groups and categories
... only “up to isomorphism.” This is so e.g. for products A × B and coproducts A + B, as well as many other operations like tensor products A ⊗ B of vector spaces, modules, algebras over a ring, etc. (the category of proofs in linear logic provides more examples). We will return to this more general no ...
... only “up to isomorphism.” This is so e.g. for products A × B and coproducts A + B, as well as many other operations like tensor products A ⊗ B of vector spaces, modules, algebras over a ring, etc. (the category of proofs in linear logic provides more examples). We will return to this more general no ...
Finite model property for guarded fragments, and extending partial
... Herwig ` s.i.p. for universal homogeneous Knfree graphs, and for Henson digraphs. 2. Finite model theory: hierarchy theorems for fixedpoint logics (Grohe, 1996). Other work of Grohe too. 3. Finite model property for guarded fragments of first-order logic, and classes of ‘relativised’ algebras in alg ...
... Herwig ` s.i.p. for universal homogeneous Knfree graphs, and for Henson digraphs. 2. Finite model theory: hierarchy theorems for fixedpoint logics (Grohe, 1996). Other work of Grohe too. 3. Finite model property for guarded fragments of first-order logic, and classes of ‘relativised’ algebras in alg ...
Theorem 1. Every subset of a countable set is countable.
... We draw attention to a simple principle, which can be used to prove many of the usual and important theorems on countabilty of sets. We formulate it as the Countability Lemma. Suppose to each element of the set A there is assigned, by some definite rule, a unique natural number in such a manner that ...
... We draw attention to a simple principle, which can be used to prove many of the usual and important theorems on countabilty of sets. We formulate it as the Countability Lemma. Suppose to each element of the set A there is assigned, by some definite rule, a unique natural number in such a manner that ...
Textbook
... secondary education majors, with the second semester available as an elective. For most students in these classes this is a first serious course requiring them to prove theorems on their own. Students with prior experience would be likely to move through it more quickly. This guide was also used at ...
... secondary education majors, with the second semester available as an elective. For most students in these classes this is a first serious course requiring them to prove theorems on their own. Students with prior experience would be likely to move through it more quickly. This guide was also used at ...
Honors Algebra 4, MATH 371 Winter 2010
... 8. Now let F be a field and V a finite dimensional F -vector space equipped with a linear transformation A : V → V . We consider V as an F [X]-module via (a0 + a1 X + · · · + an X n )v := a0 v + a1 Av + · · · + an An v, where Ai denotes the composition of A with itself i-times. Since F [X] is a PID, ...
... 8. Now let F be a field and V a finite dimensional F -vector space equipped with a linear transformation A : V → V . We consider V as an F [X]-module via (a0 + a1 X + · · · + an X n )v := a0 v + a1 Av + · · · + an An v, where Ai denotes the composition of A with itself i-times. Since F [X] is a PID, ...
Pure Extensions of Locally Compact Abelian Groups
... abelian groups (see [F]). In the category 2, a corresponding result need not hold: for groups A and C in 2, Ext(C, A) 1 is a (possibly proper) subgroup of Pext(C,A), and it coincides with Pext(C,A) if (a) A and C are compactly generated, or (b) A and C have no small subgroups (see Theorem 2.4). If G ...
... abelian groups (see [F]). In the category 2, a corresponding result need not hold: for groups A and C in 2, Ext(C, A) 1 is a (possibly proper) subgroup of Pext(C,A), and it coincides with Pext(C,A) if (a) A and C are compactly generated, or (b) A and C have no small subgroups (see Theorem 2.4). If G ...
Variations on the Bloch
... 2.2 Notation. Let U be a k-scheme. Denote by Cp(U ) a category whose objects are couples (X, Z) consisting of an U -scheme X of finite type over U and a closed subset Z of the scheme X (we assume the empty set is a closed subset of X). Morphisms from (X, Z) to (X ′ , Z ′ ) are those morphisms f : X ...
... 2.2 Notation. Let U be a k-scheme. Denote by Cp(U ) a category whose objects are couples (X, Z) consisting of an U -scheme X of finite type over U and a closed subset Z of the scheme X (we assume the empty set is a closed subset of X). Morphisms from (X, Z) to (X ′ , Z ′ ) are those morphisms f : X ...
Universal exponential solution of the Yang
... Lemma 4 [F]. Under assumptions of Lemma 3, any two “balanced” words in a and b (that is, words containing as many a’s as b’s) commute. Conjecture. The solution (19) is a universal exponential solution of the YBE. In other words, the associative algebra generated by the elements ei is isomorphic to ...
... Lemma 4 [F]. Under assumptions of Lemma 3, any two “balanced” words in a and b (that is, words containing as many a’s as b’s) commute. Conjecture. The solution (19) is a universal exponential solution of the YBE. In other words, the associative algebra generated by the elements ei is isomorphic to ...
