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Bínovský Ondrej - One of the best experts on this subject based on the ideXlab platform.

  • Imaginary quadratic fields with Class number 1
    2021
    Co-Authors: Bínovský Ondrej
    Abstract:

    We give an exposition of Heegner's and Siegel's proofs that there are exactly 9 imaginary quadratic fields with Class number equal to 1. In particular, we discuss Weber's original method of determining the Class Invariants of an imaginary quadratic field. Finally, we give an elementary proof of a sufficient condition, due to Alice Gee, for a value of a modular function to be a Class Invariant.

  • Imaginary quadratic fields with Class number 1
    Univerzita Karlova Matematicko-fyzikální fakulta, 2021
    Co-Authors: Bínovský Ondrej
    Abstract:

    Prezentujeme expozíciu Heegnerovho a Siegelovho dôkazu, že existuje práve 9 imaginárnych kvadratických telies s triedovým číslom 1. Zaoberáme sa pôvodnou Weberovou metódou určovania triedových Invariantov imaginárnych kvadra- tických telies. Tiež podáme elementárny dôkaz postačujúceho kritéria pre trie- dové Invarianty, ktorý pôvodne dokázala Alice Gee. 1We give an exposition of Heegner's and Siegel's proofs that there are exactly 9 imaginary quadratic fields with Class number equal to 1. In particular, we discuss Weber's original method of determining the Class Invariants of an imaginary quadratic field. Finally, we give an elementary proof of a sufficient condition, due to Alice Gee, for a value of a modular function to be a Class Invariant. 1Department of AlgebraKatedra algebryMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Victor Tan - One of the best experts on this subject based on the ideXlab platform.

Tal Arbel - One of the best experts on this subject based on the ideXlab platform.

  • detection over viewpoint via the object Class Invariant
    International Conference on Pattern Recognition, 2006
    Co-Authors: Matthew Toews, Tal Arbel
    Abstract:

    In this article, we present a new model of object Class appearance over viewpoint, based on learning a relationship between scale-Invariant image features (e.g. SIFT) and a geometric structure that we refer to as an OCI (object Class Invariant). The OCI is a perspective Invariant defined across instances of an object Class, and thereby serves as a common reference frame relating features over viewpoint change and object Class. A single probabilistic OCI model can be learned to capture the rich multimodal nature of object Class appearance in the presence of viewpoint change, providing an efficient alternative to the popular approach of training a battery of detectors at separate viewpoints and/or poses. Experimentation demonstrates that an OCI model of faces can be learned from a small number of natural, cluttered images, and used to detect faces exhibiting a large degree of appearance variation due to viewpoint change and intra-Class variability (i.e. (sun)glasses, ethnicity, expression, etc.).

Anthony Savidis - One of the best experts on this subject based on the ideXlab platform.

  • Application Invariants: Design by Contract augmented with deployment correctness logic
    Software - Practice and Experience, 2006
    Co-Authors: Anthony Savidis
    Abstract:

    Design by Contract is a method for the development of robust object-oriented software, introducing Class Invariants as conditions corresponding to the design axioms that should be satisfied by every valid instance of a Class. Additionally, the method states formally the way client programs should correctly utilize supplier Classes, so that the composition of correct programs may be accomplished. However, the contextual correctness of supplier instances within client programs, only reflected in the client-specific semantics for supplier-Class deployment, cannot be expressed through Design by Contract. For instance, supplier instances satisfying the supplier Class Invariant may not constitute plausible supplier instances in the context of a particular client program. In this context, we introduce application Invariants as an extension to Design by Contract, for hosting the contextual-correctness logic for supplier instances, as conditionally defined by client programs. This allows stronger validation of supplier instances, through the dynamic encapsulation of client-specific acceptance filtering, enabling more intensive defect detection. Application Invariants are implemented in the context of client Classes as methods utilizing correctness condition expressions, are dynamically hosted within supplier instances, while always called by supplier instances when the basic supplier-Class Invariant test is performed. Copyright © 2005 John Wiley & Sons, Ltd.

Joana Correia - One of the best experts on this subject based on the ideXlab platform.

  • Adding dependent types to Class-based mutable objects
    2017
    Co-Authors: Joana Correia
    Abstract:

    Tese de doutoramento, Informática (Ciência da Computação), Universidade de Lisboa, Faculdade de Ciências, 2018In this thesis, we present an imperative object-oriented language featuring a dependent type system designed to support Class-based programming and inheritance. The system brings Classes and dependent types into play so as to enable types (Classes) to be refined by value parameters (indices) drawn from some constraint domain. This combination allows statically checking interesting properties of imperative programs that are impossible to check in conventional static type systems for objects. From a pragmatic point of view, this work opens the possibility to combine the scalability and modularity of object orientation with the safety provided by dependent types in the form of index refinements. These may be used to provide additional guarantees about the fields of objects, and to prevent, for example, a method call that could leave an object in a state that would violate the Class Invariant. One key feature is that the programmer is not required to prove equations between indices issued by types, but instead the typechecker depends on external constraint solving. From a theoretic perspective, our fundamental contribution is to formulate a system that unifies the three very different features: dependent types, mutable objects and Class-based inheritance with subtyping. Our approach includes universal and existential types, as well as union types. Subtyping is induced by inheritance and quantifier instantiation. Moreover, dependent types require the system to track type varying objects, a feature missing from standard type systems in which the type is constant throughout the object’s lifetime. To ensure that an object is used correctly, aliasing is handled via a linear type discipline that enforces unique references to type varying objects. The system is decidable, provided indices are drawn from some decidable theory, and proved sound via subject reduction and progress. We also formulate a typechecking algorithm that gives a precise account of quantifier instantiation in a bidirectional style, combining type synthesis with checking. We prove that our algorithm is sound and complete. By way of example, we implement insertion and deletion for binary search trees in an imperative style, and come up with types that ensure the binary search tree Invariant. To attest the relevance of the language proposed, we provide a fully functional prototype where this and other examples can be typechecked, compiled and run. The prototype can be found at http://rss.di.fc.ul.pt/tools/dol/