Dimensional Continuum Research Articles

Infection Kinetics of COVID-19 and Lockdown Implications

Background: Covid-19 has raked a devastating trail across continents with the highest mortality to infected ratio ever for a pandemic. The key containment strategy identified has been social seclusion. Analyzing extensive live statistics with a fully predictive global Covid-19 model, we quantify implications of lockdown and strategic quarantine on Covid-containment. Methods: We analyze Covid-19 infection kinetics using a 6 dimensional (healthy, pre-existing, infected, recovered, immuned, dead) continuum model using a probabilistic Machine Learned (ML) kernel for parameter prediction using data from the Johns Hopkins repository. The ML platform itself uses a double filtration process, first using statistics for infected only, followed by statistics for the infected and dead combined. We use this multi-scale, multivariate ML architecture to categorize 19 countries into 4 different infection classes according to mortality-to-infected ratio. Finding: The model almost unerringly predicts the number of infected and dead (within 1 s.d.) for all infection classes, consistently up to 30 days beyond the last ML training date (10 May 2020). The Reproductive Number R0 estimated from the model makes catastrophic predictions (1.5 < R0 < 2.7) for countries resorting to categorical de-prioritization of lockdown (such as UK, US, Sweden and India). Countries in other infection classes show smaller R0 (1.06 < R0 < 1.5) with faster recovery time, reflecting the impact of variable lockdown measures, with numbers matching almost perfectly with real data. The model also predicts the date of secondary relapse. Interpretation: The model provides robust future estimation for the number of dead for all 19 infected countries that we studied, unfailingly up to 4 following weeks. Early versus later lockdown clearly has major impact on mortality statistics as also on the timelining of secondary relapse, both accurately predicted by the model. The critical importance of correct lockdown span and timing is thus accentuated and quantified. Funding Statement: All authors have been resourced through their affiliating organizations. Declaration of Interests: There is no known conflict of interest. Ethics Approval Statement: This is an epidemiological study with data collected from open access sources. Ethical approval is not required.

Conservation laws of some classes of models involving oscillons

In this paper, we construct and analyse the conservation laws (conserved densities) of two models that lead to oscillons; viz., in a system of two nonlinearly coupled scalar fields in (1+1)-dimensional spatiotemporal continuum and analysed a particular class of sixth-degree polynomial potentials and one in which an interaction of ‘two one-dimensional spatial vector oscillons in a trap with significant transverse dimensions’ occurs.

Simplified spring models for concrete segmental lining longitudinal joints with gaskets

To investigate the full mechanical behavior of concrete segmental lining longitudinal joints with gaskets, a numerical bending test is performed by a detailed three dimensional (3D) continuum numerical model for the segment joint of the Yellow River Crossing Tunnel lining. Simplified spring models including a simplified plate-spring analytical model and a simplified solid-spring numerical model are proposed to analyze the rotational stiffness of the longitudinal joint. The bending test reveals that the mechanical behavior of the longitudinal joint with gaskets can be divided into four stages. The proposed simplified plate-spring analytical model and simplified solid-spring numerical model allow the effect of the properties of bolts and various gaskets, the concrete contact force, the geometry of the joint section and the subjected loads to be taken into consideration. The results of the simplified spring models are well consistent with these of the detailed 3D continuum numerical model, and the proposed models can be adopted for the simulation of the segment longitudinal joints with various gaskets.

Application of a mixed variational higher order plate theory towards understanding the deformation behavior of hybrid laminates

AbstractHybrid laminates containing an elastomer layer in addition to fiber reinforced polymer as well as metal layers have been found beneficial in compensating issues frequently found with traditional fiber metal laminates. Commonly used equivalent single‐layer shell and plate theories, however, are unable to account for the strong heterogeneous stiffness distribution of the constituents within the laminate. Furthermore, the transverse shear and normal deformations in the elastomer layer are expected to significantly influence the deformation of the neighboring laminae. An accurate depiction of these transverse stresses requires a multi‐layer shell theory as opposed to commonly used single‐layer formulations. Hence, a higher order mixed variational plate theory is applied in order to study and predict the mechanical behavior of such laminates, especially on a structural level where the computational effort forbids the use of a three dimensional continuum formulation.

