Prime Numbers in Quadratic Fields
... The next three sections deal with theory of quadratic elds, prime numbers and quadratic reciprocity, needed for Section 5 in which we present our algorithm to determine prime numbers in the elds considered. Appendix 1 gives a Pascal program describing our algorithm in the form of a subprogram and ...
... The next three sections deal with theory of quadratic elds, prime numbers and quadratic reciprocity, needed for Section 5 in which we present our algorithm to determine prime numbers in the elds considered. Appendix 1 gives a Pascal program describing our algorithm in the form of a subprogram and ...
On the Domination and Total Domination Numbers of Cayley Sum
... In [6], Lev proved that if S is a subset of a finite Abelian group G, then Cay+ (G, S) is connected if and only if S is not contained in a coset of a proper subgroup of G, except, perhaps, for the non-zero coset of a subgroup of index 2. In the following theorem, we find a bound for the total domina ...
... In [6], Lev proved that if S is a subset of a finite Abelian group G, then Cay+ (G, S) is connected if and only if S is not contained in a coset of a proper subgroup of G, except, perhaps, for the non-zero coset of a subgroup of index 2. In the following theorem, we find a bound for the total domina ...
164 B—B- T = H2+H\`B, and H2- C = 0, contrary to
... integral sets for algebras Q with a= — 1. By a further transformation on the basis of algebras Q, Dickson showed, for his case a= — 1, that /3 could be taken to have no prime factors p = 4n + l. He then obtained the sets 5. Latimer considered algebras Q with a = / 3 = l (mod 2), and with a = 2, (3 = ...
... integral sets for algebras Q with a= — 1. By a further transformation on the basis of algebras Q, Dickson showed, for his case a= — 1, that /3 could be taken to have no prime factors p = 4n + l. He then obtained the sets 5. Latimer considered algebras Q with a = / 3 = l (mod 2), and with a = 2, (3 = ...
Composition algebras of degree two
... two possibilities just by comparing the factorizations that we obtain from the identity (x, x2)2 = n(x)\ So we are left with the case n(x) = p(x)2. Linearizing this expression we get (x, y) = p(x)p(y), which says that the kernel of the linear form p(x) is contained in the radical of the restriction ...
... two possibilities just by comparing the factorizations that we obtain from the identity (x, x2)2 = n(x)\ So we are left with the case n(x) = p(x)2. Linearizing this expression we get (x, y) = p(x)p(y), which says that the kernel of the linear form p(x) is contained in the radical of the restriction ...
HIGHER CATEGORIES 4. Model categories, 2: Topological spaces
... 4.5. Minimal fibrations. We will now prove two results. The first generalizes 4.4.4 as follows. 4.5.1. Proposition. Let f : X → Y be a fibration, such that all its fibers have trivial homotopy groups. Then f satisfies RLP wrt I. The second claim concludes the verification of conditions of Theorem 4. ...
... 4.5. Minimal fibrations. We will now prove two results. The first generalizes 4.4.4 as follows. 4.5.1. Proposition. Let f : X → Y be a fibration, such that all its fibers have trivial homotopy groups. Then f satisfies RLP wrt I. The second claim concludes the verification of conditions of Theorem 4. ...
tale Fundamental Groups
... Also applies to n-fold Galois covering of connected and locally connected spaces. Lemma 2.9. Let (C, F ) be a Galois category. The action of Aut(F ) on F (X) is transitive for all X ∈ Ob(C) connected. Idea of proof. We need to introduce some notations. Let I be the set of isomorphism classes of Galo ...
... Also applies to n-fold Galois covering of connected and locally connected spaces. Lemma 2.9. Let (C, F ) be a Galois category. The action of Aut(F ) on F (X) is transitive for all X ∈ Ob(C) connected. Idea of proof. We need to introduce some notations. Let I be the set of isomorphism classes of Galo ...
Full Article
... Proof:Let U be R*-open subset of Z. Since g is R*-irresolute function, g-1(U) is R*-open subset of Y. Since f is totally R*-irresolute function, f-1(g-1(U)) is R* -clopen in X. Hence g o f is totally R*-irresolute. Theorem: 4.18. Let f: X Y and g: Y Z be two functions, such that f is quasi R*-irreso ...
... Proof:Let U be R*-open subset of Z. Since g is R*-irresolute function, g-1(U) is R*-open subset of Y. Since f is totally R*-irresolute function, f-1(g-1(U)) is R* -clopen in X. Hence g o f is totally R*-irresolute. Theorem: 4.18. Let f: X Y and g: Y Z be two functions, such that f is quasi R*-irreso ...
A descending chain condition for groups definable in o
... has been a long series of papers highlighting the connections and analogies between groups definable in o-minimal structures and Lie groups. See for example papers by Pillay, Peterzil, Starchenko, Steinhorn, Edmundo, Razenj, Strzebonski, Berarducci and Otero ([1], [2], [3], [4], [6], [7], [10], [11 ...
... has been a long series of papers highlighting the connections and analogies between groups definable in o-minimal structures and Lie groups. See for example papers by Pillay, Peterzil, Starchenko, Steinhorn, Edmundo, Razenj, Strzebonski, Berarducci and Otero ([1], [2], [3], [4], [6], [7], [10], [11 ...
(pdf).
... The proof of the next theorem is due to D. Katz. (2.5) Theorem. Let (R, m, k) be a Cohen-Macaulay local ring and let M be an R-module. If I is an m-primary ideal, then M is syzygetically Artin-Rees with respect to I. For the proof we need two lemmas. (2.6) Lemma. Let F be an R-module, K be a submodu ...
... The proof of the next theorem is due to D. Katz. (2.5) Theorem. Let (R, m, k) be a Cohen-Macaulay local ring and let M be an R-module. If I is an m-primary ideal, then M is syzygetically Artin-Rees with respect to I. For the proof we need two lemmas. (2.6) Lemma. Let F be an R-module, K be a submodu ...
Lie algebra cohomology and Macdonald`s conjectures
... Since e · v = v if e is the unit element of G, we always have V 0 ⊂ G · V 0 . We call V 0 a G-submodule of V if G · V 0 ⊂ V 0 , or equivalently if G · V 0 = V 0 . Similarly V 0 is a g-submodule of V if g · V 0 ⊂ V 0 . In these cases V /V 0 is a submodule in a natural way; it is called a quotient mo ...
... Since e · v = v if e is the unit element of G, we always have V 0 ⊂ G · V 0 . We call V 0 a G-submodule of V if G · V 0 ⊂ V 0 , or equivalently if G · V 0 = V 0 . Similarly V 0 is a g-submodule of V if g · V 0 ⊂ V 0 . In these cases V /V 0 is a submodule in a natural way; it is called a quotient mo ...
Toroidal deformations and the homotopy type of Berkovich spaces
... Since the topology is totally discontinuous, analycity is not a local property on kn : there are too many locally analytic functions on Ω. J. TATE (60’) introduced the notion of a rigid analytic function by restricting the class of open coverings used to check local analycity. V. B ERKOVICH (80’) ha ...
... Since the topology is totally discontinuous, analycity is not a local property on kn : there are too many locally analytic functions on Ω. J. TATE (60’) introduced the notion of a rigid analytic function by restricting the class of open coverings used to check local analycity. V. B ERKOVICH (80’) ha ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.