On the least squares estimator asymptotic normality of the multivariate symmetric textured surface parameters

A multivariate trigonometric regression model is considered. Various discrete modifications of the similar bivariate model received serious attention in the literature on signal and image processing due to multiple applications in the analysis of symmetric textured surfaces. In the paper asymptotic normality of the least squares estimator for amplitudes and angular frequencies is obtained in multivariate trigonometric model assuming that the random noise is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly dependent random field on R M , M > 2. \mathbb {R}^M,\,\, M>2.

Nutrigonometry I: using right-angle triangles to quantify nutritional trade-offs in multidimensional performance landscapes

Animals regulate their diet in order to maximise the expression of fitness traits that often have different nutritional needs. These nutritional trade-offs have been experimentally uncovered using the Geometric framework for nutrition (GF). However, current analytical methods to measure such responses rely on either visual inspection or complex models applied to multidimensional performance landscapes, making these approaches subjective, or conceptually difficult, computationally expensive, and in some cases inaccurate. This limits our ability to understand how animal nutrition evolved to support life-histories within and between species. Here, we introduce a simple trigonometric model to measure nutritional trade-offs in multidimensional landscapes ("Nutrigonometry"). Nutrigonometry is both conceptually and computationally easier than current approaches, as it harnesses the trigonometric relationships of right-angle triangles instead of vector calculations. Using landmark GF datasets, we first show how polynomial (Bayesian) regressions can be used for precise and accurate predictions of peaks and valleys in performance landscapes, irrespective of the underlying structure of the data (i.e., individual food intakes vs fixed diet ratios). Using trigonometric relationships, we then identified the known nutritional trade-off between lifespan and reproductive rate both in terms of nutrient balance and concentration. Nutrigonometry enables a fast, reliable and reproducible quantification of nutritional trade-offs in multidimensional performance landscapes, thereby broadening the potential for future developments in comparative research on the evolution of animal nutrition.

Trajectory Planning Method for Mixed Vehicles Considering Traffic Stability and Fuel Consumption at the Signalized Intersection

Traffic lights force vehicles to stop frequently at signalized intersections, which leads to excessive fuel consumption, higher emissions, and travel delays. To address these issues, this study develops a trajectory planning method for mixed vehicles at signalized intersections. First, we use the intelligent driver car-following model to analyze the string stability of traffic flow upstream of the intersection. Second, we propose a mixed-vehicle trajectory planning method based on a trigonometric model that considers prefixed traffic signals. The proposed method employs the proportional-integral-derivative (PID) model controller to simulate the trajectory when connected vehicles (equipped with internet access) follow the optimal advisory speed. Essentially, only connected vehicle trajectories need to be controlled because normal vehicles simply follow the connected vehicles according to the Intelligent Driver Model (IDM). The IDM model aims to minimize traffic oscillation and ensure that all vehicles pass the signalized intersection without stopping. The results of a MATLAB simulation indicate that the proposed method can reduce fuel consumption and NOx, HC, CO2, and CO concentrations by 17%, 22.8%, 17.8%, 17%, and 16.9% respectively when the connected vehicle market penetration is 50 percent.

Radiological Evaluation of TATE Elbow Cartridge Position in Dogs

Objective The main purpose of this study was to evaluate the effect of varying elbow flexion angle and elbow orientation on the radiological measurement of component position for first (1G) and second-generation (2G) TATE elbow cartridges, and to test intra/inter-observer variability of measurements. Materials and Methods A cadaveric thoracic limb was implanted with a 1G then 2G cartridge, and mounted in lateral recumbency on an acrylic platform. The platform was tilted by set increments up to 10° in both planes, and radiographs were performed at each angle before repeating with the limb in caudocranial positioning. A deterministic trigonometric model was used to show how component angles should vary with changes in orientation, and these were compared with those measured by two observers. Humeral component angle (HCA), radioulnar component angle, varus/valgus cartridge alignment angle and the cartridge height:isthmus width ratio (CIR) were evaluated. Angles within 5° of the zero degrees inclination angle and ratios within 0.2 of the zero degrees inclination ratio were defined as acceptable. Results Observer component angles for both cartridges were accurate and precise for inclinations up to 10° except for HCA during adduction/abduction. The CIR values were within the acceptable limit for inclinations up to 7.5° in both planes. Clinical Significance Acceptable limits of limb inclination during positioning for TATE elbow replacement cartridge assessment were defined. All component measurements were sufficiently accurate and precise to be considered for evaluation of component position in clinical cases.

