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Related Concept Videos

Weighted Mean00:57

Weighted Mean

While taking the arithmetic, geometric, or harmonic mean of a sample data set, equal importance is assigned to all the data points. However, all the values may not always be equally important in some data sets. An intrinsic bias might make it more important to give more weightage to specific values over others.
For example, consider the number of goals scored in the matches of a tournament. While computing the average number of goals scored in the tournament, it may be more important to...
Determination of Molar Masses of Polymers I01:24

Determination of Molar Masses of Polymers I

Polymerization produces macromolecules with a range of chain lengths due to the random nature of molecular growth processes. As chains form and terminate at different stages, a single polymer sample contains molecules of varying sizes rather than a uniform structure. This variability is described using average molar masses and distribution-related parameters, which together provide a comprehensive understanding of polymer characteristics.The distribution of molar masses plays a critical role in...
Polymers: Molecular Weight Distribution01:10

Polymers: Molecular Weight Distribution

For any given polymer, the weight average molecular weight (Mw) is higher than, if not equal to, the number average molecular weight (Mn). The only situation in which the weight average molecular weight and the number average molecular weight are equal is when a polymer consists only of chains with equal molecular weight. However, this never happens in a synthetic polymer, since it is difficult to control the polymerization process up to a molecular level with accuracy to a hundred percent.
Determination of Molar Masses of Polymers II01:27

Determination of Molar Masses of Polymers II

Polymer samples typically consist of macromolecular chains with a distribution of lengths, resulting in a range of molar masses rather than a single discrete value. Conventional descriptors such as the number-average molar mass and weight-average molar mass quantify this distribution but do not fully capture polymer behavior in solution..The viscosity-average molar mass provides a more realistic description of polymer behavior in solution because it accounts for the enhanced contribution of...
Polymers: Defining Molecular Weight01:01

Polymers: Defining Molecular Weight

Unlike small molecules with definite molecular weights, polymers are a mixture of individual polymer chains of varying lengths, each with a unique molecular weight. So, the molecular weight of a polymer is expressed as an average value based on the average size of the polymer chains. The two most common forms of averages used for polymers are the number average molecular weight and weight average molecular weight.
The number average molecular weight (Mn) is the summation of the number...
Central Limit Theorem01:14

Central Limit Theorem

The central limit theorem, abbreviated as clt, is one of the most powerful and useful ideas in all of statistics. The central limit theorem for sample means says that if you repeatedly draw samples of a given size and calculate their means, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. In other words, as sample sizes increase, the distribution of means follows the normal distribution more closely.
The sample size, n, that...

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Mass-Sensitive Particle Tracking to Characterize Membrane-Associated Macromolecule Dynamics
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Minimum Data Rate for Consensus of Linear MASs Under Weighted-Average Protocols.

Can Zhao, Liwei An, Lili Zhang

    IEEE Transactions on Cybernetics
    |June 3, 2026
    PubMed
    Summary
    This summary is machine-generated.

    This study finds the minimum data rate for multiagent systems to reach consensus is inherent to the system itself. This rate is independent of specific network designs, enabling efficient consensus protocols.

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    Area of Science:

    • Control Theory
    • Networked Systems
    • Distributed Computing

    Background:

    • Multiagent systems require effective communication for coordinated behavior.
    • Data rate limitations pose significant challenges to achieving consensus in networked systems.
    • Existing consensus protocols may not be optimal under strict data rate constraints.

    Purpose of the Study:

    • To determine the minimum data rate essential for discrete-time linear multiagent systems to achieve consensus.
    • To establish whether this minimum data rate is dependent on system-specific parameters or network configurations.
    • To develop and validate an encoding-decoding scheme that operates at this minimum data rate.

    Main Methods:

    • Analysis of discrete-time linear systems under weighted-average consensus protocols.
    • Derivation of the theoretical minimum data rate required for consensus.
    • Development of a novel encoding-decoding scheme tailored to the minimum data rate.
    • Mathematical proof of consensus achievement using the proposed scheme.
    • Derivation of minimum data rates for achieving asymptotic consensus with specified convergence rates.

    Main Results:

    • The minimum data rate for consensus is an intrinsic property of the multiagent system.
    • This minimum rate is independent of communication topology, controller gains, and encoding-decoding schemes.
    • A proposed encoding-decoding scheme successfully achieves consensus at the minimum data rate.
    • The minimum data rate for asymptotic consensus, considering convergence speed, was derived.

    Conclusions:

    • Consensus in discrete-time linear multiagent systems is achievable at a fundamental, system-intrinsic minimum data rate.
    • The proposed scheme offers an efficient method for consensus under data rate constraints.
    • The findings provide a theoretical basis for designing robust and efficient networked multiagent systems.