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In this paper we discuss the relevance of continuity equation based aortic valve area (AVA) calculation as a robust parameter suitable for accurate grading of aortic stenosis (AS) irrespective of flow conditions. Combining the AVA-based grading and echocardiography-based staging, can provide with the most comprehensive clinical assessment of patients with AS and preserved left ventricular systolic function to streamline management decisions. [ABSTRACT FROM AUTHOR]

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique η -Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang. [ABSTRACT FROM AUTHOR]

The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields. These equations contain derivatives of the first order with respect to time. The derivation of the equations of continuum mechanics uses the limit transitions of the tendency of the volume increment and the time increment to zero. Derivatives are used to derive the wave equation. The differential wave equation is second order in time. Therefore, increments of volume and increments of time in continuum mechanics should be considered as small but finite quantities for problems of wave formation. This is important for calculating the generation of sound waves and water hammer waves. Therefore, the Euler continuity equation with finite time increments is of interest. The finiteness of the time increment makes it possible to take into account the quadratic and cubic invariants of the strain rate tensor. This is a new branch in hydrodynamics. Quadratic and cubic invariants will be used in differential wave equations of the second and third order in time. [ABSTRACT FROM AUTHOR]

Reservoir routing has been a routine procedure in hydrology, hydraulics and water management. It is typically based on the mass balance (continuity equation) and a conceptual equation relating storage and outflow. If the latter is linear, then there exists an analytical solution of the resulting differential equation, which can directly be utilized to find the outflow from known inflow and to obtain macroscopic characteristics of the process, such as response and residence times, and their distribution functions. Here we refine the reservoir routing framework and extend it to find approximate solutions for nonlinear cases. The proposed framework can also be useful for climatic tasks, such as describing the mass balance of atmospheric carbon dioxide and determining characteristic residence times, which have been an issue of controversy. Application of the theoretical framework results in excellent agreement with real-world data. In this manner, we easily quantify the atmospheric carbon exchanges and obtain reliable and intuitive results, without the need to resort to complex climate models. The mean residence time of atmospheric carbon dioxide turns out to be about four years, and the response time is smaller than that, thus opposing the much longer mainstream estimates. [ABSTRACT FROM AUTHOR]

Abstract In this paper we discuss the relevance of continuity equation based aortic valve area (AVA) calculation as a robust parameter suitable for accurate grading of aortic stenosis (AS) irrespective of flow conditions. Combining the AVA-based grading and echocardiography-based staging, can provide with the most comprehensive clinical assessment of patients with AS and preserved left ventricular systolic function to streamline management decisions.

The vertically integrated zonal momentum balance of the Antarctic Circumpolar Current (ACC) is dominated by wind stress at the surface and topographic form stress (TFS) at the bottom. It has been argued that wind stress is transferred from the surface to the bottom by transient baroclinic eddies, via interfacial form stress, to establish the balance between wind stress and TFS. However, ocean models indicate TFS responds rapidly to changes in wind stress, suggesting that barotropic processes play a role in this balance. We investigate the dynamics governing the wind–TFS balance of the ACC and its response to wind using an idealized, wind- and buoyancy-driven channel model. We show that the balance is established and maintained at equilibrium by the barotropic dynamics. The balance results from the continuity of the flow, in which the Ekman transport at the surface, balanced by wind stress, is compensated by a return flow at depth, balanced by TFS. This leads to a match between wind stress and TFS which is independent of momentum stresses in the interior. Transient baroclinic eddies oppose the wind-driven isopycnal steepening via eddy buoyancy fluxes, which act to flatten the isopycnals. The eddy-driven isopycnal flattening corresponds to a reduction in the zonal geostrophic shear and thus a redistribution of the zonal momentum in the interior via eddy momentum stresses. The maintenance of the vertically integrated ACC momentum balance by the barotropic dynamics explains the fast response of the wind–TFS balance to changes in wind forcing. [ABSTRACT FROM AUTHOR]

