scholarly journals The Direction of Erosion

2021 ◽  
Author(s):  
Colin Peter Stark ◽  
Gavin John Stark
Keyword(s):  
Variational Principle ◽  
Rate Of Change ◽  
Erosion Process ◽  
Surface Geometry ◽  
Orthogonal Direction ◽  
Hamiltonian Description ◽  
Surface Normal ◽  
Simplifying Assumptions ◽  
Definition Of ◽  
Metric Function

Abstract. The rate of erosion of a geomorphic surface depends on its local gradient and on the material fluxes over it. Since both quantities are functions of the shape of the catchment surface, this dependence constitutes a mathematical straitjacket, in the sense that – subject to simplifying assumptions about the erosion process, and absent variations in external forcing and erodibility – the rate of change of surface geometry is solely a function of surface geometry. Here we demonstrate how to use this geometric self-constraint to convert an erosion model into its equivalent Hamiltonian, and explore the implications of having a Hamiltonian description of the erosion process. To achieve this conversion, we recognize that the rate of erosion defines the velocity of surface motion in its orthogonal direction, and we express this rate in its reciprocal form as the surface-normal slowness. By rewriting surface tilt in terms of normal slowness components, and by deploying a substitution developed in geometric mechanics, we extract what is known as the fundamental metric function of the model phase space; its square is the Hamiltonian. Such a Hamiltonian provides several new ways of solving for the evolution of an erosion surface: here we use it to derive Hamilton's ray tracing equations, which describe both the velocity of a surface point and the rate of change of the surface-normal slowness at that point. In this context, erosion involves two distinct directions: (i) the surface-normal direction, which points subvertically downwards, and (ii) the erosion ray direction, which points upstream at a generally small angle to horizontal with a sign controlled by the scaling of erosion with slope. If the model erosion rate scales faster than linearly with gradient, the rays point obliquely upwards; but if erosion scales sublinearly with gradient, the rays point obliquely downwards. Analysis of the Hamiltonian shows that these rays carry boundary-condition information upstream, and that they are geodesics, meaning that erosion takes the path of least erosion time. This constitutes a definition of the variational principle governing landscape evolution. In contrast with previous studies of network self-organization, neither energy nor energy dissipation is invoked in this variational principle, only geometry.

scholarly journals Laser Assisted Dirac Electron in a Magnetized Annulus

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 642
Author(s):  
Emilio Fiordilino
Keyword(s):  
Magnetic Field ◽  
Analytical Form ◽  
Relative Energy ◽  
Radial Force ◽  
Rate Of Change ◽  
Orthogonal Direction ◽  
Linearly Polarized ◽  
Dirac Electron ◽  
A Charge ◽  
Stretching Effect

We study the behaviour of a charge bound on a graphene annulus under the assumption that the particle can be treated as a massless Dirac electron. The eigenstates and relative energy are found in closed analytical form. Subsequently, we consider a large annulus with radius ρ∈[5000,10,000]a0 in the presence of a static magnetic field orthogonal to its plane and again the eigenstates and eigenenergies of the Dirac electron are found in both analytical and numerical form. The possibility of designing filiform currents by controlling the orbital angular momentum and the magnetic field is shown. The currents can be of interest in optoelectronic devices that are controlled by electromagnetic radiation. Moreover, a small radial force acts upon the annulus with a stretching effect. A linearly polarized electromagnetic field propagating in the orthogonal direction is added; the time evolution of the operators show that the acceleration of the electron is proportional to the rate of change of the spin of the particle.

scholarly journals The effective mass problem for the Landau-Pekar equations

2021 ◽  
Author(s):  
Dario Feliciangeli ◽  
Simone Rademacher ◽  
Robert Seiringer
Keyword(s):  
Variational Principle ◽  
Effective Mass ◽  
Initial Velocity ◽  
Energy Functional ◽  
Mass Problem ◽  
Definition Of

Abstract We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar.

scholarly journals A VARIATIONAL FORMULATION OF SYMPLECTIC NONCOMMUTATIVE MECHANICS

2007 ◽  
Vol 04 (05) ◽  
pp. 789-805 ◽  
Author(s):  
IGNACIO CORTESE ◽  
J. ANTONIO GARCÍA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principle ◽  
Configuration Space ◽  
Variational Formulation ◽  
Noncommutative Space ◽  
First Order ◽  
Dynamical Properties ◽  
Definition Of ◽  
Space Noncommutativity

The standard lore in noncommutative physics is the use of first order variational description of a dynamical system to probe the space noncommutativity and its consequences in the dynamics in phase space. As the ultimate goal is to understand the inherent space noncommutativity, we propose a variational principle for noncommutative dynamical systems in configuration space, based on results of our previous work [18]. We hope that this variational formulation in configuration space can be of help to elucidate the definition of some global and dynamical properties of classical and quantum noncommutative space.

