35,008 research outputs found
Universality Problem for Unambiguous VASS
We study languages of unambiguous VASS, that is, Vector Addition Systems with States, whose transitions read letters from a finite alphabet, and whose acceptance condition is defined by a set of final states (i.e., the coverability language). We show that the problem of universality for unambiguous VASS is ExpSpace-complete, in sheer contrast to Ackermann-completeness for arbitrary VASS, even in dimension 1. When the dimension d ? ? is fixed, the universality problem is PSpace-complete if d ? 2, and coNP-hard for 1-dimensional VASSes (also known as One Counter Nets)
Rigidity and a common framework for mutually unbiased bases and k-nets
Many deep, mysterious connections have been observed between collections of
mutually unbiased bases (MUBs) and combinatorial designs called -nets (and
in particular, between complete collections of MUBs and finite affine - or
equivalently: finite projective - planes). Here we introduce the notion of a
-net over an algebra and thus provide a common framework for
both objects. In the commutative case, we recover (classical) -nets, while
choosing leads to collections of MUBs.
A common framework allows one to find shared properties and proofs that
"inherently work" for both objects. As a first example, we derive a certain
rigidity property which was previously shown to hold for -nets that can be
completed to affine planes using a completely different, combinatorial
argument. For -nets that cannot be completed and for MUBs, this result is
new, and, in particular, it implies that the only vectors unbiased to all but
bases of a complete collection of MUBs in are
the elements of the remaining bases (up to phase factors). In general, this
is false when is just the next integer after ; we present an
example of this in every prime-square dimension, demonstrating that the derived
bound is tight.
As an application of the rigidity result, we prove that if a large enough
collection of MUBs constructed from a certain type of group representation
(e.g. a construction relying on discrete Weyl operators or generalized Pauli
matrices) can be extended to a complete system, then in fact every basis of the
completion must come from the same representation. In turn, we use this to show
that certain large systems of MUBs cannot be completed
Representations of Conformal Nets, Universal C*-Algebras and K-Theory
We study the representation theory of a conformal net A on the circle from a
K-theoretical point of view using its universal C*-algebra C*(A). We prove that
if A satisfies the split property then, for every representation \pi of A with
finite statistical dimension, \pi(C*(A)) is weakly closed and hence a finite
direct sum of type I_\infty factors. We define the more manageable locally
normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest
ideal vanishing in all locally normal representations and we investigate its
structure. In particular, if A is completely rational with n sectors, then
C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact
operators has nontrivial K-theory, and we prove that the DHR endomorphisms of
C*(A) with finite statistical dimension act on K_A, giving rise to an action of
the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this
action corresponds to the regular representation of the associated fusion
algebra.Comment: v2: we added some comments in the introduction and new references.
v3: new authors' addresses, minor corrections. To appear in Commun. Math.
Phys. v4: minor corrections, updated reference
Explicit constructions of point sets and sequences with low discrepancy
In this article we survey recent results on the explicit construction of
finite point sets and infinite sequences with optimal order of
discrepancy. In 1954 Roth proved a lower bound for the
discrepancy of finite point sets in the unit cube of arbitrary dimension. Later
various authors extended Roth's result to lower bounds also for the
discrepancy and for infinite sequences. While it was known
already from the early 1980s on that Roth's lower bound is best possible in the
order of magnitude, it was a longstanding open question to find explicit
constructions of point sets and sequences with optimal order of
discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite
point sets and recently by the authors of this article for infinite sequences.
These constructions can also be extended to give optimal order of the
discrepancy of finite point sets for . The
main aim of this article is to give an overview of these constructions and
related results
Percolation in Hierarchical Scale-Free Nets
We study the percolation phase transition in hierarchical scale-free nets.
Depending on the method of construction, the nets can be fractal or small-world
(the diameter grows either algebraically or logarithmically with the net size),
assortative or disassortative (a measure of the tendency of like-degree nodes
to be connected to one another), or possess various degrees of clustering. The
percolation phase transition can be analyzed exactly in all these cases, due to
the self-similar structure of the hierarchical nets. We find different types of
criticality, illustrating the crucial effect of other structural properties
besides the scale-free degree distribution of the nets.Comment: 9 Pages, 11 figures. References added and minor corrections to
manuscript. In pres
De Rham compatible Deep Neural Network FEM
On general regular simplicial partitions of bounded polytopal
domains , , we construct \emph{exact
neural network (NN) emulations} of all lowest order finite element spaces in
the discrete de Rham complex. These include the spaces of piecewise constant
functions, continuous piecewise linear (CPwL) functions, the classical
``Raviart-Thomas element'', and the ``N\'{e}d\'{e}lec edge element''. For all
but the CPwL case, our network architectures employ both ReLU (rectified linear
unit) and BiSU (binary step unit) activations to capture discontinuities. In
the important case of CPwL functions, we prove that it suffices to work with
pure ReLU nets. Our construction and DNN architecture generalizes previous
results in that no geometric restrictions on the regular simplicial partitions
of are required for DNN emulation. In addition, for CPwL
functions our DNN construction is valid in any dimension . Our
``FE-Nets'' are required in the variationally correct, structure-preserving
approximation of boundary value problems of electromagnetism in nonconvex
polyhedra . They are thus an essential ingredient
in the application of e.g., the methodology of ``physics-informed NNs'' or
``deep Ritz methods'' to electromagnetic field simulation via deep learning
techniques. We indicate generalizations of our constructions to higher-order
compatible spaces and other, non-compatible classes of discretizations, in
particular the ``Crouzeix-Raviart'' elements and Hybridized, Higher Order (HHO)
methods
Fractal and Transfractal Recursive Scale-Free Nets
We explore the concepts of self-similarity, dimensionality, and
(multi)scaling in a new family of recursive scale-free nets that yield
themselves to exact analysis through renormalization techniques. All nets in
this family are self-similar and some are fractals - possessing a finite
fractal dimension - while others are small world (their diameter grows
logarithmically with their size) and are infinite-dimensional. We show how a
useful measure of "transfinite" dimension may be defined and applied to the
small world nets. Concerning multiscaling, we show how first-passage time for
diffusion and resistance between hub (the most connected nodes) scale
differently than for other nodes. Despite the different scalings, the Einstein
relation between diffusion and conductivity holds separately for hubs and
nodes. The transfinite exponents of small world nets obey Einstein relations
analogous to those in fractal nets.Comment: Includes small revisions and references added as result of readers'
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