820,457 research outputs found
Decomposition of multiple packings with subquadratic union complexity
Suppose is a positive integer and is a -fold packing of
the plane by infinitely many arc-connected compact sets, which means that every
point of the plane belongs to at most sets. Suppose there is a function
with the property that any members of determine
at most holes, which means that the complement of their union has at
most bounded connected components. We use tools from extremal graph
theory and the topological Helly theorem to prove that can be
decomposed into at most (-fold) packings, where is a constant
depending only on and .Comment: Small generalization of the main result, improvements in the proofs,
minor correction
On -Gons and -Holes in Point Sets
We consider a variation of the classical Erd\H{o}s-Szekeres problems on the
existence and number of convex -gons and -holes (empty -gons) in a set
of points in the plane. Allowing the -gons to be non-convex, we show
bounds and structural results on maximizing and minimizing their numbers. Most
noteworthy, for any and sufficiently large , we give a quadratic lower
bound for the number of -holes, and show that this number is maximized by
sets in convex position
Walks on the slit plane: other approaches
Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is
a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i
belongs to S for all i, and none of the points w_i, i>0, lie on the half-line
H= {(k,0): k =< 0}.
In a recent paper, G. Schaeffer and the author computed the length generating
function S(t) of walks on the slit plane for several sets S. All the generating
functions thus obtained turned out to be algebraic: for instance, on the
ordinary square lattice,
S(t) =\frac{(1+\sqrt{1+4t})^{1/2}(1+\sqrt{1-4t})^{1/2}}{2(1-4t)^{3/4}}.
The combinatorial reasons for this algebraicity remain obscure.
In this paper, we present two new approaches for solving slit plane models.
One of them simplifies and extends the functional equation approach of the
original paper. The other one is inspired by an argument of Lawler; it is more
combinatorial, and explains the algebraicity of the product of three series
related to the model. It can also be seen as an extension of the classical
cycle lemma. Both methods work for any set of steps S.
We exhibit a large family of sets S for which the generating function of
walks on the slit plane is algebraic, and another family for which it is
neither algebraic, nor even D-finite. These examples give a hint at where the
border between algebraicity and transcendence lies, and calls for a complete
classification of the sets S.Comment: 31 page
Resonant inelastic x-ray scattering probes the electron-phonon coupling in the spin-liquid kappa-(BEDT-TTF)2Cu2(CN)3
Resonant inelastic x-ray scattering at the N K edge reveals clearly resolved
harmonics of the anion plane vibrations in the kappa-(BEDT-TTF)2Cu2(CN)3
spin-liquid insulator. Tuning the incoming light energy at the K edge of two
distinct N sites permits to excite different sets of phonon modes. Cyanide CN
stretching mode is selected at the edge of the ordered N sites which are more
strongly connected to the BEDT-TTF molecules, while positionally disordered N
sites show multi-mode excitation. Combining measurements with calculations on
an anion plane cluster permits to estimate the sitedependent electron-phonon
coupling of the modes related to nitrogen excitation
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