text:
## Introduction

Water consumption prediction is one of the main goals in managing water supply infrastructure and water resources. Increasing urbanization, industrialization, population growth and unavoidable climate change cause the demand for drinking water to increase constantly1. Freshwater resources for water supply are constantly decreasing, and although we are demonstrating more and more conscious and pro-ecological attitudes in terms of water saving, studies show significant water scarcity of water globally and locally, especially in countries with dry climate2. Forecasting water demand over different time horizons is crucial in the management of water resources. Depending on the purpose of the observations, developed Water Demand Forecasting (WDF) systems allow predicting water consumption in short- and long-term perspectives. The aim of long-term analyses, considering the horizon of 20–30 years, is to support making decisions related to designing and developing water supply systems. Short-term simulations, usually hourly, daily, or weekly3,4, are used to optimize the work and energy costs of pump stations (Pump Scheduling Optimization, PSO)5 and to solve current operational problems6. Tiwari and Adamowski7 as well as Candelieri et al.8 also distinguish medium-term predicting with respect to weekly time ranges. It is used in supply network maintenance and developing failure prevention procedures. Future water consumption is always estimated based on historical time series. The methodology of their analyses consists in identifying the current pattern of water demand and predicting future consumption.

Apart from the purpose of water demand prediction that was mentioned above, its key aim is to ensure the appropriate amount of water to users in a long-term perspective, which is a difficult task, mainly in dry climate countries. This problem is especially intensified in large urban centers with a continuously increasing number of residents9,10,11. The analysis of the aquifer content that changes in time depending on external factors allows to determine their quantitative potential in terms of future water intake, water treatment and pumping it into the water supply system12. To be able to assess the sufficiency of water resources, it is necessary to implement procedures that are used in long-term forecasting.

Both short-term and long-term predictions require knowledge of the influence of factors that result in irregularities in water consumption from the water supply network. Short-term predictions require mainly the meteorological parameters analysis, whereas long-term forecasts require additionally analyzing demographic and economic factors10. The current weather conditions, geographical location, and socio-demographic aspects, referred to as anthropogenic11, determine the existence of hourly, daily, weekly or seasonal variability. The influence of anthropogenic factors is globally enhanced on holidays (Sundays, public holidays, and school holidays), in periods preceding religious holidays and locally, for example, during major cultural and leisure events.

One component of smart cities, an idea that is constantly being implemented and improved as part of the city vision of the future13 is Intelligent Water Systems (IWSs). Intelligence in the water sector, according to the Water Environment Federation, is evidenced by the use of advanced technologies in decision making and management. In practice, this means, among other things, the need to reduce operational costs, manage and mitigate risks, which should be served by risk assessment solutions, failure prediction, performance prediction and decision support systems14. Smart water supply systems take into consideration the variability in water consumption in the prediction sector, for systems of residential water saving15 and in solutions aimed at detecting abnormal states. Modelling their behavior requires advanced methods because of water supply system complexity16. The implementation of meteorological data enables to reduce the predicting error even by 11%17. The knowledge of the meteorological factors is necessary to obtain the best quality of water consumption prediction model18. Not taking into account the variability in water consumption in failure detecting systems results in an increased number of generated false alarms19. Including meteorological factors in the input data used for water consumption forecasts is therefore crucial, especially during short- and medium-term analyses13,20,21.

The purpose of the study was to create a methodology for extracting weekly seasonality in water consumption time series using evolutionary algorithms, which were implemented to improve the daily forecast accuracy. To the best of the authors' knowledge, this study is one of the few attempts to application of the evolutionary approach for water consumption time series implemented to improve prediction accuracy. The paper presents the results of short-term predictions of water consumption in one of the District Metered Area (DMA) zones located in Wroclaw (Poland), characterized by multifamily buildings. The applied models are based on evolutionary algorithms and on standard approach: Support Vector Regression (SVR), Multilayer Perceptron (MLP) and Random Forest (RF). Advanced methods in the field of computational intelligence are used in the research. These methods include evolutionary algorithms that discover seasonality, which is hard to determine when using simple methods due to the additional overlapping of, e.g., the development trends of DMA zones and annual seasonality. The implemented procedure allowed the authors to minimize the MAPE forecast error to even less than 2%, taking into account various case study variants. Regular evolutionary strategies may provide a tool that will support water demand prediction and thus improve the ongoing decision-making processes on the level of water supply networks managed and operating. The presented approach towards forecasting water demand levels, based on applying classical evolutionary strategies, shows that getting to know the factors determining water consumption variability is a key aspect. According to the research22, not only the analyses of meteorological parameters should be implemented in WDF models, but they should also take into consideration the periodicity of water consumption resulting from anthropogenic behavior, for instance, the different property characteristics, various household composition in terms of number and age, socioeconomic level and education level of the consumers. Finally, it may also become a component of systems that warn about abnormal states, such as failures, which are a widely analyzed aspect of managing water supply networks23.

The first section of the article contains a literature review concerning the methods used to predict water consumption. The following sections present the problem statement of the research, the methods as well as the results of the research conducted by the authors, and their interpretation and discussion. The final section of the paper provides some conclusions.

## Literature review of water demand forecasting methods

The optimization of water supply system operation is an important and up-to-date problem in civil engineering concerning increasingly scientific research24,25. The great progress in predicting water consumption that has been noted in recent years was caused by the intensive development and implementation of algorithms embedded in advanced computational intelligence. The subject of analyses and interpretation are not only historical hydraulic parameters, but also meteorological conditions.

The methods are currently applied for water demand forecasting have been presented in the form of a table (Table 1). Different characteristics for every case study and using various kinds of errors for evaluation cause that it is difficult to compare the model implemented by the authors. For this reason, the research compares the proprietary model with standard methods. Moreover, a review of the literature shows that appropriate techniques are rarely used in data preprocessing. Discrete wavelet transform (DWT) is most often used for this purpose26.

The knowledge of the nature of water consumption should also constitute the basis for expert systems warning about failures, so-called Leak Detection Systems (LDS), and identifying the location of leaks. Unfortunately, due to the complexity of these solutions, these issues are often neglected. According to Brentan et al.6, short-term prediction is the key aspect of the detection of abnormal states. These states may be identified at the moment when the estimated and observed measurement values of hydraulic parameters differ. As far as this category of applications of WDF is concerned, the ARIMA models and ANN were implemented in the past. Okeya et al.43 supplemented their research on leak detection with the WDF method, thus reducing the number of false alarms warning about potential failures. Wu et al.44 developed a solution based on the Cluster Analysis of historical flow rate measurements. According to the authors, clusterization-based methods may be effectively applied in the analysis of nonstationary time series. Apart from that, they also minimize the number of positive false alarms and they may be implemented to detect large flow leaks. To ensure that the application of water consumption prediction systems will improve the results of detecting water supply network failures, short- and medium-term analyses should be conducted based on data recorded at time steps not exceeding 15 min8.

## Research problem definition

As the literature review in the article indicates, none of the methods that have been used so far is universal enough to be used for water consumption forecasts under all water network operating conditions. The non-stationarity of time series, related to the variable character of water consumption and other factors that affect it, makes it necessary to determine the long-term trend, annual and weekly seasonality. The aforementioned conditions result in the necessity to implement analytical tools connected with the variability in water consumption as part of advanced WDF systems45,46.

Despite the fact that there are many scientific approaches to the problem of water consumption forecasting, none of them is universal enough to be directly implemented in various operating conditions of water supply networks. It is because of the spatial variability of the weather, the specific operational characteristics of water supply systems and their users. The most frequently mentioned problems related to a reliable comparison of the quality of forecasting models include: different characteristics for every case study, forecasting horizon, size and types of samples, demand periodicity, and various, disproportionate forecast errors5,18. In addition, creating long-term forecasts is subject to bigger errors than short-term forecasts, i.e., daily forecasts are more reliable than weekly ones40.

The key issues raised in 6-year research concern the creation of a reliable forecasting model for water consumption based on measurements carried out on the actual water supply network. The obtained time series are difficult to interpret due to the fact that according to the classification, it is a large water supply company with many inhabitants and additionally students in the October-June/July period, which generates dynamic seasonality. According to the data of the Statistical Office in Wroclaw, approximately 120 thousand students resided in Wroclaw, which accounts for nearly 19% of the whole population47. During holidays, most of these students leave the city and return in October to continue education. Additionally, in July and August also permanent residents of Wroclaw are out of town on holiday. This phenomenon was highlighted in the works by Candelieri and Archetti28 and Candelieri et al.9 and it justifies considering different characteristics for holiday periods in predicting the consumption of tap water. The analyzed area is specific mainly because while other studies indicate an increased water consumption during the summer period48, in Wroclaw, a significant decrease is observed Table 1, presented in the previous section, shows the existence of many analytical approaches used in water demand forecasting. Therefore, an evolutionary approach was proposed to identify seasonality in the time series and the created model was compared with the standard methods (SVR, MLP, RF, and CART).

The input information for short-term forecasting, apart from water consumption, usually describes the weather conditions. Opinions on the impact of individual meteorological parameters on the quality of the model are strongly divided. Some scientists do not take them into account at all5,37,45,49, although the literature review by Sebri18 indicates that models of better accuracy were obtained for short-term and medium-term forecasts using meteorological data. Typically, the input base is based on weather data including air temperature, occurrence of precipitation, and air humidity. As far as air humidity is concerned, literature reports are contradictory, because some studies show the greatest dependence of water consumption on air humidity50, while Ambrosio et al.20 proved that including this parameter in the input data significantly worsens the prediction quality. The quality changes of the model are analyzed in this research, taking into account various input data.

Comprehensive knowledge of the problem results in improving the prediction systems in the analyzed DMA zones, however, it should be noted that both water consumption and meteorological time series are burdened with noise and errors, so working with their use is not a simple process5. Models based on actual measurement data are subject to uncertainty resulting from deficiencies caused, among others, by failure of measuring devices, meanwhile, reliable long-term forecasts are based on long-term measurement sequences38. According to the research42, for short-term forecasting, a 4-month measurement series is sufficient to obtain reliable water demand forecasts. Another goal of the research is the analysis of critical events in the operation of the water supply network, which largely contributed to the deterioration of the quality of prediction and the development of weekly periodicity patterns in water demand.

