Topological methods to solve equations over groups
... knot complements amount to the attachment of an ”arc” and a ”disc”. The resulting effect on fundamental groups is exactly Γ ...
... knot complements amount to the attachment of an ”arc” and a ”disc”. The resulting effect on fundamental groups is exactly Γ ...
notes on the subspace theorem
... various types of Diophantine equations. We show that the Subspace Theorem implies Roth’s Theorem. Subspace Theorem =⇒ Roth’s Theorem. Let (x, y) (with y > 0) be a pair of integers satisfying |α − x/y| 6 y −2−δ . Multiplying with y 2 gives |y(x − αy)| 6 y −δ . Since the linear forms Y and X − αY are ...
... various types of Diophantine equations. We show that the Subspace Theorem implies Roth’s Theorem. Subspace Theorem =⇒ Roth’s Theorem. Let (x, y) (with y > 0) be a pair of integers satisfying |α − x/y| 6 y −2−δ . Multiplying with y 2 gives |y(x − αy)| 6 y −δ . Since the linear forms Y and X − αY are ...
A SHORT PROOF OF ZELMANOV`S THEOREM ON LIE ALGEBRAS
... is also a polynomial identity for L, where y is a new variable, y 6= xiji for all 1 ≤ i ≤ m and 1 ≤ ji ≤ ri . Then, by Lemma 2.1, L satisfies a polynomial identity which is a linear combination of monomials of the form [xj1 , . . . xjr , y]. Replacing in each of these monomials, xjk by [xjk , a] yie ...
... is also a polynomial identity for L, where y is a new variable, y 6= xiji for all 1 ≤ i ≤ m and 1 ≤ ji ≤ ri . Then, by Lemma 2.1, L satisfies a polynomial identity which is a linear combination of monomials of the form [xj1 , . . . xjr , y]. Replacing in each of these monomials, xjk by [xjk , a] yie ...
LECTURES ON ALGEBRAIC VARIETIES OVER F1 Contents
... For any k-algebra R the map X(R) → X Ω (RΩ ) is injective. (2) For any scheme S over Ω and any natural transformation ϕ : X → S ◦ β, there exists a unique algebraic morphism ϕΩ : XΩ → S such that ϕ = ϕΩ ◦ i. In other words the following diagram is commutative: O ...
... For any k-algebra R the map X(R) → X Ω (RΩ ) is injective. (2) For any scheme S over Ω and any natural transformation ϕ : X → S ◦ β, there exists a unique algebraic morphism ϕΩ : XΩ → S such that ϕ = ϕΩ ◦ i. In other words the following diagram is commutative: O ...
![arXiv:math/0609622v2 [math.CO] 9 Jul 2007](http://s1.studyres.com/store/data/014863311_1-2ba4c2a5c0b25ba9b8b8b609b66b2122-300x300.png)
arXiv:math/0609622v2 [math.CO] 9 Jul 2007
... even powers of primes of the form 4k + 3 (See [5], p.116). If we can write (p1 /q1 )2 + (p2 /q2 )2 = n for integers p1 , q1 , p2 , and q2 , then p21 + p22 = q12 q22 n; hence q12 q22 n is a sum of two integral squares. This implies that q12 q22 n (and n itself) only contain primes of the form 4k + 3 ...
... even powers of primes of the form 4k + 3 (See [5], p.116). If we can write (p1 /q1 )2 + (p2 /q2 )2 = n for integers p1 , q1 , p2 , and q2 , then p21 + p22 = q12 q22 n; hence q12 q22 n is a sum of two integral squares. This implies that q12 q22 n (and n itself) only contain primes of the form 4k + 3 ...
8. Group algebras and Hecke algebras
... it existed. Pick g ∈ G the subspace g(V H ) is easily seen to be invariant under gHg −1 . Thus if H = gHg −1 , v → gv deÞnes a linear transformation on V H . However if we want to deÞne this for all of G we need H = gHg −1 for all g ∈ G, i.e., H is normal in G. Obviously, this is an atypical situati ...
... it existed. Pick g ∈ G the subspace g(V H ) is easily seen to be invariant under gHg −1 . Thus if H = gHg −1 , v → gv deÞnes a linear transformation on V H . However if we want to deÞne this for all of G we need H = gHg −1 for all g ∈ G, i.e., H is normal in G. Obviously, this is an atypical situati ...
