This study utilized structural equation modeling to examine the associations among parental guilt induction (a form of psychological control), youth cognitive style, and youth internalizing symptoms, with parents and youth participating in a randomized controlled trial of a family-based group cognitive–behavioral preventive intervention targeting families with a history of caregiver depression. The authors present separate models utilizing parent report and youth report of internalizing symptoms. Findings suggest that families in the active condition (family-based group cognitive–behavioral group) relative to the comparison condition showed a significant decline in parent use of guilt induction at the conclusion of the intervention (6 months postbaseline). Furthermore, reductions in parental guilt induction at 6 months were associated with significantly lower levels of youth negative cognitive style at 12 months. Finally, reductions in parental use of guilt induction were associated with lower youth internalizing symptoms 1 year following the conclusion of the intervention (18 months postbaseline).

A fairly well-known problem is that of counting the number of different colourings of the set of faces of a polyhedron, using n colours. Of course it depends on what is meant by ‘different’. As perhaps the simplest non-trivial example, let n = 2 (red and blue say) and consider the regular tetrahedron. If we regard the tetrahedron as static, then perhaps the answer is 24 = 16, since each of the 4 faces can be red or blue. However we are in the habit of picking up objects and rotating them around, so maybe a better (and more interesting) interpretation of ‘different’ is to regard two paintings as distinct if one cannot be rotated into the other. We are left with the following 5 distinct possibilities :

The distribution of the prime numbers is of course a well-trodden path. The casual enquirer, working with small numbers, soon finds that the distribution has no obvious regularities, and sees that the primes thin out. The Prime number theorem (which is not elementary) helps to firm up this observation:

Prime number theorem: If π(n) is the number of primes no greater than n, then i.e. π(n)/(n/ln(n)) → 1 as n → ∞. Thus for the proportion of prime numbers upto to n, π(n)/n, we have that which tends slowly to 0.

A semi-regular tessellation is one consisting of regular polygons of the same length of side, with the same ‘behaviour’ at each vertex. By this we mean that the polygons appear in the same order (though different senses are allowed) at each vertex. The three regular tessellations are for convenience included here as special cases of semi-regular ones. An example of a semi-regular tessellation is that with triangle–triangle–square–triangle–square in cyclic order, at each vertex. A shorthand notation (‘Schläfli symbol’) for this is 3.3.4.3.4.