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The oscillon is a highly localized dynamical phenomenon occurring in a thin horizontal layer of granular material, which rests on a rigid plate oscillating in the vertical direction. The geometry is axially symmetric and physically resembles a splash of liquid due to a falling drop, except that it continually perpetuates itself and does not generate a spreading wave, as is the case for a liquid splash. The oscillon moves from "peak" to "crater" and "crater" to "peak" such that the time from "peak" to "peak" or "crater" to "crater" is twice the period of the oscillating plate. The physics of granular phenomena is not properly understood, and there is no continuum mechanical theory of granular materials which is widely accepted as accurately describing their behaviour. Here for a free-flowing (cohesion-less) granular material, under axially symmetric conditions, we present a partial continuum mechanical analysis assuming the Coulomb-Mohr yield function and non-dilatant double-shearing theory. We examine small perturbations superimposed upon a purely vertical vibration, and make the assumption that throughout the motion, the lower surface of the layer remains in contact with the rigid metal plate. We show how the temporal dependence, which decouples from the spatial structure, is governed by Mathieu's equation for the physically relevant case of the rigid plate oscillating sinusoidally, and therefore stability is determined by certain key parameters. We explore a variety of possible forms for spatial dependence. The present axially symmetric analysis complements that presented by the authors for plane strain conditions, and we find, quite remarkably, that apart from constants, both flows are governed by similar fourth-order systems of ordinary differential equations. This means that for both plane strain and axially symmetry, analogous pattern forming conditions can operate. Refereed/Peer-reviewed

Refereed/Peer-reviewed The elastic and thermal properties of graphene were investigated by illuminating graphen bubbles with a laser spot. Tempertures above 1000°C were obtained in large (>10μm) graphene bubbles. The formation of standing optical waves lead to laser heating depending on the height of the garphene bubble, which results in Raman band oscillations when scanning the laser spot across the bubble. The profile of the bubble under laser illumination can be deduced from the Raman G band oscillations. A distinct swelling at the center of the bubble is observed which is attributed to the strong softening of graphene above 1000°C. From the size and height of the swelling it is deduced that the elastic modulus is reduced by at least 40% at 1000°C. On solving the heat equation for the heat dissipation through the graphene only, analytical expressions are obtained for the isotherms on the bubble for both symmetrical and asymmetrical positions of the laser spot. It was found that a large fraction of the absorbed heat dissipates through the gas in the bubble using the finite volume method. Analytical expressions for the temperature distribution in the bubble are deduced from the numerical results. Heat conduction through the gas in the bubble influences the temperature distribution and needs to be taken into account in the heat dissipation in graphene bubbles.

The dark issues of cosmological mechanics imply that our accounting for mass and energy at this scale is incorrect. On the other hand, existing theory not only accounts for atomic physics, but does so to a very high degree of accuracy. de Broglie was first to propose a concrete physical picture of the co-existence of both particle and its associated wave. We have previously proposed a Lorentz invariant modification of Newton’s second law that applies when the particle energy itself is of comparable magnitude to the potential energy of the applied external field. A dual particle–wave formulation is developed that allows exceptions to the law that matter cannot be created or destroyed, and accommodates both particle and wave energies. Here we examine this formulation in a centrally or spherically symmetric gravitating environment. A logical analysis of the integrated rate-of-working equation gives rise to four distinct states of matter; positive and negative energies with either non-zero rest mass or zero rest mass. This identification is meaningful only within the extended theory, and it has no meaning within conventional theory. We propose that dark matter and dark energy arise from a particular alignment of the spatial physical force f and the force g in the direction of time, such that the particle and wave energies coincide, and a consistent mathematical framework supports this proposal. Allowable potentials V( r, t) arise as the solutions of certain partial differential equations depending upon the assumed state of matter, and the mathematical framework corresponding to each state is briefly described and, where possible, some of the simpler and physically more interesting solutions are examined. The Newtonian gravitational potential arises from a variety of asymptotic limits, and can be identified as predominantly wave energy, which is entirely consistent with the post-Newtonian approximation.

