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Related Concept Videos

Laws of Logarithms I01:30

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Logarithms are fundamental mathematical operations that serve as the inverse of exponentiation. They provide a means to express how many times a base must be raised to yield a given number. For base 10, often referred to as the common logarithm, the notation is written simply as log. Thus, if 10n = x, then log⁡(x) = n. This relationship makes logarithms especially valuable in simplifying complex calculations involving multiplication, division, and exponentiation.Logarithmic expressions...
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Introduction to Logarithmic Functions01:14

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Logarithmic functions are the inverses of exponential functions and are used to solve for exponents. The general form is y = logₐ(x), where a > 0 and a ≠ 1. This function returns the power to which the base a must be raised to obtain x. The logarithmic function is only defined for x > 0, and its range includes all real numbers.Graphically, logarithmic and exponential functions are reflections of each other across the line y = x. The graph of y = logₐ(x) passes through...
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In a three-phase circuit, line loss is an indicator of energy dissipated as heat due to the resistance of transmission lines. To address this, incorporating transformers into the system—a step-up transformer at the source and a step-down transformer at the load—is a strategic solution. Two three-phase transformers are introduced to improve this.
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Logarithmic and piecewise functions play central roles in mathematical modeling, particularly when capturing nonlinear or segmented behaviors in real-world phenomena. Although these functions differ fundamentally in structure and application, both serve to represent complex relationships in simplified mathematical terms.A logarithmic function is defined as the inverse of an exponential function, expressed as These functions grow quickly for small values of x but slow down as x increases,...
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Applications of Logarithms01:28

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Logarithmic functions are powerful tools for simplifying the mathematical representation of phenomena involving exponential changes. Their ability to convert multiplicative relationships into additive ones is especially valuable in various scientific and engineering contexts. One notable application of logarithms is measuring sound intensity, specifically through the decibel (dB) scale used in acoustics.Sound intensity levels vary over an extensive range, from the faintest audible whisper to...
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The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
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Universality of Logarithmic Loss in Successive Refinement.

Albert No1

  • 1Department of Electronic and Electrical Engineering, Hongik University, Seoul 04066, Korea.

Entropy (Basel, Switzerland)
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Logarithmic loss enables successive refinement for any discrete memoryless source under arbitrary distortion criteria. This finding leads to a novel, low-complexity lossy compression algorithm.

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logarithmic lossrate-distortionsuccessive refinability

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Area of Science:

  • Information Theory
  • Data Compression
  • Signal Processing

Background:

  • Successive refinement is a key concept in information theory for multi-resolution data representation.
  • Logarithmic loss is a common metric in evaluating compression algorithms.
  • Previous methods faced limitations in universally applying refinement criteria.

Purpose of the Study:

  • To establish a universal property of logarithmic loss in successive refinement.
  • To demonstrate the refinability of discrete memoryless sources under arbitrary distortion criteria.
  • To develop a practical, low-complexity lossy compression algorithm.

Main Methods:

  • Mathematical analysis of information theory principles.
  • Derivation of a universal property for logarithmic loss.
  • Algorithmic design based on theoretical findings.

Main Results:

  • A universal property of logarithmic loss is established for successive refinement.
  • Any discrete memoryless source is shown to be successively refinable with a logarithmic loss first decoder.
  • The proposed algorithm achieves low complexity for lossy compression.

Conclusions:

  • Logarithmic loss provides a powerful tool for successive refinement in data compression.
  • The developed algorithm offers an efficient solution for compressing discrete memoryless sources.
  • This work advances the theory and practice of lossy compression.