## Materials and methods

### Study area and datasets preparation

The input data for the model, related both to the hydraulic parameters of water flow inside water supply networks and weather elements, were obtained from the monitoring service of the Municipal Water and Sewage Company S.A. in Wroclaw, MPWiK (Lower Silesian Region, Poland). The water supply network of Wroclaw consists of more than 30 DMAs equipped with more than 80 measurement points. The analyzed period covered the daily time series from September 2014 to May 2020.

Water flow supplied to the area was measured with the use of an electromagnetic flow meter that records data at 10 min time intervals. It is installed in the zone measurement well in the multifamily building DMA. Time series recorded with a time step of 10 min are part of a reporting and network control system. A GSM module is used to transmit data to the water supply network operator in real time. Water is supplied to the discussed DMA through a water supply pipe of the diameter of DN500. The analyzed area is inhabited by more than 22 thousand people. To eliminate additional factors that might influence the prediction, data concerning abnormal situations, resulting, among others, from pipe failures and temporary breaks in data recording, were removed from the time series. During such events, zero pressure and flow are recorded.

The automatic meteorological station that provided the measurements is located in the city center. Similarly as in other large urban and industrial agglomerations, land development and land use lead to an Urban Heat Island forming, which results in the modification of temperature and precipitation in the city. The measured basic meteorological parameters included: air humidity and temperature, atmospheric pressure, wind speed, solar irradiation, and precipitation. The data is sent to a datalogger via a wireless GSM/GPRS connection. Although the automatic measurement station was programmed for sampling at 1-min intervals, the analyses used average values and daily sums of the particular meteorological parameters. This procedure was used because weather conditions measured with such a high-frequency measurement interval are not a factor in the ongoing variability of water use. The variation in minutes of weather is never significant in short-term prediction of water demand51.

To maintain the consistency of the analyses and the subsequent interpretation of the results, both during the creation of the model using standard methods and the evolutionary approach, the same datasets were used. To avoid the overfitting phenomenon, 10 random datasets were selected from the databases containing information on water demand and meteorological parameters. Each time, the training dataset was 365 days of measurement, while the testing set was 30 successive days. In some experiments, in addition to the data containing water consumption and meteorological parameters, characteristic variables were added to the input dataset. Most often, additional variables contain information about the day of the week, split into working days and weekends9. The authors' research included one water consumption of the last two weeks (history0—previous 7 days and history1—7 days before history0) and one hot encoding of the day of the week.

### Reference methods

In this article, several standard machine learning algorithms were used. To make this article self-contained, a very brief review of the implemented methods was presented. More details about these methods in general can be found in Murphy52 and Goodfellow et al.53.

#### Support vector regression

The idea of Support Vector Regression is similar to Support Vector Machines, but SVR concerns regression problems, while SVM—classification problems. SVR is a type of regressor, but more tolerant to outliers than regular linear regression as it does not count as error values which are close enough to the correct value. It can model nonlinear dependencies by mapping the original data space to a new data space, usually of a higher dimension, by some nonlinear kernel functions and using the kernel trick, which simplifies the transformation by allowing to express the inner products in the new data space by some precomputed values related to the kernels. In our experiments, various kernels were used, such as the linear kernel, the polynomial kernel, the sigmoid kernel or the RBF kernel, each of them with a different parameter setting leading to the best performance on a selected dataset, and the best results were obtained with the simplest variant (the linear kernel).

#### A multilayer perceptron

Nowadays, artificial neural networks are one of the most promising and most popular methods in machine learning. They are loosely inspired by biological neural networks, but can be treated as a way of describing a computation of a function values using well-defined, differentiable operations: matrix multiplication and addition, function composition, and some nonlinear functions (such as tanh, or Rectified Linear Units, ReLU). The authors used the popular architecture of a multilayer perceptron with two layers of hidden neurons (as more hidden layers did not lead to significant improvements in the computational experiments, only to overfitting). Several hidden layer sizes were tested in this research.

#### Linear regression

In this approach, the authors assume that the predicted value is a linear combination of all features (all categorical features, such as days of the week, should be represented by several binary variables, using one-hot encoding). The optimal values of the coefficients can be found using linear algebra methods or gradient-based methods (such as Stochastic Gradient Descent, SGD). LR is as one of the simplest machine learning methods.

#### Classification and regression trees (CART)

The authors construct a tree which describes in a concise manner what conditions the feature values should meet to give certain output values. CART method allows to describe very precisely the training set, but if it is necessary to achieve good generalization properties, some pruning techniques should be used.

#### Random Forests

Random Forests are an example of ensemble learning. In these methods, the authors try to combine the results of many rather weak classifiers/regressors. In random forests, CART trees (with limited depth) were used. These trees use different subsets of learning examples and different subsets of features as well. The authors used two depth limits: 5 and 10.

For all these methods, the scikit-learn library for Python54 programming language was used.

### Evaluating forecast accuracy

The main aim is to minimize the objective function, which is the Mean Absolute Percentage Error (MAPE) (using to evaluate models’ accuracy, among others by Bakker et al.17; Candelieri and Archetti28; Candelieri et al.9; Xu et al.37) of the forecasts obtained with use of machine learning. This kind of error is scale-independent, so it is especially recommended for use in the predictive model evaluation18. MAPE may be described by the following equation55:

$$MAPE=\frac{1}{n}\sum_{t=1}^{n}\frac{\left|{\widehat{y}}_{t}-{y}_{t}\right|}{{y}_{t}}\cdot 100 \%$$
(1)

where: $$n$$—the size of the sample, $${\widehat{y}}_{t}$$—the value predicted by the model for time point $$t$$, $${y}_{t}$$—the value observed at time point $$t$$.

## Evolutionary approach

Most of the prediction models require stationary time series to provide accurate predictions. However, time series of water consumption usually reveal a type of annual and weekly periodicity, which makes the time series not stationary and thus difficult to predict by regular prediction models. The annual periodicity cannot be directly extracted from the data, because it is usually irregular and related to the characteristic of a particular season in a particular year (early or late spring, dry or wet autumn, etc.), but it is strongly related to the weather conditions, so its influence on the prediction accuracy in more complex models can be reduced by adding the data describing current weather conditions. The weekly periodicity is more difficult to reduce, because it is also noised by the irregular annual periodicity.

In our approach, the input data for the model were water consumption and meteorological parameters (Fig. 1). First, the water consumption data (Q), obtained from measuring equipment and resampled to one-day frequency, were preprocessed by extracting anomalies and the global trend (defined by the 365-days moving average). Second, the water consumption data (Q) were expressed as a fraction (Q′) of the 7-days moving average (i.e., the average flow rate from the 7 days preceding each of the analyzed days), so that:

$$Q{^{\prime}}=\frac{Q}{{Q}_{7}}$$
(2)

where: $$Q{^{\prime}}$$—the current value of water consumption in relation to the previous 7 days average, $$Q$$—water consumption, $${\mathrm{Q}}_{7}$$—the daily average of water consumption in the last 7 days.

Next, the relative water consumption data (Q′) were further preprocessed by extracting a weekly seasonality (evaluated either by the regular approach as a simple average of Mondays, Tuesdays, etc., or by the proposed evolutionary approach described further). Afterwards, the relative water consumption (Q′) was predicted using a linear regression model with the input data consisting of the values of Q′ from the previous 7 days and the weather data from the last day. Predictions were transformed to the original data scale by adding the weekly seasonality and the global trend and compared to the original data to evaluate the MAPE.

In the research, the prediction models were enhanced with evolutionary algorithms that discover the weekly periodicity in the optimization process. From the technical point of view, the weekly periodicity is a real number vector of length 7, and it is defined in a type of learning process as a solution to the optimization problem with the objective function being the accuracy of the prediction over a predefined training dataset and the search space consisting of real number vectors of length 7. To solve the optimization problem, an evolutionary algorithm based on an Evolution Strategy56 is proposed. It evolves a population of candidate solutions in the following process: first, the population includes a given number of random candidate solutions (random vectors from the search space drawn from the uniform probability distribution over the search space). Afterwards, the evolution starts and, in each iteration, a given number of parent candidate solutions is selected, then each pair of parent candidate solutions produces a pair of child candidate solutions, and finally, the selected child candidate solution replaces the previous candidate solution in the current population. Parent candidate solutions are selected with the probability proportional to their value of the objective function. Children are produced by local intermediary recombination and mutation operators, as in regular Evolution Strategies. The current population is updated with the best candidate solutions from the union of the current population and the child population. Finally, the evolution terminates after a given number of iterations.

In the experiments, different configuration settings of the evolutionary algorithm were studied to calibrate the approach and the best results were obtained with the parent population size of 200, the offspring population size of 400, the number of iterations of 50 and other parameters set according to the general heuristics for Evolutionary Strategies56. Figure 2 presents the general schema of the algorithm.

### Consent for publication

All authors have given their permission to publish.

## Results and discussion

### Reference methods

In the first part of the research that was based on the standard approaches (SVR, MLP, LR, CART, and RF), it was assessed which of them allow to obtain the lowest possible value of the MAPE error, taking into account various variants of the input data. The learning process was carried out for all 10 datasets with information on historical water consumption (history0 and history1) and taking into account the optimal set of features. The implementation of all variables in the learning process could result in overfitting the model, therefore the aim of this part of the research is also to select the optimal set of input data. Since 12 features and extra features are blocked in 3 groups, there were 16,384 combinations and for every such combination the learning and evaluation procedure was performed.

Table 2 presents the results of the preliminary MAPE errors for the models created with the use of linear regression and for example, one dataset, taking into account various input data combinations.

Forecast errors ranging from 2.38 to 3.66% were obtained with the use of linear regression models. Out of all possible combinations of input data sets, the model with the best learning quality (MAPE equal to 2.38%) was obtained for the historical data, taking into account the water consumption level from the preceding 7 and 14 days, as well as a set of meteorological data: air humidity, wind speed and soil temperature. Due to the fact that some of the meteorological data are strongly correlated, e.g., soil temperature and air temperature, the use of all data does not improve the learning quality (MAPE error is 2.81%) and can lead to overfitting, even with such a simple approach as linear regression. The study results clearly show that the wind speed and historical values of water consumption are important in each case. Nevertheless, it should be emphasized that the most important factor is including the historical 7 and 14-day time series in the input data. The MAPE error for modelling with and without weather data differs by 0.08% (between the best two features datasets and the best without weekdays datasets) and 0.29% (between the best one feature and the best without weekday datasets). The meteorological parameters are strongly correlated with each other, therefore their selection for forecasting is biased.

