QUESTION BANK OF DISCRETE MATHEMATICS
... 9. What is the number of solutions of the equation x + y + z + w = 20, if x, y, z and w are nonnegative integers. 10. How many solutions are there to the equation a = b + c +d + e + f = 21, where each variable is non-negative integer such that (i) 1 ≤ x, (ii) all variables are ≥2. 11. Prove using co ...
... 9. What is the number of solutions of the equation x + y + z + w = 20, if x, y, z and w are nonnegative integers. 10. How many solutions are there to the equation a = b + c +d + e + f = 21, where each variable is non-negative integer such that (i) 1 ≤ x, (ii) all variables are ≥2. 11. Prove using co ...
1 Valuations of the field of rational numbers
... On the other hand, Afin contains Q, for a rational number α can be represented diagonally as a finite adèle (αp ) with αp = α. For all finite adèles x ∈ Afin , there exists an n ∈ N so that nx ∈ Ẑ; in other words the relation ...
... On the other hand, Afin contains Q, for a rational number α can be represented diagonally as a finite adèle (αp ) with αp = α. For all finite adèles x ∈ Afin , there exists an n ∈ N so that nx ∈ Ẑ; in other words the relation ...
ON THE PRIME SPECTRUM OF MODULES
... in studying of algebraic properties of an R-module M when we a have a related topology. For an example, when Spec.M / is homeomorphic to Spec.S /, where S is a commutative ring, we can transfer some of known topological properties of Spec.S / to Spec.M / and then by using these properties explore so ...
... in studying of algebraic properties of an R-module M when we a have a related topology. For an example, when Spec.M / is homeomorphic to Spec.S /, where S is a commutative ring, we can transfer some of known topological properties of Spec.S / to Spec.M / and then by using these properties explore so ...
CHAPTER 9: COMPLEX NUMBERS 1. Introduction Although R is a
... is also unique. We call θ the angle of (x, y). In what follows, we will assume that r ≥ 0, but we will not restrict θ unless specifically indicated. So r is unique, but the angle θ is not necessarily unique. If we view R2 as being C, then (x, y) is usually written x + yi, so we have the formula z = ...
... is also unique. We call θ the angle of (x, y). In what follows, we will assume that r ≥ 0, but we will not restrict θ unless specifically indicated. So r is unique, but the angle θ is not necessarily unique. If we view R2 as being C, then (x, y) is usually written x + yi, so we have the formula z = ...
Multiplying Polynomials
... Multiplying Polynomials • To multiply polynomials together each term of the first polynomial must be multiplied by each term of the second polynomial. • There are several methods for multiplying polynomials. • The three methods we will be using are the box method, using the memory device FOIL, and ...
... Multiplying Polynomials • To multiply polynomials together each term of the first polynomial must be multiplied by each term of the second polynomial. • There are several methods for multiplying polynomials. • The three methods we will be using are the box method, using the memory device FOIL, and ...
Group Theory: Basic Concepts Contents 1 Definitions
... • For G to be a group three conditions must be satisfied: (i) γ is associative: γ(f, γ(g, h)) = γ(γ(f, g), h), or f (gh) = (f g)h. (ii) There is an identity e ∈ G: eg = ge = e for all g ∈ G. (iii) Every g in G has an inverse ḡ, usually denoted by g −1 , such that gḡ = ḡg = e. ⋆ If (i) is satisfie ...
... • For G to be a group three conditions must be satisfied: (i) γ is associative: γ(f, γ(g, h)) = γ(γ(f, g), h), or f (gh) = (f g)h. (ii) There is an identity e ∈ G: eg = ge = e for all g ∈ G. (iii) Every g in G has an inverse ḡ, usually denoted by g −1 , such that gḡ = ḡg = e. ⋆ If (i) is satisfie ...
Quasi-Minuscule Quotients and Reduced Words for Reflections
... in a finite crystallographic group, these elements turn out to be “dominant minuscule” in the sense of Dale Peterson (see [7, 8] or [13]). Using an unpublished product formula of Peterson for counting reduced expressions of dominant minuscule elements, we obtain an explicit formula for the number of ...