In clinical practice, dogs are screened for subaortic stenosis (SAS) using two-dimensional (2DE) and Doppler echocardiography. There is no accepted antemortem diagnostic criterion to distinguish between mild SAS and unaffected, therefore additional means of evaluating the left ventricular outflow tract (LVOT) and aorta may be desirable. This study sought to determine and compare LVOT and aortic orifice areas using 2DE and three-dimensional echocardiography (3DE) in apparently healthy dogs of various breeds and somatotypes. Sixty-nine healthy, privately-owned dogs. The LVOT and aortic orifice areas were determined using 2DE aortic valve (AV) diameter-derived area; the continuity equation (CE); and 3DE planimetry of the LVOT, AV, sinus of Valsalva, and sinotubular junction. Orifice areas were indexed to body surface area (BSA). Obtaining 3DE images and performing planimetry were feasible in all dogs. The mean indexed area measured using the 2DE AV diameter (2.85 cm2/m2) was significantly lower than that derived from 3DE AV planimetry (3.85 cm2/m2; mean difference, 1.00 cm2/m2; P<0.001). There was poor agreement between the effective area calculated using the CE and the anatomic areas calculated using 2DE AV diameter and 3DE planimetry. The area calculated using the CE was less than all other calculations of area. Interobserver and intraobserver repeatability and reproducibility for 3DE planimetry were excellent. Methods for determining aortic orifice areas in dogs are not interchangeable, and this must be taken into account if these methods are investigated in the evaluation of dogs with SAS in the future. [ABSTRACT FROM AUTHOR]

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.'s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system's structure invariant, a system with the structure of Maxwell's field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations. [ABSTRACT FROM AUTHOR]

Lethal untrackable debris objects pose the highest risk to the sustainability of the space environment, and thus, shall be included in the assessment of the long-term effect of mitigation and remediation measures to the space debris problem. The introduction of centimetre-sized particles in the debris evolutionary models represents a challenge from a computational cost point of view. To answer this need, this work proposes a novel probabilistic debris environment propagator. The method classifies the objects population into intact objects and fragmentation debris. The evolution of the former population is retrieved through an individual definition of each object's mission profile. A continuum approach is adopted for the characterisation of the fragments, whose density distribution in orbital elements is propagated in time through the continuity equation. The intrinsic computational efficiency of the density-based fragments cloud models is leveraged to make the method agnostic to the lowest fragments size considered. A second classification of the population of intact objects into species, such as payloads, rocket bodies, mission related objects and constellations, ensures a faithful replication of their orbit evolution. Fragmentation debris caused by intact objects explosion and accidental fragments-intact object collision are included in a probabilistic fashion at the detected fragmentation epoch, to account for their feedback effect onto the environment. The model is applied to estimate the evolution of the space debris population in low-Earth orbit up to 200 years from the reference epoch, with and without the inclusion of a future launch traffic pattern, and considering a different fulfilment of the post-mission disposal phase. • Density-based methods are introduced into a complex space debris evolutionary model. • Intact objects are divided in species to observe debris environment dependency on mission design. • Effect of different adherence to post-mission disposal guidelines is analysed. • Predictions on the evolution of the space environment 200 years from now are presented. [ABSTRACT FROM AUTHOR]

Sediment leads to problems with navigation, agricultural productivity, and water pollution. The study of sediment flow in river reaches, which is a non-linear and complex process, is, thus, essential to addressing these issues. The application of artificial neural networks (ANN) to such problems needs to be investigated. For unsteady flow in a river system, river reach storage is an important variable that needs to be considered in data-driven models. However, previous research on sediment modeling did not involve the explicit use of storage variables in such models as is investigated in the current study. In the current study, storage variables have been explicitly (Model 2) used to predict the output state of the system at time ' t + 1' from the input state at time ' t ' using ANNs. Sediment discharge at six gaging stations on the Mississippi River system, USA, has been considered as the state variable. The model has been compared with a model considering implicit variation of the storage parameter in the river system (Model 1). Dynamic ANNs are used for time-series datasets, which are more suitable for incorporating the sequential information within the dataset. Focussed gamma memory neural networks have been used in the current study. The numbers of hidden layers and hidden nodes, activation function, and learning rate have been varied step by step to obtain the optimal ANN configurations. The best selected input–output variables are those used in Model 2 as it performed slightly better than the other model in terms of Nash–Sutcliffe efficiency coefficient (CE) values. Model performance evaluated using normalized root mean square error (NRMSE) and CE shows satisfactory results. NRMSE was < 10% for all the outputs except for the Venedy and Murphysboro locations and CE values for sediment loads were > 0.45 for all locations except Murphysboro indicating acceptable performance by both the models. The models proved highly efficient (CE > 0.80, i.e., very good predictions) for predicting sediment discharge at locations along the main river channel with acceptable accuracy (CE > 0.45) for other locations and the storage change for the river system. These models can be used for real-time forecasting and management of sediment-related problems. [ABSTRACT FROM AUTHOR]