Dimension theory
... we have �(M ) = �(M � ) + �(M �� ). Any function on f.g. A0 -modules with this property is called an additive function. Some trivial observations about the Poincaré series: (1) 0 ≤ PM (1)�≤ ∞ in general. (2) PM (1) = �(Mn ) < ∞ iff Mn = 0 for all but finitely many n. (We are given that �(Mn ) < ∞ fo ...
... we have �(M ) = �(M � ) + �(M �� ). Any function on f.g. A0 -modules with this property is called an additive function. Some trivial observations about the Poincaré series: (1) 0 ≤ PM (1)�≤ ∞ in general. (2) PM (1) = �(Mn ) < ∞ iff Mn = 0 for all but finitely many n. (We are given that �(Mn ) < ∞ fo ...
Part II Permutations, Cosets and Direct Product
... 1. We say, ϕ is a homomorphism if ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ G. 2. Given a subgroup H of G, the image of H under ϕ is defined to be ϕ(H) := {ϕ(x) : x ∈ H} Lemma 8.5. Suppose ϕ : G −→ G′ is a homomorphism of groups. Assume ϕ is injective. Then the image ϕ(G) is a subgroup of G′ and ϕ induces an ...
... 1. We say, ϕ is a homomorphism if ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ G. 2. Given a subgroup H of G, the image of H under ϕ is defined to be ϕ(H) := {ϕ(x) : x ∈ H} Lemma 8.5. Suppose ϕ : G −→ G′ is a homomorphism of groups. Assume ϕ is injective. Then the image ϕ(G) is a subgroup of G′ and ϕ induces an ...
Math 3121 Lecture 11
... – Theorem: The image of a group homomorphism is isomorphic to the group modulo its kernel. – Properties of normal subgroups – Theorem: For a subgroup of a group, left coset multiplication is welldefined if and only if the subgroup is normal. – Theorem: The canonical map is a homomorphism. ...
... – Theorem: The image of a group homomorphism is isomorphic to the group modulo its kernel. – Properties of normal subgroups – Theorem: For a subgroup of a group, left coset multiplication is welldefined if and only if the subgroup is normal. – Theorem: The canonical map is a homomorphism. ...
Logical Bilattices and Inconsistent Data 1 Introduction 2 Logical
... Note that we are using di erent implication connectives according to the strength we attach to each entailment: Penguins never y. This is a characteristic feature of penguins, and there are no exceptoins to that, hence we use the strongest implication (!) in the third asertion in order to express t ...
... Note that we are using di erent implication connectives according to the strength we attach to each entailment: Penguins never y. This is a characteristic feature of penguins, and there are no exceptoins to that, hence we use the strongest implication (!) in the third asertion in order to express t ...
Topology of Open Surfaces around a landmark result of C. P.
... By a complex affine algebraic variety X in Cn , we mean the subspace of common zeros of a finite set of polynomials in n variables with coefficients in C. The set of all C-valued maps on X which can be represented by polynomials is denoted by k[X] and is called the coordinate ring of X. One may say ...
... By a complex affine algebraic variety X in Cn , we mean the subspace of common zeros of a finite set of polynomials in n variables with coefficients in C. The set of all C-valued maps on X which can be represented by polynomials is denoted by k[X] and is called the coordinate ring of X. One may say ...
A Note on Roth`s Theorem Robert Gross Abstract We give a
... Because these constants are independent of [K : Q] = d, our result is stronger than Silverman’s statement. This type of result over Q at the archimedean place is nearly as old as Roth’s original theorem. The first statement is in Davenport and Roth [2], with the best result using Siegel’s lemma in M ...
... Because these constants are independent of [K : Q] = d, our result is stronger than Silverman’s statement. This type of result over Q at the archimedean place is nearly as old as Roth’s original theorem. The first statement is in Davenport and Roth [2], with the best result using Siegel’s lemma in M ...
The expected number of random elements to generate a finite
... p-groups for various primes p. That is, the theorem holds for all finite nilpotent groups. This observation should be compared with the discussion in [1]. We also remark that the methods of this paper may be used to compute higher moments for the random generation of finite abelian (or nilpotent) gr ...
... p-groups for various primes p. That is, the theorem holds for all finite nilpotent groups. This observation should be compared with the discussion in [1]. We also remark that the methods of this paper may be used to compute higher moments for the random generation of finite abelian (or nilpotent) gr ...
Dynamical systems: Multiply recurrent points
... call a point x of X e-recurrent if for every neighbourhood U of x, there is some n ≥ 1 such that {an (x) : a ∈ e} ⊆ U . (Note that the same n ∈ ω is required to work for all a ∈ e.) The following remarks shed some light on this notion. 12.2. Remark Given a single continuous map t : X → X on a compac ...
... call a point x of X e-recurrent if for every neighbourhood U of x, there is some n ≥ 1 such that {an (x) : a ∈ e} ⊆ U . (Note that the same n ∈ ω is required to work for all a ∈ e.) The following remarks shed some light on this notion. 12.2. Remark Given a single continuous map t : X → X on a compac ...