On the Similarity Search With Hamming Space Sketches

2021 ◽  
pp. 97-127
Author(s):  
Vladimir Mic ◽  
Pavel Zezula
Keyword(s):  
Computational Complexity ◽  
Metric Space ◽  
Similarity Search ◽  
Space Model ◽  
Similarity Query ◽  
Speed Up ◽  
Hamming Space ◽  
Definition Of ◽  
Metric Function ◽  
Selection Of

This chapter focuses on data searching, which is nowadays mostly based on similarity. The similarity search is challenging due to its computational complexity, and also the fact that similarity is subjective and context dependent. The authors assume the metric space model of similarity, defined by the domain of objects and the metric function that measures the dissimilarity of object pairs. The volume of contemporary data is large, and the time efficiency of similarity query executions is essential. This chapter investigates transformations of metric space to Hamming space to decrease the memory and computational complexity of the search. Various challenges of the similarity search with sketches in the Hamming space are addressed, including the definition of sketching transformation and efficient search algorithms that exploit sketches to speed-up searching. The indexing of Hamming space and a heuristic to facilitate the selection of a suitable sketching technique for any given application are also considered.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

Turbulent structure in free-surface jet flows

1995 ◽  
Vol 291 ◽  
pp. 223-261 ◽  
Author(s):  
D. T. Walker ◽  
C.-Y. Chen ◽  
W. W. Willmarth
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Froude Number ◽  
Tangential Velocity ◽  
Rate Of Change ◽  
Normal Velocity ◽  
Reynolds Stresses ◽  
Velocity Fluctuations ◽  
Surface Normal ◽  
Surface Disturbances

Results of an experimental study of the interaction of a turbulent jet with a free surface when the jet issues parallel to the free surface are presented. Three different jets, with different exit velocities and jet-exit diameters, all located two jet-exit diameters below the free surface were studied. At this depth the jet flow, in each case, is fully turbulent before significant interaction with the free surface occurs. The effects of the Froude number (Fr) and the Reynolds number (Re) were investigated by varying the jet-exit velocity and jet-exit diameter. Froude-number effects were identified by increasing the Froude number from Fr = 1 to 8 at Re = 12700. Reynolds-number effects were identified by increasing the Reynolds number from Re = 12700 to 102000 at Fr = 1. Qualitative features of the subsurface flow and free-surface disturbances were examined using flow visualization. Measurements of all six Reynolds stresses and the three mean velocity components were obtained in two planes 16 and 32 jet diameters downstream using a three-component laser velocimeter. For all the jets, the interaction of vorticity tangential to the surface with its ‘image’ above the surface contributes to an outward flow near the free surface. This interaction is also shown to be directly related to the observed decrease in the surface-normal velocity fluctuations and the corresponding increase in the tangential velocity fluctuations near the free surface. At high Froude number, the larger surface disturbances diminish the interaction of the tangential vorticity with its image, resulting in a smaller outward flow and less energy transfer from the surface-normal to tangential velocity fluctuations near the surface. Energy is transferred instead to free-surface disturbances (waves) with the result that the turbulence kinetic energy is 20% lower and the Reynolds stresses are reduced. At high Reynolds number, the rate of evolution of the interaction of the jet with the free surface was reduced as shown by comparison of the rate of change with distance downstream of the local Reynolds and Froude numbers. In addition, the decay of tangential vorticity near the surface is slower than for low Reynolds number so that vortex filaments have time to undergo multiple reconnections to the free surface before they eventually decay.

Exploration of local surface geometry with minimum number of contact points and surface normal information

2012 ◽  
Vol 10 (2) ◽  
pp. 383-395 ◽  
Author(s):  
Tri Cong Phung ◽  
Min Jeong Kim ◽  
Hyungpil Moon ◽  
Ja Choon Koo ◽  
Hyouk Ryeol Choi
Keyword(s):  
Local Surface ◽  
Surface Geometry ◽  
Contact Points ◽  
Surface Normal ◽  
Minimum Number