Days of the week data were entered as the only input, apart from water consumption for modelling by, e.g., Bakker et al.45, where MAPE errors ranging from 1.44 to 5.12% were obtained for 24-h forecasts. Further research results, however, indicate that weather data significantly improve the model accuracy17. Antunes et al.57 based on their research indicate that the most important meteorological parameters when forecasting water consumption are: temperature, rain level, and rain occurrence, whereas Piasecki et al.50 claim, it is air humidity. Comparing the research results of other authors is not entirely reliable due to the fact that each of them has a different set of data, which depends on the local capabilities related to the possession of measuring devices as well as the forecasting scope. Pesantez et al.51 showed that for hourly forecasts, it is not necessary to have weather condition parameters due to their low variability within this time frame, but the research was carried out in relation to the users of the water supply network and not the entire DMA zone. The results of this research indicate that for daily or monthly forecasts, the information on the weather improves the accuracy of the model. The input data should include not only meteorological parameters, but most of all data informing about the historical variability of water consumption (7 and/or 14 days). They contain seasonality and variability caused mainly by human activity and its repetitive behavior in this area.

Due to the fact that the learning process was carried out on 10 fixed data sets, and one of the objectives was to investigate to what extent the selection of optimal features is random, a random forest regressor was used and the feature selection was performed multiple times. It was also checked if the feature selection is generally a good approach by comparing the performance of the regressors with all features and with a selected set. At the same time, it should be noted that the choice was made for one classifier and then tested on another. In the next stage of analyses, a water consumption model was created for the optimal set of features (history0, history1, days of the week, precipitation, wind speed, ground temperature) and the set of all features using methods more advanced than linear regression, i.e. with SVR, MPL and RF. This stage was aimed at indicating which of the methods will minimize MAPE and thus be implemented in the authors' evolutionary approach, with the optimal set of input data already selected. RF-k random forests with depth limited to K, Classification and Regression Trees (CART), MLPk-n Multi-Layer Perception regressor with k and n neurons in two hidden layers, and SVR-linear Support Vector regressor with linear kernel were used. Table 3 presents the obtained values of the MAPE forecast errors for the previously selected optimal variables and all features. Results are averaged over all 10 datasets, detailed results for each are presented in Appendix 1.

The obtained study results indicate that the lowest MAPE error in the forecast of water consumption for all 10 datasets was obtained through Linear Regression and the largest for Multilayer Perceptron. This regularity applies both to the situation where all features were included in the input data and where those selected in the previous step were included. Random Forest method is the second that gives the lowest MAPE error after linear regression. The use of the most optimal set of features in the input model (history0, history1, days of the week, precipitation, wind speed, ground temperature) resulted in obtaining a smaller forecast error for practically most of the methods.

The obtained research results indicate that models based on linear regression are suitable for creating water demand forecasts without the need to implement more advanced solutions. After selecting the optimal set of features, even before implementing the evolutionary approach, which is the main goal of the research, the MAPE error is obtained at a satisfactory level of 2.2%.

### Evolutionary approach

This part of the article presents the results of research into the creation of a prognostic model containing implemented evolutionary algorithms, in which 10 selected datasets were used as testing data (similar to those in “Reference methods” section). The training data was a period of 1 year before each of the test datasets. To improve the quality of prediction, the evolutionary approach was implemented in conjunction with Linear Regression as the method through which the lowest MAPE error values were achieved.

Table 4 presents the prediction error for the prediction algorithm used in this experiment with the weekly periodicity vector determined by the evolutionary algorithm. MAPE values obtained during the implementation of the evolutionary approach for each of the ten datasets are presented. It includes both the quality of the forecasts for the training and testing sets.

The MAPE error for the training sets was 2.00%, while for the testing sets it was 2.12%. The results of the research show that the model with the implemented evolutionary approach allows to obtain water consumption forecasts with an error of only 1.31%. According to predictive model quality standards, a MAPE error below 10% should be regarded as a determinant of highly accurate forecasting58. Although it is not possible to directly compare the models created by other authors due to the different conditions and the scope of the conducted research, there are a number of results in which the MAPE values of forecasts range from 2.0 to 3.0%17,37,50 or they were containing larger error49,57,59. To make the obtained results more reliable, water consumption forecasts were intentionally made using standard methods (LR, SVR, MLP, RF, CART) on analogous data sets, which was described in the previous chapter. The implementation of evolutionary procedures allows to create short-term models of water consumption forecasts that are much more reliable, although the final value of the MAPE error is influenced by the measurement period and the range in which weekly periodicity was extracted from the time series.

The results and diagrams for all ten datasets are included in the attachments, which constitute an appendix to the article (Appendix 2). The maximum likelihood model with a testing error of 1.31% was obtained for dataset no. 3 (Figs. 3. and 4). The testing process (30 successive days) ended with the greatest error in the case of September 24, 2016, when water consumption in the selected zone was almost 2.5 times higher than normal. It was caused by additional supply to the neighboring zone via the analyzed DMA. Similarly, on September 29, 2016, and October 3, 2016, the error value was significant due to a failure of the measuring device. All above-mentioned situations do not diminish the quality of the forecasting model due to the fact that they are random events.

The research results indicate that the worst accuracy of testing was obtained for dataset no. 7 from 2018-07-21 to 2018-08-20 (Figs. 5 and 6). The increase in the forecast error was caused by the record-breaking hot period, which resulted in increased consumption of tap water. This is another proof that weather conditions affect the forecasts of water consumption and its anomalies significantly affect the deterioration of the prediction.

As defined in “Evolutionary approach” section, the weekly periodicity vector is discovered by an evolutionary algorithm as a solution to an optimization problem with the objective function that evaluates the accuracy of the candidate weekly periodicity vector on the training dataset. Figure 7 shows the distribution of weekly periodicity vector in water consumption. The evolutionary approach makes it possible to formulate some rules for water consumption by network users. It can be noticed that the greatest variability in water consumption is visible on Saturdays, while the lowest on Wednesdays. The results from end of the working week (Thursday–Friday) indicate that users of the water supply network are probably postponing household chores until Saturday. The water consumption habits of the inhabitants of DMA zones constitute a fairly large part of the research conducted by other scientists22 and the evolutionary approach may indirectly contribute to them and facilitate the creation of water demand patterns28,60.

Figure 8 presents the objective function in the successive iterations of the evolutionary algorithm. It is easy to see that the evolutionary algorithm is capable of solving the optimization problem and discover a quasi-solution after about 5 iterations. A good learning algorithm should minimize the number of calculations required to achieve a good accuracy of the model61. For the learning process that models water demand, which is shaped by so many random factors, this is not a high number of iterations, which demonstrates that the proposed predicting methodology is effective.

Figure 9 shows the evolution (changes in successive iterations of learning) of the weekly seasonality vector. In the chart, they are responsible for the consecutive days of the week. In the first iteration, the vector is random (not learned), while in the last iteration, it is the final vector. In intermediate iterations, it is noticeable how the evolution algorithm adjusts the vector values. For the first 10 or so iterations, the changes are quite large, then small, which means that the final vector was found quite quickly, later it was only slightly adjusted. There is an increased consumption of water on Saturdays, so the weight of the average water demand on a given weekday in relation to the weekly average is higher for Saturday.

## Conclusions and future work

Modern management of water supply infrastructure widely uses the development of information technologies that enable to enhance the processes connected with its operation. Time series of basic hydraulic parameters support increasingly advanced analytical and diagnostic tools. Water consumption prediction in various time horizons enables, among others, planning and performing maintenance and renovation works effectively, optimizing the pump station operations and detecting failures.

The research involved the implementation of a classical evolutionary approach for improving short-term prediction of water demand in one of the Wroclaw DMA zones with multifamily buildings. Additionally, an evolutionary strategy, which had not been used previously in this field of studies, was employed to determine the influence of weekly seasonality on water consumption distribution. The studies demonstrated the advantage of evolutionary strategies in combination with LR over standard methods of forecasting (SVR, MLP, RF, CART). The solution proposed by the authors enables to predict the volume of water consumption, expressed as the Mean Absolute Percentage Error (MAPE) more efficiently. The obtained final average values of MAPE errors were 2.12% (for the best 1.31% and worst 3.31% variant of forecasting, respectively). The research also demonstrated that evolutionary strategies are useful for data preprocessing and improving the knowledge of the anthropogenic behavior of water supply network users. Moreover, the results of the research indicate that the inclusion of additional variables in the input data in the form of meteorological parameters and water consumption for the preceding two weeks improves the accuracy of prediction.

In further studies, the authors intend to tackle the issues of predicting water demand in other DMA zones managed by the Municipal Water and Sewage Company, but of different consumption types, e.g., industrial consumption. An important element of the planned studies will be the spatial correlation of observations obtained from monitoring individual water supply areas.


text:
### Collecting like terms

After brackets have been expanded, the like terms can be collected.

#### Example 5

Expand the brackets and collect like terms.

1. $$2(x-6) + 5x$$
2. $$3 + 3(x-1)$$
3. $$3(2x + 4) + 6(x-1)$$

#### Solution

1. $$2(x-6)+5x=2x-12+5x=7x-12$$
2. $$3+3(x-1)=3+3x-3=3x$$
3. $$3(2x+4)+6(x-1)=6x+12+6x-6=12x+6$$

format: plain


text:
# State bug in IBM Quantum Experience

I run many tests with VQC and QSVM algorithms. In last times I can't get final result because some job hang on state Creating, Validating or Running. It can continue many hours and It doesn't finish. For example, I run test with some parameters and get result but after I run test with the same parameters and not get result because this problem. It happens with real devices and qasm_simulator. For QSVM These problems happen after long time (>1000 jobs), for VQC - in the beginning. Does it are problems of site or my code? Sorry for my english.

