A physical equation

The principal aim of this paper was to test the nearness of possible approach to complete osmotic efficiency for strong solutions. To this end the experimental verification of the exact physical equation given by A. W. Porter was undertaken, a membrane having been constructed which could withstand osmotic pressures of calcium ferrocyanide up to 150 atmospheres without any sensible percolation of the solution. It was found, notwithstanding many precautions, that the formula would not verify within about 3 per cent. But further consideration showed that this formula must refer to osmotic pressures in vacuo , whereas the experiments were necessarily conducted in air at atmospheric pressure. Reconstructing the argument in terms of ideal osmotic partitions impermeable to air but permeable to the solution, the equation was modified so as to apply strictly to the quantities involved in the experimental determinations, which required the addition of the atmospheric pressure to the limits in the first and third of the integrals concerned in it. From the concordance of these numbers, it may fairly be deduced that the membrane establishes, unambiguously, even with concentrated solutions, the full theoretical osmotic pressures, for the thermodynamic relations, at these high pressures, are completely verified.

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Fractional derivatives in electrical circuit theory – critical remarks

Archives of Electrical Engineering ◽

10.1515/aee-2017-0011 ◽

2017 ◽

Vol 66 (1) ◽

pp. 155-163 ◽

Cited By ~ 7

Author(s):

Ryszard Sikora

Keyword(s):

Maxwell Equations ◽

Fractional Derivatives ◽

Circuit Theory ◽

Electrical Circuit ◽

Physical Equation ◽

Dimensional Uniformity ◽

Potential Impact

Abstract A number of critical remarks related to the application of fractional derivatives in electrical circuit theory have been presented in this paper. Few cases have been pointed out that refer to observed in selected publications violations of dimensional uniformity of physical equation rules as well as to a potential impact on the Maxwell equations.

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About Maxwell’s equations on fractal subsets of ℝ3

Open Physics ◽

10.2478/s11534-013-0192-6 ◽

2013 ◽

Vol 11 (6) ◽

Cited By ~ 1

Author(s):

Alireza Golmankhaneh ◽

Ali Golmankhaneh ◽

Dumitru Baleanu

Keyword(s):

Differential Form ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Measure Theory ◽

Computer Programs ◽

Physical Equation ◽

Fractional Differential ◽

Local Derivative ◽

Fractional Differential Form

AbstractIn this paper we have generalized $$F^{\bar \xi }$$-calculus for fractals embedding in ℝ3. $$F^{\bar \xi }$$-calculus is a fractional local derivative on fractals. It is an algorithm which may be used for computer programs and is more applicable than using measure theory. In this Calculus staircase functions for fractals has important role. $$F^{\bar \xi }$$-fractional differential form is introduced such that it can help us to derive the physical equation. Furthermore, using the $$F^{\bar \xi }$$-fractional differential form of Maxwell’s equations on fractals has been suggested.

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A Smoke Simulation Technique Based on Particle System

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.230-232.26 ◽

2011 ◽

Vol 230-232 ◽

pp. 26-30 ◽

Cited By ~ 1

Author(s):

Yong Song Zhan ◽

Wen Zhao Liu

Keyword(s):

Virtual Reality ◽

Real Time ◽

Particle System ◽

Object Oriented ◽

Simulation Technique ◽

Program Scheme ◽

Special Effect ◽

Physical Equation ◽

Smoke Simulation ◽

Stable Scheme

To achieve photorealistic special effect for the industry of virtual reality, this paper proposed a real-time smoke simulation technique using particle system. Firstly, the component of particle system is discussed according to the requirement of virtual reality. Secondly, a stable scheme is employed to solve the physical equation regulating the behavior of smoke, which is modeled by the particle system. Thirdly, the object-oriented program scheme of particle system is presented in detail. Experiment results show the robustness and feasibility of the proposed technique.

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Lattice QCD calculations for the physical equation of state

Physics Reports ◽

10.1016/j.physrep.2007.02.012 ◽

2007 ◽

Vol 443 (2) ◽

pp. 55-96 ◽

Cited By ~ 60

Author(s):

David E. Miller

Keyword(s):

Equation Of State ◽

Lattice Qcd ◽

Physical Equation

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Inverse Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks

Machine Learning: Science and Technology ◽

10.1088/2632-2153/ac3712 ◽

2021 ◽

Author(s):

Suryanarayana Maddu Maddu ◽

Dominik Sturm ◽

Christian L. Müller ◽

Ivo F. Sbalzarini

Keyword(s):

Neural Networks ◽

Scale Model ◽

Seamless Integration ◽

Multi Scale ◽

Physical Equation ◽

Active Turbulence ◽

Data Fidelity ◽

Catastrophic Interference ◽

Sequential Training ◽

Magnitude Improvement

Abstract We characterize and remedy a failure mode that may arise from multi-scale dynamics with scale imbalances during training of deep neural networks, such as Physics Informed Neural Networks (PINNs). PINNs are popular machine-learning templates that allow for seamless integration of physical equation models with data. Their training amounts to solving an optimization problem over a weighted sum of data-fidelity and equation-fidelity objectives. Conflicts between objectives can arise from scale imbalances, heteroscedasticity in the data, stiffness of the physical equation, or from catastrophic interference during sequential training. We explain the training pathology arising from this and propose a simple yet effective inverse Dirichlet weighting strategy to alleviate the issue. We compare with Sobolev training of neural networks, providing the baseline of analytically ε-optimal training. We demonstrate the effectiveness of inverse Dirichlet weighting in various applications, including a multi-scale model of active turbulence, where we show orders of magnitude improvement in accuracy and convergence over conventional PINN training. For inverse modeling using sequential training, we find that inverse Dirichlet weighting protects a PINN against catastrophic forgetting.

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