Similarity Invariants Research Articles

Numerical exploration of MHD bioconvective Williamson-Maxwell nanoliquid flow due to an exponentially elongated porous sheet

This research delves into the intriguing realm of non-Newtonian fluids in conjunction with microorganisms, presenting a mathematical model tailored to analyze heat and mass transfer within Williamson-Maxwell nanofluids hosting gyrotactic microbes. The study investigates how these fluids behave under the influence of multiple factors such as magnetic fields, thermal radiation, chemical reactions, and dissipation effects. Employing a set of similarity invariants, the governing equations are transformed into ordinary differential equations, which are then solved using a fourth-order R-K scheme. The findings, presented graphically, offer insights into various flow parameters and are complemented by pertinent physical explanations. The influence of magnetic flux ([Formula: see text]), Buoyancy ratio ([Formula: see text]), Peclet number ([Formula: see text]), and Schmidt number ([Formula: see text]) on various physical parameters are shown graphically. Notably, the research reveals that while increasing the external magnetic field impedes fluid motion, it enhances thermal and density layers. Additionally, a higher bioconvective Schmidt number is shown to reduce microbial density. These observations hold significant implications for applications involving nanofluids and microorganisms across biomedical, pharmaceutical, biofuels, and other sectors. Overall, this study contributes valuable knowledge to the understanding and potential utilization of complex fluid systems in diverse industrial contexts.

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

An invariance constrained deep learning network for partial differential equation discovery

The discovery of partial differential equations (PDEs) from datasets has attracted increased attention. However, the discovery of governing equations from sparse data with high noise is still very challenging due to the difficulty of derivatives computation and the disturbance of noise. Moreover, the selection principles for the candidate library to meet physical laws need to be further studied. The invariance is one of the fundamental laws for governing equations. In this study, we propose an invariance constrained deep learning network (ICNet) for the discovery of PDEs. Considering that temporal and spatial translation invariance (Galilean invariance) is a fundamental property of physical laws, we filter the candidates that cannot meet the requirement of the Galilean transformations. Subsequently, we embedded the fixed and possible terms into the loss function of neural network, significantly countering the effect of sparse data with high noise. Then, by filtering out redundant terms without fixing learnable parameters during the training process, the governing equations discovered by the ICNet method can effectively approximate the real governing equations. We select the one-dimensional (1D) Kuramoto–Sivashinsky equation, the two-dimensional (2D) Burgers equation, the equation of two-dimensional channel flow over an obstacle, and the equation of three-dimensional (3D) intracranial aneurysm as examples to verify the superiority of the ICNet for fluid mechanics. Furthermore, we extend similar invariance methods to the discovery of wave equation (Lorentz Invariance) and verify it through single and coupled Klein–Gordon equation. The results show that the ICNet method with physical constraints exhibits excellent performance in governing equations discovery from sparse and noisy data.

Similarity signature curves for forming periodic orbits in the Lorenz system

In this paper, the short-periodic orbits of the Lorenz system are systematically investigated by the aid of the similarity signature curve, and a novel method to find the short-period orbits of the Lorenz system is proposed. We derive the similarity invariants by the equivariant moving frame theory, and then the similarity signature curve occurs along with them. Our analysis shows that the trajectory of the Lorenz system can be described by two completely different states. One is a stable state where the trajectory rotates around an equilibrium point. The other is a mutation state where the trajectory transitions to another equilibrium point. In particular, the similarity signature curve of the Lorenz system presents a more regular behavior than its trajectories. Additionally, with the assistance of the sliding window method, the quasi-periodic orbits can be detected numerically. Furthermore, all periodic orbits with period p⩽8 in the Lorenz system are found, and their period lengths and symbol sequences are calculated.