Refereed/Peer-reviewed Advanced biochar blended nanoparticles substances, such as nano biochar or nanocomposites, have provided long-term solutions to a wide range of modern-day problems. Biochar blended nano-composites can be created to create better composite materials that combine the benefits of biochar and nanoparticles. Such materials have been typically improved with active functional groups, porous structure, active surface area, catalytic deterioration ability, as well as easy recovery or separation of pollutants. Such biochar-basednanocomposites have good adsorption properties for a variety of pollutants in various form of polluted medium (soil and water contamination). Catalytic nanoparticle encapsulated biochar, can perform concurrently the adsorption (by biochar) as well as catalytic degradation (nanoparticles) functions for pollutants removal from polluted sites. In this review, the advanced and practically feasible techniques involved in the biochar blended nanoparticles-based nanocomposites have been discussed with environmental applications. Furthermore, the mechanisms involved in this composite material in remediation, as well as the advantages and disadvantages of biochar blended nanoparticles-based nanocomposites, were discussed, and future directions for study in this field were suggested. In this review, the advanced and practically feasible techniques involved in the biochar blended nanoparticles-based nanocomposites have been discussed with environmental applications. Furthermore, the mechanisms involved in this composite material in remediation, as well as the advantages and disadvantages of biochar blended nanoparticles-based nanocomposites, were discussed, and future directions for study in this field were suggested.

Refereed/Peer-reviewed The three-halves Bagnold profile for granular flow down an incline, assuming no-slip at the base, is a generally accepted velocity profile that applies in many instances. In the intermediate dense regime, the material behaviour resembles a Bingham fluid and the widely accepted friction model - μ(I ) rheology originally defined within the realms of visco-plasticity applies. Here for steady-state dense granular flow, we derive a generalization of the Bagnold profile which applies to the abovementioned inter-particle friction approach, for which the Navier slip boundary condition applies at the base, and the Bagnold profile is included as a special case. We extend the Bagnold profile for a Navier slip base and we provide an expression for the slip length which demonstrates the dependence on the fundamental physical constants appearing in the μ(I ) framework. The given velocity profile provides a simple formula that captures the major flow characteristics of dense granular chute flow.

Refereed/Peer-reviewed Nanomaterials have many advantages over bulk materials, including enhanced surface-to-volume proportion as well as magnetic traits. It has been a steady rise in research with using nanomaterials in various biomedical fields in the past few decades. Constructing nanomaterials has emerged as a leading research primary concern in order to discover specialized biomedical applications. Since, their advantageous properties including chemical stability, non-toxicity, bio - compatibility, relatively high magnetization, and strong magnetic vulnerability, nanoparticles of iron oxide had already influenced implementations in different biomedical fields. Nanomaterials can be divided up into four nanomaterials such as metallic nanomaterials, bimetallic or alloy nanomaterials, metal oxide nanomaterials, as well as magnetic nanomaterials. Hence, the purpose of this review is to conduct such in discussion on emerging advancements in nanomaterials for biomedical, with such a special emphasis upon those options of nanomaterials including metallic nanomaterials: Au and Ag, bimetallic nanomaterials: Fe–Co and Fe–Pt, and metal oxides: TiO2 and CeO2. Securing this information gap will result in a better comprehension of the contribution of nanomaterial type and subsequent huge-scale applications in aspects of both their potential and challenges. Hence, the purpose of this review is to conduct such in discussion on emerging advancements in nanomaterials for biomedical, with such a special emphasis upon those options of nanomaterials including metallic nanomaterials: Au and Ag, bimetallic nanomaterials: Fe–Co and Fe–Pt, and metal oxides: TiO2 and CeO2. Securing this information gap will result in a better comprehension of the contribution of nanomaterial type and subsequent huge-scale applications in aspects of both their potential and challenges.

