Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular class of sets that can be described entirely in terms of simpler sets. L is the union of the constructible hierarchy L_\alpha. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result. What L is L

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The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions.

Set Theory as a foundational system for mathematics. ZF, ZFC and ZF with atoms. Relative consistency of the Axiom of Choice, the Continuum Hypothesis, the reals as a countable union of countable sets, the existence of a countable family of pairs without any choice function.

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Compactly-Supported Smooth Interpolators for Shape Modeling with Varying Resolution

Anaïs Laure Marie-Thérèse Badoual, Julien René Fageot, Pablo Garcia-Amorena Garcia, Daniel Andreas Schmitter, Michaël Unser

In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.

2017

A Visual Query Language for Complex-Value Databases

Christoph Koch

In this paper, a visual language, VCP, for queries on complex-value databases is proposed. The main strength of the new language is that it is purely visual: (i) It has no notion of variable, quantification, partiality, join, pattern matching, regular expression, recursion, or any other construct proper to logical, functional, or other database query languages and (ii) has a very natural, strong, and intuitive design metaphor. The main operation is that of copying and pasting in a schema tree. We show that despite its simplicity, VCP precisely captures complex-value algebra without powerset, or equivalently, monad algebra with union and difference. Thus, its expressive power is precisely that of the language that is usually considered to play the role of relational algebra for complex-value databases.

2006

Mathematical logic

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research i

Axiom of choice

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory,