measure theory
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2022 ◽  
Vol Volume 18, Issue 1 ◽  
Author(s):  
Batya Kenig ◽  
Dan Suciu

Integrity constraints such as functional dependencies (FD) and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Then, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Finally, we show how some of the results in the paper can be derived using the I-measure theory, which relates between information theoretic measures and set theory. Our results recover, and sometimes extend, previously known results about the implication problem: the implication of MVDs and FDs can be checked by considering only 2-tuple relations.


Author(s):  
Costas D. Koutras ◽  
Konstantinos Liaskos ◽  
Christos Moyzes ◽  
Christos Nomikos ◽  
Christos Rantsoudis

2022 ◽  
Author(s):  
Martin Buntinas

Functional analysis deals with infinite-dimensional spaces. Its results are among the greatest achievements of modern mathematics and it has wide-reaching applications to probability theory, statistics, economics, classical and quantum physics, chemistry, engineering, and pure mathematics. This book deals with measure theory and discrete aspects of functional analysis, including Fourier series, sequence spaces, matrix maps, and summability. Based on the author's extensive teaching experience, the text is accessible to advanced undergraduate and first-year graduate students. It can be used as a basis for a one-term course or for a one-year sequence, and is suitable for self-study for readers with an undergraduate-level understanding of real analysis and linear algebra. More than 750 exercises are included to help the reader test their understanding. Key background material is summarized in the Preliminaries.


2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Qing-fu Li ◽  
Ying-qiao Yu

To accurately evaluate the durability of reinforced concrete girder bridges, a durability evaluation model was developed based on the matter element extension theory, entropy weight method, and unascertained measure theory. A total of seven indicators were selected for durability evaluation: the concrete presumed strength uniformity coefficient, reinforcement corrosion potential level, chloride ion content, average value of concrete relative carbonation depth, crack width, resistivity, and characteristic value of the reinforcement protective layer thickness. The weights of the durability evaluation indices were assigned using matter element extension combined with the entropy weight method, and the multi-indicator comprehensive evaluation vector was obtained by combining the single-indicator measurement matrix. The evaluation results were analyzed by applying the confidence criterion. The results showed that the evaluation results of this model matched with the actual conditions of the girder bridges, which indicates that this durability evaluation model has good applicability and is reasonable. Finally, a comparative study proved that the model could accurately evaluate the bridge durability.


2021 ◽  
Vol 30 (4) ◽  
pp. 899-960
Author(s):  
Camillo De Lellis ◽  
Guido De Philippis ◽  
Bernd Kirchheim ◽  
Riccardo Tione

2021 ◽  
Author(s):  
◽  
Michelle Porter

<p>Computable analysis has been well studied ever since Turing famously formalised the computable reals and computable real-valued function in 1936. However, analysis is a broad subject, and there still exist areas that have yet to be explored. For instance, Sierpinski proved that every real-valued function ƒ : ℝ → ℝ is the limit of a sequence of Darboux functions. This is an intriguing result, and the complexity of these sequences has been largely unstudied. Similarly, the Blaschke Selection Theorem, closely related to the Bolzano-Weierstrass Theorem, has great practical importance, but has not been considered from a computability theoretic perspective. The two main contributions of this thesis are: to provide some new, simple proofs of fundamental classical results (highlighting the role of ∏0/1 classes), and to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpinski, and the Blaschke Selection Theorem. This thesis focuses on classical computable analysis. It does not make use of effective measure theory.</p>


2021 ◽  
Author(s):  
◽  
Michelle Porter

<p>Computable analysis has been well studied ever since Turing famously formalised the computable reals and computable real-valued function in 1936. However, analysis is a broad subject, and there still exist areas that have yet to be explored. For instance, Sierpinski proved that every real-valued function ƒ : ℝ → ℝ is the limit of a sequence of Darboux functions. This is an intriguing result, and the complexity of these sequences has been largely unstudied. Similarly, the Blaschke Selection Theorem, closely related to the Bolzano-Weierstrass Theorem, has great practical importance, but has not been considered from a computability theoretic perspective. The two main contributions of this thesis are: to provide some new, simple proofs of fundamental classical results (highlighting the role of ∏0/1 classes), and to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpinski, and the Blaschke Selection Theorem. This thesis focuses on classical computable analysis. It does not make use of effective measure theory.</p>


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