Multiple lines of evidence currently indicate that the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) gains entry into human host cells via a high-affinity interaction with the angiotensin-converting enzyme 2 (ACE2) transmembrane receptor. Research has further shown the widespread expression of the ACE2 receptor on the surface of many different immune, non-immune and neural host cell types, and that SARS-CoV-2 has the remarkable capability to attack many different types of human-host cells simultaneously. One principal neuroanatomical region for high ACE2 expression patterns occurs in the brainstem, an area of the brain containing regulatory centers for respiration, and this may in part explain the predisposition of many COVID-19 patients to respiratory distress. Early studies also indicated extensive ACE2 expression in the whole eye and the brain’s visual circuitry in aged humans. In this study we analyzed ACE2 receptor expression at the mRNA and protein level in multiple cell types involved in human vision, including cell types of the external eye and several deep brain regions known to be involved in the processing of visual signals. Here we provide evidence: (i) that many different optical and neural cell types of the human visual system provide receptors essential for SARS-CoV-2 invasion; (ii) of the remarkable ubiquity of ACE2 presence in cells of the eye and anatomical regions of the brain involved in visual signal processing; (iii) that ACE2 receptor expression in different ocular cell types and visual processing centers of the brain provide multiple compartments for SARS-CoV-2 infiltration; and (iv) of a gradient of increasing ACE2 expression from the anterior surface of the eye to the visual signal processing areas of the occipital lobe and the primary visual neocortex. A gradient of ACE2 expression from the eye surface to the occipital lobe may provide the SARS-CoV-2 virus a novel pathway from the outer eye into deeper anatomical regions of the brain involved in vision. These findings may explain, in part, the many recently reported neuro-ophthalmic manifestations of SARS-CoV-2 infection in COVID-19 affected patients.

In this paper we exploit the fact that dense granular flow down an inclined chute exhibits a simple scaling property. This implies that the fundamental governing partial differential equation admits similarity profiles which are applicable to any general friction law $$\mu (I)$$ of the material irrespective of the particular functional form under consideration, where $$\mu (I)$$ is a function of the dimensionless parameter, called the inertial number I. We adopt a widely accepted constitutive model describing the material response that draws its original inspiration from the realms of classical visco-plasticity. The governing steady-state similarity solutions are however applicable for any friction law describing dense granular media. These steady-state solutions embody the key aspects of dense granular flow down an inclined chute that find wide usage in various industrial applications and will further aid in modelling natural geophysical phenomenon. We provide steady-state analytical solutions for some special cases and we emphasize the importance of similarity solutions as a benchmark standard for various numerical studies concerning the complex material behaviour of dense granular flows.

Background: Multiple lines of evidence currently indicate that the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)gains entry into human host cells via a high-affinity interaction with the angiotensin-converting enzyme 2 (ACE2) transmembrane receptor. Research has further shown the widespread expression of the ACE2 receptor on the surface of many different immune, non-immune and neural host cell types, and that SARS-CoV-2 has there markable capability to attack many different types of human-host cells simultaneously. One principal neuroanatomical region for highACE2 expression patterns occurs in the brainstem, an area of the brain containing regulatory centers for respiration, and this may in part explain the predisposition of many COVID-19 patients to respiratory distress. Early studies also indicated extensive ACE2 expression in the whole eye and the brain’s visual circuitry. In this study we analyzed ACE2 receptor expression at the mRNA and protein level in multiple cell types involved in human vision, including cell types of the external eye and several deep brain regions known to be involved in the processing of visual signals.Methods: ACE2 mRNA and protein analysis; multiple eye and brain cells and tissues; gamma32P-adenosine tri-phosphate ([γ-32P]dATP) radiolabeled probes; Northern analysis; ELISA.Results: The four main findings were: (i)that many different optical and neural cell types of the human visual system provide receptors essential for SARS-CoV-2 invasion; (ii)the remarkable ubiquity of ACE2 presence in cells of the eye and anatomical regions of the brain involved in visual signal processing; (iii)that ACE2 receptor expression in different ocular cell types and visual processing centers of the brain provide multiple compartments for SARS-CoV-2 infiltration; and (iv)a gradient of increasing ACE2 expression from the anterior surface of the eye to the visual signal processing areas of the occipital lobe and the primary visual neocortex.Conclusion: A gradient of ACE2 expression from the eye surface to the occipital lobe provide the SARS-CoV-2 virus a novel pathway from the outer eye into deeper anatomical regions of the brain involved in vision. These findings may explain, in part, the many recently reported neuro-ophthalmic manifestations of SARS-CoV-2infection in COVID-19 affected patients.

