15,909 research outputs found
The problem of deficiency indices for discrete Schr\"odinger operators on locally finite graphs
The number of self-adjoint extensions of a symmetric operator acting on a
complex Hilbert space is characterized by its deficiency indices. Given a
locally finite unoriented simple tree, we prove that the deficiency indices of
any discrete Schr\"odinger operator are either null or infinite. We also prove
that almost surely, there is a tree such that all discrete Schr\"odinger
operators are essentially self-adjoint. Furthermore, we provide several
criteria of essential self-adjointness. We also adress some importance to the
case of the adjacency matrix and conjecture that, given a locally finite
unoriented simple graph, its the deficiency indices are either null or
infinite. Besides that, we consider some generalizations of trees and weighted
graphs.Comment: Typos corrected. References and ToC added. Paper slightly
reorganized. Section 3.2, about the diagonalization has been much improved.
The older section about the stability of the deficiency indices in now in
appendix. To appear in Journal of Mathematical Physic
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
Spectral Theory of Infinite Quantum Graphs
We investigate quantum graphs with infinitely many vertices and edges without
the common restriction on the geometry of the underlying metric graph that
there is a positive lower bound on the lengths of its edges. Our central result
is a close connection between spectral properties of a quantum graph and the
corresponding properties of a certain weighted discrete Laplacian on the
underlying discrete graph. Using this connection together with spectral theory
of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new
results on spectral properties of quantum graphs. Namely, we prove several
self-adjointness results including a Gaffney type theorem. We investigate the
problem of lower semiboundedness, prove several spectral estimates (bounds for
the bottom of spectra and essential spectra of quantum graphs, CLR-type
estimates) and study spectral types.Comment: Dedicated to the memory of M. Z. Solomyak (16.05.1931 - 31.07.2016
Essential self-adjointness for combinatorial Schr\"odinger operators II- Metrically non complete graphs
We consider weighted graphs, we equip them with a metric structure given by a
weighted distance, and we discuss essential self-adjointness for weighted graph
Laplacians and Schr\"odinger operators in the metrically non complete case.Comment: Revisited version: Ognjen Milatovic wrote to us that he had
discovered a gap in the proof of theorem 4.2 of our paper. As a consequence
we propose to make an additional assumption (regularity property of the
graph) to this theorem. A new subsection (4.1) is devoted to the study of
this property and some details have been changed in the proof of theorem 4.
The adjacency matrix and the discrete Laplacian acting on forms
We study the relationship between the adjacency matrix and the discrete
Laplacian acting on 1-forms. We also prove that if the adjacency matrix is
bounded from below it is not necessarily essentially self-adjoint. We discuss
the question of essential self-adjointness and the notion of completeness
Ensemble of causal trees
We discuss the geometry of trees endowed with a causal structure using the
conventional framework of equilibrium statistical mechanics. We show how this
ensemble is related to popular growing network models. In particular we
demonstrate that on a class of afine attachment kernels the two models are
identical but they can differ substantially for other choice of weights. We
show that causal trees exhibit condensation even for asymptotically linear
kernels. We derive general formulae describing the degree distribution, the
ancestor-descendant correlation and the probability a randomly chosen node
lives at a given geodesic distance from the root. It is shown that the
Hausdorff dimension d_H of the causal networks is generically infinite.Comment: 11 Pages, published in proceedings of Random Geometry Workshop May
15-17, 2003 Krako
Toric dynamical systems
Toric dynamical systems are known as complex balancing mass action systems in
the mathematical chemistry literature, where many of their remarkable
properties have been established. They include as special cases all deficiency
zero systems and all detailed balancing systems. One feature is that the steady
state locus of a toric dynamical system is a toric variety, which has a unique
point within each invariant polyhedron. We develop the basic theory of toric
dynamical systems in the context of computational algebraic geometry and show
that the associated moduli space is also a toric variety. It is conjectured
that the complex balancing state is a global attractor. We prove this for
detailed balancing systems whose invariant polyhedron is two-dimensional and
bounded.Comment: We include the proof of our Conjecture 5 (now Lemma 5) and add some
reference
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