Steady motion of conducting fluids in pipes under transverse magnetic fields

1953 ◽  
Vol 49 (1) ◽  
pp. 136-144 ◽  
Author(s):  
J. A. Shercliff
Keyword(s):  
Velocity Profile ◽  
Rectangular Channel ◽  
Steady Motion ◽  
Transverse Magnetic ◽  
Pressure Gradients ◽  
Electrically Conducting ◽  
Degenerate Solution ◽  
Circular Pipes ◽  
Induced Currents ◽  
Conducting Fluids

ABSTRACTThis paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.

The implicit function theorem and multi-bump solutions of periodic partial differential equations

2006 ◽  
Vol 136 (3) ◽  
pp. 559-583 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Implicit Function Theorem ◽  
Superposition Principle ◽  
Implicit Function ◽  
Partial Differential ◽  
Degenerate Solution ◽  
Positivity Of Solutions ◽  
Perturbation Arguments

A modification of the implicit function theorem is advanced for cases where the continuity of the derivative fails. It is applied to a superposition principle for periodicpartial differential equations. The assumption of the principle, that there should exist a non-degenerate solution, is studied and instances of it realized using perturbation arguments and scaling. The positivity of solutions is considered.

scholarly journals ENERGY–MOMENTUM DISTRIBUTION: SOME EXAMPLES

2007 ◽  
Vol 22 (10) ◽  
pp. 1935-1951 ◽  
Author(s):  
M. SHARIF ◽  
M. AZAM
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Momentum Distribution ◽  
Special Class ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Degenerate Solution ◽  
Dust Solution

In this paper, we elaborate the problem of energy–momentum in General Relativity with the help of some well-known solutions. In this connection, we use the prescriptions of Einstein, Landau–Lifshitz, Papapetrou and Möller to compute the energy–momentum densities for four exact solutions of the Einstein field equations. We take the gravitational waves, special class of Ferrari–Ibanez degenerate solution, Senovilla–Vera dust solution and Wainwright–Marshman solution. It turns out that these prescriptions do provide consistent results for special class of Ferrari–Ibanez degenerate solution and Wainwright–Marshman solution but inconsistent results for gravitational waves and Senovilla–Vera dust solution.

Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings

2019 ◽  
Vol 18 (08) ◽  
pp. 1950145
Author(s):  
Wolfgang Rump
Keyword(s):  
Residue Field ◽  
Ring Structure ◽  
Baxter Equation ◽  
Exponential Map ◽  
Exponential Form ◽  
Degenerate Solution ◽  
Terminal Object ◽  
Theoretic Solution ◽  
Initial Object ◽  
Central Residue

Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang–Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-rings with a central residue field [Formula: see text] are characterized as [Formula: see text]-braces with a compatible near-ring structure. Under this correspondence, [Formula: see text]-braces are radicals of local near-rings with radical square zero.

Asymptotic Analysis for Hyperbolic Equation with Neumann Condition

2010 ◽  
Vol 29-32 ◽  
pp. 1287-1293
Author(s):  
Xin Cai
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Initial Boundary ◽  
Computational Method ◽  
Neumann Condition ◽  
Initial Layer ◽  
Degenerate Solution ◽  
Layer Function ◽  
Order Hyperbolic Equation ◽  
Transition Points

The second-order hyperbolic equation with small parameter and Neumann con- dition is considered. This kind of problem loses the boundary conditions both in x = 0 and x = 1, while it also loses two initial boundary conditions in t = 0. The solution changes rapidly near two boundary layers and one initial layer. Firstly, the asymptotic solution was studied. The analytical solution was approximated by the degenerate solution and two boundary layer functions and one initial layer function. Secondly, three transition points were presented ac- cording to Shishkin’s idea. Non-equidistant mesh partitions both in x direction and t direction were introduced. An effective computational method is given according to non-equidistant mesh partitions. Finally, numerical experiment was given.

scholarly journals Coordinated drift of receptive fields during noisy representation learning

2021 ◽  
Author(s):  
Shanshan Qin ◽  
Shiva Farashahi ◽  
David Lipshutz ◽  
Anirvan M Sengupta ◽  
Dmitri B Chklovskii ◽  
...  
Keyword(s):  
Posterior Parietal Cortex ◽  
Effective Diffusion ◽  
Receptive Fields ◽  
Solution Space ◽  
Network Models ◽  
Representation Learning ◽  
Neural Population ◽  
Degenerate Solution ◽  
Population Codes ◽  
Posterior Parietal

Long-term memories and learned behavior are conventionally associated with stable neuronal representations. However, recent experiments showed that neural population codes in many brain areas continuously change even when animals have fully learned and stably perform their tasks. This representational "drift" naturally leads to questions about its causes, dynamics, and functions. Here, we explore the hypothesis that neural representations optimize a representational objective with a degenerate solution space, and noisy synaptic updates drive the network to explore this (near-)optimal space causing representational drift. We illustrate this idea in simple, biologically plausible Hebbian/anti-Hebbian network models of representation learning, which optimize similarity matching objectives, and, when neural outputs are constrained to be nonnegative, learn localized receptive fields (RFs) that tile the stimulus manifold. We find that the drifting RFs of individual neurons can be characterized by a coordinated random walk, with the effective diffusion constants depending on various parameters such as learning rate, noise amplitude, and input statistics. Despite such drift, the representational similarity of population codes is stable over time. Our model recapitulates recent experimental observations in hippocampus and posterior parietal cortex, and makes testable predictions that can be probed in future experiments.

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