Convenient antiderivatives for differential linear categories

Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

Inner differentiability and differential forms on tangentially locally linearly independent sets

The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold. We proved this theorem on a wider family of subsets of Euclidean space, on which we can define inner differentiability. Here we define this family of sets called tangentially locally linearly independent sets, propose inner differentiability on them, postulate usual properties of differentiable real functions and show that the integration over sets that are wider than manifolds is possible.

Interval-type theorems concerning means

Each family ℳ of means has a natural, partial order (point-wise order), that is M ≤ N iff M(x) ≤ N(x) for all admissible x. In this setting we can introduce the notion of interval-type set (a subset ℐ ⊂ℳ such that whenever M ≤ P ≤ N for some M, N ∈ℐ and P ∈ℳ then P ∈ℐ). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. In the present paper we consider this property for Gini means and Hardy means. Moreover, some results concerning L∞ metric among (abstract) means will be obtained.

Motivating the Rules of Sequent Calculus

Parallelized elimination rules in natural deduction correspond to Left rules in the sequent calculus; and introduction rules correspond to Right rules. These rules may be construed as inductive clauses in the inductive definition of the notion of sequent proof. There is a natural isomorphism between natural deductions in Core Logic and the sequent proofs that correspond to them. We examine the relations, between sequents, of concentration and dilution; and describe what it is for one sequent to strengthen another. We examine some possible global restrictions on proof-formation, designed to prevent proofs from proving dilutions of sequents already proved by a subproof. We establish the important result that the sequent rules of Core Logic maintain concentration, and we explain its importance for automated proof-search.

LOCAL HOMOLOGY AND GORENSTEIN FLAT MODULES

Let R be a commutative Noetherian ring, 𝔞 be an ideal of R and [Formula: see text] denote the derived category of R-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let X be a homologically bounded to the right complex and Q be a bounded to the right complex of Gorenstein flat R-modules such that Q and X are isomorphic in [Formula: see text]. We establish a natural isomorphism LΛ𝔞(X) ≃ Λ𝔞(Q) in [Formula: see text] which immediately asserts that sup LΛ𝔞(X) ≤ Gfd RX. This isomorphism yields several conseQuences. For instance, in the case R possesses a dualizing complex, we show that Gfd RLΛ𝔞(X) ≤ Gfd RX. Also, we establish a criterion for regularity of Gorenstein local rings.

The de Rham theorem for the noncommutative complex of Cenkl and Porter

We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.

A generalization of the trie data structure

Tries, a form of string-indexed look-up structure, are generalized, in a manner first discovered by Wadsworth, to permit indexing by terms built according to an arbitrary signature. The construction is parametric with respect to the type of data to be stored as values; this is essential, because the recursion that defines tries appeals from one value type to others. ‘Trie’ (for any fixed signature) is then a functor, and the corresponding look-up function is a natural isomorphism.The trie functor is in principle definable by the ‘initial fixed point’ semantics of Smyth and Plotkin. We simplify the construction, however, by introducing the ‘category-cpo’, a class of category within which calculations can retain some domain-theoretic flavor. Our construction of tries extends easily to many-sorted signatures.

A natural proof of the cyclotomic identity

The cyclotomic identitywhere and μ is the classical Möbius function, is shown to be a consequence of a natural isomorphism of species.

Monodromy in local groups

A monodromy theorem for homomorphisms of local groups into groups is proved. It follows that under suitable conditions the universal group of the local group depends only on the germ of the local group (up to natural isomorphism).1980 Mathematics subject classification (Amer. Math. Soc.): 22 E 05.