scholarly journals Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

10.37236/9724 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Julien Courtiel ◽  
Andrew Elvey Price ◽  
Irène Marcovici
Keyword(s):  
Time Complexity ◽  
Linear Time ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Triangular Domain ◽  
Wide Range ◽  
Maximal Height ◽  
Open Question ◽  
Motzkin Paths

This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0°, 60°, 120°, 180°, 240°, 300°) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height. We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a 2n-to-one function from generic triangular walks to triangular walks only using directions 0°, 120°, 240°. The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections. Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.

scholarly journals The magnetorotational instability prefers three dimensions

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190622
Author(s):  
Jeffrey S. Oishi ◽  
Geoffrey M. Vasil ◽  
Morgan Baxter ◽  
Andrew Swan ◽  
Keaton J. Burns ◽  
...  
Keyword(s):  
Shear Rate ◽  
Angular Velocity ◽  
Laboratory Experiments ◽  
Three Dimensional ◽  
Linear Instability ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Magnetorotational Instability ◽  
Dynamo Action ◽  
Wide Range

The magnetorotational instability (MRI) occurs when a weak magnetic field destabilizes a rotating, electrically conducting fluid with inwardly increasing angular velocity. The MRI is essential to astrophysical disc theory where the shear is typically Keplerian. Internal shear layers in stars may also be MRI-unstable, and they take a wide range of profiles, including near-critical. We show that the fastest growing modes of an ideal magnetofluid are three-dimensional provided the shear rate, S , is near the two-dimensional onset value, S c . For a Keplerian shear, three-dimensional modes are unstable above S  ≈ 0.10 S c , and dominate the two-dimensional modes until S  ≈ 2.05 S c . These three-dimensional modes dominate for shear profiles relevant to stars and at magnetic Prandtl numbers relevant to liquid-metal laboratory experiments. Significant numbers of rapidly growing three-dimensional modes remainy well past 2.05 S c . These finding are significant in three ways. First, weakly nonlinear theory suggests that the MRI saturates by pushing the shear rate to its critical value. This can happen for systems, such as stars and laboratory experiments, that can rearrange their angular velocity profiles. Second, the non-normal character and large transient growth of MRI modes should be important whenever three-dimensionality exists. Finally, three-dimensional growth suggests direct dynamo action driven from the linear instability.

scholarly journals 314 Automated left atrial volume measurement by two-dimensional speckle-tracking echocardiograpy. Feasibility, accuracy, and reproducibility

2021 ◽  
Vol 23 (Supplement_G) ◽  
Author(s):  
Diana Ruxandra Florescu ◽  
Luigi Paolo Badano ◽  
Michele Tomaselli ◽  
Camilla Torlasco ◽  
Georgica Tartea ◽  
...  
Keyword(s):  
Left Atrial ◽  
Speckle Tracking ◽  
Three Dimensional ◽  
Strain Analysis ◽  
Left Atrial Volume ◽  
Volume Measurement ◽  
Two Dimensional ◽  
Automated Measurement ◽  
Maximal Volume ◽  
Wide Range

Abstract Aims A by-product of left atrial (LA) strain analysis is the automated measurement of LA maximal volume (LAVmax), which may decrease the time of echocardiography reporting, and increase the reproducibility of the LAVmax measurement. However, the automated measurement of LAVmax by two-dimensional speckle-tracking analysis (2DSTE) has never been validated. Accordingly, we sought to: (i) assess the feasibility of automated LAVmax measurement by 2DSTE; (ii) compare the automated LAVmax by 2DSTE with conventional two-dimensional (2DE) biplane and three-dimensional echocardiography (3DE) measurements; and (iii) evaluate the accuracy and reproducibility of the three echocardiography techniques. Methods and results LAVmax (34–197 ml) were obtained from 198/210 (feasibility 94%) consecutive patients with various cardiac diseases (median age 67 years, 126 men) by 2DSTE, 2DE, and 3DE. 2DE and 2DSTE measurements resulted in similar LAVmax values (bias = 1.5 ml, limits of agreement, LOA ± 7.5 ml), and slightly underestimated 3DE LAVmax (biases = −5 ml, LOA ± 17 ml, and −6 ml, LOA ± 16 ml, respectively). LAVmax by 2DSTE and 2DE were strongly correlated to those obtained by cardiac magnetic resonance (CMR) (r = 0.946, and r = 0.935, respectively; P < 0.001). However, LAVmax obtained by 2DSTE (bias = −9.5 ml, LOA ± 16 ml) and 2DE (bias = −8 ml, LOA ± 17 ml) were significantly smaller than those measured by CMR. Conversely, 3DE LAVmax were similar to CMR (bias = −2 ml, LOA ± 10 ml). Excellent intra- and inter-observer intraclass correlations were found for 3DE (0.995 and 0.995), 2DE (0.990 and 0.988), and 2DSTE (0.990 and 0.989). Conclusions Automated LAVmax measurement by 2DSTE is highly feasible, highly reproducible, and provided similar values to conventional 2DE calculations in consecutive patients with a wide range of LAVmax.

