Designing Illumination Lenses and Mirrors by the Numerical Solution of Monge-Amp\`ere Equations

We consider the inverse refractor and the inverse reflector problem. The task is to design a free-form lens or a free-form mirror that, when illuminated by a point light source, produces a given illumination pattern on a target. Both problems can be modeled by strongly nonlinear second-order partial differential equations of Monge-Amp\`ere type. In [Math. Models Methods Appl. Sci. 25 (2015), pp. 803--837, DOI: 10.1142/S0218202515500190] the authors have proposed a B-spline collocation method which has been applied to the inverse reflector problem. Now this approach is extended to the inverse refractor problem. We explain in depth the collocation method and how to handle boundary conditions and constraints. The paper concludes with numerical results of refracting and reflecting optical surfaces and their verification via ray tracing.Comment: 16 pages, 6 figures, 2 tables; Keywords: Inverse refractor problem, inverse reflector problem, elliptic Monge-Amp\`ere equation, B-spline collocation method, Picard-type iteration; OCIS: 000.4430, 080.1753, 080.4225, 080.4228, 080.4298, 100.3190. Minor revision: two typos have been corrected and copyright note has been adde

Solving the Monge-Amp\`ere Equations for the Inverse Reflector Problem

The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the target, e.g., a projected image on a screen. This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge-Amp\`ere equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Amp\`ere equation is numerically solved using a collocation method based on tensor-product B-splines, in which nested iteration techniques are applied to ensure the convergence of the nonlinear solver and to speed up the calculation. In the numerical method special care has to be taken for the constraint: It enters the discrete problem formulation via a Picard-type iteration. Numerical results are presented as well for benchmark problems for the standard Monge-Amp\`ere equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.Comment: 28 pages, 8 figures, 2 tables; Keywords: Inverse reflector problem, elliptic Monge-Amp\`ere equation, B-spline collocation method, Picard-type iteration; Minor revision: reference [59] to a recent preprint has been added and a few typos have been correcte

Empirical Study of the 1-2-3 Trend Indicator

In this paper we study automatically recognized trends and investigate their statistics. To do that we introduce the notion of a wavelength for time series via cross correlation and use this wavelength to calibrate the 1-2-3 trend indicator of Maier-Paape [Automatic One Two Three, Quantitative Finance, 2013] to automatically find trends. Extensive statistics are reported for EUR-USD, DAX-Future, Gold and Crude Oil regarding e.g. the dynamic, duration and extension of trends on different time scales.Comment: 21 pages, 13 figures, 5 tables; Keywords: market technical trends, automatic trend analysis, wavelength of time series, autocorrelatio

Backtest of Trading Systems on Candle Charts

In this paper we try to design the necessary calculation needed for backtesting trading systems when only candle chart data are available. We lay particular emphasis on situations which are not or not uniquely decidable and give possible strategies to handle such situations.Comment: 12 pages, 19 figures; Keywords: backtest evaluation, historical simulation, trading system, candle chart, imperfect dat

List of endangered spider species of Germany (Red Data Book) (Arachnida: Araneae)

Seit der Bearbeitung der 1. Fassung der Roten Liste der Spinnen Deutschlands durch HARMS (1984) sind mehr als 10 Jahre vergangen. Die Kenntnisse über Verbreitung, Ökologie und Gefährdung dieser Tiergruppe haben sich seither erheblich vermehrt, Überwiegend durch die Ergebnisse faunistisch-ökologischer Untersuchungen im Rahmen von Gutachten, z. B. für Umweltverträglichkeitsprüfungen oder Unterschutzstellungsverfahren

Associations between depressive symptoms and disease progression in older patients with chronic kidney disease: results of the EQUAL study