Simple Correlation Models for Estimation of Horizontal Global Solar Radiation for Oran, Northwest Algeria

In the literature, several correlation models have been developed to predict solar radiation on horizontal surfaces. In this paper, the daily solar radiation data and sunshine duration data measured during the period of 2006-2010, were used to calculate the monthly mean values of daily total solar radiation using five modified Angstrom–Prescott model and three day of the year based empirical models of Oran city (35°42 N and 00°36 W) in Algeria. Furthermore, those modified models are compared with three existing models. In order to evaluate the accuracy of the models, the statistical testing error such as R2, RMSE, rRMSE, MAPE and MBE are used. According to statistical test results and from the sunshine duration models, the polynomial model (#4) showed the best estimation results with a coefficient of determination R2=0.991, rRMSE=4.129%, and MAPE=3.635%. Furthermore, the accurate model from the day of the year models is the model (7), with R2=0.987, rRMSE=4.067%, and MAPE=3.5194%. Moreover, this paper finds that the best accuracy model to estimate the monthly mean daily solar radiation on horizontal surfaces in Oran city is the sine and cosine wave trigonometric model (#7).

THE PREDICTION OF POSITION AND ORIENTATION PARAMETERS OF UAV FOR VIDEO IMAGING

The paper presents the results of the prediction for the parameters of the position and orientation of the unmanned aerial vehicle (UAV) equipped with compact digital camera. Issue focus in this paper is to achieve optimal accuracy and reliability of the geo-referenced video frames on the basis of data from the navigation sensors mounted on UAV. In experiments two mathematical models were used for the process of the prediction: the polynomial model and the trigonometric model. The forecast values of position and orientation of UAV were compared with readings low cost GPS and INS sensors mounted on the unmanned Trimble UX-5 platform. Research experiment was conducted on the preview of navigation data from 23 measuring epochs. The forecast coordinate values and angles of the turnover and the actual readings of the sensor Trimble UX-5 were compared in this paper. Based on the results of the comparison it was determined that: the best results of co-ordinate comparison of an unmanned aerial vehicle received for the storage with, whereas worst for the coordinate Y on the base of both prediction models, obtained value of standard deviation for the coordinate XYZ from both prediction models does not cross over a admissible criterion 10 m for the term of the exactitudes of the position of a unmanned aircraft. The best results of the comparison of the angles of the turn of a unmanned aircraft received for the angle Pitch, whereas worst for the angles Heading and Roll on the base of both prediction models. Obtained value of standard deviation for the angles of turn HPR from both prediction models does not exceed a admissible exactitude 5° only for the angle Pitch, however crosses over this value for the angles Heading and Roll.

Trigonometric Functions Above and Below of the Double-Slit Barrier

The Double-Slit Experiment with individual electrons represents the central mystery of quantum mechanics and was voted by readers of Physics World as the most beautiful experiment in physics. In our trigonometric model we have divided the Young-Feynman double-slit interferometer into two zones: Aristotelian mixing chamber (the space between the entrance slit S0 and above the double-slit barrier with slits S1 and S2 where particles have been reflected (Euclid) and/or randomly deviated – parenklisis of Epicurus) and Plato’s projection chamber (the space below the double-slit barrier and above the projection screen). Aristotelian mixing chamber represents the classical hidden variable: individual particles have been reflected and/or randomly deviated in this zone and later pass through the double-slit barrier in such a way that they create probability distributions of single particles described by the very well known trigonometric relations. The double-slit barrier serves as a “self-organizing filter” for reflected and/or deviated individual particles and creates an order in the population densities among individual particles. For the experimental evaluation of this trigonometric model we propose to insert an internal and external “sfumato mirrors” for the quantitative manipulation of probability distributions of individual particles. The concept “sfumato” was inspired by the Old Master Leonardo da Vinci (see his picture Mona Lisa) (“sfumato” means “smoke”). Sfumato mirrors represent the classical observable variable and their influence on the resulting probability densities of individual particles can be trigonometrically described. Sfumato mirrors must not absorb any particles. In another experimental modification we can insert below one slit of the double-slit barrier an “absorption chiaroscuro mask” to fully remove those penetrating particles from the system and thus manipulate with the probability distribution observed with one opened slit (“chiaroscuro” means “light-dark”). It could be of interest to observe the resulting probability distribution of individual particles registered on the projection screen from one opened slit S1 or S2. The absorption chiaroscuro mask must not reflect particles back to the Aristotelian mixing chamber or to the Plato´s projection chamber. The results of these experiments will decide if we are just going up in a blind alley”.

On the Trigonometric Description of the Michelson-Morley Experiment

<p class="1Body">One formula with two trigonometric corrections describing the round trip of the beams in the Michelson-Morley experiment is presented. The first trigonometric correction describes the round trip path of those beams, while the second trigonometric correction describes the trigonometric geometric mean of the two-way speed of those beams. This formula gives the null fringe shift result for the first order experiments (Fizeau experiment, Hoek experiment), the null fringe shift result for the second order experiment (Michelson-Morley experiment), and predicts a measurable fringe shift result for the fourth order experiment. This trigonometric model can be tested experimentaly by the advanced LIGO (Laser Interferometer Gravitational-Waves Observatory) technology with three arms separated by the angle π/4 and the longitudinal arm directed to the CMB rest frame in the direction to the constellation Crater (known in the Greek mythology as the Cup of the god Apollo). This proposed fourth order experiment can be named as the advanced LIFE (Laser Interferometer Fringe Enigma) experiment. The published predictions before the arrival of experimental data from the advanced LIFE experiment can estimate the power of our models.</p>