On a Possibly Pure Set-Theoretic Contribution to Black Hole Entropy

Continuity as appears to us immediately by intuition (in the flow of time and in motion) differs from its current formalization, the arithmetical continuum or equivalently the set of real numbers used in modern mathematical analysis. Motivated by the known mathematical and physical problems arising from this formalization of the continuum, our aim in this paper is twofold. Firstly, by interpreting Chaitin’s variant of Gödel’s first incompleteness theorem as an inherent uncertainty or fuzziness of the arithmetical continuum, a formal set-theoretic entropy is assigned to the arithmetical continuum. Secondly, by analyzing Noether’s theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a single, stationary, asymptotically flat black hole is modeled by the arithmetical continuum in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole (proportional to the area of its “instantaneous” event horizon), indicating that this apparently physical quantity might have a pure set-theoretic origin, too.

Leaving the Slip System ‐ Cross Slip in Continuum Dislocation Dynamics

AbstractDislocations are the main contributors to plastic deformation of crystalline materials. An important step towards the description of hardening behavior is the consideration of cross slip, as it drives the exchange of dislocations between slip systems, thus playing a major role in dislocation multiplication or annihilation. In density based continuum theories of dislocations, dislocations are usually considered closed curves within their slip planes. Recent discrete dislocation dynamics simulations, however, suggest that only a small fraction of dislocations is actually closed on a single slip system, due to cross slip and dislocation reactions. We therefore investigate how the kinematics of moving open curves can be considered in dislocation density based models. The assumption of open planar curves leads to modified evolution equations for the dislocation state variables. These extended evolution equations are presented for the theory of geometrically necessary dislocations (GND) and for the higher dimensional continuum dislocation dynamics theory (hdCDD). The resulting equations are checked for plausibility by numerical calculations using the finite volume method.

Heat Capacity of an Ordered Bundle of Single-Walled Carbon Nanotubes

A quantum-statistical model of the thermodynamic properties of an ordered bundle of the single-walled carbon nanotubes is proposed. Generalization of the Debye heat capacity theory for the d dimensional phonon continuum is used to calculate the heat capacity. A formula for isochoric heat capacity is obtained; it contains two characteristic temperatures related to the macro- and the microstructural vibrational contributions. The calculations are in good agreement with the experimental data.

Longitudinal and transverse elastic waves in 1D granular materials modeled as micromorphic continua

In this paper, the granular micromechanics approach proposed by Misra and Poorsolhjouy (2016) is used to study the dispersive behavior of granular materials in response to elastic deformation waves. This study is motivated by the typical lack of connection between the mathematical models, the parameters involved, and the physics of granular materials. Therefore, extensive parametric studies are carried out in order to understand how each intergranular stiffness coefficient contributes to the dispersive behavior of the material. Two cases of one dimensional wave propagation problems have been investigated. Case 1 focuses upon longitudinal wave propagation in a one dimensional continuum, while case 2 considers transverse wave propagation in a one dimensional continuum that has a two-dimensional micro-structure. Results predict the emergence of frequency band gaps and negative group velocities for certain values of the parameters involved. Such phenomena can be produced by starting from the micro-structure and producing materials for which the inter-granular stiffness parameters are the ones the granular micromechanics approach predict. This, however, is not a one to one mapping, and therefore, sets of solutions to achieve a particular behavior might exist. Therefore, granular micromechanics provides a systematic material design process, eliminating ad-hoc processes and potentially leading to large data libraries.

Multi-critical behaviour of 4-dimensional tensor models up to order 6

Tensor models generalize the matrix-model approach to 2-dimensional quantum gravity to higher dimensions. Some models allowing a 1/N expansion have been explored, most of them generating branched-polymer geometries. Recently, enhancements yielding an additional 2d pure gravity (planar) phase and an intermediate regime of proliferating baby universes have been found. It remains an open issue to find models escaping these lower-dimensionality universality classes.Here we analyse the dominant regime and critical behaviour of a range of new models which are candidates for such effective geometries, in particular interactions based on the utility graph K3,3. We find that, upon proper enhancement, the two-phase structure of a branched-polymer and a 2d gravity regime is the common case in U(N)-invariant rank D=4 tensor models of small orders. Not only the well known so-called necklace interactions but also K3,3-type interactions turn out as the source for the planar regime. We give a systematic account of the enhancement scaling, the counting of leading-order diagrams and the multi-critical behaviour of a wide range of interactions, in particular for all order-6 interactions of rank 3 and 4. These findings support the claim of universality of such mixtures of branched-polymer and planar diagrams at criticality. In particular, this hints at the necessity to consider new ingredients, or interactions of higher order and rank, in order to obtain higher dimensional continuum geometry from tensor models.

