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Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

Gauss's Law: Problem-Solving

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...

Gaussian Elimination: Problem Solving

Gaussian Elimination: Problem Solving

Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

Linearization and Approximation

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...

Normal Distribution

Normal Distribution

The normal, a continuous distribution, is the most important of all the distributions. Its graph is a bell-shaped symmetrical curve, which is observed in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some instructors may use the normal distribution to help determine students’ grades. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is...

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Related Experiment Video

Updated: Apr 1, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

Quantification via gaussian latent space representations.

Olaya Pérez-Mon¹, Juan José Del Coz¹, Pablo González¹

¹Artificial Intelligence Center, University of Oviedo, Gijón, 33204, Asturias, Spain.

Neural Networks : the Official Journal of the International Neural Network Society

|March 30, 2026

Summary

This summary is machine-generated.

This study introduces a novel deep learning method for quantification, or prevalence estimation. The new approach achieves state-of-the-art results by directly optimizing the quantification problem without an intermediate classifier.

Keywords:

Deep learning Machine learning Neural networks Prevalence estimation Quantification

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Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data

Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data

Published on: June 26, 2013

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Last Updated: Apr 1, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

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Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data

Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data

Published on: June 26, 2013

Area of Science:

Machine Learning
Computer Science

Background:

Quantification estimates class prevalence in data.
Current methods often rely on classifier predictions and assume prior probability shifts.

Purpose of the Study:

To develop a deep learning model for direct quantification.
To overcome limitations of traditional quantification approaches.

Main Methods:

An end-to-end neural network utilizing Gaussian distributions in latent spaces.
Learning invariant representations of data bags.
Direct optimization of quantification-specific loss functions.

Main Results:

Achieved state-of-the-art performance in quantification tasks.
Outperformed traditional methods and existing deep learning quantification approaches.
Demonstrated effectiveness without relying on an intermediate classifier.

Conclusions:

The proposed deep learning method offers a powerful new approach to quantification.
Direct optimization in deep learning is effective for prevalence estimation.
The method provides invariant representations for improved quantification accuracy.