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HISTORICAL NOTES The original ideas of Kolmogorov complexity were put forth independently and almost simultaneously by Kolmogorov [321, 322], Solomonoff [504], and Chaitin [89]. These ideas were developed further by students of Kolmogorov such as Martin-L f [374], who de ned the notion of algoritho mically random sequences and algorithmic tests for randomness, and by Levin and Zvonkin [353], who explored the ideas of universal probability and its relationship to complexity. A series of papers by Chaitin [90] [92] developed the relationship between algorithmic complexity and mathematical proofs. C. P. Schnorr studied the universal notion of randomness and related it to gambling in [466] [468]. The concept of the Kolmogorov structure function was de ned by Kolmogorov at a talk at the Tallin conference in 1973, but these results were not published. V yugin pursues this in [549], where he shows that there are some very strange sequences x n that reveal their structure arbitrarily slowly in the sense that Kk (x n |n) = n k, k < K(x n |n). Zurek [606] [608] addresses the fundamental questions of Maxwell s demon and the second law of thermodynamics by establishing the physical consequences of Kolmogorov complexity.

other. Group support not only is helpful for psychological health but has been associated with longevity among cancer patients and others (Forsyth & Corazzini, 2000; Spiegal, 1990, 1992). Recently, much attention has been given to forgiveness groups (Luskin, 2002). These groups include people who may experience both mental and physical health problems associated with chronic feelings of anger and bitterness associated with the inability to forgive themselves or others. Other forms of group psychotherapy may focus on interpersonal skill development rather than psychoeducation. These groups generally provide a forum for support and peer feedback as well as for the development of interpersonal skills. Patients can gain new insights and practice new ways of interacting with others during group sessions. Group cohesion and a sense of hope often emerges through these sessions. Patients frequently feel better knowing that they are not alone and that others, like them, struggle with similar issues and concerns. Advice, suggestions, and general feedback from peer group members can often be more powerful than those provided by the group leader(s). Table 9.1 provides a list of curative factors associated with group psychotherapy.

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summary of a distribution of data; these numbers are the median, the rst and second hinges, and either the lowest and highest number or a similar measure of range number arrived at by separating very extreme values The median is equal to the 50th percentile of the distribution The hinges are either equal to or very close to the 25th and 75th percentiles although they are found using a simpler rank-based formula for computation To construct a box-plot, a scale is drawn, and a box is placed on the scale with one end of the box indicating the scale value of the lower hinge (25th percentile) and the other end of the box occurring at the scale position of the upper hinge (75th percentile) An additional line is drawn in the middle to indicate the scale value of the median.

It is also clear that Gj 1 and Fj are not treated symmetrically. We see that, as we get closer to having some elements in Gj 1 with membership grade one, the process of adding information slows. The form of the HEU essentially implements a prioritization of the rules. The rules at the highest level of the hierarchy are explored rst; if they nd a good solution, we look no further at the rules. Figure 1.2 provides an alternative view of the HPS structure.

The scale-value difference between the two hinges is called either the hinge spread or the interquartile range (often abbreviated IQR), and in a normal distribution corresponds to approximately 067 standard deviations on each side of the mean Rules for the construction of the whisker portion of the display vary In the most common version, lines are extended along the scale from the hinges to the farthest data value in each direction up to 15 hinge spreads If there are data past that point, the whiskers extend to the farthest data point prior to the 15 hinge-spread cutoff Data points beyond the whiskers are usually identi ed individually to bring attention to their extremeness and potential characterization as outliers An example of multiple box-plots is presented in Figure 25, panel A, with individual raw data values presented in panel B for comparison.

(Costa & McCrae, 1985, 1989, 1992) is a 240-item questionnaire that uses a 5-point rating system. A brief 60-item version of

These graphics depict the distributions of effect sizes from the meta-analysis of social memory conducted by Stangor and McMillan (1992) The categorical variable on the horizontal axis is the length of stimulus presentation in seconds The continuous variable on the vertical axis is the size of the effect for each study included in the meta-analysis As the reader may see, the box-plots provide a compact description of each distribution and allow relatively easy comparison of both the level and spread of each distribution The distribution farthest to the left represents all the studies for which no presentation speed is reported The range is approximately from 2 to +2, with a median slightly below zero The second box-plot depicts the distribution of effect sizes from studies that used a 2-s presentation speed It is the highest distribution of all, with some positive skew.

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