scholarly journals Use of a Two-Dimensional Simulation Model in the Thermal Analysis of a Multi-Board Electronic Module

1994 ◽  
Vol 116 (2) ◽  
pp. 126-133 ◽  
Author(s):  
C. Beckermann ◽  
T. F. Smith ◽  
B. Pospichal
Keyword(s):  
Heat Transfer ◽  
Simulation Model ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Circuit Boards ◽  
Depth Direction ◽  
Wide Range ◽  
Local Board ◽  
Dimensional Simulation ◽  
Effective Component

A study is reported of heat transfer and air flow in an electronic module consisting of an array of narrowly spaced vertical circuit boards with highly-protruding components contained in a naturally vented chassis. A two-dimensional simulation model is developed that accounts for heat transfer by conduction, convection, and radiation, and sensitivity studies are performed. Experiments are conducted using a specially constructed test module. Comparisons with the experiments reveal the need to calibrate the model by selecting an effective component height that represents the drag properties of the actual three-dimensional component geometry. The need to account in the model for heat losses in the depth direction is also discussed. The importance of accurate thermophysical properties and of multi-dimensional radiation is shown. Good agreement with measured velocities and local board temperatures is obtained over a wide range of power levels, and it is concluded that the calibrated model is capable of representing the thermal behavior of the present module.

The Modeling of a Three-Dimensional Reservoir with a Two-Dimensional Reservoir Simulator-The Use of Dynamic Pseudo Functions

1973 ◽  
Vol 13 (03) ◽  
pp. 175-185 ◽  
Author(s):  
Hugh H. Jacks ◽  
Owen J.E. Smith ◽  
C.C. Mattax
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Relative Permeability ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Relative Permeabilities ◽  
Large Reservoir ◽  
Model Simulations ◽  
Wide Range ◽  
Section Model

Abstract Dynamic pseudo-relative permeabilities derived from cross-section models can be used to simulate three-dimensional flow accurately in a two-dimensional areal model of a reservoir Techniques are presented for deriving and using dynamic pseudos that are applicable over a wide range of rates and initial fluid saturations. Their validity is demonstrated by showing calculated results from comparable runs in a vertical cross-section model and in a one-dimensional areal model using the dynamic pseudo-relative permeabilities and vertical equilibrium (VE) pseudo-capillary pressures. Further substantiation is provided by showing the close agreement in calculated performance for a three-dimensional model and corresponding two-dimensional areal model representing a typical pattern on the flanks of a large reservoir. The areal pattern on the flanks of a large reservoir. The areal model gave comparable accuracy with a substantial savings in computing and manpower costs. Introduction Meaningful studies can be made for almost all reservoirs now that relatively efficient three-dimensional reservoir simulators are available. In many instances, however, less expensive two-dimensional areal (x-y) models can be used to solve the engineering problem adequately, provided the nonuniform distribution and flow of fluids in the implied third, or vertical, dimension of the areal model is properly described. This is accomplished through the use of special saturation-dependent functions that have been labeled pseudo-relative permeability (k ) and pseudo-capillary pressure permeability (k ) and pseudo-capillary pressure (P ) or, for simplicity "pseudo functions", to distinguish them from the conventional laboratory measured values that are used in their derivation. Two types of reservoir models have been suggested in the past to derive pseudo functions: the vertical equilibrium (VE) model of Coats et al., which is based on gravity-capillary equilibrium in the vertical direction; and the stratified model of Hearn, which assumes that viscous forces dominate vertical fluid distribution. Neither of these models accounts for the effects of large changes in flow rate that take place as a field is developed, approaches and place as a field is developed, approaches and maintains its peak rate, and then falls into decline. This paper presents an alternative method for developing pseudo functions that are applicable over a wide range of flow rates and over the complete range of initial fluid saturations. The functions may be both space and time dependent and, again for clarity and convenience in nomenclature, we have labeled them "dynamic pseudo functions". DESCRIPTION OF PSEUDO-RELATIVE PERMEABILITY FUNCTIONS PERMEABILITY FUNCTIONS Methods for developing pseudo functions have been presented in the literature. The distinction between our method and those used by others lies in the technique for deriving the vertical saturation distribution upon which the pseudo-relative permeabilities are based. In our approach, the permeabilities are based. In our approach, the vertical saturation distribution is developed through detailed simulation of the fluid displacement in a vertical cross-section (x-z) model of the reservoir. The simulation is run under conditions that are representative of those to be expected during the period to be covered in the areal model simulations. period to be covered in the areal model simulations. Results of the cross-section simulation are then processed to give depth-averaged fluid saturations processed to give depth-averaged fluid saturations (S ) and "dynamic" pseudo-relative permeability values (k ) for each column of blocks in the cross-section model at each output time. The above approach can result in a different set of dynamic pseudo functions for each column of blocks due to differences in initial saturation, rate of displacement, reservoir stratification, and location. However, differences between columns are frequently minor or they can be accounted for by correlation of the data. In this and several other reservoir studies, it was possible to reduce the complexity of the pseudo function sets through correlations with initial fluid saturations and fluid velocities. SPEJ P. 175

