58 research outputs found
On the Two q-Analogue Logarithmic Functions
There is a simple, multi-sheet Riemann surface associated with e_q(z)'s
inverse function ln_q(w) for 0< q < 1. A principal sheet for ln_q(w) can be
defined. However, the topology of the Riemann surface for ln_q(w) changes each
time "q" increases above the collision point of a pair of the turning points of
e_q(x). There is also a power series representation for ln_q(1+w). An
infinite-product representation for e_q(z) is used to obtain the ordinary
natural logarithm ln{e_q(z)} and the values of sum rules for the zeros "z_i" of
e_q(z). For |z|<|z_1|, e_q(z)=exp{b(z)} where b(z) is a simple, explicit power
series in terms of values of these sum rules. The values of the sum rules for
the q-trigonometric functions, sin_q(z) and cos_q(z), are q-deformations of the
usual Bernoulli numbers.Comment: This is the final version to appear in J.Phys.A: Math. & General.
Some explict formulas added, and to update the reference
Hopf algebras in dynamical systems theory
The theory of exact and of approximate solutions for non-autonomous linear
differential equations forms a wide field with strong ties to physics and
applied problems. This paper is meant as a stepping stone for an exploration of
this long-established theme, through the tinted glasses of a (Hopf and
Rota-Baxter) algebraic point of view. By reviewing, reformulating and
strengthening known results, we give evidence for the claim that the use of
Hopf algebra allows for a refined analysis of differential equations. We
revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern
approach involving Lie idempotents. Approximate solutions to differential
equations involve, on the one hand, series of iterated integrals solving the
corresponding integral equations; on the other hand, exponential solutions.
Equating those solutions yields identities among products of iterated Riemann
integrals. Now, the Riemann integral satisfies the integration-by-parts rule
with the Leibniz rule for derivations as its partner; and skewderivations
generalize derivations. Thus we seek an algebraic theory of integration, with
the Rota-Baxter relation replacing the classical rule. The methods to deal with
noncommutativity are especially highlighted. We find new identities, allowing
for an extensive embedding of Dyson-Chen series of time- or path-ordered
products (of generalized integration operators); of the corresponding Magnus
expansion; and of their relations, into the unified algebraic setting of
Rota-Baxter maps and their inverse skewderivations. This picture clarifies the
approximate solutions to generalized integral equations corresponding to
non-autonomous linear (skew)differential equations.Comment: International Journal of Geometric Methods in Modern Physics, in
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Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Many non-equivalent realizations of the associahedron
Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two
different ways of constructing exponential families of realizations of the
n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the
constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify
the associahedra obtained by these constructions modulo linear equivalence of
their normal fans and show, in particular, that the only realization that can
be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002)
associahedron.
For the Hohlweg-Lange associahedra our classification is a priori coarser
than the classification up to isometry of normal fans, by
Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As
a consequence, we get that two Hohlweg-Lange associahedra have linearly
equivalent normal fans if and only if they are isometric.
The Santos construction, which produces an even larger family of
associahedra, appears here in print for the first time. Apart of describing it
in detail we relate it with the c-cluster complexes and the denominator fans in
cluster algebras of type A.
A third classical construction of the associahedron, as the secondary
polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to
never produce a normal fan linearly equivalent to any of the other two
constructions.Comment: 30 pages, 13 figure
Phase II evaluation of MGBG in non-small cell carcinoma of the lung
One hundred and eight patients with non-small cell lung cancer were treated in a Phase II trial with MGBG at a dose of 600 mg/m 2 i.v. weekly. Partial responses were noted in 3/43 patients with adenocarcinoma and 1/40 with squamous cell carcinoma. No responses were noted in 24 patients with large cell carcinoma. Overall, the drug was reasonably well-tolerated. At this dosage and schedule, MGBG has no substantial antitumor activity for patients with non-small cell lung cancer.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45128/1/10637_2004_Article_BF00180196.pd
Mobilise-D insights to estimate real-world walking speed in multiple conditions with a wearable device
This study aimed to validate a wearable device’s walking speed estimation pipeline, considering complexity, speed, and walking bout duration. The goal was to provide recommendations on the use of wearable devices for real-world mobility analysis. Participants with Parkinson’s Disease, Multiple Sclerosis, Proximal Femoral Fracture, Chronic Obstructive Pulmonary Disease, Congestive Heart Failure, and healthy older adults (n = 97) were monitored in the laboratory and the real-world (2.5 h), using a lower back wearable device. Two walking speed estimation pipelines were validated across 4408/1298 (2.5 h/laboratory) detected walking bouts, compared to 4620/1365 bouts detected by a multi-sensor reference system. In the laboratory, the mean absolute error (MAE) and mean relative error (MRE) for walking speed estimation ranged from 0.06 to 0.12 m/s and − 2.1 to 14.4%, with ICCs (Intraclass correlation coefficients) between good (0.79) and excellent (0.91). Real-world MAE ranged from 0.09 to 0.13, MARE from 1.3 to 22.7%, with ICCs indicating moderate (0.57) to good (0.88) agreement. Lower errors were observed for cohorts without major gait impairments, less complex tasks, and longer walking bouts. The analytical pipelines demonstrated moderate to good accuracy in estimating walking speed. Accuracy depended on confounding factors, emphasizing the need for robust technical validation before clinical application.
Trial registration: ISRCTN – 12246987
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