The Importance of Cognition in Education

Cognitive or mental architectures are models made up of input - output type modules of the central nervous system activity, formed with the help of white, black or grey cybernetic “boxed” and of the determination connection. Despite the great existing differences between the biochemical and electronic processing of information, the mental architectures aid in comparing the human intelligence with the artificial one and even contribute to the creation of a common language between the human and the animal intelligence (taking into account static and dynamic representations). Based on that cultural and social representations share a „situational“semiotics, we use a common language with connection to referentiality or not, for physical or social representations. In this work I will create the formal model as a representation more or less simplified of a “material world”. We will, therefore, admit the presence of some physical-mathematical, psychic and semiotic models (depending on the language). The physical - mathematical models refer to the image of the world starting from the 4th dimensional continuum perceived by our senses. The models are referential rather than imaginary because physics inhibits the mathematical tendency of generating too many possible worlds. On the contrary, the psychological models can be also reported to imaginary luxurious worlds due to the outcome of our imagination which can be hardly quelled.

Wave equation in fractional Zener-type viscoelastic media involving Caputo–Fabrizio fractional derivatives

We investigate propagation of waves in the Zener-type viscoelastic media through a model which involves fractional derivatives with a regular kernel. The restrictions on the coefficients in the constitutive equation that follow from the weak form of the dissipation principle are obtained. We formulate a problem of motion of a spatially one dimensional continuum in a dimensionless form. Then, it is considered in the frame of distribution theory. The existence and the uniqueness of a distributional solution as well as the analysis of its regularity are presented. Numerical results provide the illustration of our approach.

Opposing patterns of neuronal variability in the sensorimotor network mediate cyclothymic and depressive temperaments.

Affective temperaments have been described since the early 20th century and may play a central role in psychiatric illnesses, such as bipolar disorder (BD). However, the neuronal basis of temperament is still unclear. We investigated the relationship of temperament with neuronal variability in the resting state signal-measured by fractional standard deviation (fSD) of Blood-Oxygen-Level Dependent signal-of the different large-scale networks, that is, sensorimotor network (SMN), along with default-mode, salience and central executive networks, in standard frequency band (SFB) and its sub-frequencies slow4 and slow5, in a large sample of healthy subject (HC, n = 109), as well as in the various temperamental subgroups (i.e., cyclothymic, hyperthymic, depressive, and irritable). A replication study on an independent dataset of 121 HC was then performed. SMN fSD positively correlated with cyclothymic z-score and was significantly increased in the cyclothymic temperament compared to the depressive temperament subgroups, in both SFB and slow4. We replicated our findings in the independent dataset. A relationship between cyclothymic temperament and neuronal variability, an index of intrinsic neuronal activity, in the SMN was found. Cyclothymic and depressive temperaments were associated with opposite changes in the SMN variability, resembling changes previously described in manic and depressive phases of BD. These findings shed a novel light on the neural basis of affective temperament and also carry important implications for the understanding of a potential dimensional continuum between affective temperaments and BD, on both psychological and neuronal levels.

Spectrum or subtypes? A latent profile analysis of restricted and repetitive behaviours in autism

BackgroundAutism Spectrum Disorder (ASD) is a heterogeneous condition. One way of understanding this heterogeneity is by investigating whether homogenous subgroups within the autism population exist. Some studies have attempted to do this by looking at social and communication skills. However, few studies have looked at subtyping using restricted and repetitive behaviours. While restricted and repetitive behaviours form part of the core features of autism, their presentation is diverse across different individuals on the spectrum. The aim of this study was to determine if restricted and repetitive behaviours could be used to identify potential subtypes of autism. MethodThis study used unsupervised clustering algorithms to differentiate subgroups of individuals on the autism spectrum based on their scores on the Repetitive Behaviour Scale-Revised (RBS-R). ResultsThree groups were found that reported low, medium and high levels of restricted and repetitive behaviours. These groups also differed on a range of clinical measures including problematic behaviours, autistic traits and adaptive behaviours. ConclusionsOur findings indicate that subgroups of individuals with autism can be identified based on their level of restricted and repetitive behaviours. This highlights that restricted and repetitive behaviours may be best understood under a dimensional continuum of severity. This has implications for our understanding of the non-social characteristics of autism.