My code:

import warnings
import numpy as np
from qiskit.aqua import QuantumInstance
from qiskit.aqua.algorithms import VQC
from qiskit.aqua.components.optimizers import SPSA, COBYLA
from qiskit.aqua.components.feature_maps import RawFeatureVector
from qiskit.circuit.library import TwoLocal, ZZFeatureMap
from qiskit import IBMQ

#I read file where data look so:
#1234
#28313
#2145
#27216
#...

def get_bd(str):
f = open(str,'r')
_true = []
_false = []
for i in range(3):
m = []
a,none,data = data.partition('\n')
b,none,data = data.partition('\n')
m.append(int(a))
m.append(int(b))
_false.append(m)

for i in range(3):
m = []
a,none,data = data.partition('\n')
b,none,data = data.partition('\n')
m.append(int(a))
m.append(int(b))
_true.append(m)
return _false, _true

def get_tr(name,n):
f = open(name,'r')
test = []
for i in range(n):
m = []
a,none,data = data.partition('\n')
b,none,data = data.partition('\n')
m.append(int(a))
m.append(int(b))
test.append(m)
return test

train_false,train_true = get_bd('spectrain.txt')
test_false,test_true = get_bd('spectest.txt')

warnings.filterwarnings("ignore", category=DeprecationWarning)

provider = IBMQ.get_provider(hub='ibm-q')

shots = 8192
feature_dim = 2
backend = provider.get_backend('ibmq_qasm_simulator')
optimizer = SPSA(max_trials=1000, c0=4.0, skip_calibration=True)
optimizer.set_options(save_steps=1)
#feature_map = RawFeatureVector(feature_dimension=feature_dim)
feature_map = ZZFeatureMap(feature_dimension=feature_dim, reps=2,entanglement='linear')
#feature_map = SecondOrderExpansion(feature_dimension=2,depth=2,entanglement='linear')

var_form = TwoLocal(feature_dim, ['ry', 'rz'], 'cz', reps=3)

training_data = {'A': np.asarray(train_false),'B': np.asarray(train_true)}
#test_false,test_true = get_bd('test_data')
testing_data = {'A': np.asarray(test_false),'B': np.asarray(test_true)}

vqc = VQC(optimizer, feature_map, var_form, training_data, testing_data)
quantum_instance = QuantumInstance(backend, shots=shots,skip_qobj_validation=False)

result = vqc.run(quantum_instance)
print("testing success ratio: ", result['testing_accuracy'])

Hmm. I run it today on trainning samples in 3-norm,3-anomaly samples and It doesn't hang... while. So yesterday It worked in first time until I run training for 1000-norm,1000-anomaly samples where problem happens. I try to run 1000-norm,1000-anomaly samples later for checking.

Update 21.11.2020 I try run VQC algorithm 1000-norm,1000-anomaly samples and get this:

Some jobs were completed but It hangs now. I get good result with this test yesterday. Today I set 8192 shots (It was 2 shots in last time) and We see this image.

• If you can put your code here then I can run it myself to see whether I will run into the same issue or not. I have had a particular job hang in queue for a long time and wasn't going anywhere... I usually just cancel it and Aqua will resubmit my job for me. – KAJ226 Nov 19 '20 at 22:47
• I try to cancel it but It repeated with other jobs again and again. Validating error happens sometimes after it so. – Kirill Nov 20 '20 at 9:02


text:
## Stream: general

### Topic: subset or subtype?

#### Kevin Buzzard (Mar 04 2018 at 17:15):

I wrote presheaves of types twice in my life and I see now that my definitions differ.

First is

#### Kevin Buzzard (Mar 04 2018 at 17:16):

(F : Π U : set α, T.is_open U → Type*)

Second is

#### Kevin Buzzard (Mar 04 2018 at 17:17):

(F : {U // topological_space.is_open T U} → Type*)

#### Kevin Buzzard (Mar 04 2018 at 17:17):

Which is "better"?

#### Kevin Buzzard (Mar 04 2018 at 17:17):

I seem to be able to work with either

#### Kevin Buzzard (Mar 04 2018 at 17:21):

Is the only difference that with the first I have two different names U and HU, and with the second I have V.val and V.property?

#### Gabriel Ebner (Mar 04 2018 at 17:21):

Yes, it is analogous to the difference between A → B → C and A ∧ B → C.

#### Kevin Buzzard (Mar 04 2018 at 17:22):

But with those two choices, functional program people prefer the first, right?

#### Patrick Massot (Mar 04 2018 at 17:22):

Proof assistant people seem to always prefer A → B → C, so which is better for presheaves?

#### Kevin Buzzard (Mar 04 2018 at 17:22):

which is which ;-)

#### Kevin Buzzard (Mar 04 2018 at 17:22):

Presumably H.1 and H.2 is V.val and V.property so this is the one I should perhaps avoid

#### Reid Barton (Mar 04 2018 at 17:22):

The second one also gives you the name "V", which for doing abstract sheaf theory things (that don't care about the particular site) seems like it would be more convenient.

#### Kevin Buzzard (Mar 04 2018 at 17:23):

oh but here's a difference: A and B is a type

#### Kevin Buzzard (Mar 04 2018 at 17:23):

so it can be used more easily as a target

#### Kevin Buzzard (Mar 04 2018 at 17:27):

This doesn't seem to matter in practice. I am going for set because it seems to be analogous to the "more functional" A -> B -> C

#### Reid Barton (Mar 04 2018 at 17:36):

If you imagine defining a sheaf of sets on a general site, then the first way seems really unnatural (what would be the analogue of is_open?) So from a "math" perspective, the second way looks better.
I don't know what the practical implications are, though.

#### Reid Barton (Mar 04 2018 at 17:40):

(I would also set it up so that I can write U : T.open_set, and use $\cap$ and $\cup$ and $\subset$ directly on open sets, etc.)

Last updated: May 10 2021 at 00:31 UTC


text:
Type 3 Questions: Graph and Function Analysis

The long name is “Here’s the graph of the derivative, tell me things about the function.”

Students are given either the equation of the derivative of a function or a graph identified as the derivative of a function but no equation is given. It is not expected that students will write the equation (although this may be possible); rather, students are expected to determine important features of the function directly from the graph of the derivative. They may be asked for the location of extreme values, intervals where the function is increasing or decreasing, concavity, etc. They may be asked for function values at points.

The graph may be given in context and student will be asked about that context. The graph may be identified as the velocity of a moving object and questions will be asked about the motion. See Type 2 questions – Linear motion problems

Less often the function’s graph may be given and students will be asked about its derivatives.

What students should be able to do:

• Read information about the function from the graph of the derivative. This may be approached by derivative techniques or by antiderivative techniques.
• Find and justify where the function is increasing or decreasing.
• Find and justify extreme values (1st and 2nd derivative tests, Closed interval test aka.  Candidates’ test).
• Find and justify points of inflection.
• Find slopes (second derivatives, acceleration) from the graph.
• Write an equation of a tangent line.
• Evaluate Riemann sums from geometry of the graph only.
• FTC: Evaluate integral from the area of regions on the graph.
• FTC: The function, g(x), maybe defined by an integral where the given graph is the graph of  the integrand, f(t), so students should know that if,  $\displaystyle g\left( x \right)=g\left( a \right)+\int_{a}^{t}{f\left( t \right)dt}$ then  ${g}'\left( x \right)=f\left( x \right)$  and  ${{g}'}'\left( x \right)={f}'\left( x \right)$. In this case students should write ${g}'(t)=f\left( t \right)$ on their answer paper, so it is clear to the reader that they understand this.

Not only must students be able to identify these things, but they are usually asked to justify their answer and reasoning. See Writing on the AP Exams for more on justifying and explaining answers.

The ideas and concepts that can be tested with this type question are numerous. The type appears on the multiple-choice exams as well as the free-response. Between multiple-choice and free-response this topic may account for 15% or more of the points available on recent tests. It is very important that students are familiar with all the ins and outs of this situation.

As with other questions, the topics tested come from the entire year’s work, not just a single unit. In my opinion many textbooks do not do a good job with integrating these topics, so be sure to use as many actual AP Exam questions as possible. Study past exams; look them over and see the different things that can be asked.

For some previous posts on this subject see October 1517192426 (my most read post), 2012 and  January 2528, 2013

Free-response questions:

• Function given as a graph, questions about its integral (so by FTC the graph is the derivative): 2014 AB 3/BC 3
• Function given as an equation: 2016 AB 6
• Function given as a graph 2016 AB 3/BC 3
• Table and graph of function given, questions about related functions: 2017 AB 6,
• Derivative given as a graph: 2016 AB 5 and 2017 AB 3

Multiple-choice questions from non-secure exam. Notice the number of questions all from the same year; this is in addition to one free-response question (~25 points on AB and ~23 points on BC out of 108 points total)

• 2012 AB: 2, 5, 15, 17, 21, 22, 24, 26, 76, 78, 80, 83, 82, 84, 85, 87
• 2012 BC 3, 11, 12, 15, 12, 18, 21, 76, 78, 80, 81, 84, 88, 89

A good activity on this topic is here. The first pages are the teacher’s copy and solution. Then there are copies for Groups A, B, and C. Divide your class into 3 or 6 or 9 groups and give one copy to each. After they complete their activity have the students compare their results with the other groups.

