On prime factors of subset sums
... and A is a subset of {1, ..., n} of cardinality t then at least half of the primes between (t/C0 )2 /2 and (t/C0 )2 divide s(A). Thus, provided that (11) holds, both (6) and (7) also hold. We shall prove Theorem 1 by combining Szemerédi’s Theorem about the representation of zero in abelian groups w ...
... and A is a subset of {1, ..., n} of cardinality t then at least half of the primes between (t/C0 )2 /2 and (t/C0 )2 divide s(A). Thus, provided that (11) holds, both (6) and (7) also hold. We shall prove Theorem 1 by combining Szemerédi’s Theorem about the representation of zero in abelian groups w ...
FUNDAMENTAL GROUPS AND THE VAN KAMPEN`S THEOREM
... and g with h, where f and g are path homotopic loops based at x0 . Furthermore, the induced homomorphism has two crucial properties. Theorem 1.36. If h : (X, x0 ) → (Y, y0 ) and k : (Y, y0 ) → (Z, z0 ) are continuous, then (k ◦ h)∗ = k∗ ◦ h∗ . If i : (X, x0 ) → (X, x0 ) is the identity map, then i∗ ...
... and g with h, where f and g are path homotopic loops based at x0 . Furthermore, the induced homomorphism has two crucial properties. Theorem 1.36. If h : (X, x0 ) → (Y, y0 ) and k : (Y, y0 ) → (Z, z0 ) are continuous, then (k ◦ h)∗ = k∗ ◦ h∗ . If i : (X, x0 ) → (X, x0 ) is the identity map, then i∗ ...
Universiteit Leiden Super-multiplicativity of ideal norms in number
... When we are studying a number ring R, that is a subring of a number field K, it can be useful to understand “how big” its ideals are compared to the whole ring. The main tool for this purpose is the norm map: N : I(R) −→ Z>0 I 7−→ #R/I where I(R) is the set of non-zero ideals of R. It is well known ...
... When we are studying a number ring R, that is a subring of a number field K, it can be useful to understand “how big” its ideals are compared to the whole ring. The main tool for this purpose is the norm map: N : I(R) −→ Z>0 I 7−→ #R/I where I(R) is the set of non-zero ideals of R. It is well known ...
INDEPENDENCE, MEASURE AND PSEUDOFINITE FIELDS 1
... Fact 3.2. Let G be a locally compact group. There exists a left invariant positive measure µ on G. All the other left invariant measures on G are proportional to it. It is called (the) left Haar measure. When G is compact, µ is also right invariant, we can normalise it so that µ(G) = 1 and we speak ...
... Fact 3.2. Let G be a locally compact group. There exists a left invariant positive measure µ on G. All the other left invariant measures on G are proportional to it. It is called (the) left Haar measure. When G is compact, µ is also right invariant, we can normalise it so that µ(G) = 1 and we speak ...
Intersection Theory course notes
... In particular, generic conic has genus zero and generic cubic curve has genus one. Note also that according to this theorem a generic plane curve can not have genus two. To prove the theorem one can show that all generic curves of degree d are homeomorphic and hence, have the same genus. Then it is ...
... In particular, generic conic has genus zero and generic cubic curve has genus one. Note also that according to this theorem a generic plane curve can not have genus two. To prove the theorem one can show that all generic curves of degree d are homeomorphic and hence, have the same genus. Then it is ...
1 Vector Spaces
... Theorem 1.19 (Exchange Property). Let I be a linearly independent set and let S be a spanning set. Then (∀x ∈ I)(∃y ∈ S) such that y 6∈ I and (I − {x} ∪ {y}) is independent. Consequently, |I| ≤ |S| Definition 1.10 (Finite Dimensional). V is said to be finite dimensional if it has a finite spanning s ...
... Theorem 1.19 (Exchange Property). Let I be a linearly independent set and let S be a spanning set. Then (∀x ∈ I)(∃y ∈ S) such that y 6∈ I and (I − {x} ∪ {y}) is independent. Consequently, |I| ≤ |S| Definition 1.10 (Finite Dimensional). V is said to be finite dimensional if it has a finite spanning s ...
THE NUMERICAL FACTORS OF ∆n(f,g)
... i.e., the numbers ordq (Lnpt ) are stable when t is sufficiently large. (4) Let Sn,p (t) be the set of all primes which divide Lnpt . Then ]Sn,p (t) −→ +∞ as t −→ +∞. In this paper, we generalize Pierce’s recurring series {∆n }n∈N defined by (3) and {gn,m (π1 , π2 , · · · , π2g )}m∈N defined above t ...