Thermality from a Rindler quench

Ultracold fermionic atoms in an optical lattice, with a sudden position-dependent change (a quench) in the effective dispersion relation, have been proposed by Rodríguez-Laguna et al as an analogue spacetime test of the Unruh effect. We provide new support for this analogue by analysing a massless scalar field on a (1+1)-dimensional continuum spacetime with a similar quench: an early time Minkowski region is joined at a constant time surface, representing the quench, to a late time static region in which left and right asymptotically Rindler domains are connected by a smooth negative curvature bridge. We show that the quench is energetically mild, and late time static observers, modelled as a derivative-coupling Unruh–DeWitt detector, see thermality, in a temperature that equals the Unruh temperature for observers in the asymptotic Rindler domains. The Unruh effect hence prevails, despite the energy injected into the field by the quench and despite the absence of a late time Killing horizon. These results strengthen the motivation to realise the experimental proposal.

Quantum Field Theory of Nematic Transitions in Spin-Orbit-Coupled Spin-1 Polar Bosons

We theoretically study an ultracold gas of spin-1 polar bosons in a one-dimensional continuum, which are subject to linear and quadratic Zeeman fields and a Raman induced spin orbit coupling. Concentrating on the regime in which the background fields can be treated perturbatively, we analytically solve the model in its low-energy sector; i.e., we characterize the relevant phases and the quantum phase transitions between them. Depending on the sign of the effective quadratic Zeeman field ε, two superfluid phases with distinct nematic order appear. In addition, we uncover a spin-disordered superfluid phase at strong coupling. We employ a combination of renormalization group calculations and duality transformations to access the nature of the phase transitions. At ε=0, a line of spin-charge separated pairs of Luttinger liquids divides the two nematic phases, and the transition to the spin-disordered state at strong coupling is of the Berezinskii-Kosterlitz-Thouless type. In contrast, at ε≠0, the quantum critical theory separating nematic and strong coupling spin-disordered phases contains a Luttinger liquid in the charge sector that is coupled to a Majorana fermion in the spin sector (i.e., the critical theory at finite ε maps to a quantum critical Ising model that is coupled to the charge Luttinger liquid). Because of an emergent Lorentz symmetry, both have the same logarithmically diverging velocity. We discuss the experimental signatures of our findings that are relevant to ongoing experiments in ultracold atomic gases of ^{23}Na.

Vertical strong ground motion effects on seismic damage propagations of historical masonry rectangular minarets

Abstract Masonry towers and minarets are slender tall structures and highly vulnerable to strong ground motions due to their structural and material characteristics. The paper aims to investigate the effect of vertical ground motion on damage propagations of historical masonry slender tall rectangular stone minarets numerically. The Ulu Mosque minaret with height of 42.90 m constructed in 639 in Diyarbakir, Turkey, is selected as an application. Firstly, three dimensional solid and continuum finite element models of the minaret are obtained from the surveys. Since the foundation of the minaret is sitted on the hard soil, soil-structure interaction is not considered in the finite element model. The finite element model of the minaret is updated by modal analyses results and an empirical frequency formulation developed using the measurement results. Mechanical material properties of the masonry unit are determined using the properties of the stone and mortar used in the minaret. Concrete damaged plasticity (CDP) material model adjusted to masonry structures is considered in the nonlinear analyses. The acceleration records of horizontal (N-S) and vertical (V) components of May 1, 2003 Bingol earthquake (Mw = 6.4), which was recorded in hard soil and occurred in the near region of Diyarbakir city, are chosen as a strong ground motion. Then, linear time history and nonlinear step by step seismic analyses of the minaret are implemented for only horizontal, and combined horizontal and vertical components of the earthquake. The time histories of displacements, minimum and maximum principal strains and stresses, and damage propagations on the minaret are compared for only horizontal, and combined horizontal and vertical load cases including self-weight. It can be stated that consideration of vertical ground motion component in combination with horizontal component affects damage percentages and propagations on slender tall rectangular stone minarets.

Blood-Based Biomarker Screening with Agnostic Biological Definitions for an Accurate Diagnosis Within the Dimensional Spectrum of Neurodegenerative Diseases.

