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**1 - 4**of**4**### Affine Constellations Without Mutually Unbiased Counterparts

, 2010

"... It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations. The observed discrepan ..."

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It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations. The observed discrepancies make a deeper relation between the two existence problems unlikely. Two orthonormal basesB andB of a d-level quantum system are mutu-ally unbiased (MU) if |〈b|b′〉|2 = 1/d for any two states |b 〉 ∈ B and |b′ 〉 ∈ B ′. This means that the probabilities for a transition of a quantum system pre-pared in the state |b 〉 ∈ B into a state |b′ 〉 ∈ B ′ are independent of both the initital and the final state [1]. It is known how to construct triples of MU bases in Cd for all values of d ≥ 2 [2]. The construction of (d + 1)-tuples of MU bases [3, 4, 5, 6] can be

### Generalized Lattices Express Parallel Distributed Concept Learning

"... Abstract—Concepts have been expressed mathematically as propositions in a distributive lattice. A more comprehensive formulation is that of a generalized lattice, or category, in which the concepts are related in hierarchical fashion by lattice-like links called concept morphisms. A concept morphism ..."

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Abstract—Concepts have been expressed mathematically as propositions in a distributive lattice. A more comprehensive formulation is that of a generalized lattice, or category, in which the concepts are related in hierarchical fashion by lattice-like links called concept morphisms. A concept morphism describes how a more abstract concept is used within a more specialized concept, as the color ”red ” is used in describing ”apples”. Often, an abstract concept can be used in a more specialized concept in more than one way as with ”color”, which can appear in ”apples ” as either ”red”, ”yellow ” or ”green”. Further, ”color” appears in ”apples ” because it appears in ”red”, ”yellow” and ”green”, which in turn appear in ”apples”, expressed via the composition of concept morphisms. Using categorical constructs based upon composition together with structure-preserving mappings that preserve compositional structure, a recently-developed semantic theory shows how abstract and specialized concepts are learned by a neural network. I.

### ONTOLOGIES AND WORLDS IN CATEGORY THEORY: IMPLICATIONS FOR NEURAL SYSTEMS

"... ABSTRACT. We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a sub-category of a cate ..."

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ABSTRACT. We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a sub-category of a category of theories expressed in a formal logic. In addition to providing mathematical rigor, this approach has several advantages. It allows the incremental analysis of ontologies by basing them in an interconnected hierarchy of theories, with an operation on the hierarchy that expresses the formation of complex theories from simple theories that express first principles. Another operation forms abstractions expressing the shared concepts in an array of theories. The use of categorical model theory makes possible the incremental analysis of possible worlds, or instances, for the theories, and the mapping of instances of a theory to instances of its more abstract parts. We describe the theoretical approach by applying it to the semantics of neural networks.