Let K be a compact subset of R d and write C(K ) for the family of continuous functions on K . In this paper we study different fractal and multifractal dimensions of functions in C(K ) that are generic in the sense of prevalence. We first prove a number of general results, namely, for arbitrary "dimension" functions : C(K ) → R satisfying various natural scaling conditions, we obtain formulas for the "dimension"( f ) of a prevalent function f in C(K ); this is the contents of Theorems 1.1-1.3. By applying Theorems 1.1-1.3 to appropriate choices of we obtain the following results: we compute the (lower and upper) local dimension of a prevalent function f in C(K ); we compute the (lower or upper) Hölder exponent at a point x of a prevalent Communicated by P. Friz.

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About a decade ago, it was realised that the satisfaction of a given identity (or equation) of the form ffalse(x1,…,xnfalse)≈ffalse(y1,…,ynfalse) in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called loop conditions, and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a Siggers identity s(x,y,z,x)≈s(y,x,y,z) in any arbitrary non‐trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form ffalse(x1,1,…,x1,nfalse)≈⋯≈ffalse(xm,1,…,xm,nfalse), which we call loop conditions of width m. We show that their satisfaction in an algebra is equivalent to any action of the algebra on a certain type of relation forcing a constant tuple into the relation. Proving that for each fixed width m there is a weakest loop condition (that is, one entailed by all others), we obtain a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non‐trivial idempotent, possibly infinite, algebra. The framework of classical (width 2) loop conditions is insufficient for such proof. We then consider pseudo‐loop conditions of finite width, a generalisation suitable for non‐idempotent algebras; they are of the form u1∘ffalse(x1,1,…,x1,nfalse)≈⋯≈um∘ffalse(xm,1,…,xm,nfalse), and of central importance for the structure of algebras associated with ω‐categorical structures. We show that for the latter, satisfaction of a pseudo‐loop condition is characterised by pseudo‐loops, that is, loops modulo the action of the automorphism group, and that a weakest pseudo‐loop condition exists (for ω‐categorical cores). This way we obtain a new and short proof of the theorem that the satisfaction of any non‐trivial identities of height 1 in such algebras implies the satisfaction of a fixed single identity.

In this paper, we consider the group Aut(Q, ) of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut(Q, ) is contained in an N -generated subgroup of Aut(Q, ) for some fixed N ∈ N. We show that the least such N is 2. Moreover, for every countable subset of Aut(Q, ), we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that a and b freely generate the free semigroup {a, b} + consisting of the non-empty words over a and b. Then we show that there exists a sequence of words w 1, w2, . . . over {a, b} such that for every sequence f1, f2, . . . ∈ Aut(Q, ) there is a homomorphism φ : {a, b} + → Aut(Q, ) where (wi)φ = fi for every i.As a corollary to the main theorem in this paper, we obtain a result of Droste and Holland showing that the strong cofinality of Aut(Q, ) is uncountable, or equivalently that Aut(Q, ) has uncountable cofinality and Bergman's property.