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Related Concept Videos

Properties of the z-Transform I01:17

Properties of the z-Transform I

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The z-transform is a fundamental tool in digital signal processing, enabling the analysis of discrete-time systems through its various properties. It is an invaluable tool for analyzing discrete-time systems, offering a range of properties that simplify complex signal manipulations. One fundamental property is linearity. For any two discrete-time signals, the z-transform of their linear combination equals the same linear combination of their individual z-transforms. This property is essential...
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Introduction to z Scores01:05

Introduction to z Scores

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A z score (or standardized value) is measured in units of the standard deviation. It indicates how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z scores, and values of x that are smaller than the mean have negative z scores. If x equals the mean, then x has a zero z score. It is important to note that the mean of the z scores is zero, and the standard deviation is one.
z scores...
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Region of Convergence01:17

Region of Convergence

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The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is a crucial tool in the analysis of discrete-time systems, but its convergence is limited to specific values of the complex variable z. This range of values, known as the Region of Convergence (ROC), is fundamental in determining the behavior and stability of a system or signal. The ROC defines the region in the complex plane where the z-transform converges, which can take various...
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Definition of z-Transform01:26

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The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is an essential analytical tool, analogous to the Laplace transform used in continuous-time systems. It plays a crucial role in the analysis of signals and systems, complementing the discrete-time Fourier transform. Both the z-transform and the Laplace transform convert differential or difference equations into algebraic equations, simplifying the process of solving complex problems.
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z Scores and Unusual Values01:07

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The z score is one of the three measures of relative standing. It describes the location of a value in a dataset relative to the mean. z scores are obtained after the standardization of the values in a dataset. The z score for the mean is 0.
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z scores are the standardized values obtained after converting a normal distribution into a standard normal distribution. A z score is measured in units of the standard deviation. The z score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z scores, and values of x that are smaller than the mean have negative z scores. If x equals the mean, then x has a z score of...
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Hyper-Zagreb index in fuzzy environment and its application.

Sk Rabiul Islam1,2, Bandar Bin Mohsin3, Madhumangal Pal2,4

  • 1Department of Mathematics, Brainware University, 398, Ramkrishnapur Road, Jagadighata Market, Barasat, Kolkata-700125, India.

Heliyon
|September 9, 2024
PubMed
Summary

This study introduces the hyper- Zagreb index (hyper-ZI) for fuzzy graphs, calculating its bounds and relationships with other graph indices. The hyper-ZI for fuzzy graphs shows superior realism in crime analysis compared to its crisp graph version.

Keywords:
05C0905C72First Zagreb indexFuzzy graphHyper-Zagreb indexSecond Zagreb indexTopological indices

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Area of Science:

  • Mathematics
  • Graph Theory
  • Fuzzy Graph Theory

Background:

  • Zagreb indices (ZIs) are crucial graph invariants with applications in mathematics and chemistry.
  • Fuzzy graph theory (FGT) extends graph theory to model uncertainty.
  • Topological indices quantify molecular structure for chemical and network analysis.

Purpose of the Study:

  • Introduce and define the hyper-Zagreb index (hyper-ZI) specifically for fuzzy graphs (FGs).
  • Determine the bounds of the hyper-ZI for various types of FGs and FG operations.
  • Compare the effectiveness of the hyper-ZI for FGs with other topological indices in a real-world application.

Main Methods:

  • Computation of hyper-ZI bounds for paths, cycles, stars, complete FGs, and partial fuzzy subgraphs.
  • Analysis of hyper-ZI behavior under various FG operations (direct product, Cartesian product, composition, join, union, strong product, semi-strong product).
  • Comparative analysis of hyper-ZI for FGs against hyper-ZI for crisp graphs, first ZI for FGs, and F-index for FGs.

Main Results:

  • Isomorphic FGs yield identical hyper-ZI values.
  • Established connections between the hyper-ZI and the second Zagreb index for FGs.
  • The hyper-ZI for FGs demonstrated superior realism in analyzing crime data compared to its crisp graph counterpart.

Conclusions:

  • The hyper-ZI is a valuable topological index for fuzzy graphs.
  • The hyper-ZI for FGs offers enhanced applicability in real-world scenarios, such as crime analysis.
  • This index provides a more nuanced understanding of graph structures in uncertain environments.