scholarly journals Turbulent 2.5-dimensional dynamos

2016 ◽  
Vol 799 ◽  
pp. 246-264 ◽  
Author(s):  
K. Seshasayanan ◽  
A. Alexakis
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Two Dimensional Flow

We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components $(u(x,y,t),v(x,y,t),w(x,y,t))$ that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier–Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers $Re$, magnetic Reynolds numbers $Rm$ and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows $Pm=Rm/Re$, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of $Re$ and the asymptotic behaviour in the large $Rm$ limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.

scholarly journals Beyond the Rigid Lid: Baroclinic Modes in a Structured Atmosphere

2017 ◽  
Vol 74 (11) ◽  
pp. 3551-3566 ◽  
Author(s):  
Jacob P. Edman ◽  
David M. Romps
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mode Decomposition ◽  
Wide Range ◽  
Tropospheric Heating ◽  
Baroclinic Modes

Abstract The baroclinic-mode decomposition is a fixture of the tropical-dynamics literature because of its simplicity and apparent usefulness in understanding a wide range of atmospheric phenomena. However, its derivation relies on the assumption that the tropopause is a rigid lid that artificially restricts the vertical propagation of wave energy. This causes tropospheric buoyancy anomalies of a single vertical mode to remain coherent for all time in the absence of dissipation. Here, the authors derive the Green’s functions for these baroclinic modes in a two-dimensional troposphere (or, equivalently, a three-dimensional troposphere with one translational symmetry) that is overlain by a stratosphere. These Green’s functions quantify the propagation and spreading of gravity waves generated by a horizontally localized heating, and they can be used to reconstruct the evolution of any tropospheric heating. For a first-baroclinic two-dimensional right-moving or left-moving gravity wave with a characteristic width of 100 km, its initial horizontal shape becomes unrecognizable after 4 h, at which point its initial amplitude has also been reduced by a factor of 1/π. After this time, the gravity wave assumes a universal shape that widens linearly in time. For gravity waves on a periodic domain the length of Earth’s circumference, it takes only 10 days for the gravity waves to spread their buoyancy throughout the entire domain.

scholarly journals Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

2016 ◽  
Vol 802 ◽  
pp. 5-36 ◽  
Author(s):  
A. Kalogirou ◽  
D. T. Papageorgiou
Keyword(s):  
Stokes Equations ◽  
Travelling Waves ◽  
Type Theorem ◽  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Wide Range ◽  
Time Periodic ◽  
Two Fluid ◽  
Permanent Form