We present a mechanical model that identifies dark energy and dark matter as singular states. The model introduces the notions of “forces in the direction of time”, and that every particle has a particle energy e and a wave energy $$\mathscr {E}$$ and is based upon an idea due to de Broglie for which he had a concrete physical picture of the coexistence of both particle and its associated wave. The model is a properly Lorentz invariant extension of special relativity and is intermediate between special and general relativity. A logical analysis of the energy statement gives rise to four states of matter, which is meaningful because it is interpreted within the proposed extension of special relativity. This interpretation is not available within the existing special relativity since it lacks the notion of force in the direction of time. Dark matter and dark energy arise as backward and forward waves, respectively, such that there is a particular alignment of the spatial physical force f and the force g in the direction of time, namely $$f = \pm \, gc$$ and for which $$e = \mathscr {E}$$ , respectively. In an actual circumstance, we might expect a situation comparable to a “fuzzy region” where the key equalities are constantly switching on and off dependent upon varying local conditions. The latest Planck data for the relative fractions for a three-component universe are exploited to obtain numerical values for the relative fractions for a four-component universe comprising baryonic matter, dark matter, dark energy, and a fourth state termed here anti-matter. The values so obtained are consistent with expected allowable occupancies for four states given that the higher energy states have a ready access to each of the lower energy states.

Quantitative ultrasound has a great potential for the non-destructive evaluation of tissue engineered constructs, where the local attenuation and the integrated backscatter coefficient (IBC) can be used for monitoring the development of biological processes. The local determination of both parameters can be achieved using the reference phantom method (RPM). However, its accuracy can be affected when evaluating constructs of evolving sound speed, attenuation and thickness, for example, when evaluating biodegradable hydrogels developing neocartilage. To assess the feasibility of using the RPM under such dynamic conditions while employing a 50-MHz transducer, we conducted a series of experiments on 3-mm-thick acellular hydrogels laden with microspheres. The ultrasonic evaluation procedure used was validated by detecting and compensating for large attenuation variations occurring in the construct, up to 20-fold with respect to the reference phantom, with estimations errors below 1%. We found that sound speed mismatch does not affect the local attenuation estimation, but causes a strong diffraction effect by reducing the backscatter intensity. Such intensity reduction was compensated by determining the IBC percentage change (IBCΔ) as function of sound speed mismatch with respect to the reference phantom (ΔSS), with the equation IBCΔ = (0.63 ± 0.07) ΔSS + (8.54 ± 0.76) 10–3 ΔSS2. The investigated ΔSS interval was up to 120 m/s and using two different concentrations of microspheres, with estimation errors below 7% relative to the construct's actual IBC. Finally, we found that the spectral difference method is sufficient to measure within a few millimetres in depth mismatch, and when combining with sound speed mismatch, we found negligible additional effects. These results pave the way for the use of a generic reference phantom for the evaluation of thin dynamic constructs, thus simplifying the need for using different phantoms depending on the construct's properties. Refereed/Peer-reviewed

