Almost Eigenvalues and Eigenvectors of Almost Mathieu Operators

The almost Mathieu operator is the discrete Schr\"odinger operator $H_{\alpha,\beta,\theta}$ on $\ell^2(\mathbb{Z})$ defined via $(H_{\alpha,\beta,\theta}f)(k) = f(k + 1) + f(k - 1) + \beta \cos(2\pi \alpha k + \theta) f(k)$. We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator. We furthermore show that the (properly rescaled) $m$-th Hermite function $\phi_m$ is an approximate eigenvector of this operator, and that it satisfies the same properties that characterize the true eigenvector associated to the $m$-th largest eigenvalue. Moreover, a properly translated and modulated version of $\phi_m$ is also an approximate eigenvector of this operator, and it satisfies the properties that characterize the true eigenvector associated to the $m$-th largest (in modulus) negative eigenvalue. The results hold at the edge of the spectrum, for any choice of $\theta$ and under very mild conditions on $\alpha$ and $\beta$. We also give precise estimates for the size of the "edge", and extend some of our results to the infinite dimensional case. The ingredients for our proofs comprise Taylor expansions, basic time-frequency analysis, Sturm sequences, and perturbation theory for eigenvalues and eigenvectors. Numerical simulations demonstrate the tight fit of the theoretical estimates

Dynamic relational contracts under complete information

This paper considers a long-term relationship between two agents who both undertake a costly action or investment that together produces a joint benefit. Agents have an opportunity to expropriate some of the joint benefit for their own use. Two cases are considered: (i) where agents are risk neutral and are subject to limited liability constraints and (ii) where agents are risk averse, have quasi-linear preferences in consumption and actions but where limited liability constraints do not bind. The question asked is how to structure the investments and division of the surplus over time so as to avoid expropriation. In the risk-neutral case, there may be an initial phase in which one agent overinvests and the other underinvests. However, both actions and surplus converge monotonically to a stationary state in which there is no overinvestment and surplus is at its maximum subject to the constraints. In the risk-averse case, there is no overinvestment. For this case, we establish that dynamics may or may not be monotonic depending on whether or not it is possible to sustain a first-best allocation. If the first-best allocation is not sustainable, then there is a trade-off between risk sharing and surplus maximization. In general, surplus will not be at its constrained maximum even in the long run

The Moderating Effects of Power Distance and Individualism/Collectivism on Empowering Leadership, Psychological Empowerment, and Self-Leadership in International Development Organizations

The importance of finding appropriate leadership styles to use in cross-cultural situations is paramount. Development organizations and multinational organizations both struggle to find forms of leadership that are effective in mobilizing the workforce in highly diverse cultural contexts. In this article, the effects of empowering leadership on psychological empowerment and self-leadership are measured in two cultural contexts—Rwanda and the United States, representing both high and low power distance and individualism/ collectivism—to explore how empowering leadership behaviors affects the empowerment of subordinates. First, hierarchical regression analysis shows that empowering leadership has a significant positive effect on both psychological empowerment and self-leadership in both cultural contexts. Second, hierarchical regression analysis with tests for moderation shows that power distance moderates these relationships, especially in high power distance cultures, while individualism/collectivism moderates these relationships only occasionally. This article provides evidence that empowering leadership is an effective form of leadership that produces employee empowerment in diverse cultural contexts. It also provides new insights into appropriate forms of leadership for international development organizations when working in different countries

Some properties of the spinor fourier transform

In this paper, the theory of the spinor Fourier transform introduced in [Batard T, Berthier M, Saint-Jean C, Clifford-Fourier Transform for Color Image Processing, Geometric Algebra Computing for Engineering and Computer Science (E. Bayro-Corrochano and G. Scheuermann Eds.), Springer, London, 2010, pp. 135–161] is further developed. While in the original paper, the transform was determined for vector-valued functions only, it now will be extended to functions taking values in the entire Clifford algebra. Next, two bases are determined under which this Fourier transform is diagonalizable. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. This problem will be tackled in the final section of this paper

The Moderating Effects of Power Distance and Individualism/Collectivism on Empowering Leadership, Psychological Empowerment, and Self-Leadership in International Development Organizations

The importance of finding appropriate leadership styles to use in cross-cultural situations is paramount. Development organizations and multinational organizations both struggle to find forms of leadership that are effective in mobilizing the workforce in highly diverse cultural contexts. In this article, the effects of empowering leadership on psychological empowerment and self-leadership are measured in two cultural contexts—Rwanda and the United States, representing both high and low power distance and individualism/ collectivism—to explore how empowering leadership behaviors affects the empowerment of subordinates. First, hierarchical regression analysis shows that empowering leadership has a significant positive effect on both psychological empowerment and self-leadership in both cultural contexts. Second, hierarchical regression analysis with tests for moderation shows that power distance moderates these relationships, especially in high power distance cultures, while individualism/collectivism moderates these relationships only occasionally. This article provides evidence that empowering leadership is an effective form of leadership that produces employee empowerment in diverse cultural contexts. It also provides new insights into appropriate forms of leadership for international development organizations when working in different countries

Foreign direct investment and the risk of expropriation

Foreign direct investment accounts for a considerable proportion of international capital flows. In 1986 the flow of foreign direct investment from developed market economies to developing countries was $12.5 billion or roughly one-half of all private capital flows from the developed to the developing nations (and roughly one-quarter of the flow of all foreign direct investments). Its significance for developing countries may even grow in the future as debt is swapped for equity (see Pollio and Riemschneider, 1988). The most important sector in volume term is the manufacturing sector, the concern of this paper. In 1978 total stocks of manufacturing foreign direct investment accounted for roughly two-thirds of the total in less developed countries, with just one-eighth devoted to the extractive industries (see Stopford and Dunning, 1983, p.22).