21,355 research outputs found
Limit theorems for von Mises statistics of a measure preserving transformation
For a measure preserving transformation of a probability space
we investigate almost sure and distributional convergence
of random variables of the form where (called the \emph{kernel})
is a function from to and are appropriate normalizing
constants. We observe that the above random variables are well defined and
belong to provided that the kernel is chosen from the projective
tensor product with We establish a form of the individual ergodic theorem for such
sequences. Next, we give a martingale approximation argument to derive a
central limit theorem in the non-degenerate case (in the sense of the classical
Hoeffding's decomposition). Furthermore, for and a wide class of
canonical kernels we also show that the convergence holds in distribution
towards a quadratic form in independent
standard Gaussian variables . Our results on the
distributional convergence use a --\,invariant filtration as a prerequisite
and are derived from uni- and multivariate martingale approximations
A general realization theorem for matrix-valued Herglotz-Nevanlinna functions
New special types of stationary conservative impedance and scattering
systems, the so-called non-canonical systems, involving triplets of Hilbert
spaces and projection operators, are considered. It is established that every
matrix-valued Herglotz-Nevanlinna function of the form
V(z)=Q+Lz+\int_{\dR}(\frac{1}{t-z}-\frac{t}{1+t^2})d\Sigma(t) can be realized
as a transfer function of such a new type of conservative impedance system. In
this case it is shown that the realization can be chosen such that the main and
the projection operators of the realizing system satisfy a certain
commutativity condition if and only if L=0. It is also shown that with
an additional condition (namely, is invertible or L=0), can be realized as
a linear fractional transformation of the transfer function of a non-canonical
scattering -system. In particular, this means that every scalar
Herglotz-Nevanlinna function can be realized in the above sense.
Moreover, the classical Livsic systems (Brodskii-Livsic operator
colligations) can be derived from -systems as a special case when
and the spectral measure is compactly supported. The realization
theorems proved in this paper are strongly connected with, and complement the
recent results by Ball and Staffans.Comment: 28 page
A generalization of the injectivity condition for Projected Entangled Pair States
We introduce a family of tensor network states that we term semi-injective
Projected Entangled-Pair States (PEPS). They extend the class of injective PEPS
and include other states, like the ground states of the AKLT and the CZX models
in square lattices. We construct parent Hamiltonians for which semi-injective
PEPS are unique ground states. We also determine the necessary and sufficient
conditions for two tensors to generate the same family of such states in two
spatial dimensions. Using this result, we show that the third cohomology
labeling of Symmetry Protected Topological phases extends to semi-injective
PEPS.Comment: 63 page
Algebraic K-theory of quasi-smooth blow-ups and cdh descent
We construct a semi-orthogonal decomposition on the category of perfect
complexes on the blow-up of a derived Artin stack in a quasi-smooth centre.
This gives a generalization of Thomason's blow-up formula in algebraic K-theory
to derived stacks. We also provide a new criterion for descent in Voevodsky's
cdh topology, which we use to give a direct proof of Cisinski's theorem that
Weibel's homotopy invariant K-theory satisfies cdh descent.Comment: 24 pages; to appear in Annales Henri Lebesgu
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