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique η-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang.

Introduction: An arteriovenous fistula (AVF) in patients with end‐stage kidney disease (ESKD) can influence flow states. We sought to evaluate if assessment of aortic stenosis (AS) by transthoracic echocardiographic (TTE) differs in the presence of AVF compared to other dialysis accesses in patients on dialysis. Methods: We identified consecutive ESKD patients on dialysis and concomitant AS from a single center between January 2000 and March 2021. We analyzed TTE parameters of AS severity (velocities, gradients, aortic valve area [AVA]) and hemodynamics (cardiac output [CO], valvuloarterial impedance [Zva]) and compared AS parameters in patients with AVF versus other dialysis access. Results: The cohort included 94 patients with co‐prevalent ESKD and AS; mean age 66 years, 71% male; 43% Black, 24% severe AS. Dialysis access: 53% AVF, 47% others. In the overall cohort, no significant differences were noted between AVF versus non‐AVF in AVA/CO/Zva, but with notable subgroup differences. In mild AS, CO was significantly higher in AVF versus non‐AVF (6.3 vs. 5.2 L/min; p =.04). In severe AS, Zva was higher in the AVF versus non‐AVF (4.6 vs. 3.6 mm Hg/mL/m2). With increasing AS severity in the AVF group, CO decreased, coupled with increase in Zva, likely counterbalancing the net hemodynamic impact of the AVF. Conclusion: Among ESKD patients with AS, TTE parameters of flow states and AS severity differed in those with AVF versus other dialysis accesses and varied with progression in severity of AS. Future longitudinal assessment of hemodynamic parameters in a larger cohort of co‐prevalent ESRD and AS would be valuable. [ABSTRACT FROM AUTHOR]

Deformation theory is common to the theories of elasticity, hydrodynamics, and electrodynamics. The law of conservation for the deformation of a control figure contains linear, quadratic, and cubic invariants. Moving to the limit of deriving the continuity equation destroys the quadratic and cubic invariants in the formula for the coefficient of volume expansion and the equation of continuity. This simplification can result in the loss of some modes of fluid motion and descriptions of the possible behavior of electromagnetic fields. A way of considering the three invariants in solutions to equations of the hydrodynamics and electrodynamics is presented in this work. It is proposed that second- and third-order wave differential equations of time be used to describe the motion of a fluid and the behavior of a magnetic field's strength with allowance for quadratic and cubic invariants. [ABSTRACT FROM AUTHOR]

Reservoir routing has been a routine procedure in hydrology, hydraulics and water management. It is typically based on the mass balance (continuity equation) and a conceptual equation relating storage and outflow. If the latter is linear, then there exists an analytical solution of the resulting differential equation, which can directly be utilized to find the outflow from known inflow and to obtain macroscopic characteristics of the process, such as response and residence times, and their distribution functions. Here we refine the reservoir routing framework and extend it to find approximate solutions for nonlinear cases. The proposed framework can also be useful for climatic tasks, such as describing the mass balance of atmospheric carbon dioxide and determining characteristic residence times, which have been an issue of controversy. Application of the theoretical framework results in excellent agreement with real-world data. In this manner, we easily quantify the atmospheric carbon exchanges and obtain reliable and intuitive results, without the need to resort to complex climate models. The mean residence time of atmospheric carbon dioxide turns out to be about four years, and the response time is smaller than that, thus opposing the much longer mainstream estimates.

