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Variation: Normal Distribution, Range, and Standard Deviation02:32

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In the field of psychology, there are several ways to organize measurements of a trait, feature, or characteristic (i.e., variables). Qualitative data, such as ethnicity, can be tabulated into a frequency count to provide information about the proportion, as well as the variety of groups in a sample or population. On the other hand, researchers can perform a wider set of calculations on quantitative data. The mean, mode, and median, for instance, are central tendency measures to identify a...
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The range is one of the measures of variation. It can be defined as the difference between a dataset's highest and lowest values. For example, in the study of seven 16-ounce soda cans, the filled volume of soda was measured, thus producing the following amount (in ounces) of soda:
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A Range Condition for Polyconvex Variational Regularization.

Clemens Kirisits1, Otmar Scherzer1,2

  • 1Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria.

Numerical Functional Analysis and Optimization
|September 25, 2018
PubMed
Summary
This summary is machine-generated.

This study extends variational regularization results to polyconvex settings. It demonstrates that variational inequalities imply range conditions for polyconvex regularization, with a condition for the converse.

Keywords:
Convergence ratesinverse problemspolyconvex functionsregularization theorysource conditions

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Area of Science:

  • Mathematics
  • Applied Mathematics
  • Optimization Theory

Background:

  • Convex variational regularization has established relationships between source and range conditions.
  • The converse implication in convex regularization requires additional operator restrictions.

Purpose of the Study:

  • To investigate analogous source and range conditions for polyconvex regularization.
  • To establish implications between variational inequalities and range conditions in the polyconvex context.

Main Methods:

  • Mathematical analysis of polyconvex regularization functionals.
  • Derivation and analysis of variational inequalities.
  • Investigation of operator properties and their dual-adjoints.

Main Results:

  • The variational inequality in polyconvex regularization implies that the regularization functional's derivative is in the dual-adjoint operator's range.
  • A specific operator restriction is identified to achieve the converse implication.

Conclusions:

  • The findings generalize known results from convex to polyconvex regularization.
  • The study provides a complete characterization of source and range conditions for polyconvex regularization.