Schedule of review posts

• Tuesday February 27 – AP Exam Review
• Friday, March 2 – Resources for reviewing
• Tuesday March 6 – Type 1 questions – Rate and accumulation questions
• Friday March 9 – Type 2 questions – Linear motion problems
• Tuesday March 13 – Type 3 questions – Graph analysis problems (this post)
• Friday March 16 – Type 4 questions – Area and volume problems
• Tuesday Match 20 Type 5 questions – Table and Riemann sum questions
• Friday March 23 Type 6 questions – Differential equation questions
• Tuesday March 27 – Type 7 questions – miscellaneous
• Friday March 30 Type 8 questions – Parametric and vector questions (BC topic)
• Tuesday April 3 Type 9 questions – Polar equations
• Friday April 6 Type 10 questions – Sequences and Series


text:
# Nehe's Quaternion camera question

This topic is 3092 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Can anyone here tell me where did he get his formula? Did he make it or is it the standard way to convert for OpenGL use. http://nehe.gamedev.net/data/lessons/lesson.asp?lesson=Quaternion_Camera_Class http://www.opengl.org/sdk/docs/man/xhtml/glMultMatrix.xml
// First row
pMatrix[ 0] = 1.0f - 2.0f * ( m_y * m_y + m_z * m_z );
pMatrix[ 1] = 2.0f * (m_x * m_y + m_z * m_w);
pMatrix[ 2] = 2.0f * (m_x * m_z - m_y * m_w);
pMatrix[ 3] = 0.0f;

// Second row
pMatrix[ 4] = 2.0f * ( m_x * m_y - m_z * m_w );
pMatrix[ 5] = 1.0f - 2.0f * ( m_x * m_x + m_z * m_z );
pMatrix[ 6] = 2.0f * (m_z * m_y + m_x * m_w );
pMatrix[ 7] = 0.0f;

// Third row
pMatrix[ 8] = 2.0f * ( m_x * m_z + m_y * m_w );
pMatrix[ 9] = 2.0f * ( m_y * m_z - m_x * m_w );
pMatrix[10] = 1.0f - 2.0f * ( m_x * m_x + m_y * m_y );
pMatrix[11] = 0.0f;

// Fourth row
pMatrix[12] = 0;
pMatrix[13] = 0;
pMatrix[14] = 0;
pMatrix[15] = 1.0f;

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That's the standard method for building a matrix from a quaternion - you'll find it in pretty much any math library, and in any reference on the subject.

The algorithm itself isn't specific to OpenGL or any other API, but for a given implementation you do have to make sure that the code matches the conventions of the API being used (e.g. row- or column-vector notation, row- or column-major matrix layout, etc.).

##### Share on other sites
Now, I got it. Thank you jyk. I found it here. http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/index.htm

I have another question. Does including it on my project will really make my ass pain defending it? What if they ask me on how did I derive that formula? Is there any book or site that will help me how the formula went out.

##### Share on other sites
Quote:
 I have another question. Does including it on my project will really make my ass pain defending it? What if they ask me on how did I derive that formula? Is there any book or site that will help me how the formula went out.
Are you asking if it's ok to use the formula in your projects without getting explicit permission from anyone? If so, the answer is yes, it's fine: it's common knowledge, and can be found in literally hundreds (maybe even thousands) of math libraries and game code bases. I'm sure there's nothing for you to worry about.

Deriving it oneself can still be a good exercise though. There's more than one way to derive the formula, but here's a fairly straightforward method. Start with the formula for rotating a vector v by a quaternion q (using standard quaternion multiplication order):
v' = q * v * conjugate(q);
Note that for the purpose of this formula, the vector v must be represented as a quaternion (the x, y, and z elements of v are assigned to the x, y, and z elements of the quaternion, and the 'w' element of the quaternion is typically given a value of zero).

Now, work through the multiplications longhand until you have a value for each element of v'. If you then express the result in the form of a matrix-vector multiplication, you will have the matrix that you posted earlier.

[Edit: A couple of places you can find derivations are in the book '3D Math Primer', and in the articles on quaternions in one of the first two Game Programming Gems books.]

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text:
## Chemistry: Molecular Approach (4th Edition)

Limiting reactant: Ethanol $(C_2H_5OH)$ Theoretical yield: $4.28$ g Percent yield: 87.0 %
1. Find the masses of ethanol and water: $$4.62 \space mL \space C_2H_5OH \times \frac{0.789 \space g}{1 \space mL} = 3.64\underline{52} \space g \space C_2H_5OH$$ $$3.72 \space mL \space H_2O \times \frac{1.00 \space g}{1 \space mL} =3.72 \space g \space H_2O$$ 2. This is a combustion reaction; ethanol will react with oxygen to produce water and carbon dioxide. $$C_2H_5OH + O_2 \longrightarrow CO_2 + H_2O$$ - Balance the amount of carbon atoms, then the amount of hydrogen, and the amount of oxygen atoms. $$C_2H_5OH + \frac 72 O_2 \longrightarrow 2CO_2 + 3H_2O$$ - Multiply all coefficients by 2 to remove the fraction: $$2C_2H_5OH + 7O_2 \longrightarrow 4CO_2 + 6H_2O$$ 3. Find the limiting reactant, and calculate the theoretical and percent yield. - Calculate or find the molar mass for $C_2H_5OH$: $C_2H_5OH$ : ( 1.008 $\times$ 6 )+ ( 12.01 $\times$ 2 )+ ( 16.00 $\times$ 1 )= 46.07 g/mol - Using the molar mass as a conversion factor, find the amount in moles: $$3.6452 \space g \times \frac{1 \space mole}{ 46.07 \space g} = 0.07912 \space mole$$ - Calculate or find the molar mass for $O_2$: $O_2$ : ( 16.00 $\times$ 2 )= 32.00 g/mol - Using the molar mass as a conversion factor, find the amount in moles: $$15.55 \space g \times \frac{1 \space mole}{ 32.00 \space g} = 0.4859 \space mole$$ Find the amount of product if each reactant is completely consumed. $$0.07912 \space mole \space C_2H_5OH \times \frac{ 6 \space moles \ H_2O }{ 2 \space moles \space C_2H_5OH } = 0.2374 \space mole \space H_2O$$ $$0.4859 \space mole \space O_2 \times \frac{ 6 \space moles \ H_2O }{ 7 \space moles \space O_2 } = 0.4165 \space mole \space H_2O$$ Since the reaction of $C_2H_5OH$ produces less $H_2O$ for these quantities, it is the limiting reactant. - Calculate or find the molar mass for $H_2O$: $H_2O$ : ( 1.008 $\times$ 2 )+ ( 16.00 $\times$ 1 )= 18.02 g/mol - Using the molar mass as a conversion factor, find the mass in g: $$0.2374 \space mole \times \frac{ 18.02 \space g}{1 \space mole} = 4.27\underline{8} \space g = 4.28 \space g$$ $$Percent \space yield = \frac{ 3.72 }{ 4.278 } \times 100\% = 87.0 \%$$


text:
# The IEEE Communications Society Heinrich Hertz Award for Best Communications & Wireless Communications Letter Award

Prize
Plaque and honorarium of $500 per author (up to a maximum total of$2000).

Description
Letters published in any letter journal financially sponsored or co-sponsored by the Communications Society which open new lines of research, envision bold approaches to communication, formulate new problems to solve, and essentially enlarge the field of communications engineering are eligible for this annual award.

Eligibility
This award is given to an outstanding manuscript published in any letter journal financially sponsored or co-sponsored by the Communications Society during the previous 3 calendar years which opens new lines of research, envisions bold approaches to communication, formulates new problems to solve, and essentially enlarges the field of communications engineering.  Recipients need not be IEEE members.

Basis for Judging
Quality,exposition, novelty and impact.

Nominee Solicitation
Nominations will be solicited by the Editor-in-Chief of the letter journal from the Editorial Board.  Each associate editor may submit no more than one paper to the Editor-in-Chief for consideration. A sub-committee comprised of the Editor-in-Chief and five Associate Editors will screen the nominated papers.  Nominations will also be solicited from ComSoc Technical Committee chairs and readership (through published notices).

No more than five manuscripts will be submitted to the Awards Committee for final selection by March 1 of each year.  Selection will be based on rankings given to each nominated paper by each member of the sub-committee.  Each paper submitted to the Awards Committee will be accompanied by a summary statement.

The IEEE Communications Society will administer the award covered out of the Society’s annual operating budget until a suitable endowment is established.  Each recipient receives an honorarium of $500 up to maximum of$2000 (plus the minimal cost of plaques) and will be budgeted annually to cover the total cost of up to \$2000 per year.


text:
# Electron energy levels

by Pengwuino
Tags: electron, energy, levels
PF Gold
P: 7,120
The question is:

 We can approximate an electron moving in a nanowire as a one-dimensional infinte square-well potential. Let the wire be 2 um long. The nanowire is cooled to a temperature of 13K, and we assume the electron’s average kinetic energy is that of a gas molecule at this temperature ( = 3/2 kT). (a) What are the three lowest possible energy levels of the electrons? (b) What is the approximate quantum number of the electrons moving in the wire?
So I have the equation to find the energy in an infinite well…

$$$\begin{array}{l} E_n = n^2 \frac{{\pi ^2 (\hbar c)^2 }}{{2mc^2 L^2 }} \\ L = 2*10^3 nm \\ m = 5.11*10^5 eV \\ \hbar c = 197.33eV*nm \\ \end{array}$$$

Using this, I find n=1 to equal 9.40*10^-8 eV

The book says im good there.

The problem is when I have to find the second energy level. I figured you use n=2 which means its just 2^2 or 4 times the energy but the book says its just 2 times the energy of the first energy level. What am I not understanding here?
 P: 1 I'm pretty sure the book made a mistake. For n=2, E2 = 4*E1 = 3.8 X 10^-7. I double checked with cramster, and that's how it was done there also.

 Related Discussions Atomic, Solid State, Comp. Physics 2 Atomic, Solid State, Comp. Physics 0 Advanced Physics Homework 0 Introductory Physics Homework 1


text:
An $n$-dimensional topological manifold $M$ is orientable if for each compact set $K \subset M$, there exists a homology class $\langle o_K \rangle \in H_n(M, M \setminus K)$ with the property that for any $x \in K$, the natural map $H_n(M, M \setminus K) \to H_n(M , M \setminus x)$ induced by inclusion maps $\langle o_K \rangle$ to a generator of $H_n(M , M \setminus x) \cong \mathbb{Z}$. This is equivalent to the fibre bundle $\mathcal{O}(M)$ from last lecture being a trivial bundle.

We being this lecture we begin by proving the main result from last time: that the two functors $H_n(M, \square ;A)$ and $\Gamma_c(M \setminus \square ; A)$ are naturally isomorphic, and then use this to define orientability as above.

