4,455 research outputs found
Stable domination and independence in algebraically closed valued fields
We seek to create tools for a model-theoretic analysis of types in
algebraically closed valued fields (ACVF). We give evidence to show that a
notion of 'domination by stable part' plays a key role. In Part A, we develop a
general theory of stably dominated types, showing they enjoy an excellent
independence theory, as well as a theory of definable types and germs of
definable functions. In Part B, we show that the general theory applies to
ACVF. Over a sufficiently rich base, we show that every type is stably
dominated over its image in the value group. For invariant types over any base,
stable domination coincides with a natural notion of `orthogonality to the
value group'. We also investigate other notions of independence, and show that
they all agree, and are well-behaved, for stably dominated types. One of these
is used to show that every type extends to an invariant type; definable types
are dense. Much of this work requires the use of imaginary elements. We also
show existence of prime models over reasonable bases, possibly including
imaginaries
Geometric local invariants and pure three-qubit states
We explore a geometric approach to generating local SU(2) and
invariants for a collection of qubits inspired by lattice
gauge theory. Each local invariant or 'gauge' invariant is associated to a
distinct closed path (or plaquette) joining some or all of the qubits. In
lattice gauge theory, the lattice points are the discrete space-time points,
the transformations between the points of the lattice are defined by parallel
transporters and the gauge invariant observable associated to a particular
closed path is given by the Wilson loop. In our approach the points of the
lattice are qubits, the link-transformations between the qubits are defined by
the correlations between them and the gauge invariant observable, the local
invariants associated to a particular closed path are also given by a Wilson
loop-like construction. The link transformations share many of the properties
of parallel transporters although they are not undone when one retraces one's
steps through the lattice. This feature is used to generate many of the
invariants. We consider a pure three qubit state as a test case and find we can
generate a complete set of algebraically independent local invariants in this
way, however the framework given here is applicable to mixed states composed of
any number of level quantum systems. We give an operational interpretation
of these invariants in terms of observables.Comment: 9 pages, 3 figure
Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term Institutions
Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the -institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-Férez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-Férez pertaining to arbitrary multi-term -institutions. The approach places hidden multi-sorted deductive systems in a more general framework and bypasses the laborious reuse of well-known proof techniques from traditional abstract algebraic logic by using “off the shelf” tools
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