1,703 research outputs found
Higher-Order Results for the Relation between Channel Conductance and the Coulomb Blockade for Two Tunnel-Coupled Quantum Dots
We extend earlier results on the relation between the dimensionless tunneling
channel conductance and the fractional Coulomb blockade peak splitting
for two electrostatically equivalent dots connected by an arbitrary number
of tunneling channels with bandwidths much larger than the
two-dot differential charging energy . By calculating through second
order in in the limit of weak coupling (), we illuminate
the difference in behavior of the large- and
small- regimes and make more plausible extrapolation to the
strong-coupling () limit. For the special case of
and strong coupling, we eliminate an apparent ultraviolet
divergence and obtain the next leading term of an expansion in . We show
that the results we calculate are independent of such band structure details as
the fraction of occupied fermionic single-particle states in the weak-coupling
theory and the nature of the cut-off in the bosonized strong-coupling theory.
The results agree with calculations for metallic junctions in the
limit and improve the previous good
agreement with recent two-channel experiments.Comment: 27 pages, 1 RevTeX file with 4 embedded Postscript figures. Uses eps
Subsampling Mathematical Relaxations and Average-case Complexity
We initiate a study of when the value of mathematical relaxations such as
linear and semidefinite programs for constraint satisfaction problems (CSPs) is
approximately preserved when restricting the instance to a sub-instance induced
by a small random subsample of the variables. Let be a family of CSPs such
as 3SAT, Max-Cut, etc., and let be a relaxation for , in the sense
that for every instance , is an upper bound the maximum
fraction of satisfiable constraints of . Loosely speaking, we say that
subsampling holds for and if for every sufficiently dense instance and every , if we let be the instance obtained by
restricting to a sufficiently large constant number of variables, then
. We say that weak subsampling holds if the
above guarantee is replaced with whenever
. We show: 1. Subsampling holds for the BasicLP and BasicSDP
programs. BasicSDP is a variant of the relaxation considered by Raghavendra
(2008), who showed it gives an optimal approximation factor for every CSP under
the unique games conjecture. BasicLP is the linear programming analog of
BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional
constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of
unique games type. 3. There are non-unique CSPs for which even weak subsampling
fails for the above tighter semidefinite programs. Also there are unique CSPs
for which subsampling fails for the Sherali-Adams linear programming hierarchy.
As a corollary of our weak subsampling for strong semidefinite programs, we
obtain a polynomial-time algorithm to certify that random geometric graphs (of
the type considered by Feige and Schechtman, 2002) of max-cut value
have a cut value at most .Comment: Includes several more general results that subsume the previous
version of the paper
Goussarov-Habiro theory for string links and the Milnor-Johnson correspondence
We study the Goussarov-Habiro finite type invariants theory for framed string
links in homology balls.
Their degree 1 invariants are computed: they are given by Milnor's triple
linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and
Rochlin's invariant. These invariants are seen to be naturally related to
invariants of homology cylinders through the so-called Milnor-Johnson
correspondence: in particular, an analogue of the Birman-Craggs homomorphism
for string links is computed.
The relation with Vassiliev theory is studied.Comment: 23 pages. New exposition. One new section (relation with Vassiliev
theory). To appear in Topology & its Application
Graph product structure for non-minor-closed classes
Dujmovi\'c et al. (FOCS 2019) recently proved that every planar graph is a
subgraph of the strong product of a graph of bounded treewidth and a path.
Analogous results were obtained for graphs of bounded Euler genus or
apex-minor-free graphs. These tools have been used to solve longstanding
problems on queue layouts, non-repetitive colouring, -centered colouring,
and adjacency labelling. This paper proves analogous product structure theorems
for various non-minor-closed classes. One noteable example is -planar graphs
(those with a drawing in the plane in which each edge is involved in at most
crossings). We prove that every -planar graph is a subgraph of the
strong product of a graph of treewidth and a path. This is the first
result of this type for a non-minor-closed class of graphs. It implies, amongst
other results, that -planar graphs have non-repetitive chromatic number
upper-bounded by a function of . All these results generalise for drawings
of graphs on arbitrary surfaces. In fact, we work in a much more general
setting based on so-called shortcut systems that are of independent interest.
This leads to analogous results for map graphs, string graphs, graph powers,
and nearest neighbour graphs.Comment: v2 Cosmetic improvements and a corrected bound for
(layered-)(tree)width in Theorems 2, 9, 11, and Corollaries 1, 3, 4, 6, 12.
v3 Complete restructur
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