The lecture concludes with the introduction of (yet another) product, called the cap product. This works for arbitrary topological space and is a "mixed" product:

$$H^{ \bullet}(X) \otimes H_{ \bullet}(X) \to H_{ \bullet}(X), \qquad \langle \alpha \rangle \otimes \langle c \rangle \mapsto \langle \alpha \rangle \frown \langle c \rangle.$$

Although useful in its own right, for us the most important property of the cap product is for Poincaré Duality, which states the following: if $M$ is a closed oriented topological manifold then $\square \mapsto \square \frown \langle o_M \rangle$ is an isomorphism. We will prove this in Lecture 39.


text:
# How do you differentiate f(x)=1/sqrtsec(e^(x) )  using the chain rule?

Jan 30, 2018

$f ' \left(x\right) = - \frac{{e}^{x} \tan \left({e}^{x}\right)}{2 {\left(\sec \left({e}^{x}\right)\right)}^{\frac{1}{2}}}$

$\textcolor{w h i t e}{f ' \left(x\right)} = - \frac{{e}^{x} \tan \left({e}^{x}\right) \sec {\left({e}^{x}\right)}^{- \frac{1}{2}}}{2}$

#### Explanation:

Here we have $f \left(x\right) = {\left(\sec \left({e}^{x}\right)\right)}^{- \frac{1}{2}}$

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left[{\left(\sec \left({e}^{x}\right)\right)}^{- \frac{1}{2}}\right]$

$\textcolor{w h i t e}{f ' \left(x\right)} = {\left(\sec \left({e}^{x}\right)\right)}^{- \frac{1}{2} - 1} \cdot - \frac{1}{2} \cdot \frac{d}{\mathrm{dx}} \left[\sec \left({e}^{x}\right)\right]$

$\textcolor{w h i t e}{f ' \left(x\right)} = {\left(\sec \left({e}^{x}\right)\right)}^{- \frac{3}{2}} \cdot - \frac{1}{2} \cdot \frac{d}{\mathrm{dx}} \left[\sec \left({e}^{x}\right)\right]$

$\textcolor{w h i t e}{f ' \left(x\right)} = {\left(\sec \left({e}^{x}\right)\right)}^{- \frac{3}{2}} \cdot - \frac{1}{2} \cdot \sec \left({e}^{x}\right) \tan \left({e}^{x}\right) \cdot \frac{d}{\mathrm{dc}} \left[{e}^{x}\right]$

$\textcolor{w h i t e}{f ' \left(x\right)} = - \frac{{e}^{x} \sec \left({e}^{x}\right) \tan \left({e}^{x}\right)}{2 {\left(\sec \left({e}^{x}\right)\right)}^{\frac{3}{2}}}$

$\textcolor{w h i t e}{f ' \left(x\right)} = - \frac{{e}^{x} \tan \left({e}^{x}\right)}{2 {\left(\sec \left({e}^{x}\right)\right)}^{\frac{1}{2}}}$

$\textcolor{w h i t e}{f ' \left(x\right)} = - \frac{{e}^{x} \tan \left({e}^{x}\right) \sec {\left({e}^{x}\right)}^{- \frac{1}{2}}}{2}$


text:
# How does an MCU pin detect a voltage?

If I have an MCU pin connected in between 2 resistors (a voltage divider) how does the MCU pin actually detect the voltage? I am using an ATMega328 and reading the voltage from the divider by an ADC enabled pin.

I believe my understanding is broken but here is what I'm thinking...

• the pin is floating but is set for input (has internal pull-up enabled)
• the top resistor in the divider takes some voltage (voltage drop)
• now, the wire that connects the middle of the voltage divider to the pin has a potential of x volts (it could potentially push by x volts)
• BUT, no current will flow through the connecting wire to the pin as the pin isn't at ground (there is nothing to "pull" the current towards it)
• if no current flows to the pin, how is voltage detected? If the current is 0 then according to Ohm's Law voltage would be 0 too
• Your question is unclear. What kind of pin? A pin connected internally to an ADC? or to a low-speed GPIO? Or to a high-speed serial link? Or to a reset line? – Marcus Müller Nov 21 '16 at 15:11
• This will all depend on your pin's function, and on the microcontroller used. I'm afraid there's no way around reading your microcontroller's datasheet. – Marcus Müller Nov 21 '16 at 15:11
• @MarcusMüller The pin would be one that is connected to an internal ADC. The MCU I use is an ATMega328. Sorry for being unclear but I'm very much a beginner – CS Student Nov 21 '16 at 15:16
• That's OK, we all began somewhere. Thanks for the edit! I think this is a great question now! – Marcus Müller Nov 21 '16 at 15:18
• So - are you really asking "how does an ADC work", or are you asking "how does an MCU interpret a voltage on a pin as a 1 or 0"? – brhans Nov 21 '16 at 15:29

the pin is floating but is set for input (has internal pull-up enabled)

is a paradox. Either it's floating, or it's pulled up. These things are opposites. "Pull-up" means there is a resistor that connects the pin to a fixed voltage; floating means that the pin is not connected to either ground or any non-zero voltage.

For ADC pins, pull-up doesn't make sense – you don't want to measure your supply voltage, you want to measure an external voltage!

Hence, there's no resistor voltage divider.

So, let's have a look at the datasheet, p.310:

It says there, the input resistance of the ADC is 100MOhm.

So, yes: Practically no current will flow into the ADC. That's a good thing, too, because if current flows into the the ADC, then that would change the voltage on the input, thus destroying the measurability of the signal of interest.

So, the question is: How does the actual measurement take place?

Now, that is a different question, and depends on the architecture of the integrated ADC. In case of your ATMega328, you're dealing with a successive approximation ADC. Which means, that via means of an analog circuit, the voltage at the moment you measure is "buffered", saved to a capacitor, and then, a DAC's output value is changed successively until it matches the stored voltage (very rough overview). I'd like to point you to the internet on successive approximation ADCs.

• Ah of course the ADC pin wouldn't be pulled up, that makes sense. So if the pin is floating but NOT at ground, how come a small current does flow? My understanding is that a current will flow from a higher potential to a lower potential but this may not be the case if the pin floats to a potential that is the same or more than the voltage being measured. I can see why a current would flow using Ohm's Law (we have a voltage and resistance of the ADC input) but I can't quite comprehend it in terms of flowing from a higher potential to a lower one. Thanks for the answer – CS Student Nov 21 '16 at 15:42
• The pin is not floating. If it was floating then nothing is connected to it. If nothing is connected to it then you aren't trying to measure anything sensible and so shouldn't expect a sensible result. The pin is being driven to a defined voltage by the input source, a voltage divider in this case. To to outside world the ADC pin looks like a 100 Mohm resistor to ground. This will have a slight impact on the voltage being measured however this impact is small enough that it can normally be ignored, most of the time it will be less than the measurement noise level of the ADC. – Andrew Nov 21 '16 at 16:02
• @Andrew Thanks Andrew that has cleared it up for me, I'm basically connecting a voltage source to a large resistor which is then connected to ground, therefore there is a potential difference between the source and ground allowing for a current to flow. The pin can't be floating as I've connected something to it! Thanks again!! – CS Student Nov 21 '16 at 16:06


text:
# Sidon Sets and Diophantine Equation

Suppose $$X$$ is a subset of $$\{1, \cdots, n\}$$ such that the equation $$ax_i+bx_j=cx_k+dx_{\ell}$$ where $$a+b=c+d,$$ $$a,b,c,d \in \mathbb{N}$$ and $$x_i, x_j, x_k, x_{\ell} \in X,$$ has only trivial solution. A solution is trivial if $$x_i=x_j=x_{k}=x_{\ell}.$$

What can we say about the size of $$X?$$ Is this possible that $$|X|\geq n^{1-o(1)}$$?

I think the answer is related to Sidon Sets, but I could not find any references. Any help is greatly appreciated.

• are $a,b,c,d$ fixed? Else there are solutions like $a=c,b=d$, $x_i=x_k$, $x_j=x_l$. – Fedor Petrov Oct 7 '18 at 19:53
• also maybe you need the reverse inequality for $|X|$? – Fedor Petrov Oct 7 '18 at 20:01
• Then "is it true" must be read as "is it possible"? And what is $\epsilon$? – Fedor Petrov Oct 7 '18 at 20:07
• what does it mean "$\epsilon$ is a positive constant depends on $n$?" – Fedor Petrov Oct 7 '18 at 20:53
• in any case: you may construct the set by adding the elements one by one, this allows to get about $c\cdot n^{1/3}$ elements for free. Is it enough? – Fedor Petrov Oct 7 '18 at 20:54

Such $$X$$ do indeed exist, and are explicitly constructed in I.Z. Ruzsa, Solving a linear equation in a set of integers, Acta Arith., LXV.3 (1993), pp. 259-282, Theorem 7.5. The whole paper is devoted to upper and lower bounds on sizes of solution-free sets to equations such as the one you are asking about. A mathscinet forward search from that paper should yield further results.


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# The Unapologetic Mathematician

## Cardinal numbers

I’ve said a bunch about natural numbers, but I seem to have ignored what we’re most used to doing with them: counting things! The reason is that we actually don’t use natural numbers to count, we use something called cardinal numbers.

So let’s go back and think about sets and functions. In fact, for the moment let’s just think about finite sets. It seems pretty straightforward to say there are three elements in the set $\{a,b,c\}$, and that there are also three elements in the set $\{x,y,z\}$. Step back for a moment, though, and consider why there are the same number of elements in these two sets. Try to do it without counting the numbers first. I’ll wait.

The essential thing that says there’s something the same about these two sets is that there is a bijection between them. For example, I could define a function $f$ by $f(a)=x$, $f(b)=z$, and $f(c)=y$. Every element of $\{x,y,z\}$ is hit by exactly one element of $\{a,b,c\}$, so this is a bijection. Of course, it’s not the only one, but we’ll leave that alone for now.

So now let’s move back to all (possibly infinte) sets and define a relation. Say that sets $X$ and $Y$ are “in bijection” — and write $X\leftrightarrow Y$ — if there is some bijection $f:X\rightarrow Y$. This is an equivalence relation! Any set is in bijection with itself, using the identity function. If $X$ is in bijection with $Y$ then we can use the inverse function to see that $Y\leftrightarrow X$. Finally, if $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are bijections, then $g\circ f:X\rightarrow Z$ is a bijection.

Any time we have an equivalence relation we can split things up into equivalence classes. Now I define a cardinal number to be an bijection class of sets — every set in the class is in bijection with every other, and with none outside the class.