Disentangling inhibition and prediction in negation processing

Negation as a universal feature of human language is used effortlessly in everyday communication. However, experimental research has shown that the comprehension of negated sentences seems to require additional cognitive resources compared to affirmative sentences. Many studies investigating the processing of negation report longer reading and reaction times for negative compared to affirmative sentences and many studies report a Polarity by Truth interaction: false affirmative sentences lead to longer response times and larger N400 event-related potentials (ERPs) than true affirmative sentences, whereas the pattern is reversed for negative sentences where it is the true sentence that elicits longer reaction times and higher N400 ERPs compared to false negative sentences. These interactions have been discussed in the light of lexical associations, predictability, and the need to construct two subsequent mental representations. Furthermore, recent studies have shown that the comprehension of negated sentences seems to make use of neural resources that are typically involved in cognitive control and inhibitory mechanisms. As both processes have been associated with two different and temporally overlapping ERP components (the N400 and the P300), we focus on studies with high temporal resolution. We discuss linguistic aspects of negation, such as semantic similarity and contextual invariance of negation. We furthermore discuss the role of the verb as well as the position of the negative marker with respect to the verb, and their respective relevance for predictive and inhibitory mechanisms in negated sentences.

Analysis of Cattaneo-Christov heat diffusion on MHD Casson-Williamson bioconvective nanofluid flow across an exponential porous stretching sheet

ABSTRACT The present work analyzes the magnetized Casson-Williamson nanofluid flow past permeable and exponentially stretching devices. Microorganisms are immersed to improve the stability of nanoparticles. The novel effect of frictional forces working between the fluid layers is described using the Cattaneo – Christov heat flux model. The energy equation consists of viscous dissipation and Joule heating effect in the system. Further, thermal radiation and chemical reaction effects are considered during the mathematical framing. The nanofluid characteristics are elaborated using the Buongiorno nanofluid model. Thus, the obtained governing system of equation is further renovated using the similarity invariants. The graphical illustrations of the numerical outcomes are obtained in MATLAB. The code validation is carried out with a numerical comparison between the recent publications. Our results show that the fluid temperature enhances with higher values of thermophoresis, Brownian motion parameter, Rayleigh number, Eckert number, and Joule heating. The velocity declines with an increase in the strength of the magnetic field.

Higher-order curvature operators in causal set quantum gravity

We construct higher-order curvature invariants in causal set quantum gravity. The motivation for this work is twofold: First, to characterize causal sets, discrete operators that encode geometric information on the emergent spacetime manifold, e.g., its curvature invariants, are indispensable. Second, to make contact with the asymptotic-safety approach to quantum gravity in Lorentzian signature and find a second-order phase transition in the phase diagram for causal sets, going beyond the discrete analog of the Einstein–Hilbert action may be critical. Therefore, we generalize the discrete d’Alembertian, which encodes the Ricci scalar, to higher orders. We prove that curvature invariants of the form R^{2} -2 Box R (and similar invariants at higher powers of derivatives) arise in the continuum limit.

Dimensionality and Measurement Invariance of the 5-Item Coronavirus Fear Scale in Brazilian General Population

IntroductionThe COVID-19 Fear Scale (FCV-19S) is the most widely used instrument to assess fear of coronaviruses. Although preliminary analyses of the Brazilian–Portuguese version showed promising data for the 7-item version, several studies in Latin America suggest that the 5- and 6-item versions present better psychometric indicators. ObjectiveTo replicate and compare the Brazilian–Portuguese version of the (FCV-5S), studying its homogeneity and dimensionality. MethodsA total of 1003 adults between 18 and 78 voluntarily participated. The data were analyzed through exploratory factorial analysis and structural equations modeling. A Multiple Indicators and Multiples Causes model (MIMIC) was used to check the differential functioning of each item regressed on age. Likewise, Cronbach's alpha and McDonald's omega were calculated for FCV-5S. Finally, as a test of nomological validity, the mean scores and standard deviation between men and women were compared after testing similarity invariance. Results73.3% were younger adults (18–44 years old), 71.3% were women, and 59.7% had a university education. The 5-item version (FCV-5S) of the COVID-19 Fear Scale has better goodness-of-fit indicators than the 6-item version for a one-factor structure. FCV-5S accomplish with invariance by gender and partial invariance by age in the general population of Brazil. ConclusionsThe FCV-5S has a dimensional structure with partial invariance by gender and age and can be used to assess COVID-19 fear in the general population in Brazil.

