Related Experiment Video
Updated: Jul 10, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
Mathematics and complex systems
1Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA. foote@math.uvm.edu
Science (New York, N.Y.)
|October 20, 2007
Summary
This study explores complex systems, proposing that even foundational mathematics exhibits emergent macroscopic behavior and profound implications. Mathematics itself may be the ultimate complex system, surpassing human verification capabilities.
Area of Science:
- Complex Systems Science
- Foundations of Mathematics
- Mathematical Physics
Background:
- Contemporary research faces challenges in understanding complex physical phenomena with numerous constituents and forces.
- Complex systems often manifest macroscopically from underlying systems with their own complex behaviors and elemental laws.
Purpose of the Study:
- To propose that areas of mathematics, irrespective of their simple axiomatic foundations, possess discernible layers.
- To highlight unexpected macroscopic outcomes in mathematics.
- To explore the profound mathematical and physical ramifications of mathematical concepts beyond their origins.
Main Methods:
- Conceptual analysis of mathematical structures.
- Comparative study of emergent properties in physical and mathematical systems.
- Exploration of the limits of human verification in advanced mathematics.
Main Results:
- Identification of layered structures within mathematical fields.
- Observation of emergent macroscopic behaviors in mathematics analogous to physical complex systems.
- Recognition of far-reaching mathematical and physical consequences stemming from fundamental mathematical principles.
Conclusions:
- Mathematical fields, like physical complex systems, exhibit emergent properties and layered complexity.
- The study of mathematics itself represents a potentially ultimate complex system, challenging traditional verification methods.
Related Concept Videos
01:29Complex Numbers
The real number system cannot represent the square root of a negative number, which restricts solutions for certain equations, such as quadratics with negative discriminants. To address this, the complex number system was developed, introducing the imaginary unit i, where i = √(-1). This extension allows for the representation of all roots, including those involving negative radicands.A complex number is written in the form x + yi, where x and y are real numbers. Here, x represents the real...
01:22Multi-input and Multi-variable systems
Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
In the absence of...
In the absence of...
01:16SFG Algebra
In Signal Flow Graph (SFG) algebra, the value a node represents is determined by the sum of all signals entering that node. This summed value is then transmitted through every branch leaving the node, making the SFG a powerful tool for visualizing and analyzing control systems.
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
01:26Classification of Systems-I
Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
01:20Mason's Rule
Mason's rule is a powerful tool in control systems and signal processing. It simplifies the calculation of transfer functions from signal-flow graphs. This method leverages various elements, including loop gains, forward-path gains, and non-touching loops, to determine the transfer function efficiently.
Loop gain is determined by identifying and tracing a path from a node back to itself. This involves computing the product of branch gains along the loop. Each loop's gain is crucial for further...
Loop gain is determined by identifying and tracing a path from a node back to itself. This involves computing the product of branch gains along the loop. Each loop's gain is crucial for further...
01:30Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the Complete Factorization...