So what does this have to do with natural numbers? Well, let’s focus in on finite sets again. There’s only one empty set $\{\}$, so let’s call its cardinal number ${}0$. Now given any finite set $X$ with cardinal number — bijection class — $c$, there’s something not in $X$. Pick any such something, call it $x$, and look at the set $X\cup\{x\}$. If I took any other set $Y$ in bijection with $X$ and anything $y$ not in $Y$ then there is a bijection between $x\cup\{x\}$ and $Y\cup\{y\}$. Just apply the bijection from $X$ to $Y$ on those elements from $X$, and send $x$ to $y$. This shows that the bijection class — the cardinal number — doesn’t depend on what choices we made along the way. Since it’s well-defined we can call it the successor $S(c)$.

We look at the set of all bijection classes of finite sets. We’ve got an identified element ${}0$, and a successor function. In fact, this satisfies the universal property for natural numbers. The set of cardinal numbers of finite sets is (isomorphic to) the set of natural numbers!

And that’s how we count things.

April 13, 2007 - Posted by | Fundamentals, Numbers

## 3 Comments »

1. […] We use cardinal numbers to count how many elements are in a set. Another thing we think of numbers for is listing elements. […]

Pingback by Ordinal numbers « The Unapologetic Mathematician | April 26, 2007 | Reply

2. […] and Ordinals as Categories We can import all of what we’ve said about cardinal numbers and ordinal numbers into […]

Pingback by Cardinals and Ordinals as Categories « The Unapologetic Mathematician | May 24, 2007 | Reply

3. […] to do the same thing. The term we tell them is “counting”, and the more technical term is cardinality. We can set up a bijection between two finite sets exactly when they have the same number of […]

Pingback by Reading Hume’s Principle Eliminatively? « if-then knots | December 10, 2008 | Reply


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# Group Which Isn't Solvable

For a recent qualifier problem, I was to show that if a group $G$ has two solvable subgroups $H,K$ such that $HK=G$ and $H$ is normal, then $G$ is solvable. This is simply a matter of showing that $G/H$ is solvable, and I think is not too difficult. The next part of the question was to find an example of a group with all of the above conditions except the normality condition (i.e. for any two such solvable subgroups, neither is normal in $G$) and show that $G$ is no longer solvable. Does anyone know of a good example? I don't even know that many groups which aren't solvable. I have been told $A_5$ is not solvable, but that is quite a large group, and it seems like it would take a long time to show this in 20 minutes (the time I would have if I was doing a qualifier) if it is even true for $A_5$. I'd love to know what group to look at, so I can prove it myself.

Thanks!

-
$A_5$ may appear large, but it is in fact the smallest non-solvable group... –  Mariano Suárez-Alvarez Apr 20 '11 at 16:20
So I see, (just read that any group of order less than 60 is solvable). My current strategy consists of showing that $S_5$ is the product of two cyclic groups. –  Jon Beardsley Apr 20 '11 at 17:13

Here's a hint - I don't know what you already know, so if you don't understand, just ask for clarification!

OK, so we're looking for two solvable subgroups $H$ and $K$ of a non-solvable group $G$, such that $HK = G$. The smallest non-solvable subgroup is indeed $A_5$, and every smaller group is solvable. In particular, $A_4$ and $C_5$ are solvable.

Can you find two subgroups of $A_5$, one isomorphic to $A_4$ and another isomorphic to $C_5$ which together generate $A_5$? Can you then show that not only $\langle H,K \rangle = G$ but also $HK = G$?

EDIT: In the previous wrong version I was hinting towards two copies of $A_4$ which together generate $A_5$, but do not have $HK = G$.

-
I don't think HK=G in your A4*A4=A5 example. A5=A4*C5 though. –  Jack Schmidt Apr 20 '11 at 17:36
@Jack You're completely right! Should've checked before leaving out the right answer. –  yatima2975 Apr 20 '11 at 18:18
Thanks @Jack and @Yatima. This is very helpful. –  Jon Beardsley Apr 21 '11 at 15:25


text:
Inverse trig integration

• May 5th 2010, 06:34 PM
Mattpd
Inverse trig integration
I have the function

$1/\sqrt{4-3x^2}$

and I see that it is in the form of arcsin. Where do I go from here? I know arctan has a little trick where the integral is

$(1/a) arctan (x/a)$

Does arcsin have a similar trick? How would you solve this?
• May 5th 2010, 06:51 PM
skeeter
Quote:

Originally Posted by Mattpd
I have the function

$1/\sqrt{4-3x^2}$

and I see that it is in the form of arcsin. Where do I go from here? I know arctan has a little trick where the integral is

$(1/a) arctan (x/a)$

Does arcsin have a similar trick? How would you solve this?

6. Integration: Inverse Trigonometric Forms
• May 5th 2010, 07:45 PM
Mattpd
Quote:

Originally Posted by skeeter

Thanks. It's even easier than the arctan form (Clapping)


text:
# Welcome.

## January 19, 2011

### Harmonic Analysis Lecture Notes 7

Filed under: 2011 Harmonic Analysis Course — xuhmath @ 6:50 pm

Weak-${L^p}$ and Lorentz spaces

For further information on this topic, see Chapter 1 of Grafakos’s text, “Classical Fourier Analysis”.

Definition For ${1 \leq p < \infty}$ and integer ${d \geq 1}$, we define the weak-${L^p({\mathbb R}^d)}$ space as the vector space of measurable functions ${f}$ on ${{\mathbb R}^d}$ such that

$\displaystyle ||f||^*_{L^{p,\text{weak}}({\mathbb R}^d)} = \sup_{0<\lambda<\infty} \lambda \cdot \big|\{ x : |f(x)| > \lambda \}\big|^{1/p} < \infty.$

Equivalently, the space consists of ${f}$ such that ${|\{ |f| > \lambda\}| \lesssim \lambda^{-p}}$.

Note: The “${*}$” is used in the notation to emphasize that this expression is not a norm (cf discussion on ${|| \:\:||^*_{L^{p,q}}}$).

For comparison’s sake, note that if ${f}$ is a measurable function on ${{\mathbb R}^d}$, then

$\displaystyle \begin{array}{rcl} ||f||_{L^p({\mathbb R}^d)} &=& \displaystyle \left( \int_{{\mathbb R}^d} \int_{0 \leq \lambda < |f(x)|} p\lambda^{p-1} \; d\lambda \; dx \right)^{1/p} \\ \\ &=& \displaystyle \left(\int_0^\infty \big|\{ |f| > \lambda \} \big| p\lambda^{p-1} \; d\lambda \right)^{1/p} \\ \\ &=& p^{1/p} \Big|\Big| \lambda |\{ |f| > \lambda \} |^{1/p} \Big|\Big|_{L^p\big( (0,\infty), \frac{d\lambda}{\lambda} \big)} \end{array}$

strongly resembles

$\displaystyle ||f||_{L^{p,\text{weak}}} = \underbrace{p^{1/\infty}}_{=1} \Big|\Big| \lambda |\{ |f| > \lambda\} |^{1/p} \Big|\Big|_{L^\infty\left( (0,\infty), \frac{d\lambda}{\lambda}\right)}$

and so suggests the following definition.

Definition For ${1 \leq p,q < \infty}$ and integer ${d \geq 1}$, we define the Lorentz space ${L^{p,q}({\mathbb R}^d)}$ as the vector space of measurable functions ${f}$ on ${{\mathbb R}^d}$ for which

$\displaystyle ||f||_{L^{p,q}}^* := p^{1/q} \Big|\Big| \lambda |\{ |f| > \lambda \}|^{1/p} \Big|\Big|_{L^q\left( (0,\infty), \frac{d\lambda}{\lambda}\right)}$

is finite. For the remainder of this discussion, we will write ${L^q\left(\frac{d\lambda}{\lambda}\right)}$ to mean ${L^q\left( (0,\infty), \frac{d\lambda}{\lambda}\right)}$.

Remark Indeed, ${L^{p,p} = L^p}$ and ${L^{p,\infty} = L^{p,\text{weak}}}$ by the above discussion.

Again ${|| \: \:\:||_{L^{p,q}}^*}$ is not a norm in general. Nevertheless, for all ${a \in {\mathbb C}}$,

$\displaystyle ||af||^*{L^{p,q}} = \Big|\Big| \lambda \big| \{ |f| > \frac{\lambda}{|a|} \} \big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)} = |a| \: \big|\big| f \big|\big|^*_{L^{p,q}}$

and

$\displaystyle \begin{array}{rcl} ||f+g||^*_{L^{p,q}} &=& \displaystyle \Big|\Big| \lambda \big|\{ |f+g| > \lambda\} \big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)}\\ &\leq& \displaystyle \Big|\Big| \lambda \left( \big| \{ |f| > \frac{\lambda}{2} \} \big| + \big| \{ |g| > \frac{\lambda}{2} \} \big| \right)^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)}\\ \\ \scriptsize\begin{bmatrix} \text{concavity of fractional}\\ \text{powers and Minkowski} \end{bmatrix} &\leq& \displaystyle \Big|\Big| \lambda \big| \{ |f| > \frac{\lambda}{2} \}\big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)} + \Big|\Big| \lambda \big| \{ |g| > \frac{\lambda}{2} \}\big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)} \\ \\ &\leq& 2 ||f||_{L^{p,q}}^* + 2||g||^*_{L^{p,q}}. \end{array}$

Thus, ${|| \:||_{L^{p,q}}^*}$ is a quasi-norm. When ${p \neq 1}$, this quasi-norm is equivalent to an actual norm, ${|| f||_{L^{p,q}}^* \lesssim_{p,q} || f||_{L^{p,q}} \lesssim_{p,q} || f||_{L^{p,q}}^*}$. When ${p=1}$ and ${q \neq 1}$, there cannot be a norm that is equivalent to ${|| \:||_{L^{p,q}}^*}$. Nevertheless, there is a metric which generates the same topology. In either case, we obtain a complete metric space.


text:
# What is the probability of having a solar eclipse on a planet?

If we go by our planets 5/8 of our planets have solar eclipses.

What is the probability of a planet having a solar eclipse based off other simulations?

Is there a more accurate way to calculate the probability of a solar eclipse than taking the averages of our current planets statistics? Related: What is the formula to predict lunar and solar eclipses accurately?