MHD-driven chemically active and thermally radiative Prandtl hybrid nanofluid flow on stretching device with Ohmic heating, dissipation, and diffusion effects

This study explores the magnetohydrodynamics of reactive hybrid nanofluid flow with the Prandtl fluid model. The flow is past a linear stretching device and its physical characteristic are studied for the novel effects of viscous dissipation, Joule heating, chemical reaction, thermal radiation, and double diffusion. Furthermore, the multiple slip conditions are considered while framing the governing system of equations. The mathematical set of equations is transmuted with similarity invariants and obtained from the nonlinear set of ordinary differential equations. The numerical simulation is carried out via the Runge–Kutta fourth-order method with a shooting scheme. The graphical illustration of numerical outcomes is sketched through MATLAB. The code validation is achieved via comparative analysis with recent publications. It is observed that the applied magnetic field retard the fluid motion and increases the thermal boundary. Dissipation and radiation effects also boost the thermal boundary regime.

The reflection invariance problems in stereochemical nomenclature for absolute configuration

Abstract Genesis of chirality and its link to the origin of life continue to intrigue scientists. Chirality also plays an important role in medicine with the advent of chiral drugs and the consequent importance of asymmetric synthesis. The configuration of a chiral molecule gets inverted in its mirror image, as is manifested by (i) sign of specific rotation and (ii) normally, through the absolute configurational notation (R/S). While the first parameter is always true, the second one, that is the mirror image relationship of R and S configurations, called the reflection variance, may not always hold good. In this manuscript, we have highlighted examples of molecules with combinations of central and axial or planar chirality where the mirror image relationship between R and S configurations does not hold. This reflection invariance originates from the CIP rule of R precedes S. Similar reflection invariance can be observed in P or M forms of conformations of some particular molecules with chiral ligands. Proposal to avoid such reflection invariances, has been put forward. An ignorance about the possibility of reflection invariance in enantiomeric molecules can lead to common mistakes about stereoisomeric relationship or reactivity profile.

Coarse-grained numerical simulation for compressible fluid–particle two-phase flows

Compressible fluid–particle two-phase flows broadly exist in engineering problems, and the Eulerian–Lagrangian method is a popular branch of simulation studies. Usually, the coarse-grained strategy is adopted to reduce computational costs, and the coarse-grained criterion becomes critical for maintaining accuracy. In this study, a coarse-grained criterion was proposed for simulating compressible particulate two-phase flows by considering similarity invariants and regime transition behaviors. Based on our developed computation framework, in which the particle phase is solved using the discrete element method, a series of benchmark cases, including shock impacting granular column, shock impacting granular layer, and shock impacting granular ring cases, were considered to investigate the validity of the proposed criterion. It was proven that the stiffness coefficient should be scaled to the parcel size to maintain the invariance of the spreading velocity of the particle stress wave and the restitution coefficient should be reduced to help recover the internal energy dissipation inside the parcels. Furthermore, to describe more accurately the regime transition behaviors, which are common phenomena in compressible particulate two-phase flows, an adaptive interpolation operator was introduced to adjust the influencing range of the Lagrangian parcels dynamically.

Geometric similarity invariants of Cowen-Douglas operators

In 1978, M. J. Cowen and R.G. Douglas introduced a class of operators B_n(\Omega) (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such an operator. They gave a complete set of unitary invariants in terms of the curvature and its covariant derivatives. At the same time they asked whether one can use geometric ideas to find a complete set of similarity invariants of Cowen-Douglas operators. We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely determine the similarity orbit of a norm dense class of Cowen-Douglas operators. As an application we show that uncountably many (non-similar) strongly irreducible operators in B_n(\mathbb{D}) can be constructed from a given operator in B_1(\mathbb{D}) . We also characterize a class of strongly irreducible weakly homogeneous operators in B_n(\mathbb{D}) .