The gateway for invasion by the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) into human host cells is via the angiotensin-converting enzyme 2 (ACE2) transmembrane receptor expressed in multiple immune and nonimmune cell types. SARS-CoV-2, that causes coronavirus disease 2019 (COVID-19; CoV-19) has the unusual capacity to attack many different types of human host cells simultaneously via novel clathrin- and caveolae-independent endocytic pathways, becoming injurious to diverse cells, tissues and organ systems and exploiting any immune weakness in the host. The elicitation of this multipronged attack explains in part the severity and extensive variety of signs and symptoms observed in CoV-19 patients. To further our understanding of the mechanism and pathways of SARS-CoV-2 infection and susceptibility of specific cell- and tissue-types and organ systems to SARS-CoV-2 attack in this communication we analyzed ACE2 expression in 85 human tissues including 21 different brain regions, 7 fetal tissues and 8 controls. Besides strong ACE2 expression in respiratory, digestive, renal-excretory and reproductive cells, high ACE2 expression was also found in the amygdala, cerebral cortex and brainstem. The highest ACE2 expression level was found in the pons and medulla oblongata in the human brainstem, containing the medullary respiratory centers of the brain, and may in part explain the susceptibility of many CoV-19 patients to severe respiratory distress.

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix. Refereed/Peer-reviewed

The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that individually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between individual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018; 69:133-145; Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019; 70: 5–14; Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019; 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy (Formula presented.). These papers develop expressions for the de Broglie wave energy (Formula presented.) by making a distinction between particle energy (Formula presented.) and the total work done by the particle (Formula presented.), that which accumulates from both a spatial physical force (Formula presented.) and a force (Formula presented.) in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of (Formula presented.) or (Formula presented.) is physically measured, and particles appear for (Formula presented.) and de Broglie waves occur for (Formula presented.), but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that (Formula presented.) vanishes and (Formula presented.) becomes purely imaginary. If both (Formula presented.) and (Formula presented.) are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner Refereed/Peer-reviewed

The first successful attempt to obtain purified aluminum metal was accomplished by the Danish physicist and chemist Hans Christian Orsted in 1824, however it was not until about ~140 years later that aluminum's capacity for neurological disruption and neurotoxicity was convincingly established. The earliest evidence of the possible involvement of this biosphere-rich metallotoxin in Alzheimer's disease (AD) originated in the early-to-mid-1960's from animal and human research investigations that arose almost simultaneously from independent laboratories in the United States and Canada. This short communication pays tribute to the pioneering research work on aluminum in susceptible species, in AD animal models and in AD patients by the early investigators Drs. Robert D. Terry, Igor Klatzo and Henryk M. Wisniewski with special acknowledgement to the late Dr. Donald RC McLachlan, and their contemporary physician-scientist colleagues and collaborators. Together these researchers established the groundwork and foundation towards our understanding of the potential contribution of aluminum to progressive, age-related and lethal neurodegenerative diseases of the human central nervous system.

Gram-negative bacteria of the human gastrointestinal (GI) tract microbiome: (i) are capable of generating a broad-spectrum of highly neurotoxic, pro-inflammatory and potentially pathogenic molecules; and (ii) these include a highly immunogenic class of amphipathic surface glycolipids known as lipopolysaccharide (LPS). Bacteroides fragilis (B. fragilis), a commensal, Gram negative, non-motile, non-spore forming obligatory anaerobic bacillus, and one of the most abundant bacteria found in the human GI tract, produces a particularly pro-inflammatory and neurotoxic LPS (BF-LPS). BF-LPS: (i) is known to be secreted from the B. fragilis outer membrane into the external-medium; (ii) can damage biophysiological barriers via cleavage of zonula adherens cell-cell adhesion proteins, thereby disrupting both the GI-tract barrier and the blood-brain barrier (BBB); (iii) is able to transit GI-tract barriers into the systemic circulation and cross the BBB into the human CNS; and (iv) accumulates within CNS neurons in neurodegenerative disorders such as Alzheimer’s disease (AD). This short communication provides evidence that the incubation of B. fragilis with aluminum sulfate [Al(2)(SO(4))(3)] is a potent inducer of BF-LPS. The results suggest for the first time that the pro-inflammatory properties of aluminum may not only be propagated by aluminum itself, but by a stimulation in the production of microbiome-derived BF-LPS and other pro-inflammatory pathogenic microbial products normally secreted from human GI-tract-resident microorganisms. GRAPHICAL ABSTRACT: Gram-negative bacteria such as B. fragilis of the human gastrointestinal (GI) tract microbiome generate pro-inflammatory glycolipids known as lipopolysaccharide (LPS) and other neurotoxic species, and are amongst the most pro-inflammatory molecules known. Incubation of aluminum sulfate [Al(2)(SO(4))(3)] with B. fragilis in culture resulted in a significant increase in the release of LPS.

