423 research outputs found
The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy
We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik
hierarchy with complex-valued initial data and prove unique solvability
globally in time for a set of initial (Dirichlet divisor) data of full measure.
To this effect we develop a new algorithm for constructing stationary
complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy,
which is of independent interest as it solves the inverse algebro-geometric
spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators,
starting from a suitably chosen set of initial divisors of full measure.
Combined with an appropriate first-order system of differential equations with
respect to time (a substitute for the well-known Dubrovin-type equations), this
yields the construction of global algebro-geometric solutions of the
time-dependent Ablowitz-Ladik hierarchy.
The treatment of general (non-unitary) Lax operators associated with general
coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties
that, to the best of our knowledge, are successfully overcome here for the
first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but
applies generally to (1+1)-dimensional completely integrable soliton equations
of differential-difference type.Comment: 47 page
Unusual Non-Fermi Liquid Behavior of CeLaNiGe Analyzed in a Single Impurity Anderson Model with Crystal Field Effects
CeNiGe exhibits unusual non-Fermi liquid behavior with the largest
ever recorded value of the electronic specific heat
JKmol without showing any evidence of magnetic order. Specific
heat measurements show that the logarithmic increase of the Sommerfeld
coefficient flattens off below 200 mK. In marked contrast, the local
susceptibility levels off well above 200 mK and already becomes
constant below 1 K. Furthermore, the entropy reaches 2ln2 below 20 K
corresponding to a four level system. An analysis of and was
performed in terms of an single impurity Anderson model with
additional crystal electric field (CEF) splitting. Numerical renormalization
group calculations point to a possible consistent description of the different
low temperature scales in and stemming from the
interplay of Kondo effect and crystal field splitting.Comment: 2 pages, 2 figure
Un-reduction
This paper provides a full geometric development of a new technique called
un-reduction, for dealing with dynamics and optimal control problems posed on
spaces that are unwieldy for numerical implementation. The technique, which was
originally concieved for an application to image dynamics, uses Lagrangian
reduction by symmetry in reverse. A deeper understanding of un-reduction leads
to new developments in image matching which serve to illustrate the
mathematical power of the technique.Comment: 25 pages, revised versio
Mathematical Modeling Links Pregnancy-Associated Changes and Breast Cancer Risk
Recent debate has concentrated on the contribution of bad luck to cancer development. The tight correlation between the number of tissue-specific stem cell divisions and cancer risk of the same tissue suggests that bad luck has an important role to play in tumor development, but the full extent of this contribution remains an open question. Improved understanding of the interplay between extrinsic and intrinsic factors at the molecular level is one promising route to identifying the limits on extrinsic control of tumor initiation, which is highly relevant to cancer prevention. Here, we use a simple mathematical model to show that recent data on the variation in numbers of breast epithelial cells with progenitor features due to pregnancy are sufficient to explain the known protective effect of full-term pregnancy in early adulthood for estrogen receptor-positive (ER+) breast cancer later in life. Our work provides a mechanism for this previously ill-understood effect and illuminates the complex influence of extrinsic factors at the molecular level in breast cancer. These findings represent an important contribution to the ongoing research into the role of bad luck in human tumorigenesis
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
We study Sobolev-type metrics of fractional order on the group
\Diff_c(M) of compactly supported diffeomorphisms of a manifold . We show
that for the important special case the geodesic distance on
\Diff_c(S^1) vanishes if and only if . For other manifolds we
obtain a partial characterization: the geodesic distance on \Diff_c(M)
vanishes for and for ,
with being a compact Riemannian manifold. On the other hand the geodesic
distance on \Diff_c(M) is positive for and
.
For we discuss the geodesic equations for these metrics. For
we obtain some well known PDEs of hydrodynamics: Burgers' equation for ,
the modified Constantin-Lax-Majda equation for and the
Camassa-Holm equation for .Comment: 16 pages. Final versio
Poisson bracket in classical field theory as a derived bracket
We construct a Leibniz bracket on the space of
all differential forms over the finite-dimensional jet bundle . As
an example, we write Maxwell equations with sources in the covariant
finite-dimensional hamiltonian form.Comment: 4 page
Eradication of chronic myeloid leukemia stem cells: a novel mathematical model predicts no therapeutic benefit of adding G-CSF to imatinib
Imatinib mesylate induces complete cytogenetic responses in patients with chronic myeloid leukemia (CML), yet many patients have detectable BCR-ABL transcripts in peripheral blood even after prolonged therapy. Bone marrow studies have shown that this residual disease resides within the stem cell compartment. Quiescence of leukemic stem cells has been suggested as a mechanism conferring insensitivity to imatinib, and exposure to the Granulocyte-Colony Stimulating Factor (G-CSF), together with imatinib, has led to a significant reduction in leukemic stem cells in vitro. In this paper, we design a novel mathematical model of stem cell quiescence to investigate the treatment response to imatinib and G-CSF. We find that the addition of G-CSF to an imatinib treatment protocol leads to observable effects only if the majority of leukemic stem cells are quiescent; otherwise it does not modulate the leukemic cell burden. The latter scenario is in agreement with clinical findings in a pilot study administering imatinib continuously or intermittently, with or without G-CSF (GIMI trial). Furthermore, our model predicts that the addition of G-CSF leads to a higher risk of resistance since it increases the production of cycling leukemic stem cells. Although the pilot study did not include enough patients to draw any conclusion with statistical significance, there were more cases of progression in the experimental arms as compared to continuous imatinib. Our results suggest that the additional use of G-CSF may be detrimental to patients in the clinic
On the monotonicity of scalar curvature in classical and quantum information geometry
We study the statistical monotonicity of the scalar curvature for the
alpha-geometries on the simplex of probability vectors. From the results
obtained and from numerical data we are led to some conjectures about quantum
alpha-geometries and Wigner-Yanase-Dyson information. Finally we show that this
last conjecture implies the truth of the Petz conjecture about the monotonicity
of the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.Comment: 20 pages, 2 .eps figures; (v2) section 2 rewritten, typos correcte
Evolution of Quantum Criticality in CeNi_{9-x}Cu_xGe_4
Crystal structure, specific heat, thermal expansion, magnetic susceptibility
and electrical resistivity studies of the heavy fermion system
CeNi_{9-x}Cu_xGe_4 (0 <= x <= 1) reveal a continuous tuning of the ground state
by Ni/Cu substitution from an effectively fourfold degenerate non-magnetic
Kondo ground state of CeNi_9Ge_4 (with pronounced non-Fermi-liquid features)
towards a magnetically ordered, effectively twofold degenerate ground state in
CeNi_8CuGe_4 with T_N = 175 +- 5 mK. Quantum critical behavior, C/T ~ \chi ~
-ln(T), is observed for x about 0.4. Hitherto, CeNi_{9-x}Cu_xGe_4 represents
the first system where a substitution-driven quantum phase transition is
connected not only with changes of the relative strength of Kondo effect and
RKKY interaction, but also with a reduction of the effective crystal field
ground state degeneracy.Comment: 15 pages, 9 figure
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