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence

2007 ◽  
Vol 579 ◽  
pp. 383-412 ◽  
Author(s):  
ANDRÉ THESS ◽  
OLEG ZIKANOV
Keyword(s):  
Magnetic Field ◽  
Three Dimensional ◽  
Magnetic Interaction ◽  
Vortex Sheet ◽  
Triaxial Ellipsoid ◽  
Two Dimensional ◽  
Vortex Sheets ◽  
Mhd Flows ◽  
Open Question ◽  
The Magnetic Field

We report a theoretical investigation of the robustness of two-dimensional inviscid magnetohydrodynamic (MHD) flows at low magnetic Reynolds numbers with respect to three-dimensional perturbations. We use a combination of linear stability analysis and direct numerical simulations to analyse three problems, namely the flow in the interior of a triaxial ellipsoid, and two unbounded flows: a vortex with elliptical streamlines and a vortex sheet parallel to the magnetic field. The flow in a triaxial ellipsoid is found to present an exact analytical model which demonstrates both the existence of inviscid unstable three-dimensional modes and the stabilizing role of the magnetic field. The nonlinear evolution of the flow is characterized by intermittency typical of other MHD flows with long periods of nearly two-dimensional behaviour interrupted by violent three-dimensional transients triggered by the instability. We demonstrate, using the second model, that motion with elliptical streamlines perpendicular to the magnetic field becomes unstable with respect to the elliptical instability once the magnetic interaction parameter falls below a critical magnitude whose value tends to infinity as the eccentricity of the streamlines increases. Furthermore, the third model indicates that vortex sheets parallel to the magnetic field, which are unstable for any velocity and any magnetic field, emit eddies with vorticity perpendicular to the magnetic field. Whether the investigated instabilities persist in the presence of small but finite viscosity, in which case two-dimensional turbulence would represent a singular state of MHD flows, remains an open question.

SHEAR VISCOSITY OF STRONGLY-COUPLED TWO-DIMENSIONAL YUKAWA LIQUIDS: EXPERIMENT AND MODELING

2007 ◽  
Vol 21 (21) ◽  
pp. 1357-1376 ◽  
Author(s):  
Z. DONKÓ ◽  
P. HARTMANN ◽  
J. GOREE
Keyword(s):  
Shear Viscosity ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Simulation Studies ◽  
Strongly Coupled ◽  
Shear Rates ◽  
High Shear Rates ◽  
Wide Range ◽  
Computer Simulation Studies

This paper reviews experimental and modeling efforts aimed at the determination of the shear viscosity of strongly-coupled Yukawa liquids. After briefly reviewing prior work on three-dimensional (3D) systems, recent experimental and computer simulation studies of two-dimensional (2D) settings are presented in detail. In the experiments two counterpropagating laser beams were used to perturb a dusty plasma monolayer and monitoring of the velocity field reconstructed from particle trajectories allowed the determination of the shear viscosity with the aid of an analytical model. Subsequent computer simulations based on the molecular dynamics approach resulted in velocity profiles which are in very good agreement with the experimental ones. Further simulation studies of idealized 2D Yukawa liquids (in which gas friction is neglected) gave results for the shear viscosity over a wide range of system parameters and demonstrated the existence of the shear thinning effect (non-Newtonian behavior) of the liquid at high shear rates.

scholarly journals DIMENSION REDUCTION USING THE INVERSE STAMPING METHOD

2021 ◽  
Vol 2021 (4) ◽  
pp. 4810-4817
Author(s):  
JAROMIR KASPAR ◽  
◽  
MARCEL SVAGR ◽  
PETR BERNARDIN ◽  
VACLAVA LASOVA ◽  
...  
Keyword(s):  
Graph Theory ◽  
Dimension Reduction ◽  
Automotive Industry ◽  
Three Dimensional ◽  
Stochastic Methods ◽  
Complex Structures ◽  
Two Dimensional ◽  
Crucial Step ◽  
Development Tool ◽  
Wide Range

The aim of this work is to improve the inverse stamping method and increase its robustness. The first, crucial step of inverse stamping is the reduction of the three-dimensional part into a two-dimensional flat plane. There are several methods for reducing the dimension. These are geometrical methods, methods based on graph theory and stochastic methods. We examine the last two methods because of their reliability. These methods can even be used for geometrically complex structures which include holes, hooks and walls perpendicular to the flat plane. An algorithm which combines several methods for dimension reduction is proposed for use for a wide range of parts. Deep drawing is a widely used technology in the automotive industry and inverse stamping is a useful development tool.

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