Targetted drug delivery is currently an important topic in bio-nanotechnology research. However, at present an improved understanding of the underlying mechanisms is required to provide accurate and effective drug transportation in humans. In particular, to increase the blood-circulation time and to increase the interacting surface area between the drugs and the targeted cells. The study here focuses on determining the optimal location for a drug encapsulated in a nanocarrier in order to enhance the blood-circulation time. To illustrate the modelling, the anti-cancer drug Doxorubicin is adopted and liposome is assumed to be the nanocapsule. The Doxorubicin molecule is modelled as a sphere, which is assumed to be encapsulated in a spherical liposome of radius 50 nm, and the model consists of three systems, depending on the number of identical Doxorubicin molecules inside the liposome; one, two or three. In all cases, we determine the offset positions of the Doxorubicin from the centre of the liposomes, and also the angle between any two Doxorubicin molecules. Our findings predict the most stable configuration of a particular drug encapsulated in a capsule, and will prove useful as a guide in the design of other drug delivery systems involving small drug molecules inside a nanocapsule. Refereed/Peer-reviewed

In this paper, we investigate a carbon nanotorus as a caged molecular structure interacting with an atom. Assuming that the atom is located along the central axis perpendicular to the torus, the interaction energy of the system is determined using the continuum approximation together with the Lennard-Jones potential. This approach avoids the intensive computational calculations that are involved in other modelling approaches. Numerical results are presented in terms of dimensionless variables. The results show that the optimal major radius of the torus has a linear relationship with its minor radius when the atom is symmetrically situated along the torus axis. When the atom is offset from this axis, the minimum energy location shifts away from the centre as the ratio of the major and minor radii exceeds the value of 0.90. Finally, the oscillatory behaviour for the carbon atom is investigated. Our findings predict a novel nano-oscillator which can produce frequencies in the gigahertz range. Refereed/Peer-reviewed

The optimal or lowest energy configuration of two carbon nanotori is investigated. With the assumption that the tori are symmetrically situated and parallel with the centres lying on their common axis, the interaction energy between the two tori is evaluated using the continuum approximation and the Lennard-Jones potential function. This mathematical modelling approach provides a fast and accurate computation as compared to other simulation methods. The numerical results obtained determine the most stable configuration corresponding to the minimum energies for five different sized tori, and a relation between the major and minor radii of the tori and the equilibrium position is demonstrated. The interaction energies between two nanotori determined here might be utilised to model many promising devices for future developments in nano and biotechnology Refereed/Peer-reviewed