... in a finite crystallographic group, these elements turn out to be “dominant minuscule” in the sense of Dale Peterson (see [7, 8] or [13]). Using an unpublished product formula of Peterson for counting reduced expressions of dominant minuscule elements, we obtain an explicit formula for the number of ...
the farrell-jones isomorphism conjecture for finite covolume
... manifold to construct a new manifold which is compact. This modification of our original manifold changes its sectional curvature, but we are able to control it in such a way that it stays nonpositive. The resulting extension of the Isomorphism Conjecture yields a wide variety of examples on which o ...
... manifold to construct a new manifold which is compact. This modification of our original manifold changes its sectional curvature, but we are able to control it in such a way that it stays nonpositive. The resulting extension of the Isomorphism Conjecture yields a wide variety of examples on which o ...
An Injectivity Theorem for Casson
... to introduce more notation. Let ϕ : π → H be a homomorphism to a torsion-free abelian group, let Q be a field of characteristic zero and let α : π → GL(k, Q) be a representation. Note that α and ϕ give rise to a right Z[π]-module structure on Qk ⊗Q Q[H] = Q[H]k as follows: Qk ⊗Q Q[H] × Z[π] → Qk ⊗Q Q ...
... to introduce more notation. Let ϕ : π → H be a homomorphism to a torsion-free abelian group, let Q be a field of characteristic zero and let α : π → GL(k, Q) be a representation. Note that α and ϕ give rise to a right Z[π]-module structure on Qk ⊗Q Q[H] = Q[H]k as follows: Qk ⊗Q Q[H] × Z[π] → Qk ⊗Q Q ...
My notes - Harvard Mathematics
... Remark. Since the complement of Uφ is U6φ , each Uφ is also closed. In particular, the topology is totally disconnected. Here’s the real theorem: 1.7 Theorem (Godel). This topology makes the class of all fields into a compact space. This is where the math happens. ...
... Remark. Since the complement of Uφ is U6φ , each Uφ is also closed. In particular, the topology is totally disconnected. Here’s the real theorem: 1.7 Theorem (Godel). This topology makes the class of all fields into a compact space. This is where the math happens. ...
§9 Subgroups
... complex numbers under multiplication by Lemma 9.3 as it is closed under multiplication. The same lemma shows that {1, 1} is a subgroup of E. Also, E has no other nontrivial proper subgroup, for any subgroup of E that contains i or i must contain i 2,i 3,i 4 or ( i)2,( i)3,( i)4 and thus must be E it ...
... complex numbers under multiplication by Lemma 9.3 as it is closed under multiplication. The same lemma shows that {1, 1} is a subgroup of E. Also, E has no other nontrivial proper subgroup, for any subgroup of E that contains i or i must contain i 2,i 3,i 4 or ( i)2,( i)3,( i)4 and thus must be E it ...
GROUP ACTIONS ON SETS 1. Group Actions Let X be a set and let
... that if xH = yH are the same coset, then (gx)−1 (gy) = x−1 g −1 gy = x−1 y ∈ H, so (gx)H = (gy)H. In other words, this action is well defined. It is also clear that 1(xH) = xH and (gh)(xH) = g(h(xH)), so it does define an action. Let 1 denote the trivial coset H. Then clearly X = G1 so the action is ...
... that if xH = yH are the same coset, then (gx)−1 (gy) = x−1 g −1 gy = x−1 y ∈ H, so (gx)H = (gy)H. In other words, this action is well defined. It is also clear that 1(xH) = xH and (gh)(xH) = g(h(xH)), so it does define an action. Let 1 denote the trivial coset H. Then clearly X = G1 so the action is ...
ON BOREL SETS BELONGING TO EVERY INVARIANT
... are equivalent: (1) C is perfectly 2-small, (2) C − C does not contain a neighbourhood of the neutral element of G, (3) C − C has empty interior. Leaving aside interrelations between κ-small and perfectly κ-small sets (see, however, Proposition 3.13), in the next two results we show that all Borel s ...
... are equivalent: (1) C is perfectly 2-small, (2) C − C does not contain a neighbourhood of the neutral element of G, (3) C − C has empty interior. Leaving aside interrelations between κ-small and perfectly κ-small sets (see, however, Proposition 3.13), in the next two results we show that all Borel s ...
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.