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Vector Forms of Green’s Theorem01:26

Vector Forms of Green’s Theorem

The study of fluid motion often involves understanding how local rotational behavior relates to global circulation. In the context of a pond with pollutants, direct measurement of water movement along an irregular shoreline can be impractical. Green’s Theorem in vector form provides an alternative by relating the circulation around a closed boundary to properties of the flow within the enclosed region.Measurements of water velocity at different points define a continuous vector field that...
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Electric potential can be pictorially represented as a three-dimensional surface. On such a surface, the electric potential is constant everywhere. The equipotential surface is always perpendicular to the electric field lines, and while it is three-dimensional, it can be treated as an equipotential line in a two-dimensional case. These equipotential lines are also always perpendicular to electric field lines. The term equipotential is often used as a noun, referring to an equipotential line or...
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A vector-valued function describes position as a function of time. For example, in Cartesian coordinates, the position of a car moving along a curved road can be written as\begin{equation*}\textbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}Secant Vector and Average Velocity:This secant vector captures the overall change in position during the interval and provides a crude estimate of the direction of motion.At time t, the car is at point P, with position r(t). After a short interval h,...
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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Gauge fields, membranes, and subdeterminant vector models.

Robert G Leigh1, Andrea Mauri, Djordje Minic

  • 1Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801, USA.

Physical Review Letters
|September 28, 2010
PubMed
Summary
This summary is machine-generated.

We introduce new N-vector models in 3 and 4 dimensions with potentials derived from matrix subdeterminants. These models display unique large-N scaling behaviors in their effective potentials.

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Area of Science:

  • Theoretical Physics
  • High-Energy Physics
  • Condensed Matter Physics

Background:

  • N-vector models are crucial in quantum field theory.
  • Scalar potentials in (3+1) and (2+1) dimensions are of significant interest.
  • Generalizations of the Bagger-Lambert-Gustavsson model offer new theoretical avenues.

Purpose of the Study:

  • To introduce and analyze a novel class of classically marginal N-vector models.
  • To explore scalar potentials representable as subdeterminants of symmetric matrices.
  • To investigate the large-N scaling behavior of their effective potentials.

Main Methods:

  • Formulation of N-vector models in d=4 and d=3 dimensions.
  • Representation of scalar potentials using subdeterminants of symmetric matrices.
  • Application of the Hubbard-Stratonovich transformation to calculate effective potentials.

Main Results:

  • Classically marginal N-vector models were constructed.
  • The scalar potentials were expressed as subdeterminants.
  • Effective potentials revealed intriguing large-N scaling behaviors.

Conclusions:

  • The presented models generalize aspects of the Bagger-Lambert-Gustavsson model.
  • The findings suggest potential connections to string theory, membrane theory, and novel spin systems.
  • The large-N scaling behaviors offer insights into the collective properties of these systems.