By Catherine Cavagnaro, William T. Haight II

Containing greater than 1,000 entries, the Dictionary of Classical and Theoretical arithmetic specializes in mathematical phrases and definitions of severe significance to practising mathematicians and scientists. This single-source reference presents operating definitions, meanings of phrases, comparable references, and an inventory of other phrases and definitions. The dictionary is certainly one of 5 constituent works that make up the casebound CRC accomplished Dictionary of arithmetic.

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**Extra resources for Dictionary of Classical and Theoretical Mathematics (Comprehensive Dictionary of Mathematics)**

**Sample text**

Degree of mapping Let f : (S n , x0 ) → n (S , x0 ) be a continuous map where n ≥ 1 and let f∗ : Hn (S n , x0 ) → Hn (S n , x0 ) be the induced map. If z0 is the generator of Hn (S n , x0 ), then f∗ (z0 ) = d · z0 for some integer d. In this case d is the degree of the map f and is denoted deg(f ). An important consequence of this is the Brouwer Fixed-Point Theorem: Every continuous map f : B n → B n has a fixed point. 36 derived set The set of accumulation points of a subset A of a topological space X.

See convergence of a continued convergent sequence A sequence of points {xn : n ∈ N} in a topological space X converges to x ∈ X if, for any open set U containing x, there is an N ∈ N with xn ∈ U for all n ≥ N . A sequence is convergent if it converges to some x ∈ X. For example, the sequence of reals { n1 } converges to 0, while the sequence {n} does not converge. convex (1) A non-empty subset X of Rn such that, for any elements x, y ∈ X, and any number c such that 0 ≤ c ≤ 1, the element cx +(1−c)y of Rn belongs to X.

A function f has a Dirichlet inverse if and only if f (1) = 0. When it exists, the inverse is unique. For example, μ, the Möbius function, and u, the unit function, are Dirichlet inverses of one another. Dirichlet multiplication The operation under which the Dirichlet convolution of two arithmetic functions is computed. It is commutative and associative. In fact, the set of arithmetic functions f such that f (1) = 0 forms a group under this operation. See also Dirichlet convolution. Dirichlet product See Dirichlet convolution.