Geometric and Measure-Theoretic Shrinking Targets in Dynamical Systems

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described

Geometric and Measure-Theoretic Shrinking Targets in Dynamical Systems

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described

Uniform distributions on curves and quantization

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of n -means and the n th quantization errors for different values of n with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise \cite{BW}, which says that for a Borel probability measure P with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number

Optimal Quantization Via Dynamics

Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes. We are interested in how close stationary process can be to giving optimal n-means and nth optimal mean distortion errors. We also consider different ways of measuring the degree of approximation by quantization, and their advantages and disadvantages in these different contexts

Differentiating Orlicz spaces with rare bases of rectangles

In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space

Individual Differences in Holistic Processing Predict the Own-Race Advantage in Recognition Memory

Individuals are consistently better at recognizing own-race faces compared to other-race faces (other-race effect, ORE). One popular hypothesis is that this recognition memory ORE is caused by differential own- and other-race holistic processing, the simultaneous integration of part and configural face information into a coherent whole. Holistic processing may create a more rich, detailed memory representation of own-race faces compared to other-race faces. Despite several studies showing that own-race faces are processed more holistically than other-race faces, studies have yet to link the holistic processing ORE and the recognition memory ORE. In the current study, we sought to use a more valid method of analyzing individual differences in holistic processing by using regression to statistically remove the influence of the control condition (part trials in the part-whole task) from the condition of interest (whole trials in the part-whole task). We also employed regression to separately examine the two components of the ORE: own-race advantage (regressing other-race from own-race performance) and other-race decrement (regressing own-race from other-race performance). First, we demonstrated that own-race faces were processed more holistically than other-race faces, particularly the eye region. Notably, using regression, we showed a significant association between the own-race advantage in recognition memory and the own-race advantage in holistic processing and that these associations were weaker when examining the other-race decrement. We also demonstrated that performance on own- and other-race faces across all of our tasks was highly correlated, suggesting that the differences we found between own- and other-race faces are quantitative rather than qualitative. Together, this suggests that own- and other-race faces recruit largely similar mechanisms, that own-race faces more thoroughly engage holistic processing, and that this greater engagement of holistic processing is significantly associated with the own-race advantage in recognition memory.Psycholog