407,332 research outputs found
Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
The modified algebraic Bethe ansatz, introduced by Cramp\'e and the author
[8], is used to characterize the spectral problem of the Heisenberg XXZ
spin- chain on the segment with lower and upper triangular
boundaries. The eigenvalues and the eigenvectors are conjectured. They are
characterized by a set of Bethe roots with cardinality equal to the length
of the chain and which satisfies a set of Bethe equations with an additional
term. The conjecture follows from exact results for small chains. We also
present a factorized formula for the Bethe vectors of the Heisenberg XXZ
spin- chain on the segment with two upper triangular boundaries.Comment: V2: published version, some typos were corrected and one remark (5.2)
on scalar product was adde
Extremal Problems in Minkowski Space related to Minimal Networks
We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan
[FLM]: Is there an upper bound polynomial in for the largest cardinality of
a set S of unit vectors in an n-dimensional Minkowski space (or Banach space)
such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n
and that equality holds iff the space is linearly isometric to \ell^n_\infty,
the space with an n-cube as unit ball. We also remark on similar questions
raised in [FLM] that arose out of the study of singularities in
length-minimizing networks in Minkowski spaces.Comment: 6 pages. 11-year old paper. Implicit question in the last sentence
has been answered in Discrete & Computational Geometry 21 (1999) 437-44
Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser
On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the
\emph{Algebraic Eraser} scheme for key agreement over an insecure channel,
using a novel hybrid of infinite and finite noncommutative groups. They also
introduced the \emph{Colored Burau Key Agreement Protocol (CBKAP)}, a concrete
realization of this scheme.
We present general, efficient heuristic algorithms, which extract the shared
key out of the public information provided by CBKAP. These algorithms are,
according to heuristic reasoning and according to massive experiments,
successful for all sizes of the security parameters, assuming that the keys are
chosen with standard distributions.
Our methods come from probabilistic group theory (permutation group actions
and expander graphs). In particular, we provide a simple algorithm for finding
short expressions of permutations in , as products of given random
permutations. Heuristically, our algorithm gives expressions of length
, in time and space . Moreover, this is provable from
\emph{the Minimal Cycle Conjecture}, a simply stated hypothesis concerning the
uniform distribution on . Experiments show that the constants in these
estimations are small. This is the first practical algorithm for this problem
for .
Remark: \emph{Algebraic Eraser} is a trademark of SecureRF. The variant of
CBKAP actually implemented by SecureRF uses proprietary distributions, and thus
our results do not imply its vulnerability. See also arXiv:abs/12020598Comment: Final version, accepted to Advances in Applied Mathematics. Title
slightly change
Quantum and Classical Message Identification via Quantum Channels
We discuss concepts of message identification in the sense of Ahlswede and
Dueck via general quantum channels, extending investigations for classical
channels, initial work for classical-quantum (cq) channels and "quantum
fingerprinting".
We show that the identification capacity of a discrete memoryless quantum
channel for classical information can be larger than that for transmission;
this is in contrast to all previously considered models, where it turns out to
equal the common randomness capacity (equals transmission capacity in our
case): in particular, for a noiseless qubit, we show the identification
capacity to be 2, while transmission and common randomness capacity are 1.
Then we turn to a natural concept of identification of quantum messages (i.e.
a notion of "fingerprint" for quantum states). This is much closer to quantum
information transmission than its classical counterpart (for one thing, the
code length grows only exponentially, compared to double exponentially for
classical identification). Indeed, we show how the problem exhibits a nice
connection to visible quantum coding. Astonishingly, for the noiseless qubit
channel this capacity turns out to be 2: in other words, one can compress two
qubits into one and this is optimal. In general however, we conjecture quantum
identification capacity to be different from classical identification capacity.Comment: 18 pages, requires Rinton-P9x6.cls. On the occasion of Alexander
Holevo's 60th birthday. Version 2 has a few theorems knocked off: Y Steinberg
has pointed out a crucial error in my statements on simultaneous ID codes.
They are all gone and replaced by a speculative remark. The central results
of the paper are all unharmed. In v3: proof of Proposition 17 corrected,
without change of its statemen
The traveling salesman problem on cubic and subcubic graphs
We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on TeX vertices a tour of length TeX exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented an algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Mömke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs
Shape Optimization Problems for Metric Graphs
We consider the shape optimization problem where is the one-dimensional Hausdorff measure and is an
admissible class of one-dimensional sets connecting some prescribed set of
points . The cost
functional is the Dirichlet energy of defined
through the Sobolev functions on vanishing on the points . We
analyze the existence of a solution in both the families of connected sets and
of metric graphs. At the end, several explicit examples are discussed.Comment: 23 pages, 11 figures, ESAIM Control Optim. Calc. Var., (to appear
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