It is well known that interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which have recently emerged as an important mathematical structure underlying heterogenous multi-logic specification. Our main result can be used in the applications in several different ways. It can be used to establish interpolation properties for multi-logic Grothendieck institutions, but also to lift interpolation properties from unsorted logics to their many sorted variants. The importance of the latter resides in the fact that, unlike other structural properties of logics, many sorted interpolation is a non-trivial generalisation of unsorted interpolation. The concepts, results, and the applications discussed in this paper are illustrated with several examples from conventional logic and algebraic specification theory.

The traditional way of approaching placement problems in computer-aided design (CAD) tools for analog layout is to explore an extremely large search space of feasible or unfeasible placement configurations, where the cells are moved in the chip plane (being even allowed to overlap in possibly illegal ways) by a stochastic optimizer. This paper presents a novel exploration technique for analog placement operating on a subset of tree representations of the layout—called symmetric-feasible, where the typical presence of an arbitrary number of symmetry groups of devices is directly taken into account during the search of the solution space. The computation times exhibited by this novel approach are significantly better than those of the algorithms using the traditional exploration strategy. This superior efficiency is partly due to the use of segment trees, a data structure introduced by Bentley, mainly used in computational geometry.

We describe a general framework in which we can precisely compare the structures of quantum-like theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operations, and a toy theory proposed by Spekkens. We discover that viewed within our framework these theories are very similar, but differ in one key aspect- a four element group we term the phase group which emerges naturally within our framework. In the case of the stabiliser theory this group is Z4 while for Spekkens’s theory the group is Z2 × Z2. We further show that the structure of this group is intimately involved in a key physical difference between the theories: whether or not they can be modelled by a local hidden variable theory. This is done by establishing a connection between the phase group, and an abstract notion of GHZ state correlations. We go on to formulate precisely how the stabiliser theory and toy theory are ‘similar ’ by defining a notion of ‘mutually unbiased qubit theory’, noting that all such theories have four element phase groups. Since Z4 and Z2 × Z2 are the only such groups we conclude that the GHZ correlations in this type of theory can only take two forms, exactly those appearing in the stabiliser theory and those appearing in Spekkens’s theory. The results point at a classification of local/non-local behaviours by finite Abelian groups, extending beyond qubits to any finitary theory whose observables are all mutually unbiased. 1

Abstract. We develop the theory of semisimple weak Hopf algebras and obtain analogues of a number of classical results for ordinary semisimple Hopf algebras. We prove a criterion for semisimplicity and analyze the square of the antipode S 2 of a semisimple weak Hopf algebra A. We explain how the Frobenius-Perron dimensions of irreducible A-modules and eigenvalues of S 2 can be computed using the inclusion matrix associated to A. A trace formula of Larson and Radford is extended to a relation between the categorical and Frobenius-Perron dimensions of A. Finally, an analogue of the Class Equation of Kac and Zhu is established and properties of A-module algebras and their dimensions are studied. 1.

Abstract. We first construct and give basic properties of the fibered coproduct in the category of ringed spaces. We then look at some special cases where this actually gives a fibered coproduct in the category of schemes. Intuitively this is gluing a collection of schemes along some collection of subschemes. We then use this to construct a scheme without closed points. 1.

We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphesis is given to the question which properties of these categories can be seen as instances of general categorical resp. algebraic results. We obtain new results concerning the existence of limits and of factorizations of morphisms.

Abstract. It is shown that the categories of R-coalgebras for a commutative unital ring R and the category of A-corings for some R-algebra A as well as their respective categories of comodules are locally presentable.

Abstract. We give a short proof for the Clebsch- Gordan decompositions for the finite-dimensional modules over k[X]. 1. The representation ring of k[X]: the primitive case Let k be an algebraically closed field of characteristic zero. The structure of finitely generated k[X] modules is well-known: a torsion-free module is free and an indecomposable torsion module is isomorphic to Jk(µ, m): = k[X]/(X − µ) m for some µ ∈ k and some natural number m. The modules Jk(µ, m) and Jk(µ ′ , m ′) are not isomorphic if (µ, m) ̸ = (µ ′ , m ′). If the field k is fixed we shall simply write J(µ, m). The isomorphism class of this module will be denoted [J(µ, m)] and the image of this module in any representation ring of k[X] will be denoted [µ, m]. Viewed as a k-vector space, J(µ, m) has a standard basis {ei: = (X−µ) i−1}i=1,...,m. Since (X − µ)ei = ei+1 for all i ≥ 1 (assuming that em+1 = em+2 =... = 0), we have that Xei = µei + ei+1 for all i. Hence, in this basis, X acts on J(µ, m) as µ1m + Dm, where Dm is the nilpotent operator sending each ei to ei+1. Let C be the full subcategory of k[X]-mod consisting of modules which are finitedimensional over k. It is immediate that C is closed under isomorphisms, finite direct sums, and the tensor product over k. Furthermore, by the structure theorem for finite torsion modules over a PID, C has the Krull- Remak- Schmidt property. Therefore, the representation ring R(C) is a free Z-module on the elements [µ, m]. Our goal in this section is to describe the multiplicative structure of the representation ring R(C) of k[X] corresponding to the primitive product A ⊗ 1 + 1 ⊗ B. In the next section we shall solve the same problem for the Kronecker product. Given k[X]-modules M: = J(µ, m) with standard basis ei, i = 1,..., m, and N: = J(n, ν) with standard basis fj, j = 1,..., n, we define an action of X on M ⊗k N by the matrix A⊗1+1⊗B, where A and B are the matrices corresponding to M and N. In the basis ei,j: = ei ⊗ fj, it is given by the operator (µ1m + Dm) ⊗ 1n + 1m ⊗ (ν1n + Dn) = (µ + ν)1mn + Dm ⊗ 1n + 1m ⊗ Dn. Here (µ + ν)1mn is the semi-simple part of the operator X: M ⊗ N → M ⊗ N and (1.1) D: = D ′ + D ′ ′ : ei,j ↦ → ei+1,j + ei,j+1, where D ′: = Dm ⊗ 1n and D ′ ′: = 1m ⊗ Dn, is the nilpotent part of X. In short, (1.2) Xei,j = (µ + ν)ei,j + ei+1,j + ei,j+1.

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In this section we return to studying the structure of finite dimensional algebras. Throughout the section, we work over an algebraically closed field k (of any characteristic). 7.1 Projective modules Let A be an algebra, and P be a left A-module. Theorem 7.1. The following properties of P are equivalent: (i) If ϕ: M ⊃ N is a surjective morphism, and λ: P ⊃ N any morphism, then there exists a morphism µ: P ⊃ M such that ϕ ∞ µ = λ. (ii) Any surjective morphism ϕ: M ⊃ P splits, i.e., there exists µ: P ⊃ M such that ϕ∞µ = id. (iii) There exists another A-module Q such that P � Q is a free A-module, i.e., a direct sum of copies of A.