Intermediate dimensions were introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine ...Bedford–McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. In this paper, we determine a precise formula for the intermediate dimensions dimθΛ of any Bedford–McMullen carpet Λ for the whole spectrum of θ∈0,1, in terms of a certain large deviations rate function. The intermediate dimensions exist and are strictly increasing in θ, and the function θ↦dimθΛ exhibits interesting features not witnessed on any previous example, such as having countably many phase transitions, between which it is analytic and strictly concave.
We make an unexpected connection to multifractal analysis by showing that two carpets with non-uniform vertical fibres have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of the uniform Bernoulli measures on the two carpets are equal. Since intermediate dimensions are bi-Lipschitz invariant, this shows that the equality of these multifractal spectra is a necessary condition for two such carpets to be Lipschitz equivalent.
Consider (Y,ρ) as a complete metric space and S as the space of probability Borel measures on Y. Let dim‾BΨ,Φ(E) be the general upper box dimension of the set E⊂Y. We begin by proving that the ...general packing dimension of the typical compact set, in the sense of the Baire category, is at least inf{dim‾BΨ,Φ(B(x,r))|x∈Y,r>0} where B(x,r) is the closed ball in Y with center at x and radii r>0. Next, we obtain some estimates of the general upper and lower box dimensions of typical measures in the sense of the Baire category. Finally, we demonstrate that if S is equipped with the weak topology and under some assumptions then the set of measures possessing the general upper and lower correlation dimension zero are residual. Furthermore, the general upper correlation dimension of typical measures (in the sense of the Baire category) is approximated through the general local lower and upper entropy dimensions of Y.
Industry 4.0 technologies provide critical perspectives for future innovation and business growth. Technologies like Artificial Intelligence (AI), Internet of Things (IoT), Big data, Machine Learning ...(ML), and other advanced upcoming technologies are being used to implement Industry 4.0. This paper explores how Industry 4.0 technologies help create a sustainable environment in manufacturing and other industries. Industry 4.0 technologies and the crucial interrelationships through advanced technologies should impact the environment positively. In the age of Industry 4.0, manufacturing is tightly interlinked with information and communication systems, making it more scalable, competitive, and knowledgeable. Industry 4.0 provides a range of principles, instructions, and technology for constructing new and existing factories, enabling consumers to choose different models at production rates with scalable robotics, information, and communications technology. This paper aims to study the significant benefits of Industry 4.0 for sustainable manufacturing and identifies tools and elements of Industry 4.0 for developing environmental sustainability. This literature review-based research is undertaken to identify how Industry 4.0 technologies can help to improve environmental sustainability. It also details the capabilities of Industry 4.0 in dealing with environmental aspects. Twenty major applications of Industry 4.0 to create a sustainable environment are identified and discussed. Thus, it gives a better understanding of the production environment, the supply chains, the delivery chains, and market results. Overall, Industry 4.0 technology seems environmentally sustainable while manufacturing goods with better efficiency and reducing resource consumption.
The present manuscript delineates the derivation of strong solutions for the linear elasticity problem in a two dimensional rectangular beam. The materials under consideration can exhibit either ...isotropic or orthotropic properties. Additionally, the analysis is not restricted to slender beams because the ratio between the length and the height of the beam can be arbitrary. The boundary conditions fall into the Dirichlet category, implying that both horizontal and vertical displacements are specified on all four surfaces. The sole requirement is that these surface displacements are continuous functions, although they may not necessarily be smooth. Since the displacements at the surfaces can be arbitrary, there is no need to consider approximations, such as those concerning local (small) boundaries in slender beams. Nonetheless, it is demonstrated that an equivalent principle to the Saint-Venant principle exists for pure displacement boundary conditions. The partial differential equations are solved through the separation of variables method, leading to the identification of two solution types, encompassing both cosine and sine Fourier series. Particular emphasis is placed on evaluating the convergence of these solutions. For two distinct and general examples, it is confirmed that the solutions indeed exist.
•Derive strong solutions for linear elasticity in a 2D rectangular beam.•The boundary conditions are of the Dirichlet category.•The surface displacements are continuous but not necessarily smooth.•Identify two type of solutions and evaluate the convergence.•Confirm existence of solutions through two distinct general examples.
A
bstract
In this paper, we consider a coupling mechanism between the kinetic term and the gravity, in which a coupling function
F
(
R
) is introduced into the kinetic term of the five-dimensional ...scalar field. Based on this scenario, we investigate the localization of scalar fields in three specific braneworld models: the Minkowski brane, the de Sitter brane, and the Anti-de Sitter brane. The brane models considered here are regular with no singularity for scalar curvature. For the Minkowshi brane case, the zero mode can always be localized on the brane, and the massive modes can be localized or quasi-localized on the brane. For the dS
4
brane case, two forms of factor
F
(
R
) is considered. The zero mode can always be localized, and the massive modes could be quasi-localized on the brane. Besides, with the second coupling factor, the scalar zero mode could be localized on the both sides of the origin of extra dimension, while the massive modes could be quasi-localized on the origin. Lastly, for the AdS
4
brane case, the localization of the scalar zero mode requires the consideration of a coupling potential
V
(Φ,
φ
), while the massive modes can still be localized on the brane with an infinite number.
In this paper, Pakistan's energy security performance based on a study of its energy security index during 1991–2018 is assessed. The assessment, based on an analysis of the energy security and ...energy policies of Pakistan, focuses on the concepts of “Availability”, “Affordability”, “Technology”, “Governance”, and “Environment”. Twenty-two indicators, termed the Energy Security Indicators for Pakistan (ESIP), were selected. The ESIP were normalised using the z-score method and weighted based on principal component analysis (PCA). The index analysis revealed that Pakistan's energy security performance decreased between 1991 and 1999, followed by an increase until 2018. A maximum score of 8.36 was attained in 1991, and a minimum of 7.59 was achieved in 1999. During 2000–2018, the performance improved, resulting in a score of 8.29 in 2018. Supply, consumption, and import indicators played a key role in the energy security performance in the studied period.
•The energy security of Pakistan is based on Availability, Affordability, Technology, Governance, and Environment.•From 320 published indicators, seven themes identified in dimensions.•22 commonly used indicators are selected in themes and are termed as Energy Security Indicators for Pakistan (ESIP).•PCA is utilized for index formulation with z-score normalisation.•Pakistan’s energy security performance declined followed by continuous improvement.
The view that innovation is a key driver for sustainability is widely accepted among scholars, industry professionals, and government representatives. This is due to the fact that sustainable ...development is a pressing issue that requires immediate action and changes from governments, industry, and society as whole. This article reviews the literature on innovations that can lead to transformations in individuals, organizations, supply chains, and communities toward a sustainable future. Although many of the articles explored in this review report on existing urgent environmental and social issues, their findings, recommendations, and contributions are encouraging as we make progress toward a sustainable society through innovation and change. This article reviews the diversity of innovation for Sustainable Development in the literature, proposes a typology of such a phenomenon, provides an overview of key articles based on the primary subjects they address, and identifies a series of recommendations for the future development of the field.
•This article proposes a typology of innovation for sustainable development.•An organization, supply chain, or community evolves on a sustainability trajectory.•These trajectories are the paths these entities take to become more sustainable.•Learning is a crucial element for these entities to pursue sustainable innovations.•Sustainable innovation improves entities' sustainability trajectories and performance.
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