Proof theory and non-smooth analysis.
Ulrich Kohlenbach1, Nicholas Pischke1
1Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstraße 7, 64289 Darmstadt, Germany.
Summary
This study introduces a proof-theoretic framework for set-valued operators, developing quantitative methods for rectangularity. These advancements refine theorems like Brezis-Haraux and explain recent solutions to the zero displacement conjecture.
Area of Science:
- Mathematical Analysis
- Proof Theory
- Operator Theory
Background:
- Set-valued operators are crucial in various mathematical fields.
- Existing frameworks lack quantitative analysis for operator sums.
- The Brezis-Haraux theorem provides foundational results for operator theory.
Purpose of the Study:
- To establish a general proof-theoretic framework for set-valued operators.
- To introduce quantitative forms for (weakly) uniformly rectangular operators.
- To provide quantitative versions of the Brezis-Haraux theorem.
Main Methods:
- Development of a general proof-theoretic framework.
- Introduction of quantitative moduli for uniform rectangularity.
- Application of logical metatheorems for bound extractions.
Main Results:
- A unified framework for maximally, cyclically monotone, and rectangular operators.
- Quantitative forms of the Brezis-Haraux theorem using witnessing moduli.
- Demonstration that a weak uniform rectangularity modulus is sufficient for certain bounds.
Conclusions:
- The developed framework offers a unified approach to studying operator sums.
- Quantitative results simplify bounds based on proof constructivity.
- The findings illuminate recent advancements in the zero displacement conjecture.
Related Concept Videos
01:15Thevinin's Theorem
621
Thévenin's theorem plays a pivotal role in electrical circuit analysis, offering a solution to the challenges posed by variable loads within a circuit. In practical applications, it is common to encounter circuits where certain elements remain fixed while others fluctuate, often referred to as the "load." A typical household electrical outlet serves as a prime example of a variable load, as it can be connected to a variety of appliances, each with its own unique electrical...
621
01:14Norton's Theorem
664
Norton's theorem is a fundamental principle stating that a linear two-terminal circuit can be substituted with an equivalent circuit, which comprises a current source (ⅠN) in parallel with a resistor (RN). Here, ⅠN represents the short-circuit current flowing through the terminals, and RN stands for the input or equivalent resistance at the terminals when all independent sources are deactivated. This implies that the circuit illustrated in Figure (a) can be exchanged with the...
664
01:10Theorems of Pappus and Guldinus
2.0K
The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
For finding the surface area, consider a differential line element that generates a ring with surface area dA when revolved.
For finding the surface area, consider a differential line element that generates a ring with surface area dA when revolved.
2.0K
01:12Theorems of Pappus and Guldinus: Problem Solving
776
Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
776
01:18Castigliano's Theorem
458
Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.
458
01:18Parseval's Theorem
578
Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which...
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which...
578