3,314 research outputs found
A Linear Kernel for Planar Total Dominating Set
A total dominating set of a graph is a subset such
that every vertex in is adjacent to some vertex in . Finding a total
dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on
general graphs when parameterized by the solution size. By the meta-theorem of
Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total
Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how
such a kernel can be effectively constructed, and how to obtain explicit
reduction rules with reasonably small constants. Following the approach of
Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating
Set on planar graphs with at most vertices, where is the size of the
solution. This result complements several known constructive linear kernels on
planar graphs for other domination problems such as Dominating Set, Edge
Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue
Dominating Set.Comment: 33 pages, 13 figure
Quantum Thermodynamics and Canonical Typicality
We present here a set of lecture notes on quantum thermodynamics and
canonical typicality. Entanglement can be constructively used in the
foundations of statistical mechanics. An alternative version of the postulate
of equal a priori probability is derived making use of some techniques of
convex geometr
Self-adjoint extensions and unitary operators on the boundary
We establish a bijection between the self-adjoint extensions of the Laplace
operator on a bounded regular domain and the unitary operators on the boundary.
Each unitary encodes a specific relation between the boundary value of the
function and its normal derivative. This bijection sets up a characterization
of all physically admissible dynamics of a nonrelativistic quantum particle
confined in a cavity. More- over, this correspondence is discussed also at the
level of quadratic forms. Finally, the connection between this parametrization
of the extensions and the classical one, in terms of boundary self-adjoint
operators on closed subspaces, is shown.Comment: 16 page
Quantum cavities with alternating boundary conditions
We consider the quantum dynamics of a free nonrelativistic particle moving in
a cavity and we analyze the effect of a rapid switching between two different
boundary conditions. We show that this procedure induces, in the limit of
infinitely frequent switchings, a new effective dynamics in the cavity related
to a novel boundary condition. We obtain a dynamical composition law for
boundary conditions which gives the emerging boundary condition in terms of the
two initial ones
Moving Walls and Geometric Phases
We unveil the existence of a non-trivial Berry phase associated to the
dynamics of a quantum particle in a one dimensional box with moving walls. It
is shown that a suitable choice of boundary conditions has to be made in order
to preserve unitarity. For these boundary conditions we compute explicitly the
geometric phase two-form on the parameter space. The unboundedness of the
Hamiltonian describing the system leads to a natural prescription of
renormalization for divergent contributions arising from the boundary.Comment: 16 pages, 5 figure
Explicit linear kernels via dynamic programming
Several algorithmic meta-theorems on kernelization have appeared in the last
years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of
bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding
a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed
topological minor. Typically, these results guarantee the existence of linear
or polynomial kernels on sparse graph classes for problems satisfying some
generic conditions but, mainly due to their generality, it is not clear how to
derive from them constructive kernels with explicit constants. In this paper we
make a step toward a fully constructive meta-kernelization theory on sparse
graphs. Our approach is based on a more explicit protrusion replacement
machinery that, instead of expressibility in CMSO logic, uses dynamic
programming, which allows us to find an explicit upper bound on the size of the
derived kernels. We demonstrate the usefulness of our techniques by providing
the first explicit linear kernels for -Dominating Set and -Scattered Set
on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs
excluding a fixed (topological) minor in the case where all the graphs in
\mathcal{F} are connected.Comment: 32 page
Boundaries without boundaries
Starting with a quantum particle on a closed manifold without boundary, we
consider the process of generating boundaries by modding out by a group action
with fixed points, and we study the emergent quantum dynamics on the quotient
manifold. As an illustrative example, we consider a free nonrelativistic
quantum particle on the circle and generate the interval via parity reduction.
A free particle with Neumann and Dirichlet boundary conditions on the interval
is obtained, and, by changing the metric near the boundary, Robin boundary
conditions can also be accommodated. We also indicate a possible method of
generating non-local boundary conditions. Then, we explore an alternative
generation mechanism which makes use of a folding procedure and is applicable
to a generic Hamiltonian through the emergence of an ancillary spin degree of
freedom.Comment: 19 pages, 4 figure
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