Search Results

101 - 110 of 111 items :

  • Cartesian product x
  • Algebra and Number Theory x
Clear All
Submodule of free Z-module

abstraction. Formalized Mathematics, 1(3):441-444, 1990. [23] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999. [24] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990. [25] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. [26] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990. [27] Wojciech A. Trybulec

Open access
Tietze Extension Theorem for n-dimensional Spaces

References [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. [2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990. [3] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547-552, 1991. [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990

Open access
The Perfect Number Theorem and Wilson's Theorem

Mathematics , 12(1):49-58, 2004. [30] Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics , 6(3):335-338, 1997. [31] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics , 9(1):95-110, 2001. [32] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics , 1(2):329-334, 1990. [33] Andrzej Trybulec. Tuples, projections and Cartesian products

Open access
Free Term Algebras

References [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics , 1( 2 ):377-382, 1990. [2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics , 1( 2 ):421-427, 1990. [3] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics , 1( 3 ):589-593, 1990. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics , 1( 1 ):91-96, 1990. [5] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics , 2( 4 ):547-552, 1991

Open access
Program Algebra over an Algebra

References [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics , 1( 2 ):377-382, 1990. [2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics , 1( 2 ):421-427, 1990. [3] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics , 1( 3 ):589-593, 1990. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics , 1( 1 ):91-96, 1990. [5] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics , 2( 4 ):547-552, 1991

Open access
Characteristic of Rings. Prime Fields

–122, 2008. doi:10.2478/v10037-008-0017-z. [32] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1( 1 ): 115–122, 1990. [33] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics , 11( 4 ): 341–347, 2003. [34] Michał J. Trybulec. Integers. Formalized Mathematics , 1( 3 ):501–505, 1990. [35] Wojciech A. Trybulec. Groups. Formalized Mathematics , 1( 5 ):821–827, 1990. [36] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics , 1( 2 ):291–296, 1990. [37

Open access
The First Isomorphism Theorem and Other Properties of Rings

Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008. doi:10.2478/v10037-008-0017- z. [32] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990. [33] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990. [34] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003. [35] Michał J. Trybulec. Integers. Formalized

Open access
Torsion Part of ℤ-module

structures. Formalized Mathematics , 9( 3 ):559–564, 2001. [28] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics , 8( 1 ):29–34, 1999. [29] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1( 1 ): 115–122, 1990. [30] Michał J. Trybulec. Integers. Formalized Mathematics , 1( 3 ):501–505, 1990. [31] Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics , 1( 2 ):395–399, 1990. [32] Wojciech A. Trybulec. Vectors in

Open access
The construction of π0 in Axiomatic Cohesion

Abstract

We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of π0 as the image of a natural map to a product of discrete spaces. A particular case of this construction produces, out of a local and hyperconnected geometric morphism p : Ɛ → S, an idempotent monad π0 : Ɛ → Ɛ such that, for every X in E, π0X = 1 if and only if (p*Ω)! : (p*Ω)1 → (p*Ω)X is an isomorphism. For instance, if E is the topological topos (over S = Set), the π0-algebras coincide with the totally separated (sequential) spaces. To illustrate the connection with classical topology we show that the π0-algebras in the category of compactly generated Hausdorff spaces are exactly the totally separated ones. Also, in order to relate the construction above with the axioms for Cohesion we prove that, for a local and hyperconnected p : Ɛ → S, p is pre-cohesive if and only if p* : S → Ɛ is cartesian closed. In this case, p! = p*π0 : Ɛ → S and the category of π0-algebras coincides with the subcategory p* : S → E.

Open access
Optimal Quantization for Piecewise Uniform Distributions

] HAYNE, A.—KOIVUSALO, H.: Constructing bounded remainder sets and cut-andproject sets which are bounded distance to lattices , Israel J. Math. 212 (2016), no. 1, 189–201. [HS] HEWITT, E.—SAVAGE, L.: Symmetric measures on Cartesian products , Trans. Amer. Math. Soc. 80 (1955), 470–501. [K] KHINCHIN, A.: Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen , Math. Ann. 92 (1924), no. 1–2, 115–125. [KN] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences . John Wiley and Sons, New York

Open access