... i.e., the numbers ordq (Lnpt ) are stable when t is sufficiently large. (4) Let Sn,p (t) be the set of all primes which divide Lnpt . Then ]Sn,p (t) −→ +∞ as t −→ +∞. In this paper, we generalize Pierce’s recurring series {∆n }n∈N defined by (3) and {gn,m (π1 , π2 , · · · , π2g )}m∈N defined above t ...
S14 - stony brook cs
... p has some special structure like p = 2k + 1 p − 1 has many small factors However, if g is a generator, the problem is believed to be hard. Normally we want to work with a group such that order of the group, |G| ,the number of elements in G, is prime. However, Z∗p has order (p − 1), which is not pri ...
... p has some special structure like p = 2k + 1 p − 1 has many small factors However, if g is a generator, the problem is believed to be hard. Normally we want to work with a group such that order of the group, |G| ,the number of elements in G, is prime. However, Z∗p has order (p − 1), which is not pri ...
Lecture 8
... of topological spaces/groups serve the purpose with p and q being the two projection maps. 5. Discuss arbitrary products in a category generalizing the preceding exercise and discuss the existence of arbitrary products in the categories Top, Gr and AbGr. 6. We say a category C admits finite coproduc ...
... of topological spaces/groups serve the purpose with p and q being the two projection maps. 5. Discuss arbitrary products in a category generalizing the preceding exercise and discuss the existence of arbitrary products in the categories Top, Gr and AbGr. 6. We say a category C admits finite coproduc ...
LIE ALGEBRAS OF CHARACTERISTIC 2Q
... properties are then deduced by using the theory of quadratic forms and a different form of the root diagrams. Brown retains most of the common properties of root spaces and root diagrams by altering the definitions of root spaces and roots. See[l]. 2. Basic definitions and axioms. A Lie algebra over ...
... properties are then deduced by using the theory of quadratic forms and a different form of the root diagrams. Brown retains most of the common properties of root spaces and root diagrams by altering the definitions of root spaces and roots. See[l]. 2. Basic definitions and axioms. A Lie algebra over ...
KMS states on self-similar groupoid actions
... t 7→ σt (a) extends to an entire function z 7→ σz (a). ...
... t 7→ σt (a) extends to an entire function z 7→ σz (a). ...
COMPLETELY RANK-NONINCREASING LINEAR MAPS Don
... The notion of completely rank-nonincreasing maps arose from an attempt to characterize the linear maps on a linear subspace of B(H) that are point-strong limits of similarities or point-strong limits of skew-compressions introduced in [9]. Suppose S is a linear subspace of B(H) and φ : S → B(M ) is ...
... The notion of completely rank-nonincreasing maps arose from an attempt to characterize the linear maps on a linear subspace of B(H) that are point-strong limits of similarities or point-strong limits of skew-compressions introduced in [9]. Suppose S is a linear subspace of B(H) and φ : S → B(M ) is ...
On continuous images of ultra-arcs
... is an arc uniquely irreducible about {r(0D ), r(1D )} (use (i) and (iii)), the closed set r[S] is not a set of irreducibility for IR D . Let K be a proper subcontinuum containing r[S]. Then r−1 [K] is a proper subcontinuum of ID contining S, a contradiction. 4. monotone images are decomposable As ...
... is an arc uniquely irreducible about {r(0D ), r(1D )} (use (i) and (iii)), the closed set r[S] is not a set of irreducibility for IR D . Let K be a proper subcontinuum containing r[S]. Then r−1 [K] is a proper subcontinuum of ID contining S, a contradiction. 4. monotone images are decomposable As ...
Hp boundedness implies Hp ! Lp boundedness
... from H p ! H p boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([19]). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this paper, we will illustr ...
... from H p ! H p boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([19]). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this paper, we will illustr ...
FREE-BY-FREE GROUPS OVER POLYNOMIALLY GROWING
... graph Γ, with filtration ∅ = Γ0 ( Γ1 ( · · · ( Γi ( Γi+1 ( · · · ( Γr = Γ and r filtered homotopy equivalences f1 , · · · , fr with fi inducing αi as an outer automorphism of Fn (in particular hα1 , · · · , αr i = U). We may assume that, once chosen a base-point, each fi induces the automorphism αi ...
... graph Γ, with filtration ∅ = Γ0 ( Γ1 ( · · · ( Γi ( Γi+1 ( · · · ( Γr = Γ and r filtered homotopy equivalences f1 , · · · , fr with fi inducing αi as an outer automorphism of Fn (in particular hα1 , · · · , αr i = U). We may assume that, once chosen a base-point, each fi induces the automorphism αi ...
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.