In this note, we provide some new derivations and further elaboration on a recent proposal by the author Hill (Zeitschrift fur angewandte Mathematik und Physik 69: 133–145, 2018. https://doi.org/10.1007/s00033-018-1028-4 ) regarding the formal origins of dark energy. Using the framework of special relativity and validation through Lorentz invariance, Hill (2018) develops the usual relations of special relativity, but makes a distinction between particle energy $$e = mc^2$$ and the actual work done by the particle $$ \mathscr {E}$$ , and adopts the perspective that both the momentum vector $${\mathbf{p} = m\mathbf{u}}$$ and the intrinsic particle energy e itself contribute to the work done $$ \mathscr {E}$$ . The alternative formulation provides an extension of Newton’s second law that is properly invariant under the Lorentz group and yields new expressions for the work done involving the $$\log $$ function. The $$\log $$ terms can generate extremely large energies and may represent the first formal indication of the origin of dark energy. The new approach suggests that large energies might be generated for slowly moving systems and that dark energy might arise in consequence of conventional mechanical theory neglecting both the de Broglie wave energy and the work done in the direction of time. In these notes, a number of alternative derivations and connections relating to Hill (2018) are presented that were not made explicit in Hill (2018), including new derivations from the wave equation, a generalised approach and relating the solutions of Hill (2018) to a known result for uniform acceleration and to Hubble’s law for receding galaxies.

The existence of so-called dark energy and matter in the universe implies that the conventional accounting of mass and energy is incorrect. Here, we use the framework of special relativity and validation through Lorentz invariance to develop an alternative accounting of mass and energy. We assume the usual Einstein relations of special relativity, but we make the distinction between the particle energy $$e = mc^2$$ and the actual work done by the particle $$ \mathscr {E}$$ , and we adopt the perspective that it is not just the momentum vector $${\mathbf{p} = m\mathbf{u}}$$ that contributes to the work done $$ \mathscr {E}$$ , but rather the intrinsic particle energy e itself plays an important role through the combined potentials $$(\mathbf{p}, e/c)$$ as a well-defined four vector within special relativity. The resulting formulation provides a natural extension of Newton’s second law, emerges as a fully consistent development of special relativity that is properly invariant under the Lorentz group, and yields an extension of Einstein’s famous equation for the work done involving new terms. The new work done expressions can involve the $$\log $$ function and possibly generate extremely large energies that might well represent the first formal indication of the origin of dark energy. Two alternative expressions are both well defined as a limiting case for energy–mass waves travelling at the speed of light and are in complete accord with well-established theory for photons and light for which energy is known to vary linearly with momentum. The present formulation suggests that large energies might be generated even for slowly moving systems, and that dark energy might arise in consequence of conventional mechanical theory neglecting the work done in the direction of time.

The conventional rolled-up model for carbon nanocones assumes that the cone is constructed from a rolled-up graphene sheet joined seamlessly, which predicts five distinct vertex angles. This model completely ignores any effects due to the changing curvature and all bond lengths and bond angles are assumed to be those for the planar graphene sheet. Clearly curvature effects will become more important closest to the cone vertex, and especially so for the cones with the smaller apex angles. Here we construct carbon nanocones which in the assembled cone are assumed to comprise bond lengths and bond angles which are, as far as possible, equal throughout the structure at the same distance from the conical apex. Predicted bond angles and bond lengths are shown to agree well with those obtained by relaxing the conventional rolled-up model using the LAMMPS software. The major objective here is not simply to model physically realisable carbon nanocones for which numerical procedures are far superior, but rather to produce an improved model that takes into account curvature effects close to the vertex, and from which we may determine an analytical formula which represents an improvement on that for the conventional rolled-up model.

The interaction energy of a fullerene symmetrically situated inside a carbon nanotorus is studied. For these non-bonded molecules, the main interaction originates from the van der Waals energy which is modelled by the 6–12 Lennard-Jones potential. Upon utilising the continuum approximation which assumes that there are infinitely many atoms that are uniformly distributed over the surfaces of the molecules, the total interaction energy between the two structures is obtained as a surface integral over the spherical and the toroidal surfaces. This analytical energy is employed to determine the most stable configuration of the torus encapsulating the fullerene. The results show that a torus with major radius around 20–22 Å and minor radius greater than 6.31 Å gives rise to the most stable arrangement. This study will pave the way for future developments in biomolecules design and drug delivery system Refereed/Peer-reviewed