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

We show that vector fields b whose spatial derivative D x b satisfies a Orlicz summability condition have a spatially continuous representative and are well-posed. For the case of sub-exponential summability, their flows satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if D x b satisfies a suitable exponential summability condition then the flow associated to b has Sobolev regularity, without assuming boundedness of div x b. We then apply these results to the representation and Sobolev regularity of weak solutions of the Cauchy problem for the transport and continuity equations. [ABSTRACT FROM AUTHOR]

A steady, two-dimensional, incompressible couple stress fluid flow over a rigid sphere of radius 'a' surrounded by infinite porous region specifying a constant velocity away from the boundary is considered. An exact solution is found for the governing equations which leads to the expression for the stream function and shearing stress. The impact of couple stress parameter and porosity on the flow patterns is examined through streamlines. Also shear stress is computed for various values of couplestress parameter and porous parameter. The obtained results reveal that as coupling stress parameter increases for fixing the porosity, streamlines are symmetric and meandered near the rigid sphere. But for fixed coupling stress parameter and increase in porous parameter cause the streamlines to move away from the solid sphere. Also, the dimensionless shear stress increases as couple-stress parameter intensifies for fixed porous parameter and vanishes at two stagnation points. The amplitude of the shearing stress raises with raise in porous parameter for fixed coupling stress parameter. [ABSTRACT FROM AUTHOR]

Particle image velocimetry (PIV) is a widely used experimental technique in ocean engineering, for instance, to study the vortex fields near marine risers and the wake fields behind wind turbines or ship propellers. However, the flow fields measured using PIV in water tanks or wind tunnels always have low resolution; hence, it is difficult to accurately reveal the mechanics behind the complex phenomena sometimes observed. In this paper, physics-informed neural networks (PINNs), which introduce the Navier–Stokes equations or the continuity equation into the loss function during training to reconstruct a flow field with high resolution, are investigated. The accuracy is compared with the cubic spline interpolation method and a classic neural network in a case study of reconstructing a two-dimensional flow field around a cylinder, which is obtained through direct numerical simulation. Finally, the validated PINN method is applied to reconstruct a flow field measured using PIV and shows good performance. [ABSTRACT FROM AUTHOR]

A model of nonstationary flow of liquid and of a gas–liquid mixture in the reservoir–pipeline system is constructed and related equations are solved. Analytical expressions are obtained allowing one to determine the volume of the liquid flowing through any cross sections of a pipe per unit time. Numerical calculations were made at various values of the system parameters. [ABSTRACT FROM AUTHOR]

We construct divergence-free Sobolev vector fields in C([0, 1];W1,r(Td;Rd)) with r < d and d ≥ 2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. These vector fields are then shown to have at least as many integral curves starting from Ld-a.e. point of Td as the number of distinct positive solutions to the continuity equation these vector fields admit. This work uses convex integration techniques to study nonuniqueness for positive solutions of the continuity equation. Nonuniqueness for integral curves is then inferred from Ambrosio's superposition principle. [ABSTRACT FROM AUTHOR]

Hydrogen sulfide (H2S) is considered a toxic and corrosive gas, commonly found in natural gas, crude oil, and other fossil fuels. This corrosive gas may lead to stress corrosion cracking (SCC). This phenomenon is caused by the combined influence of tensile stress and a corrosive environment. This may lead to the sudden failure of normally ductile metal alloys, especially at an elevated temperature. Desulfurization is the process of removing H2S from these fuels to reduce their harmful environmental and health impacts. Ionic liquids (ILs) have shown great potential for application as liquid absorbents for H2S extraction because of their advantages such as non-volatility, functionality, high carbon solubility and low energy requirements for regeneration. The proposed hydrogen sulfide extraction system consists of a tube, membrane and shell. 1-ethyl-3-methylimidazolium (emim)-based ionic liquids with bis-(trifluoromethyl) sulfonylimide (NTf2) anion has been selected due to its high H2S diffusion coefficient. Functionalized graphene oxide (GO) advanced membranes have been employed in this design. In this research, H2S extraction with ionic liquids has been numerically studied. The COMSOL finite element and multi-physics code has been employed to solve the continuity, turbulent fluid flow (k-ε model), and transient diffusion equations. For small time periods, there is sharp gradient in H2S concentration profile inside the shell section. This is because the diffusion coefficient of H2S in the ionic liquid is very small and the shell section is much thicker than the membrane. It has been determined that H2S is absorbed almost completely by ionic liquids after a time period of 30,000 s. [ABSTRACT FROM AUTHOR]