Slow and fast collective neutrino oscillations: Invariants and reciprocity

The flavor evolution of a neutrino gas can show ''slow'' or ''fast'' collective motion. In terms of the usual Bloch vectors to describe the mean-field density matrices of a homogeneous neutrino gas, the slow two-flavor equations of motion (EOMs) are $\dot{\mathbf{P}}_\omega=(\omega\mathbf{B}+\mu\mathbf{P})\times\mathbf{P}_\omega$, where $\omega=\Delta m^2/2E$, $\mu=\sqrt{2} G_{\mathrm{F}} (n_\nu+n_{\bar\nu})$, $\mathbf{B}$ is a unit vector in the mass direction in flavor space, and $\mathbf{P}=\int d\omega\,\mathbf{P}_\omega$. For an axisymmetric angle distribution, the fast EOMs are $\dot{\mathbf{D}}_v=\mu(\mathbf{D}_0-v\mathbf{D}_1)\times\mathbf{D}_v$, where $\mathbf{D}_v$ is the Bloch vector for lepton number, $v=\cos\theta$ is the velocity along the symmetry axis, $\mathbf{D}_0=\int dv\,\mathbf{D}_v$, and $\mathbf{D}_1=\int dv\,v\mathbf{D}_v$. We discuss similarities and differences between these generic cases. Both systems can have pendulum-like instabilities (soliton solutions), both have similar Gaudin invariants, and both are integrable in the classical and quantum case. Describing fast oscillations in a frame comoving with $\mathbf{D}_1$ (which itself may execute pendulum-like motions) leads to transformed EOMs that are equivalent to an abstract slow system. These conclusions carry over to three flavors.

Buongiorno Modelled Nanoliquid Consequence of Thermal and Solutal Convection on the Magneto-Micropolar Fluid Inside an Inclined Porous Stretching Device

The objective of the present work is to investigate the chemically reacting micropolar nanofluid flow past an inclined permeable stretching sheet. The flow is exposed in the magnetic and thermally radiative environment and convective boundary conditions are taken into consideration to form the complete description of the flow model. The appropriate similarity invariants are used to obtain the system of the non-linear ordinary differential equation. The well-known Runge-Kutta fourth-order method is worked out with a shooting technique for numerical investigation. The behavior of fluid motion, temperature, concentration, and engineering quantities for arising physical parameters is sketched through graphs and discussed in detail. The tabular comparison is used to provide the validation of current work. Outputs demonstrate flow decrement, which is due to the produced Lorentz force that itself is the result of the enhancement in magnetic parameter and inclination of angle, and rise in the profiles of porosity index. The fluid temperature exceeds in conjunction with the accumulation of radiative heat, dissipation heat, as well as thermophoresis and Brownian motion. Physical engineering supporting forces like Skin friction and heat transfer rates are favorable for magnetic and material parameters but the mass transfer rate exhibits an opposite response.

Convergence of blanket times for sequences of random walks on critical random graphs

AbstractUnder the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$ -blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$ -blanket times of the random walks if the $\varepsilon$ -blanket time of the limiting diffusion is continuous at $\varepsilon$ with probability 1. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees and the Erdős-Rényi random graph in the critical window. We highlight that proving continuity of the $\varepsilon$ -blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and therefore we expect our results to hold for other examples of random graphs with a similar scale invariance property.

Optimal Extraction Method of Feature Points in Key Frame Image of Mobile Network Animation

In order to effectively extract the feature points of mobile network animation images and accurately reflect the main content of the video, an optimization method to extract the feature points of key frame images of mobile network animation is proposed. Firstly, the key frames are selected according to the content change degree of the animation video. The scale invariant feature transformation algorithm is used to describe the feature points of the key frame image of the animation video. The local feature points of the image are estimated by the constraint optimization method to realize the optimization extraction of the feature points of the key frame image of the mobile network animation. The efficiency of feature points extraction is analyzed from the number and effectiveness of feature points extraction, time-consuming and similarity invariance of feature points. The experimental results show that the proposed method has excellent adaptability, and can effectively extract feature points of mobile network animation image.

