840 research outputs found
Subproduct systems and Cartesian systems; new results on factorial languages and their relations with other areas
We point out that a sequence of natural numbers is the dimension sequence of
a subproduct system if and only if it is the cardinality sequence of a word
system (or factorial language). Determining such sequences is, therefore,
reduced to a purely combinatorial problem in the combinatorics of words. A
corresponding (and equivalent) result for graded algebras has been known in
abstract algebra, but this connection with pure combinatorics has not yet been
noticed by the product systems community. We also introduce Cartesian systems,
which can be seen either as a set theoretic version of subproduct systems or an
abstract version of word systems. Applying this, we provide several new results
on the cardinality sequences of word systems and the dimension sequences of
subproduct systems.Comment: New title; added references; to appear in Journal of Stochastic
Analysi
Algorithmic information and incompressibility of families of multidimensional networks
This article presents a theoretical investigation of string-based generalized
representations of families of finite networks in a multidimensional space.
First, we study the recursive labeling of networks with (finite) arbitrary node
dimensions (or aspects), such as time instants or layers. In particular, we
study these networks that are formalized in the form of multiaspect graphs. We
show that, unlike classical graphs, the algorithmic information of a
multidimensional network is not in general dominated by the algorithmic
information of the binary sequence that determines the presence or absence of
edges. This universal algorithmic approach sets limitations and conditions for
irreducible information content analysis in comparing networks with a large
number of dimensions, such as multilayer networks. Nevertheless, we show that
there are particular cases of infinite nesting families of finite
multidimensional networks with a unified recursive labeling such that each
member of these families is incompressible. From these results, we study
network topological properties and equivalences in irreducible information
content of multidimensional networks in comparison to their isomorphic
classical graph.Comment: Extended preprint version of the pape
Some relational structures with polynomial growth and their associated algebras II: Finite generation
The profile of a relational structure is the function which
counts for every integer the number, possibly infinite, of
substructures of induced on the -element subsets, isomorphic
substructures being identified. If takes only finite values, this
is the Hilbert function of a graded algebra associated with , the age
algebra , introduced by P.~J.~Cameron.
In a previous paper, we studied the relationship between the properties of a
relational structure and those of their algebra, particularly when the
relational structure admits a finite monomorphic decomposition. This
setting still encompasses well-studied graded commutative algebras like
invariant rings of finite permutation groups, or the rings of quasi-symmetric
polynomials.
In this paper, we investigate how far the well know algebraic properties of
those rings extend to age algebras. The main result is a combinatorial
characterization of when the age algebra is finitely generated. In the special
case of tournaments, we show that the age algebra is finitely generated if and
only if the profile is bounded. We explore the Cohen-Macaulay property in the
special case of invariants of permutation groupoids. Finally, we exhibit
sufficient conditions on the relational structure that make naturally the age
algebra into a Hopf algebra.Comment: 27 pages; submitte
Effective Scalar Products for D-finite Symmetric Functions
Many combinatorial generating functions can be expressed as combinations of
symmetric functions, or extracted as sub-series and specializations from such
combinations. Gessel has outlined a large class of symmetric functions for
which the resulting generating functions are D-finite. We extend Gessel's work
by providing algorithms that compute differential equations these generating
functions satisfy in the case they are given as a scalar product of symmetric
functions in Gessel's class. Examples of applications to k-regular graphs and
Young tableaux with repeated entries are given. Asymptotic estimates are a
natural application of our method, which we illustrate on the same model of
Young tableaux. We also derive a seemingly new formula for the Kronecker
product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2:
corrections from original submission, improved clarity; now formatted for
journal + bibliograph
On the Implicit Graph Conjecture
The implicit graph conjecture states that every sufficiently small,
hereditary graph class has a labeling scheme with a polynomial-time computable
label decoder. We approach this conjecture by investigating classes of label
decoders defined in terms of complexity classes such as P and EXP. For
instance, GP denotes the class of graph classes that have a labeling scheme
with a polynomial-time computable label decoder. Until now it was not even
known whether GP is a strict subset of GR. We show that this is indeed the case
and reveal a strict hierarchy akin to classical complexity. We also show that
classes such as GP can be characterized in terms of graph parameters. This
could mean that certain algorithmic problems are feasible on every graph class
in GP. Lastly, we define a more restrictive class of label decoders using
first-order logic that already contains many natural graph classes such as
forests and interval graphs. We give an alternative characterization of this
class in terms of directed acyclic graphs. By showing that some small,
hereditary graph class cannot be expressed with such label decoders a weaker
form of the implicit graph conjecture could be disproven.Comment: 13 pages, MFCS 201
Convergent Puiseux Series and Tropical Geometry of Higher Rank
We propose to study the tropical geometry specifically arising from
convergent Puiseux series in multiple indeterminates. One application is a new
view on stable intersections of tropical hypersurfaces. Another one is the
study of families of ordinary convex polytopes depending on more than one
parameter through tropical geometry. This includes cubes constructed by
Goldfarb and Sit (1979) as special cases.Comment: 32 pages, 3 figure
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
- …