solutions to HW#3
... forms a group isomorphic to Z under addition. (The isomorphism is either doubling of halving, depending on which way it goes.) 1.1.6(f ) The set of rational numbers with denominators equal to 1, 2, or 3: This set cannot form a group because it is not closed under addition. For example, 1/2 + 1/3 = 5 ...
... forms a group isomorphic to Z under addition. (The isomorphism is either doubling of halving, depending on which way it goes.) 1.1.6(f ) The set of rational numbers with denominators equal to 1, 2, or 3: This set cannot form a group because it is not closed under addition. For example, 1/2 + 1/3 = 5 ...
Document
... We define binary operations of + and in R[x] to be polynomial addition and multiplication. For polynomials f (x) and g(x) members of R[x], the products f (x) g(x) and g(x) f (x) are equal because the coefficients are real numbers, and we can use all the properties of real numbers under multipl ...
... We define binary operations of + and in R[x] to be polynomial addition and multiplication. For polynomials f (x) and g(x) members of R[x], the products f (x) g(x) and g(x) f (x) are equal because the coefficients are real numbers, and we can use all the properties of real numbers under multipl ...
Sets with a Category Action Peter Webb 1. C-sets
... sets sending every element onto a single element. We see various things from this example, such as that a finite category may have infinitely non-isomorphic transitive sets, and also that transitive sets need not be generated by any single element. We have available another operation on C-sets, name ...
... sets sending every element onto a single element. We see various things from this example, such as that a finite category may have infinitely non-isomorphic transitive sets, and also that transitive sets need not be generated by any single element. We have available another operation on C-sets, name ...
4 Calibration - Wageningen UR E
... environmental variable x only and suppose that the responses of the species are mutually independent for each fixed value of x. Denote the response curve of the probability of occurrence of the k-th species by Pk(x). The probability that the k-th species is absent also depends on x and equals I - Pk ...
... environmental variable x only and suppose that the responses of the species are mutually independent for each fixed value of x. Denote the response curve of the probability of occurrence of the k-th species by Pk(x). The probability that the k-th species is absent also depends on x and equals I - Pk ...
Noting the Difference: Musical Scales and
... and Myerson address is how many different-sounding progressions there are among those with the same structure. We now make this more precise. We define the diatonic set (or scale) to be {A, B, C, D, E, F, G}, the set of notes corresponding to the white keys on a piano. (We are really considering eq ...
... and Myerson address is how many different-sounding progressions there are among those with the same structure. We now make this more precise. We define the diatonic set (or scale) to be {A, B, C, D, E, F, G}, the set of notes corresponding to the white keys on a piano. (We are really considering eq ...
Category Theory Example Sheet 1
... for all a, b, c ∈ L). Show that there is a category MatL whose objects are the natural numbers, and whose morphisms n −→ m are m × n matrices with entries from L, where we define ‘multiplication’ of such matrices by analogy with that of matrices over a field, interpreting ∧ as multiplication and ∨ a ...
... for all a, b, c ∈ L). Show that there is a category MatL whose objects are the natural numbers, and whose morphisms n −→ m are m × n matrices with entries from L, where we define ‘multiplication’ of such matrices by analogy with that of matrices over a field, interpreting ∧ as multiplication and ∨ a ...
Functors and natural transformations A covariant functor F : C → D is
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
Exercise sheet 5
... 2. (a) You have 7 pieces of paper, and you apply the following procedure as many times as you want: Pick any one of your pieces of paper and cut it in 7. Show that you can never get 1997 pieces of paper. Hint: Think modulo 6. (b) Find the remainder in the division by 3 of each of the following numbe ...
... 2. (a) You have 7 pieces of paper, and you apply the following procedure as many times as you want: Pick any one of your pieces of paper and cut it in 7. Show that you can never get 1997 pieces of paper. Hint: Think modulo 6. (b) Find the remainder in the division by 3 of each of the following numbe ...
Combinatorial species
In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures. These operations correspond to equivalent manipulations of generating functions, so producing such functions for complicated structures is much easier than with other methods. The theory was introduced by André Joyal.The power of the theory comes from its level of abstraction. The ""description format"" of a structure (such as adjacency list versus adjacency matrix for graphs) is irrelevant, because species are purely algebraic. Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species.