High-accuracy four-dimensional trajectory prediction for civil aircraft

2012 ◽  
Vol 116 (1175) ◽  
pp. 45-66 ◽  
Author(s):  
W. Schuster ◽  
M. Porretta ◽  
W. Ochieng
Keyword(s):  
Flight Control ◽  
Three Dimensional ◽  
High Accuracy ◽  
Trajectory Prediction ◽  
Entire Duration ◽  
Civil Aircraft ◽  
Model Aircraft ◽  
Simplifying Assumptions ◽  
Definition Of ◽  
Better Than

AbstractCurrent state-of-the-art trajectory prediction tools typically model aircraft as three-dimensional point-masses, and make a number of simplifying assumptions about the actual and anticipated dynamics states of the aircraft. They are typically based on predefined settings obtained from existing databases such as Eurocontrol’s Bada rather than real-time information, including on the environment, available onboard the aircraft. This significantly limits trajectory prediction performance. This paper proposes a high-accuracy four-dimensional trajectory prediction model for use onboard civil aircraft, as well as by ground-based systems, which addresses these limitations. It is designed for strategic traffic capacity optimisation and conflict-detection and resolution over time-horizons covering the entire duration of a flight. The model incorporates a number of features including a novel flight-control-system and an enhanced flight-script that incorporates new taxonomy and content thereby enabling better definition of aircraft intent. The accuracy of the model is characterised using operational data acquired during a real flight trial. Results show that the performance of the proposed model is significantly better than the current models. Its accuracy is better than the required navigation performance for departure, en route and Non-Precision-Approach phases of flight.

scholarly journals THEORETICAL STUDY OF MHD MAXWELL FLUID WITH COMBINED EFFECT OF HEAT AND MASS TRANSFER VIA LOCAL AND NONLOCAL TIME DERIVATIVES

Fractals ◽  
2021 ◽  
pp. 2240010
Author(s):  
MUHAMMAD BILAL RIAZ ◽  
FAHD JARAD ◽  
DUMITRU BALEANU ◽  
MARYAM ASGIR
Keyword(s):  
Mass Transfer ◽  
Heat And Mass Transfer ◽  
Integral Transform ◽  
Previous History ◽  
Maxwell Fluid ◽  
Rate Of Change ◽  
Combined Effect ◽  
Inversion Algorithm ◽  
Previous State ◽  
Definition Of

This study highlights the combined effect of heat and mass transfer on MHD Maxwell fluid under time-dependent generalized boundary conditions for velocity, temperature, and concentration. The classical calculus due to the fact that it is assumed as the instant rate of change of the output when the input level changes. Therefore, it is not able to include the previous state of the system called the memory effect. But in the fractional calculus (FC), the rate of change is affected by all points of the considered interval, so it can incorporate the previous history/memory effects of any system. Due to this reason, we applied the modern definition of fractional derivatives (local and nonlocals kernels). Here, the order of fractional derivative will be treated as an index of memory. The exact and semi-analytical solutions are obtained using the integral transform and inversion algorithm. Several important properties of different parameters are analyzed by graphs. Interesting results are revealed by this investigation due to their vast applications in engineering and applied sciences.

scholarly journals Considerations for a "design" earthquake

1972 ◽  
Vol 5 (2) ◽  
pp. 47-50
Author(s):  
G. L. Evans
Keyword(s):  
Soil Properties ◽  
Seismic Design ◽  
Soil Mass ◽  
Soil Dynamics ◽  
Simplifying Assumptions ◽  
Acceleration Method ◽  
Risk Magnitude ◽  
Definition Of ◽  
Design Earthquake ◽  
Relative Factors

Any structure is only as good as its foundation material. Under earthquake the properties of foundation materials can change drastically. Recent advances in soil dynamics indicate that the simplifying assumptions on which our seismic building code is based, are not adequate to cater for the variations in foundation conditions. The code provides
clear definition of seismic design forces, in terms of acceleration and period, but ignores any possible effects of displacement, velocity or wavelength. Currently available methods of design and analysis can provide for calculation of ground period, displacements, velocities, accelerations and stress values at any point in a soil mass. Although not perhaps suitable
for detailed code recommendations these methods can be used by designers where needed and the code could contain simplified but conservative data on the use of such methods, The design forces to be imposed on a structure and variations of these are defined exactly in the code, but these are not matched by the definition of base motions, which are influenced by magnitude of the earthquake, distance and soil properties. It should be possible to provide for a "design earthquake" which takes into account, type of structure, nature of
 risk, magnitude of earthquake, distance from active seismic areas and soil properties. Even relative factors, initially based on overseas research, relating these things would provide a more rational basis for seismic effects on structures than the simplified structure mass acceleration method used at present.

Sign in / Sign up

Export Citation Format

Share Document