Aortic stenosis (AS) is the most prevalent degenerative valvular disease in western countries. Transthoracic echocardiography (TTE) is considered, nowadays, to be the main imaging technique for the work-up of AS due to high availability, safety, low cost, and excellent capacity to evaluate aortic valve (AV) morphology and function. Despite the diagnosis of AS being considered straightforward for a very long time, based on high gradients and reduced aortic valve area (AVA), many patients with AS represent a real dilemma for cardiologist. On the one hand, the acoustic window may be inadequate and the TTE limited in some cases. On the other hand, a growing body of evidence shows that patients with low gradients (due to systolic dysfunction, concentric hypertrophy or coexistence of another valve disease such as mitral stenosis or regurgitation) may develop severe AS (low-flow low-gradient severe AS) with a similar or even worse prognosis. The use of complementary imaging techniques such as transesophageal echocardiography (TEE), multidetector computed tomography (MDTC), or cardiac magnetic resonance (CMR) plays a key role in such scenarios. The aim of this review is to summarize the diagnostic challenges associated with patients with AS and the advantages of a comprehensive multimodality cardiac imaging (MCI) approach to reach a precise grading of the disease, a crucial factor to warrant an adequate management of patients.

The permanent power loss and the deviation of the trajectory of satellites impacted by centimetre and sub-centimetre sized debris have highlighted the need of taking into account such small fragments in the evolutionary models of the debris population and in the assessment of the in-orbit collision risk. When scaling down to the centimetre-millimetre range, deterministic models for propagating the fragments' orbit suffer from the massive computational cost required. The continuum approach for modelling the debris clouds is a well-established alternative to the piece-by-piece propagation. A density function is formulated to describe the distribution of fragments over a suitable phase space. Accurate and efficient continuum formulations have been developed to propagate clouds of fragments under atmospheric drag and J 2 perturbations, but a general model able to work under any dynamical regime has still to be found. This paper proposes a continuum approach that combines the method of characteristics with the discretisation of the domain in Keplerian elements and area-to-mass ratio into bins. The problem of using a binning approach with such a multi-dimensional phase space is addressed bounding and partitioning the domain, through probabilistic models on the way the fragments distribute over the phase space, as consequence of a fragmentation event. The proposed approach is applied to the modelling and propagation of a space debris cloud under the full set of orbital perturbations, and compared against a Monte Carlo simulation in terms of objects' number and distribution. The method proves to be accurate on the medium scale, in both space and time, and guarantees statistical validity with a reduced computational effort, leveraging its probabilistic nature. [ABSTRACT FROM AUTHOR]

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise. [ABSTRACT FROM AUTHOR]

