Given a set of positive integers Q=q1, q2, we develop a general method to find an upper bound for the function s(N), where N s(N) is the size of the largest subset A of 1,N such that A - A does not contain any elements of Q. We combine ergodic-theoretical ideas with the standard Hardy-Littlewood approach, and in particular prove that s(M)=O(1/log log log M) when qi=P(i), where P is a polynomial with integer coefficients, P(0)=0.
Authors:Ágnes Tóth, Barbara Wisse and Klára Faragó
Most people want to be happy in their lives and actively try to achieve some degree of contentment. Previous studies have shown that pursuing goals can increase peoples’ well-being and that in order to understand the role of goals in well-being, it is important to differentiate between the importance and the attainment of both extrinsic and intrinsic goals. Yet, the issue of how the congruence between goal importance on the one hand and goal attainment on the other affects well-being has rarely been addressed.
We investigated if well-being is a function of goal pursuit, or more precisely, if the extent to which people are satisfied with their lives is a result of their success in achieving goals that are relatively important to them. We expected that goal attainment would be a stronger predictor of well-being than goal importance. We also expected that the congruence between intrinsic goal attainment and importance would be positively related to subjective well-being. In addition, we explored whether the congruence between extrinsic goal attainment and importance would be negatively or positively associated with subjective well-being.
A survey of 149 Hungarian adults was conducted (75% female). To test our hypotheses we used bivariate polynomial regression and response surface analysis. This tool is ideal to measure the joint effect of two predictor variables on a third variable, such as the goal importance and goal attainment on well-being.
Intrinsic goal attainment is positively related to well-being (B = .77, p = .04), while goal importance has no such effect. We also found that the congruence between intrinsic goal importance and goal attainment is positively related to well-being (a1 = 1.29, p = .04). The polynomial regression with well-being as the dependent variable and extrinsic goal attainment and importance as the predictor variables showed that whereas extrinsic goal importance (B = –.32, p = .02) has a negative relationship with well-being, goal attainment (B = .51, p = .007) has a positive one. Moreover, we found that well-being is higher when extrinsic goal attainment is higher than extrinsic goal importance (a3 = –.84, p = .005) and that well-being increases more sharply to the extent that the degree of discrepancy increases (a4 = –.41, p = .03).
Based on our results it seems that the congruence between intrinsic goal attainment and goal importance enhances our well-being. While valuing extrinsic goals does not seem to increase happiness, attaining those goals does so.
x 3 + α 4 x 4 + … + α n x n Apolynomial function written in form of Eq. (10) can be used to express the deflection function of a rectangular plate [ 22 ]. Using this approach involves applying the non-dimensional coordinate system given in
It is a classical unsolved problem whether there is a polynomial with integral coef- ficients whose values at natural numbers form a Sidon set. In this note we prove the existence of a polynomial of degree 5, with real coeficients, such that the integer parts of the values form a Sidon set.