On the Collision between two PNG Droplets

In this article we study the interface generated by the collision between two crystals growing layer by layer on a one-dimensional substrate through random decomposition of particles. We relate this interface with the notion of β-path in an equivalent directed polymer model and, by using asymptotics results from J. Baik and E. Rains, J. Stat. Phys., 100:523–541 (2000). and some hydrodynamic tools introduced by E. Cator and P. Groeneboom, Ann. Probab., 33:879–903 (2005), we derive a law of large numbers for such a path and obtain some bounds for its fluctuations. 2000 Mathematics Subject Classification: 60C05, 60K35     

The Voice Handicap Index: Correlation Between Subjective Patient Response and Quantitative Assessment of Voice

The aim of this prospective study is to elucidate the relationship between the Voice Handicap Index (VHI) and several voice laboratory measurements in the network of the multidimensional voice assessment. Fifty-eight patients were included. Each patient replies to the questionnaire and performs a voice assessment during the same time. The following parameters were measured: minimum frequency, maximum frequency, range, minimum intensity, subglottic pressure, mean flow, maximum phonation time, jitter, and dysphonia severity index. Regarding the relationship with the scores of the VHI, poor correlations with the minimal frequency for all the scores except the emotional one (total and subscales) and with the range for only the physical one are found. Seventeen questions correlate with the voice laboratory measurements we performed, with a decreased distribution between physical, functional, and emotional subscales. We observe that acoustic parameter is correlated with the emotional subscale, the parameters of the profile range are more often involved in the emotional subscale, as is the minimal frequency, but never with the physical subscale, and all the subscales are interesting despite the smaller number of differences with the emotional one. The VHI and the laboratory measurements give independent informations in practice.     

Sister Beiter and Kloosterman: A tale of cyclotomic coefficients and modular inverses

For a fixed prime p$p$, the maximum coefficient (in absolute value) M(p)$M\left(p\right)$ of the cyclotomic polynomial _Φpqr(x)${\Phi }_{pqr}\left(x\right)$, where r$r$ and q$q$ are free primes satisfying r>q>p$r>q>p$ exists. Sister Beiter conjectured in 1968 that M(p)≤(p+1)/2$M\left(p\right)\le (p+1)/2$. In 2009 Gallot and Moree showed that M(p)≥2p(1−?)/3$M\left(p\right)\ge 2p(1-?)/3$ for every p$p$ sufficiently large. In this article Kloosterman sums (‘cloister man sums’) and other tools from the distribution of modular inverses are applied to quantify the abundancy of counter-examples to Sister Beiter’s conjecture and sharpen the above lower bound for M(p)$M\left(p\right)$.     

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities

We show that if ${{\mathcal A} \subset \mathbb{R}^N}$ is an annulus or a ball centered at zero, the homogeneous Neumann problem on ${{\mathcal A}}$ for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and ${\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}$ The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.     

Long-time behavior of an angiogenesis model with flux at the tumor boundary

This paper deals with a nonlinear system of partial differential equations modeling a simplified tumor-induced angiogenesis taking into account only the interplay between tumor angiogenic factors and endothelial cells. Considered model assumes a nonlinear flux at the tumor boundary and a nonlinear chemotactic response. It is proved that the choice of some key parameters influences the long-time behavior of the system. More precisely, we show the convergence of solutions to different semi-trivial stationary states for different range of parameters.     

Problem modification as a tool for detecting cognitive flexibility in school children

This paper presents the results of an experiment in which fourth to sixth graders with above-average mathematical abilities modified a given problem. The experiment found evidence of links between problem posing and cognitive flexibility. Emerging from organizational theory, cognitive flexibility is conceptualized through three primary constructs: cognitive variety, cognitive novelty, and changes in cognitive framing. Among these components, changes in cognitive framing could be effectively detected in problem-posing situations, giving a relevant indication of students’ creative potential. The students’ capacity to generate coherent and consistent problems in the context of problem modification may indicate the existence of a strategy of functional type for generalizations, which seems to be specific to mathematical creativity.     

From Heisenberg to Gödel via Chaitin

In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle” which implies Chaitin’s information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics.     

Transverse oriented electric field re-entrant resonator (TERR) with automatic tuning and coupling control for EPR spectroscopy and imaging of the beating heart

Sample motion, particularly that of a beating heart, induces baseline noise and spectral distortion on an EPR spectrum. In order to quench motional noise and restore the EPR signal amplitude and line-width, an L-band transverse oriented electric field re-entrant resonator (TERR) was designed and constructed with provisions for automatic tuning control (ATC) and automatic coupling control (ACC) suited for studies of isolated beating rat hearts. Two sets of electronic circuits providing DC biased voltage to two varactor diodes were implemented to electronically adjust coupling and tuning. The resonator has a rectangular cross-sectional sample arm of 25 mm diameter with a Q value of 1100 without sample. Once inserted with lossy aqueous samples of 0.45% NaCl, Q value drops to 400 with a volume of 0.5 ml and 150 with 5 ml. The ATC/ACC functions were tested with a moving phantom and isolated beating rat hearts with the improvement of signal to noise ratio (S/N, peak amplitude of signal over peak amplitude of baseline noise) of 6.7-, and 4 to 6-fold, respectively. With these improvements, EPR imaging could be performed on an isolated beating rat heart. Thus, this TERR resonator with ATC/ACC enables application of EPR spectroscopy and imaging for the measurement and imaging of radical metabolism, redox state, and oxygenation in the isolated beating rat heart.