Background Depressive symptoms are associated with adverse clinical outcomes in patients with end-stage kidney disease; however, few small studies have examined this association in patients with earlier phases of chronic kidney disease (CKD). We studied associations between baseline depressive symptoms and clinical outcomes in older patients with advanced CKD and examined whether these associations differed depending on sex. Methods CKD patients (>= 65 years; estimated glomerular filtration rate <= 20 mL/min/1.73 m(2)) were included from a European multicentre prospective cohort between 2012 and 2019. Depressive symptoms were measured by the five-item Mental Health Inventory (cut-off <= 70; 0-100 scale). Cox proportional hazard analysis was used to study associations between depressive symptoms and time to dialysis initiation, all-cause mortality and these outcomes combined. A joint model was used to study the association between depressive symptoms and kidney function over time. Analyses were adjusted for potential baseline confounders. Results Overall kidney function decline in 1326 patients was -0.12 mL/min/1.73 m(2)/month. A total of 515 patients showed depressive symptoms. No significant association was found between depressive symptoms and kidney function over time (P = 0.08). Unlike women, men with depressive symptoms had an increased mortality rate compared with those without symptoms [adjusted hazard ratio 1.41 (95% confidence interval 1.03-1.93)]. Depressive symptoms were not significantly associated with a higher hazard of dialysis initiation, or with the combined outcome (i.e. dialysis initiation and all-cause mortality). Conclusions There was no significant association between depressive symptoms at baseline and decline in kidney function over time in older patients with advanced CKD. Depressive symptoms at baseline were associated with a higher mortality rate in men

Modular portfolio theory : a general framework with risk and utility measures as well as trading strategies on multi-period markets

The following four modular building blocks are crucial in the context of portfolio theory: (a) the market model, (b) the "trading strategy" of the investor, (c) the risk and utility function and (d) the optimization problem. The setting of the so called "modern portfolio theory" by [Markowitz, John Wiley & Sons, Hoboken, NJ, 1959] and the capital asset pricing model (CAPM) by [Sharpe, The Journal of Finance, 19(3):425-442, 1964] consist of a one-period market model for (a) with the standard deviation as risk and the expected return as utility for (c). The trading strategies/portfolios are then just elements in the vector space ℝ^n whose entries represent the corresponding asset in the portfolio, i.e. there is nothing to do in block (b). The optimization problem, in general, is parameter depend, because of a trade-off between risk and utility. It is known in the modern portfolio theory, that the risk-utility values of the solutions, i.e. of the so-called efficient portfolios, for different parameters are on the boundary of a convex set within the two-dimensional risk-utility space. In the CAPM they even lie on a straight line. A generalization of such results is studied in [Rockafellar et al., Journal of Banking & Finance, 30(2):743-778, 2006] and [Maier-Paape and Zhu, Risks, 6(2):53, 2018]. In both literatures efficient portfolios are studied similar as in the modern portfolio theory and CAPM. In [Rockafellar et al., 2006] they use the same linear utility function but the risk is allowed to be a so-called generalized deviation measure, which e.g. requires convexity and also positive homogeneity. In [Maier-Paape and Zhu, 2018] a similar situation is discussed but the risk is defined with less assumptions, e.g. positive homogeneity is not required anymore. In addition, more general (concave) utility functions are allowed, which, however, still must be of a special form. Therein, block (a) again as one-period market and blocks (c) and (d) are studied. Again, the portfolios/trading strategies are just vectors in ℝ^n because a one-period model allows no trading strategies. This theory is now extended by the subdivision into the four modular building blocks (a) to (d) for general multi-period market models. The portfolios then are no longer represented by vectors in ℝ^n but can be complex time dependent trading strategies for investing the wealth. Examples are the buy and hold strategy or a strategy which reallocates the invested money after each day to ensure constant weights. Reasonable properties of (a) and (b), which are required for (c) and (d), are studied as well. Furthermore, the (concave) utility now is of general form. With just a few reasonable assumptions on the market model, the trading strategy and the risk and utility functions, the existence and uniqueness of efficient portfolios and many similar results as in the references above are shown here for the modular setup as well. The multi-period model is required e.g. to define risk functions based on the so-called drawdown. The drawdown of an equity curve is the (absolute or relative) difference between the maximum of this curve and its last value. Since this is an important value in praxis, the application of the optimization problem from above is studied also for a drawdown risk measure. However, the drawdown is difficult to evaluate. Therefore, some reasonable properties are derived in more detail for the absolute drawdown in case of an equity curve modeled by a Markov chain. For a general case of a Markov chain it is shown that the limit distribution for the absolute drawdown after N time steps as N goes to infinity exists if and only if the expectation of a win or loss after each time step is positive. For some special cases, the corresponding distribution can even be expressed at least with an implicit formula. In the random walk case, an explicit expression can be given for the distribution after just N time steps as well as in the limit case