maturity and readiness models and is expected to give future researchers and practitioners a holistic guideline to design and develop I4.0 maturity and readiness models.

We propose that the effective dimensionality of the space we live in depends on the length scale we are probing. As the length scale increases, new dimensions open up. At short scales the space is lower dimensional; at the intermediate scales the space is three-dimensional; and at large scales, the space is effectively higher dimensional. This setup allows for some fundamental problems in cosmology, gravity, and particle physics to be attacked from a new perspective. The proposed framework, among the other things, offers a new approach to the cosmological constant problem and results in striking collider phenomenology and may explain elongated jets observed in cosmic-ray data.

Background: The terminal phalanx of the fingers carries the attachment of the Flexor Digitorum Profundus (FDP) on the volar surface and the extensor on the dorsal surface. Avulsion of these tendons has traditionally been repaired with pull-through sutures. Recently, bone anchor sutures have been found to be of comparable biomechanical strength but with the added advantage of technical ease and fewer complications. However, the dimensions of the bone, at the site of insertion of the anchors, have never been studied.

Methods: Following some cases of penetration of the dorsal cortex by the anchors, we measured the antero-posterior dimensions of the terminal phalanx in 251 digits from plain radiographs and compared these with the dimensions of the commonly used bone anchors. We also compared male and female digits.

Results: The anchors were oversized in 76% of index, 78% of ring and 100% of little fingers in the female population and in 49%, 44% and 97% of index, ring and little fingers respectively in the male population.

Conclusions: This analysis of bone dimensions can be a useful guide to surgeons in choosing the appropriate implant for flexor tendon avulsions.

Online Analytical Processing, or OLAP, is an approach to answering multidimensional analytical (MDA) queries in an interactive way. However, the traditional OLAP approaches can only deal with structured data, but not unstructured textual data like tweets. To address this problem, we propose a Latent Dirichlet Allocation (LDA)-based model, called Multilayered Semantic LDA (MS-LDA), which detects the hidden layered interests from Twitter data based on LDA. The layered dimension of interests can be further used to apply OLAP techniques to Twitter data. Furthermore, MS-LDA employs the semantic similarity among words of tweets based on word2vec, and also the social relationship among twitters, to improve its effectiveness. The extensive experiments demonstrate that MS-LDA can effectively extract the dimension hierarchy of tweeters' interests for OLAP.