Background: Diagnosing severe aortic stenosis (AS) depends on flow and pressure conditions. It is suspected that concomitant aortic regurgitation (AR) has an impact on the assessment of AS severity. The aim of this study was to analyze the impact of concomitant AR on Doppler-derived guideline criteria. We hypothesized that both transvalvular flow velocity (maxVAV) and the mean pressure gradient (mPGAV) will be affected by AR, whereas the effective orifice area (EOA) and the ratio between maximum velocity of the left ventricular outflow tract and transvalvular flow velocity (maxVLVOT/maxVAV) will not. Furthermore, we hypothesized that EOA (by continuity equation), and the geometric orifice area (GOA) (by planimetry using 3D transesophageal echocardiography, TEE), will not be affected by AR. Methods and Results: In this retrospective study, 335 patients (mean age 75.9 ± 9.8 years, 44% male) with severe AS (defined by EOA < 1.0 cm2) who underwent a transthoracic and transesophageal echocardiography were analyzed. Patients with a reduced left ventricular ejection fraction (LVEF < 53%) were excluded (n = 97). The remaining 238 patients were divided into four subgroups depending on AR severity, and they were assessed using pressure half time (PHT) method: no, trace, mild (PHT 500–750 ms), and moderate AR (PHT 250–500 ms). maxVAV, mPGAV and maxVLVOT/maxVAV were assessed in all subgroups. Among the four subgroups (no (n = 101), trace (n = 49), mild (n = 61) and moderate AR (n = 27)), no differences were obtained for EOA (no AR: 0.75 cm2 ± 0.15; trace AR: 0.74 cm2 ± 0.14; mild AR: 0.75 cm2 ± 0.14; moderate AR: 0.75 cm2 ± 0.15, p = 0.998) and GOA (no AR: 0.78 cm2 ± 0.20; trace AR: 0.79 cm2 ± 0.15; mild AR: 0.82 cm2 ± 0.19; moderate AR: 0.83 cm2 ± 0.14, p = 0.424). In severe AS with moderate AR, compared with patients without AR, maxVAV (p = 0.005) and mPGAV (p = 0.022) were higher, whereas EOA (p = 0.998) and maxVLVOT/maxVAV (p = 0.243) did not differ. The EOA was smaller than the GOA in AS patients with trace (0.74 cm2 ± 0.14 vs. 0.79 cm2 ± 0.15, p = 0.024), mild (0.75 cm2 ± 0.14 vs. 0.82 cm2 ± 0.19, p = 0.021), and moderate AR (0.75 cm2 ± 0.15 vs. 0.83 cm2 ± 0.14, p = 0.024). In 40 (17%) patients with severe AS, according to an EOA < 1.0 cm2, the GOA was ≥ 1.0 cm2. Conclusion: In severe AS with moderate AR, the maxVAV and mPGAV are significantly affected by AR, whereas the EOA and maxVLVOT/maxVAV are not. These results highlight the potential risk of overestimating AS severity in combined aortic valve disease by only assessing transvalvular flow velocity and the mean pressure gradient. Furthermore, in cases of borderline EOA, of approximately 1.0 cm2, AS severity should be verified by determining the GOA. [ABSTRACT FROM AUTHOR]

We study the mixing properties of a passive scalar advected by an incompressible flow. We consider a class of cellular flows (more general than the class in [Crippa-Schulze M$ ^3 $AS 2017]) and show that, under the constraint that the palenstrophy is bounded uniformly in time, the mixing scale of the passive scalar cannot decay exponentially.

The mean-field type control problems with incomplete information are considered. There are several points of view that can be adopted to study the dynamics in probability space. Eulerian framework describes probability flows by the continuity equation. Kantorovich formulation describes each probability flows in terms of a single distribution on the set of admissible trajectories. The superposition principle connects these frameworks for uncontrolled dynamics. In this article, a probability flow in the both frameworks must be generated by a control that based on incomplete information about state and/or the probability at every time instance. This article presents some links between these frameworks in the case of incomplete information. In particular, besides the convexity condition, the assumptions are founded that guarantees the equivalence between the Kantorovich and Eulerian framework. This expands [6, Theorem 1] to mean-field type control problem with incomplete information.

In the present paper, we derive and solve the space-fractional traffic flow model which is considered as a generalization of the transport density equation. Based on the fundamental physical principles on finite-length highway where the number of vehicles is conserved, without entrances or exits, we construct a fractional continuity equation. As a limitation of the classical calculus, the continuity equation is constructed based on truncating after the first order of Taylor expansion, which means that the change in the number of vehicles is linear over the finite-length highway. However, in fractional calculus, we prove that nonlinear flow is a result of truncating the fractional Taylor polynomial after the second term with zero error. Therefore, the new fractional traffic flow model is free from being linear, and the space now is described by the fractional powers of coordinates, provided with a single variable measure. Further, some exact solutions of the fractional model are generated by the method of characteristics. Remarkably, these solutions have significant physical implications to help to make the proper decisions for constructing traffic signals in a smart city.