Application of similarity laws to dual-frequency capacitively coupled radio frequency plasmas with the electrical asymmetry effect

Similarity laws (SLs) are useful for correlation and prediction of plasma parameters at different scales, which have been verified for geometrically symmetric capacitive radio-frequency (rf) discharges in nonlocal kinetic regimes. In this work, we demonstrate the applicability of SLs to dual-frequency rf discharges and confirm that similarity relations still hold considering the electrical asymmetry effect (EAE). By simultaneously tuning the control parameters (the gas pressure p, discharge gap d, and driving frequency f), we examine the similarity relations in rf plasmas via fully kinetic particle-in-cell simulations with the external circuits coupled and solved self-consistently. The validity of the SL scalings in dual-frequency rf plasmas with the EAE is confirmed for parameters such as the electron/ion density, ion flux, dc self-bias, ion energy distribution function, and power absorption. Although adjusted by the EAE, the dc self-bias and ion energy distribution functions are identified as similarity invariants under similar discharge conditions. Furthermore, the plasma series resonance phenomenon, filamentation of power depositions of electrons and ions in bulk plasma, and electric field reversal are observed in dual-frequency discharges with the EAE, which can also be exactly replicated under similar discharge conditions. The results further extend the application of SL scaling to dual-frequency rf plasmas, providing a more comprehensive understanding of the scaling characteristics in rf plasmas.

Eyring-Powell nano liquid flow through permeable elongated sheet conveying inclined magnetic field subject to constructive chemical reaction and multiple slip effects

ABSTRACT The present study examines the multiple slip effects and constructive chemical reaction on Eyring-Powell nanoliquid through a permeable elongated sheet on an inclined magnetic field. The energy balance equation includes Heat and Joule dissipation terms. The renowned Buongiorno nanofluid model is employed extensively to explore the thermophoresis and Brownian motion phenomena. The primary partial differential equations (PDEs) of governing flow phenomena are remodeled to non-linear ordinary differential equations (ODEs) by adopting proper similarity invariants. Runge-Kutta 4th order quadrature employing shooting technique is utilized to transform boundary value problem to initial value problem to seek numerical results via graphs and tables on physically interesting parameters. Current outcomes are validated through the comparison with previously published studies. An applied magnetic field slows down the fluid and raises both thermal and solutal boundary layers. It is significant to notice that the thermal boundary layer rises, as the Brownian and Thermophoresis are amplified.

Unsteady thermally radiative Prandtl fluid flow past a magnetized inclined porous stretching device with double-diffusion, viscous dissipation, and Joule heating

The extensive applications and complex nature of non-Newtonian fluids inspire researchers to explore them so it serves more precisely humankind. In this scenario, the Prandtl fluid over the inclined porous surface has been explored and formulated the mathematical model considering the magnetic, thermal radiation, and double diffusion effect. Also, the viscous dissipation and joule heating effect are taken into account to explore more features of the flow situation. The mathematical model of flow is renovated with the effective implementation of similarity invariants. The numerical investigation is carried out using the Runge–Kutta method and the results obtained are interpreted graphically. The effects of physical parameters such as Dufour and Soret number, radiation parameter, magnetic parameter, etc., on non-dimensional quantities, are graphically illustrated with MATLAB. The thermal boundary layer gets elevation for rising values of Eckert number, magnetic parameter, unsteadiness parameter, Dufour number, and thermal radiation.

MHD Williamson nanofluid flow past a permeable stretching sheet with thermal radiation and chemical reaction

ABSTRACT Heat and mass transmission of non-Newtonian fluids get considerable attention due to their complexity and high nonlinearity in rheology. Among the various fluid models, Williamson fluid model very well describes the low as well as high-viscosity situations. With this perspective, the present work deals with heat and mass transfer of MHD Williamson nanofluid flow past a porous stretching surface with thermal radiation and chemical reaction. Highly nonlinear governing equations are transformed into ordinary ones by employing similarity invariants. The reduced systems of ODEs are numerically treated with R-K’s fourth-order method along with the shooting technique. Using MATLAB the associated nondimensional parameters are graphically interpreted and found to be in good agreement with the literature. For the higher values of magnetic and Williamson parameters, fluid motion becomes a decreasing function whereas the temperature profile is enhanced for the magnetic, Williamson, and radiation parameters. The investigation will serve in the process involving fluids such as food processing, glass manufacturing, oil extraction, etc.