232,070 research outputs found
Automata theory in nominal sets
We study languages over infinite alphabets equipped with some structure that
can be tested by recognizing automata. We develop a framework for studying such
alphabets and the ensuing automata theory, where the key role is played by an
automorphism group of the alphabet. In the process, we generalize nominal sets
due to Gabbay and Pitts
Local cloning of entangled states
We investigate the conditions under which a set \SC of pure bipartite
quantum states on a system can be locally cloned deterministically
by separable operations, when at least one of the states is full Schmidt rank.
We allow for the possibility of cloning using a resource state that is less
than maximally entangled. Our results include that: (i) all states in \SC
must be full Schmidt rank and equally entangled under the -concurrence
measure, and (ii) the set \SC can be extended to a larger clonable set
generated by a finite group of order , the number of states in the
larger set. It is then shown that any local cloning apparatus is capable of
cloning a number of states that divides exactly. We provide a complete
solution for two central problems in local cloning, giving necessary and
sufficient conditions for (i) when a set of maximally entangled states can be
locally cloned, valid for all ; and (ii) local cloning of entangled qubit
states with non-vanishing entanglement. In both of these cases, a maximally
entangled resource is necessary and sufficient, and the states must be related
to each other by local unitary "shift" operations. These shifts are determined
by the group structure, so need not be simple cyclic permutations. Assuming
this shifted form and partially entangled states, then in D=3 we show that a
maximally entangled resource is again necessary and sufficient, while for
higher dimensional systems, we find that the resource state must be strictly
more entangled than the states in \SC. All of our necessary conditions for
separable operations are also necessary conditions for LOCC, since the latter
is a proper subset of the former. In fact, all our results hold for LOCC, as
our sufficient conditions are demonstrated for LOCC, directly.Comment: REVTEX 15 pages, 1 figure, minor modifications. Same as the published
version. Any comments are welcome
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
AND-NOT logic framework for steady state analysis of Boolean network models
Finite dynamical systems (e.g. Boolean networks and logical models) have been
used in modeling biological systems to focus attention on the qualitative
features of the system, such as the wiring diagram. Since the analysis of such
systems is hard, it is necessary to focus on subclasses that have the
properties of being general enough for modeling and simple enough for
theoretical analysis. In this paper we propose the class of AND-NOT networks
for modeling biological systems and show that it provides several advantages.
Some of the advantages include: Any finite dynamical system can be written as
an AND-NOT network with similar dynamical properties. There is a one-to-one
correspondence between AND-NOT networks, their wiring diagrams, and their
dynamics. Results about AND-NOT networks can be stated at the wiring diagram
level without losing any information. Results about AND-NOT networks are
applicable to any Boolean network. We apply our results to a Boolean model of
Th-cell differentiation
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