In the work, hydrodynamics equations approximation was carried out based on the calculation of the distribution of water flow velocities in different time layers. A study of the stability of equation difference scheme for calculating the pressure by the Rote method was carried out. The error of equation approximation for pressure calculation is estimated. The accuracy of the numerical solution of the problem, built on the basis of a three-layer difference scheme, depends on the error in the approximation of the initial conditions and the error accumulated during transitions between time layers. An estimate of the approximation order of the initial conditions using a change of variables is carried out. An estimate of approximation error by the difference scheme of the hydrodynamic model was obtained by expanding the functions in a Taylor series. [ABSTRACT FROM AUTHOR]

Liouville equation is a fundamental one in statistical mechanics. It is rooted in ensemble theory. By ensemble theory, the variation of the system's microscopic state is indicated by the moving of the phase point, and the moving trajectory is believed continuous. Thus, the ensemble density is thought to be a smooth function, and it observes continuity equation. When the Hamiltonian canonical equations of the molecules are applied to the continuity equation, Liouville equation can be obtained. We carefully analyze a gas composed of a great number of molecules colliding with each other. The defects in deriving Liouville equation are found. Due to collision, molecules' momenta changes discontinuously, so that the trajectories of the phase points are actually not continuous. In statistical mechanics, infinitesimals in physics and in mathematics should be distinguished. In continuity equation that the ensemble density satisfies, the derivatives with respect to space and time should be physical infinitesimals, while in Hamiltonian canonical equations that every molecule follows, the derivatives take infinitesimals in mathematics. In the course of deriving Liouville equation, the infinitesimals in physics are unknowingly replaced by those in mathematics. The conclusion is that Liouville equation is not applicable to gases. [ABSTRACT FROM AUTHOR]

In the filtration theory there are numerous approaches to solving three-dimensional problems of fluid motion in a porous medium. Basically, the solutions of such problems are obtained by numerical methods. The question of finding an analytical solution of precisely three-dimensional problems of fluid motion is open. The first results on the use of the apparatus of four-dimensional mathematics for solving the three-dimensional model of the Navier-Stokes equations by the analytical method were obtained by the Kazakh mathematician Professor M.M. Abenov. After the author of this article with other researchers proved the theorem on the existence of a solution to the Cauchy problem for a three-dimensional model of filtration theory. This paper is devoted to the study of a three-dimensional model of the filtration theory in one of the spaces of four-dimensional numbers. The purpose of this article is to obtain an analytical solution of the three-dimensional Cauchy problem for the mathematical model of linear filtration model by the method of four-dimensional regular functions. In this study, a class of infinitely differentiable and bounded functions of the initial conditions of the Cauchy problem, satisfying the Cauchy-Riemann condition, with five degrees of freedom for a specific four-dimensional function is found, and also a class of infinitely differentiable and bounded solutions of this problem is found that satisfy the linear Darcy law.

In a basis of the space-time coordinate frame four quaternions discovered by Hamilton can be used. For subsequent reproduction of the coordinate frame these four quaternions are expanded to four 4 × 4 matrices with real-valued matrix coefficients −0 and 1. This group set is isomorphic to the SU(2) group. Such a matrix basis introduces extra six degrees of freedom of matter motion in space-time. There are three rotations about three space axes and three boosts along these axes. Next one declares the differential generating operators acting on the energy-momentum density tensor written in the above quaternion basis. The subsequent actions of this operator together with its transposed one on the above tensor lead to the emergence of the gravitomagnetic equations that are like the Maxwell equations. Wave equations extracted from the gravitomagnetic ones describe the propagation of energy density waves and their vortices through space. The Dirac equations and their reduction to two equations with real-valued functions, the quantum Hamilton-Jacobi equations and the continuity equations, are considered. The Klein-Gordon equations arising on the mass shell hints to the alternation of the paired fermion fields and boson ones. As an example, a Feynman diagram of an electron–positron time crystal is illustrated. [ABSTRACT FROM AUTHOR]