The discovery, development, and validation of novel candidate biomarkers in Alzheimer's disease (AD) and other neurodegenerative diseases (NDs) are increasingly gaining momentum. As a result, evolving diagnostic research criteria of NDs are beginning to integrate biofluid and neuroimaging indicators of pathophysiological mechanisms. More than 10% of people aged over 65 suffer from NDs. There is an urgent need for a refined two-stage diagnostic model to first initiate an early, sensitive, and noninvasive process in primary care settings. Individuals that meet detection criteria will then be channeled to more specific, costly (positron-emission tomography), and invasive (cerebrospinal fluid) assessment methods for confirmatory biological characterization and diagnosis.A reliable and sensitive blood test for AD and other NDs is not yet established; however, it would provide the golden screening gate for an efficient primary care management. A limitation to the development of a large-scale blood-screening biomarker-based test is the traditional application of clinically descriptive criteria for the categorization of single late-stage ND constructs. These are genetically and biologically heterogeneous, reflected in multiple pathophysiological mechanisms and subsequent pathologies throughout a dimensional continuum. Evidence suggests that a shared, "open-source" integrated multilevel categorization of NDs that clusters individuals based on descriptive clinical phenotypes and pathophysiological biomarker signatures will provide the next incremental step toward an improved diagnostic process of NDs. This intermediate objective toward unbiased biomarker-guided early detection of individuals at risk for NDs is currently carried out by the international pilot Alzheimer Precision Medicine Initiative Cohort Program (APMI-CP).

Large deflection analysis of a fiber reinforced composite beam

The objective of this work is to analyze large deflections of a fiber reinforced composite cantilever beam under point loads. In the solution of the problem, finite element method is used in conjunction with two dimensional (2-D) continuum model. It is known that large deflection problems are geometrically nonlinear problems. The considered non-linear problem is solved considering the total Lagrangian approach with Newton-Raphson iteration method. In the numerical results, the effects of the volume fraction and orientation angles of the fibre on the large deflections of the composite beam are examined and discussed. Also, the difference between the geometrically linear and nonlinear analysis of fiber reinforced composite beam is investigated in detail.

Analytical validation of a $2+1$ dimensional continuum model for epitaxial growth with elastic substrate

An analiyical validation is obtained for the evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h], introduced in [18] by W.T. Tekalign and B.J. Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, h denotes the surface height of the film, F is the Fourier transform, and a, E, r are positive material constants. Existence, uniqueness, and Lipschitz regularity in time for weak solutions are proved, under suitable assumptions on the initial datum. Introduction The epitaxial deposition of a thin film on a relatively thick substrate has gained much interest in the recent years due to its applications to semiconductor electronics and quantum dots. Roughly speaking, the morphology of the film is known to be the result of a competition between the elastic energy associated to the mismatch between film and substrate, and the surface mass transport due to the film deposition. An extensive mathematical analysis of the mechanism associated to epitaxial film growth has been carried out in [2, 3, 10, 11, 15, 16] in the context of plane linear elasticity, and regularity results have been established for volume-constrained minimizers. Short time existence for a surface diffusion type geometric evolution equation keeping into account elasticity has first been analyzed in [12] in a two-dimensional setting (see also [17]). The previous result has been recently extended in [13] to the three-dimensional case. The central aim of this work is to study existence and Lipschitz regularity in time of weak solutions to the 2+1 dimensional evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h] (0.1) derived by W.T. Tekalign and B.J. Spencer in [18], where h denotes the surface height of the film and F is the Fourier transform. The quantities a, E, r are positive material constants. To be precise, a (resp. r) is the wavenumber (resp. wetting coefficient) associated to the equation, and E is defined as E := 2μ F (1 + ν )(1− ν) (1− νF )μS , where μ and ν (resp. μ and ν) are the elastic shear modulus and the Poisson’s 2010 Mathematics Subject Classification. 35K55, 35K67, 44A15, 74K35.

Boundary Control of a Flexible Manipulator Based on a High Order Disturbance Observer with Input Saturation

This paper studies a new boundary control strategy for a flexible manipulator subject to unknown fast time‐varying disturbances. The flexible manipulator essentially is an infinite dimensional continuum. Hence, a continuous function of space and time can be employed to describe the position of such a distributed parameter structure, the motion of which can be described by partial differential equations (PDEs). To cope with fast time‐varying external disturbances, a high order disturbance observer is adopted. A control strategy based on such a disturbance observer is proposed for the rest‐rest maneuvering of the flexible manipulator. Moreover, a smooth hyperbolic function is included in the controller to satisfy the requirement of input saturation. The stability of the boundary control is analyzed using LaSalle’s invariance principle. Finally, the performance of the presented boundary controller is verified through comparison with that of employing a constant disturbance observer via numerical simulations.