(4/79+7/62+12/27+6/6+0)/5

32.15%

• looks good, thanks! It'll be hard to answer though, requires a model of the statistical distribution of moons about exoplanets. Different but related: How to find Exomoons? and What's still needed before we can observe orbits of exomoons thereby weighing exoplanets?
– uhoh
May 22 at 3:13
• We don't know any exo- moons. Thus there is no data to check May 22 at 5:00
• I don't know. BUT... to get a solar eclipse a planet needs a moon with a size/orbital radius that will basically cover the star. Mercury and Venus don't have moons because they're too close to the sun, a moon couldn't exist in a stable orbit. So it can be figured which exoplanets CAN'T have a solar eclipse.
– Greg
May 22 at 15:51
• Are you assuming that you have a planet that is $d$ AU from a star with radius $r$, with a moon with distance $d_m$ away from the planet and radius $r_m$? May 31 at 2:01
• @fasterthanlight my initial answer would be yes but something may be going over my head. May 31 at 2:03


text:
# Isolated Heisenberg magnet as a quantum time crystal

@article{Medenjak2020IsolatedHM,
title={Isolated Heisenberg magnet as a quantum time crystal},
author={Marko Medenjak and Berislav Bu{\vc}a and Dieter Jaksch},
journal={Physical Review B},
year={2020},
volume={102},
pages={041117}
}
• Published 20 May 2019
• Physics
• Physical Review B
Local observables of closed many-body quantum systems are generally believed to quickly equilibrate after a quench, according to the eigenstate thermalization hypothesis. Here, the authors show that the paradigmatic Heisenberg quantum magnet violates this fundamental assumption. They analytically find measurable local correlation functions that show perpetual oscillations, while at the same time all single-particle observables equilibrate. They link their existence to the presence of dynamical…
32 Citations

## Figures from this paper

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It is analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints, which is explicitly identified.
Time-Resolved Observation of Thermalization in an Isolated Quantum System.
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This work uses trapped atomic ions forming a hybrid Coulomb crystal and exploit its phonons to study an isolated quantum system composed of a single spin coupled to an engineered bosonic environment and observes the emergence of thermalization.
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This work reports significant modification of the tunneling rate induced by interactions and shows clear evidence for collective effects in the oscillatory response in an ensemble of tilted one-dimensional atomic Bose-Hubbard chains after a sudden quench.
Discrete Time Crystals in the Absence of Manifest Symmetries or Disorder in Open Quantum Systems.
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A semiclassical approach reveals the emergence of a robust discrete time-crystalline phase in the thermodynamic limit in which metastability, dissipation, and interparticle interactions play a crucial role.
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The nonequilibrium dynamics of the quantum Ising model following an abrupt quench of the transverse field is studied, and it is observed that the dependence of tau(Q)(phi) on the effective temperature fairly agrees with that obtained in thermal equilibrium as a function of the equilibrium temperature.
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A recent experiment on a 51-atom Rydberg-blockaded chain observed anomalously long-lived temporal oscillations of local observables after quenching from an antiferromagnetic initial state. This


text:
# Subpixel - what is it?

I'm studying SuperResolution techniques, and I read often "subpixel"/"subpixel registration"/"subpixel accuracy". I don't find what exactly it means; Wikipedia says that is one of the three RGB channel in the pixel, but I think it isn't the right meaning in this context. Anyone can help explain what it is it or give me any tutorial?

• I'm lost that if we have two pixels v1 and v2 as mentioned as top,and their positions are recpectively p1(10,3) and p2(33,7). So how to calculate the p(avg). How to apply the formula p(avg)=(p1v1+p2v2)/(v1+v2); Jan 10 '17 at 11:20

Images are technically limited to the original pixel resolution. Objects in an image are spatially quantized at this resolution. However, an edge for instance may be localized "between" native pixels. Traditional tools like basic gradients often stick to that initial resolution. However, using additional information (multiple data, motion vectors, prior), one can infer a more accurate object location with a sub-pixel resolution, i.e. with an accuracy finer than that of the pixel width, at a fraction of the pixel size, with non-integer coordinates.

This can also apply to track/detect objects with a size smaller than that of a pixel. See for instance: What is a subpixel?

Using clever subpixel shifts, one can for instance reveal asymmetry not visible in the original image, as shown below:

Old and recent references:

Note: subpixels is also used in subpixel rendering on displays, and refers to colored pixel components (red, green, blue, plus yellow, white or cyan) that are displayed, or viewed, like a single pixel.

Sub-pixel means that instead of getting a location in the image in terms of $(x,y)$ coordinates at the pixel level, which means integer values for $x$ and $y$, the location is calculated to possibly give fractional pixel locations.

For example, instead of getting $(10,12)$ as a location, the algorithm may give $(9.934, 12.1234)$.

Suppose you have two pixels with values $v_1$ and $v_2$ and their pixel positions are $p_1$ and $p_2$ (which are integers). The average pixel position calculation could be: $$p_{\tt avg} = \frac{p_1 v_1 + p_2 v_2}{v_1 + v_2}$$ which will, in general, not give an integer value for $p_{\tt avg}$.

• how can it give fractional numbers?? Sep 5 '16 at 13:39
• @volperossa See my update.
– Peter K.
Sep 5 '16 at 15:38

Sub-pixel is a virtual pixel defined between two actual pixels in an image capturing sensor. to enhance the resolution or image quality, sub-pixel calculation is very useful. https://baike.baidu.com/item/%E4%BA%9A%E5%83%8F%E7%B4%A0/7904986?fr=aladdin


text:
Question
A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.50-m-high rise as shown in Figure 7.39. Find her final speed at the top, given that the coefficient of friction between her skis and the snow is 0.0800. (Hint: Find the distance traveled up the incline assuming a straight-line path as shown in the figure.)
Question Image
$9.5 \textrm{ m/s}$
Solution Video

# OpenStax College Physics Solution, Chapter 7, Problem 24 (Problems & Exercises) (6:50)

View sample solution

## Calculator Screenshots

Video Transcript

This is College Physics Answers with Shaun Dychko. This skier has some initial velocity of 12.0 meters per second and then they encounter a 35 degree slope. They go up the slope—a total height of 2.5 meters—and then they are skiing horizontally at the top and the question is what speed will they have on this top level assuming that there's a certain amount of friction due to a coefficient of friction of 0.0800 along this slope here. So we are going to figure out how much work is done by the non-conservative force, namely friction, along this slope and then that work is going to be the energy that is taken away from kinetic and potential energy that the skier had initially and then when they are at the top here, they will be left with just some potential and kinetic energy but the total will not be the total initial kinetic and potential energy because some of it will be taken away by the work done by the non-conservative force. There's a plus sign here which suggests that maybe this thing is adding energy but in fact, it's going to be negative because the friction and the displacement are in opposite directions and so this work will be negative. Okay! So the distance that they travel along the slope we can figure that out from this triangle here: we know that sin of this angle Θ is the opposite divided by the hypotenuse so that's h over d and then we can solve for d by multiplying both sides by d over sin Θ. We cancel the sin Θ's there and cancel the d's over here and we have d equals h over sin Θ. Okay! This is important because we are gonna substitute in for d here in this formula for the work done by the non-conservative force which is going to be the negative and I put negative there just because, you know, work is force times displacement times cosine of the angle between them and cos of 180 degrees is negative 1 or you could think of it as the displacement and the force are in opposite directions and that's why there's a negative there. Alright. The other factor here is this friction force and it's going to be their coefficient of friction multiplied by the normal force applied by the slope and the normal force is going to equal the component of gravity that is perpendicular to the slope. So gravity is straight down but there's this component here which is the force of gravity times cos of Θ which will be the perpendicular component of gravity and these two have to be equal because the skier is not accelerating perpendicular to the slope. Okay! So we have μ kF N and then substituting in the perpendicular component of gravity in place of normal force and we have μ k is times mgcos Θ is the friction force. So we can substitute for both friction force— μ kmgcos Θ—and substitute for the distance along the slope— which is h over sin Θ— and then we have this formula for the work done by friction. It's gonna be negative times coefficient of friction times mgh divided by tangent Θ— because cos over sin... there's an identity which says that cos Θ divided by sin Θ is the reciprocal of tan Θ. Okay! So all of this gets plugged into the work done by the non-conservative force here in our conservation of energy formula and then we can start plugging in for the other terms as well. So the initial kinetic energy is one-half mass times initial velocity squared plus the initial potential energy and we'll say that that is zero because we'll define this to be our reference level where h equals 0 so mgh would be mg times 0 and then adding to that... that should be a minus or maybe a plus for the formula and then a minus for the substitution but let's just resolve it into a single operation which is minus. Okay! And that equals one-half mv f squared plus mgh— this is the potential energy at the top of the slope and the kinetic energy at the top of the slope and this v f is what we are ultimately trying to find. So we'll make things look a little bit simpler by multiplying everything by 2 over m; it's kind of messy having fractions so we'll get rid of the fractions by multiplying by 2 and then we'll also get rid of the m which is a factor in all of the terms. So we are left with, after you switch the sides around, v f squared plus 2gh—the m is canceled and this 2 is canceled but the 2 appears here because it has to get distributed among both terms into the brackets— and that equals v initial squared minus 2μ kgh over tan Θ. I made a mistake here with the plus sign but nevertheless I corrected it anyway and it's correctly written as a minus here. Okay! So we are gonna subtract 2gh from both sides to solve for v f squared and we have this line and I factored out this common factor 2gh from both of these terms and it becomes minus 2gh times 1 plus μ k over tan Θ and then take the square root of both sides and that's all the algebra that we need to do. So v f is the square root of v initial squared minus 2gh times 1 plus μ k over tan Θ. So that's the square root of 12.0 meters per second squared minus 2 times 9.80 meters per second squared times 2.5 meters up the slope—that's the height of the slope— times 1 plus 0.0800—coefficient of friction— divided by tan of the slope angle of 35 degrees and this gives 9.5 meters per second. Now we are expecting some answer that is less than the initial speed so if we had had an answer more than 12, we would have said that doesn't pass the reality test but something less than 12 does. So 9.5 is plausibly the correct answer.