We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data are not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of its physical measure. Thus, we fit physical measures by taking the Wasserstein distance from optimal transportation as a misfit function between two probability distributions. In particular, we analyze the regularity of the resulting loss function for general transportation costs and derive gradient formulas. Physical measures are approximated as fixed points of suitable PDE-based Perron--Frobenius operators. Test cases discussed in the paper include common low-dimensional dynamical systems. [ABSTRACT FROM AUTHOR]

We study the mixing properties of a passive scalar advected by an incompressible flow. We consider a class of cellular flows (more general than the class in [Crippa-Schulze M3AS 2017]) and show that, under the constraint that the palenstrophy is bounded uniformly in time, the mixing scale of the passive scalar cannot decay exponentially. [ABSTRACT FROM AUTHOR]

We consider a PDE which describes the evolution of an electric charge distribution in one spatial dimension. Due to the (singular) electrostatic interaction, the PDE is nonlocal. Moreover, the PDE describes the neutralization of charge at points where the charge distribution changes sign. Despite these complex features of the PDE, we develop a simple scheme to solve the PDE numerically. We demonstrate by means of simulations that the scheme is accurate and that it preserves the (expected) properties of the exact solution. [ABSTRACT FROM AUTHOR]

The Long-Wave Theory is applied to investigate the dynamic stability of free thin fluid films flowing down an inclined plane. We assume that thin supported films have a thickness of H and less than or equal to one hundred nm. Equations of Navier and Stokes, continuity-equation, and related boundary conditions are used to represent a two-dimensional stream demonstrated as a continuum. Under long-wave approximation, the governing equations for the film interface have been rescaled and simplified to obtain a highly non-linear condition of development for the film interface. A procedure for evaluating the magnitude of the effects of the high-order effects is also used to formulate simplified governing equations. In the future, we can study this problem by adding heat transfer over the stretching plate. In addition, we can also study the stability analysis to twodimension flow of a viscous liquid within a horizontal thin liquid film with neglecting the inertia terms of Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the investigation of absolutely continuous curves in asymmetric metric spaces induced by Finsler structures. Firstly, for asymmetric spaces induced by Finsler manifolds, we show that three different kinds of absolutely continuous curves coincide when their domains are bounded closed intervals. As an application, a universal existence and regularity theorem for gradient flow is obtained in the Finsler setting. Secondly, we study absolutely continuous curves in Wasserstein spaces over Finsler manifolds and establish the Lisini structure theorem in this setting, which characterize the nature of absolutely continuous curves in Wasserstein spaces in terms of dynamical transference plans concentrated on absolutely continuous curves in base Finsler manifolds. Besides, a close relation between continuity equations and absolutely continuous curves in Wasserstein spaces is founded. Last but not least, we also consider nonsmooth "Finsler-like" spaces, in which case most of the aforementioned results remain valid. Various model examples are constructed in this paper, which point out genuine differences between the asymmetric and symmetric settings. [ABSTRACT FROM AUTHOR]

Particle image velocimetry (PIV) is a widely used experimental technique in ocean engineering, for instance, to study the vortex fields near marine risers and the wake fields behind wind turbines or ship propellers. However, the flow fields measured using PIV in water tanks or wind tunnels always have low resolution; hence, it is difficult to accurately reveal the mechanics behind the complex phenomena sometimes observed. In this paper, physics-informed neural networks (PINNs), which introduce the Navier–Stokes equations or the continuity equation into the loss function during training to reconstruct a flow field with high resolution, are investigated. The accuracy is compared with the cubic spline interpolation method and a classic neural network in a case study of reconstructing a two-dimensional flow field around a cylinder, which is obtained through direct numerical simulation. Finally, the validated PINN method is applied to reconstruct a flow field measured using PIV and shows good performance.