11,018 research outputs found

    An Induction Principle for the Weighted p-Energy Minimality of x/|x|

    Get PDF
    In this paper, we investigate minimizing properties of the map x/xx/\|x\| from the Euclidean unit ball Bn\mathbf{B}^{n} to its boundary Sn1\mathbb{S}^{n-1}, for the weighted energy functionals En_p,α(u)=_BnxαupdxE^n\_{p,\alpha}(u)=\int\_{\mathbf{B}^{n}} \|x\|^{\alpha}\|\nabla u\|^p dx. We establish the following induction principle: if the map xx:Bn+1Sn\frac{x}{\|x\|}:\mathbf{B}^{n+1}\to \mathbb{S}^n minimizes En+1_p,αE^{n+1}\_{p,\alpha} among the maps u:Bn+1Snu: \mathbf{B}^{n+1}\to \mathbb{S}^n satisfying u(x)=xu(x)=x on Sn\mathbb{S}^n, then the map yy:BnSn1\frac{y}{\|y\|}:\mathbf{B}^n\to\mathbb{S}^{n-1} minimizes En_p,α+1E^{n}\_{p,\alpha+1} among the maps v:BnSn1v: \mathbf{B}^n\to\mathbb{S}^{n-1} satisfying v(y)=yv(y)=y on Sn1\mathbb{S}^{n-1}. This result enables us to enlarge the range of values of pp and α\alpha for which x/xx/\|x\| minimizes En_p,αE^n\_{p,\alpha}.Comment: 8 page

    On the Approximation of Toeplitz Operators for Nonparametric H\mathcal{H}_\infty-norm Estimation

    Full text link
    Given a stable SISO LTI system GG, we investigate the problem of estimating the H\mathcal{H}_\infty-norm of GG, denoted G||G||_\infty, when GG is only accessible via noisy observations. Wahlberg et al. recently proposed a nonparametric algorithm based on the power method for estimating the top eigenvalue of a matrix. In particular, by applying a clever time-reversal trick, Wahlberg et al. implement the power method on the top left n×nn \times n corner TnT_n of the Toeplitz (convolution) operator associated with GG. In this paper, we prove sharp non-asymptotic bounds on the necessary length nn needed so that Tn||T_n|| is an ε\varepsilon-additive approximation of G||G||_\infty. Furthermore, in the process of demonstrating the sharpness of our bounds, we construct a simple family of finite impulse response (FIR) filters where the number of timesteps needed for the power method is arbitrarily worse than the number of timesteps needed for parametric FIR identification via least-squares to achieve the same ε\varepsilon-additive approximation

    Conceptualizing interdependences between regulatory and monetary policies. Some preliminary considerations

    Get PDF
    In this paper we investigate the interplay between regulatory and monetary policies. We analyze how changes in institutional settings modify the functioning of various channels of monetary transmission. The paper begins with a brief presentation of the main channels of monetary transmission, including credit channel, exchange rate channel, Tobin q theory, and the credit channel. After that we define a positive institutional change and we check how such adjustments can be put into the logic of monetary transmission. We show that the most profound way institutions impact the monetary transmission is via its effect on the elasticity of investments to changes in interest rates.monetary transmission channels, monetary policy, regulatory policy, institutional change, financial globalization

    Criticizing the critique. Some methodological insights into the debate on the state of economic theory in the face of the post 2008 crisis

    Get PDF
    The aim of this paper is to investigate the current debate on the state of economics from a methodological perspective. We claim that the majority of contributions criticizing modern economics are not based on clear methodological principles and thus many of them are not correct. We show this with respect to such issues as the problem of realisticness of models and their assumptions, the role of mathematics in economics, the way we conceptualize the relation between economics (theory) and economy (empiria), as well as the general problem of comparing theories. In doing so we use the research apparatus taken form the philosophy of science and also we benefit a lot from recent developments in the philosophy of economics. Finally, we show one of the paradoxes of that debate, namely that many critics of economics accuse economists of using the wrong language (mathematics) while they do not use proper language themselves while criticizing economics, namely the apparatus taken from the philosophy of science.methodology of economics, mathematization, assumptions, theories of explanation, global economic crisis

    A note on nn-axially symmetric harmonic maps from B3B^3 to S2S^2 minimizing the relaxed energy

    Get PDF
    For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Hardt, Lin and Poon for the case n=1.Comment: 17 page

    The history of transaction cost economics and its recent developments

    Get PDF
    The emergence of transaction cost economics (TCE) in the early 1970s with Oliver Williamson’s successful reconciliation of the so called neoclassical approach with Herbert Simon’s organizational theory can be considered an important part of the first cognitive turn in economics. The development of TCE until the late 1980s was particularly marked by treating the firm as an avoider of negative frictions, i.e., of transaction costs. However, since the 1990s TCE has been enriched by various approaches stressing the role of the firm in creating positive value, e.g., the literature on modularity. Hence, a second cognitive turn has taken place: the firm is no longer only seen as an avoider of negative costs but also as a creator of positive knowledge.transaction cost economics, Oliver Williamson, theory of the firm, modularity literature, cognitive turn

    Beating Randomized Response on Incoherent Matrices

    Full text link
    Computing accurate low rank approximations of large matrices is a fundamental data mining task. In many applications however the matrix contains sensitive information about individuals. In such case we would like to release a low rank approximation that satisfies a strong privacy guarantee such as differential privacy. Unfortunately, to date the best known algorithm for this task that satisfies differential privacy is based on naive input perturbation or randomized response: Each entry of the matrix is perturbed independently by a sufficiently large random noise variable, a low rank approximation is then computed on the resulting matrix. We give (the first) significant improvements in accuracy over randomized response under the natural and necessary assumption that the matrix has low coherence. Our algorithm is also very efficient and finds a constant rank approximation of an m x n matrix in time O(mn). Note that even generating the noise matrix required for randomized response already requires time O(mn)
    corecore