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Jacques Riguet
Jacques Riguet (1921 to October 20, 2013) was a French mathematician known for his contributions to algebraic logic and category theory. According to Gunther Schmidt and Thomas Ströhlein, "Alfred Tarski and Jacques Riguet founded the modern calculus of relations". Career Already at his lycée, Riguet was impressed by the power of geometric reasoning. He studied Louis Couturat and Bourbaki, who made contributions to logic and set theory.Stephane Dugowson and othersHommage a Jacques Riguetat Google Sites Riguet studied higher mathematics with Albert Châtelet and was introduced to lattices. In 1948 he published "Relations binaires, fermetures, correspondances de Galois" which revived the calculus of binary relations. He published his thesis ''Fondements de la Theorie de Relations Binaires'' in October 1951. In 1954 Riguet gave a plenary address at the International Congress of Mathematicians in Amsterdam, speaking on the applications of binary relations to algebra and machine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Logic
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic . Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator . Calculus of relations A homogeneous binary relation is found in the power set of ''X'' × ''X'' for some set ''X'', while a heterogeneous relation is found in the power set of ''X'' × ''Y'', where ''X'' ≠ ''Y''. Whether a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition Of Relations
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions. The word uncle indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In algebraic logic it is said that the relation of Uncle (x U z) is the composition of relations "is a brother of" (x B y) and "is a parent of" (y P z). U = BP \quad \text \quad xByPz \text xUz. Beginning with Augustus De Morgan, the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition. Definition If R \subseteq X \times Y and S \subseteq Y \times Z are two binary relations, then their composition R; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Project Euclid
Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent and society publishers. It was created to provide a platform for small publishers of scholarly journals to move from print to electronic in a cost-effective way. Through a combination of support by subscribing libraries and participating publishers, Project Euclid has made 70% of its journal articles available as open access. As of 2010, Project Euclid provided access to over one million pages of open-access content. Mission and goals Project Euclid's stated mission is to advance scholarly communication in the field of theoretical and applied mathematics and statistics. Through a "mixture of open access, subscription, and hosted subscription content it provides a way for small publishers (especially societies) to host their math or statistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roland Fraisse
Roland (; frk, *Hrōþiland; lat-med, Hruodlandus or ''Rotholandus''; it, Orlando or ''Rolando''; died 15 August 778) was a Frankish military leader under Charlemagne who became one of the principal figures in the literary cycle known as the Matter of France. The historical Roland was military governor of the Breton March, responsible for defending Francia's frontier against the Bretons. His only historical attestation is in Einhard's ''Vita Karoli Magni'', which notes he was part of the Frankish rearguard killed in retribution by the Basques in Iberia at the Battle of Roncevaux Pass. The story of Roland's death at Roncevaux Pass was embellished in later medieval and Renaissance literature. The first and most famous of these epic treatments was the Old French ''Chanson de Roland'' of the 11th century. Two masterpieces of Italian Renaissance poetry, the ''Orlando Innamorato'' and ''Orlando Furioso'' (by Matteo Maria Boiardo and Ludovico Ariosto respectively), are even further ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Matrix
In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix. (In some contexts, particularly computer science, the term "Boolean matrix" implies this restriction.) Let ''U'' be a non-trivial Boolean algebra (i.e. with at least two elements). Intersection, union, complementation, and containment of elements is expressed in ''U''. Let ''V'' be the collection of ''n'' × ''n'' matrices that have entries taken from ''U''. Complementation of such a matrix is obtained by complementing each element. The intersection or union of two such matrices is obtained by applying the operation to entries of each pair of elements to obtain the corresponding matrix intersection or union. A matrix is contained in another if each entry of the first is contained in the corresponding entry of the second. The product of two Boolean matrices is expressed as follows: :(AB)_ = \bigcup_^n (A_ \ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition (number Theory)
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. Examples The seven partitions of 5 are: * 5 * 4 + 1 * 3 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferrers Diagram
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. Examples The seven partitions of 5 are: * 5 * 4 + 1 * 3 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heterogeneous Relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a multiple of , but not to an integer that is not a multiple of . In this relation, for instance, the prime number 2 is related to number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Diagonal
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned. This notion can be made more precise for an n by m matrix M by partitioning n into a collection \text, and then partitioning m into a collection \text. The original matrix is then considered as the "total" of these groups, in the sense that the (i, j) entry of the original matrix corresponds in a 1-to-1 way with some (s, t) offset entry of some (x,y), where x \in \text and y \in \text. Block matrix algebra arises in general from biproducts in categories of matrices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difunctional
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a multiple of , but not to an integer that is not a multiple of . In this relation, for instance, the prime number 2 is related to number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outer Product
In linear algebra, the outer product of two coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...s is a Matrix (mathematics), matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra. The outer product contrasts with: * The dot product (a special case of "inner product"), which takes a pair of coordinate vectors as input and produces a Scalar (mathematics), scalar * The Kronecker product, which takes a pair of matrices as input and produces a block matrix * Matrix multiplication, Standard mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
