Brownian Motion in a Weyl Chamber, Non-Colliding Particles, and Random Matrices

Let $n$ particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is $A_{n-1}$, the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time $t$ without having collided by time $t$. We show that the probability that there will be no collision up to time $t$ is asymptotic to a constant multiple of $t^{-n(n-1)/4}$ as $t$ goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group $B_n$ gives a model of $n$ independent particles with a wall at $x=0$. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point in the Weyl chamber, giving a Brownian motion conditioned never to exit the chamber. If there are $m$ roots in $n$ dimensions, this shows that the radial part of the conditioned process is the same as the $n+2m$-dimensional Bessel process. The conditioned process also gives physical models, generalizing Dyson's model for $A_{n-1}$ corresponding to ${\mathfrak s}{\mathfrak u}_n$ of $n$ particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them

Random Walk in an Alcove of an Affine Weyl Group, and Non-Colliding Random Walks on an Interval

We use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walks in the alcoves of the classical affine Weyl groups. In all cases, we get determinant formulas for the number of k-step walks. One important example is the region m>x_1>x_2>...>x_n>0, which is a rescaled alcove of the affine Weyl group C_n. If each coordinate is considered to be an independent particle, this models n non-colliding random walks on the interval (0,m). Another case models n non-colliding random walks on the circle.Comment: v.2, 22 pages; correction in a definition led to changes in many formulas, also added more background, references, and example

Weak* Properties of Weighted Convolution Algebras

Suppose that L1(ω) is a weighted convolution algebra on R+ = [0,∞) with the weight ω(t) normalized so that the corresponding space M(ω) of measures is the dual space of the space C0(1/ω) of continuous functions. Suppose that φ : L1(ω) → L1(ω0 ) is a continuous nonzero homomorphism, where L1(ω0 ) is also a convolution algebra. If L1(ω)∗f is norm dense in L1(ω), we show that L1(ω0 ) ∗ φ(f) is (relatively) weak∗ dense in L1(ω0 ), and we identify the norm closure of L1(ω0 ) ∗ φ(f) with the convergence set for a particular semigroup. When φ is weak∗ continuous it is enough for L1(ω) ∗ f to be weak∗ dense in L1(ω). We also give sufficient conditions and characterizations of weak∗ continuity of φ. In addition, we show that, for all nonzero f in L1(ω), the sequence fn/||fn|| converges weak∗ to 0. When ω is regulated, fn+1/||fn|| converges to 0 in norm

Attention demanding tasks during treadmill walking reduce step width variability in young adults

BACKGROUND: The variability of step time and step width is associated with falls by older adults. Further, step time is significantly influenced when performing attention demanding tasks while walking. Without exception, step time variability has been reported to increase in normal and pathologically aging older adults. Because of the role of step width in managing frontal plane dynamic stability, documenting the influence of attention-demanding tasks on step width variability may provide insight to events that can disturb dynamic stability during locomotion and increase fall risk. Preliminary evidence suggests performance of an attention demanding task significantly decreases step width variability of young adults walking on a treadmill. The purpose of the present study was to confirm or refute this finding by characterizing the extent and direction of the effects of a widely used attention demanding task (Stroop test) on the step width variability of young adults walking on a motorized treadmill. METHODS: Fifteen healthy young adults walked on a motorized treadmill at a self-selected velocity for 10 minutes under two conditions; without performing an attention demanding task and while performing the Stroop test. Step width of continuous and consecutive steps during the collection was derived from the data recorded using a motion capture system. Step width variability was computed as the standard deviation of all recorded steps. RESULTS: Step width decreased four percent during performance of the Stroop test but the effect was not significant (p = 0.10). In contrast, the 16 percent decrease in step width variability during the Stroop test condition was significant (p = 0.029). CONCLUSION: The results support those of our previous work in which a different attention demanding task also decreased step width variability of young subjects while walking on a treadmill. The decreased step width variability observed while performing an attention demanding task during treadmill walking may reflect a voluntary gait adaptation toward a more conservative gait pattern emphasizing frontal plane control of the trunk. Extension of the experimental paradigm to older adults and mechanistic approaches to link step width variability to dynamic stability, and falls, in a cause-effect manner are necessary

East vs. West: South Korean Student Acculturation

This paper is an ethnography that explores the acculturation of South Korean exchange students at Southern Adventist university through a triangulation of information from theory, interviews, and field observation. The paper relies on Young Yun Kim’s theory of acculturation, which closely looks at the push and pull of one’s home and host culture in the process of acculturation. In my ethnography, I observed approximately 60-100 South Korean students, ages 18-23, in different social settings such as church, social dining, and study. Full participant, participant observer, and full observer were the three methods of observation used in the various social settings. I found that the South Korean exchange students did indeed acculturate to local culture; however, those who stayed within their South Korean social group acculturated at slower pace than those who left the group

Pericope Adulterae: A Most Perplexing Passage

The account of the woman caught in adultery, traditionally found in John’s Gospel, is full of encouragement to sinners in need of forgiveness. Nevertheless, due to its textual history, this story—referred to as the Pericope Adulterae—is considered by many scholars to be an interpolation. The textual history is one of the most intriguing of any biblical passage. This article reviews that history, examines possible reasons for the passage’s inclusion or exclusion from John’s Gospel, engages discussion on the issue of its canonicity, and gives suggestions for how today’s pastors might relate to the story in their preaching