RING THEORY 1. Ring Theory - Department of Mathematics
... The following notation is useful. Let a and b be subsets of A, at least one of which is an additive subgroup of A. Let ab denote the set of all sums of elements of the form ab with a ∈ a and b ∈ b. It is not hard to see that ab is an additive subgroup of A. (Clearly, it is closed under addition. Can ...
... The following notation is useful. Let a and b be subsets of A, at least one of which is an additive subgroup of A. Let ab denote the set of all sums of elements of the form ab with a ∈ a and b ∈ b. It is not hard to see that ab is an additive subgroup of A. (Clearly, it is closed under addition. Can ...
Mathematical Problems and Games
... the smallest number of operations leading from the basis to the given family. The allowed operations could be not only the internal and the external Boolean operations but also the operation of direct product, projection and perhaps fixed allowed transformations of a given initial set E into itself. ...
... the smallest number of operations leading from the basis to the given family. The allowed operations could be not only the internal and the external Boolean operations but also the operation of direct product, projection and perhaps fixed allowed transformations of a given initial set E into itself. ...
9 Solutions for Section 2
... so, by assumption it must equal R. In particular 1 belongs to this ideal, so there is r ∈ R with ar = 1. Hence a has a right inverse. Of course the same applies to r (since certainly r 6= 0), say rs = 1. Then we have a = a1 = ars = 1s = s. Thus r is also a left inverse for a and hence every non-zero ...
... so, by assumption it must equal R. In particular 1 belongs to this ideal, so there is r ∈ R with ar = 1. Hence a has a right inverse. Of course the same applies to r (since certainly r 6= 0), say rs = 1. Then we have a = a1 = ars = 1s = s. Thus r is also a left inverse for a and hence every non-zero ...
Categories and functors
... Definition 22.5. If C, D are preadditive categories, a functor F : C → D is called additive if F induces a homomorphism F : HomC (A, B) → HomD (F A, F B) for all A, B ∈ Ob(C). Lemma 22.6. An additive functor F : CR → Ab is the same as a left Rmodule. Proof. Suppose that M is a left R-module. Then M ...
... Definition 22.5. If C, D are preadditive categories, a functor F : C → D is called additive if F induces a homomorphism F : HomC (A, B) → HomD (F A, F B) for all A, B ∈ Ob(C). Lemma 22.6. An additive functor F : CR → Ab is the same as a left Rmodule. Proof. Suppose that M is a left R-module. Then M ...
LEFT VALUATION RINGS AND SIMPLE RADICAL RINGS(i)
... Result 18. If uv = 0 and í is nilpotent, then utv = 0. Proof. Let L denote the left ideal of left annihilators of v. By the previous result, Lt <=.L. Thus ut e L if u e L, or utv = 0 if uv = 0. Definition. A ring A is called zero-divisor expandable if, whenever u, v, t e A and uv = 0, then utv = 0. ...
... Result 18. If uv = 0 and í is nilpotent, then utv = 0. Proof. Let L denote the left ideal of left annihilators of v. By the previous result, Lt <=.L. Thus ut e L if u e L, or utv = 0 if uv = 0. Definition. A ring A is called zero-divisor expandable if, whenever u, v, t e A and uv = 0, then utv = 0. ...
Revised version
... is equal to the difference between those that are given. Let the two given cubes be B 3 and D3 , the first the greater, the second the smaller. Two other cubes are to be found, the sum of which is equal to B 3 − D3 . Let B − A be the root of the first one that is to be found, and let B 2 A/D2 −D be ...
... is equal to the difference between those that are given. Let the two given cubes be B 3 and D3 , the first the greater, the second the smaller. Two other cubes are to be found, the sum of which is equal to B 3 − D3 . Let B − A be the root of the first one that is to be found, and let B 2 A/D2 −D be ...
Delta-matroids and Vassiliev invariants
... pair of corresponding vertices a, b of the graphs we obtain two graphs with nonisomorphic matroids. The goal of the present paper is to show that the situation is different for binary delta-matroids: one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce ...
... pair of corresponding vertices a, b of the graphs we obtain two graphs with nonisomorphic matroids. The goal of the present paper is to show that the situation is different for binary delta-matroids: one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce ...
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.