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# GPS Weather Forecasting for UAVs in Urban Environments

Python Aug 29, 2020

Abstract. The dominating sensor systems for outdoor localization are GNSS (Global Navigation Satellite System) receivers. Due to limited power consumption, small construction size, and low costs GNSS devices are embedded in most of non-autonomous vehicles (airplanes, cars, boats), autonomous systems (robots, drones, etc.) and mobile devices (smart phones, tablets). A single GNSS device provides a localization precision from 1 to 300 m, depending on the number of satellites, their geometrical configuration, and the current disturbance level. Unfortunately this kind of localization is most error prone in urban environments, because shading effects caused by buildings negatively influence localization accuracy.

This is especially problematic when one considers that many future applications based on UAVs (Unmanned Aerial Vehicles) are intended to operate in complex 3D environments, where precise and accurate position estimation is a prerequisite. In order to cope with vague and varying GNSS localization qualities, we have developed a system that can predict the localization uncertainty for a requested space, time, and resolution with different quality metrics.

Authors: André Dietrich & Sebastian Zug

# 1. Introduction

Most present day localization tasks for manned or unmanned intelligent vehicles are equipped with receivers for at least one Global Navigation Satellite Systems (GNSS). The ubiquity is reasonable, due to the fact that modern GNSS devices are cheap, small in size, have a low-energy consumption, and provide position information with a maximum uncertainty of 300 m for each position on or near earth. The continuous modernization of the U.S. Global Positioning System (GPS), the new European Galileo satellite system, Russian GLONASS, Chinese BeiDou, and the launch of quasi-zenith satellites by Japan illustrate the increased importance of an omnipresent position information.

Because of the previously mentioned maximum uncertainty level, most of today’s GNSS receivers are embedded in a multi-modal sensor system for (semi-)autonomous, automotive scenarios. GNSS measurements and inertial measurements are fused with camera and odometry data in order to provide more precise localization and speed information. In Section 2 we give a comprehensive overview of commonly applied approaches for improving localization quality based on GNSS measurements. These concepts provide only vague or not applicable improvements for near future scenarios in urban environments, such as delivery services based on Unmanned Aerial Vehicles (UAVs), as intended by Amazon PrimeAir [1], Google Wing [2], DHL Parcelcopter [3], the DomiCopter [4] and TacoCopter [5] projects (for fast food delivery), even first aid drones [6], or autonomous systems for law enforcement, surveillance, traffic monitoring, evaluation of accident scenes [7], etc. Here we have to guarantee a certain localization accuracy for all points of a 3D-trajectory too. UAVs operating in urban environments have to fulfill higher safety requirements, while state of the art mechanisms can only be applied partly or not at all. The power consumption and construction space are the limiting factors for additional sensors, whereby external disturbances (e. g., wind, rain, dust) and the number of degrees of freedom (depending on the UAV type) make the localization much more difficult. For this purpose we present a new kind of GNSS forecasting service for urban scenarios, which is currently implemented for GPS only. It can be used for localization oriented path planning, GNSS signal selection and filtering, and thus for minimizing localization uncertainties.

## 3.1 Our Approach

gments 3D geometrical representations of the environment with predicted GNSS information for a certain time or period of time. This new abstraction can be used to answer questions according to reliable localization quality, based on GNSS measurements, like:

• “What is the best time to perform a flight on a certain trajectory?”

• “At which height level does a UAV have to operate corresponding to a certain point in time and space?”

• “What is the optimal combination of satellite signals for a position estimation for a certain area and time?”

• “What is the average/minimal/deviation of/... localization quality in that region?”

For this purpose, we gather landscape and building geometries as well as status information about satellites from different web sources to predict the shading effects caused by the environment on GNSS measurements for a certain point in time and space, determine satellite visibility and calculate the corresponding localization quality.

The picture above shows an aerial view of Manhattan representing the GPS dilution of precision (DOP). This quality metric (which will be discussed in detail in Section 3.3) forms canyons and valleys between buildings with varying localization precisions. New kinds of 4D path planning algorithms can be applied on these representations to optimize flight trajectories, heights, energy consumptions, etc.

## 1.2 Overview

In the next section we discuss concepts for improving the quality of GNSS measurements and discuss the benefits of our approach, based on environmental context information (gained from highly accurate 3D models). In Section 3 we describe our concept in detail and outline each of the challenging tasks to be accomplished. The section contains and discusses furthermore the initial results, which is followed by a conclusion and an outlook for future works.

# 2. Related Work

Path planning algorithms considering the expected localization quality are already described for different robot domains and sensor systems [8], [9]. The scenarios described are often bound to a certain sensor type and environment. Nevertheless, the basic concept is always the same, beside the existing environment model an additional prediction and representation of the localization uncertainty is needed [10]. A comprehensive overview on trajectory planning methods for such uncertain environments is given by Jun, Chaudhry, and D’Andrea in [11].

Our mapping method includes a quality information for GNSS measurements and depends on appropriate metrics to represent them. Related errors and their sources for GNSS localization are discussed in detail in [12]. The effect on the localization quality is covered by a large number of studies [13], [14], which derive an associated statistical quality assessment. A corresponding probabilistic uncertainty is suitable to express the capabilities of a GNSS receiver in general, but it does not answer the question of the specific localization quality for a certain point.

GNSS receivers fuse individual distance measurements and calculate a position estimation. The effects of the above mentioned errors strongly depend on the geometric configuration of the satellite orbits and is described by the Dilution of Precision (DOP) in [15]. Wang, Groves, and Ziebart evaluated the DOP for small area in London. The authors furthermore revealed that a reliable localization is not possible for a lot of urban positions. The is DOP above a minimum threshold (we explain this issue in detail in Section 3.3.2) for 12 to 22 % of the evaluated positions [16]. Another study, presented in [15], discussed the fluctuation of DOP values over 24 hours and Milbert presented a detailed analysis of DOP and introduced DOP-maps in [17].

Beside the actual configuration of the GNSS, the situational context on Earth affects the quality of position estimations too. For urban scenarios in particular, the uncertainty increases due to the fact that the number of satellites in the line-of-sight is reduced [18]. The corresponding signals do not reach the receiver or get reflected from walls. Many papers address this problem and present different solutions, the most relevant paper [19] in this context was written by Wang, Groves, and Ziebart. The authors had implemented a tool-chain to evaluate the quality of GNSS signals with the help of a 3D city model. It was used to identify satellites in line-of-sight and at the end to improve GNSS predictions. An infrared camera system was applied in [20] to capture the surrounding skyline before calculating a position. The skyline was used to identify invisible satellites, which were excluded from the position calculation. The authors could show a significant improvement of the localization quality (for one specific position). Position estimations for buses were improved in [21] by context information about road map and the typical routes. Based on the assumption that devices are moving on a street, GNSS outputs can be weighted respectively. Other approaches evaluate signal characteristics like the signal strength in [22] or the different polarizations of reflected and direct line-of-sight signals [23].

All of the methods mentioned above are related to GNSS measurements and are focused on an online-improvement of the localization quality. In essence, this goal can be reached only by integrating additional sensors. In order to avoid associated costs and energy consumption we shift the effort to a pre-flight offline planning phase by predicting the GNSS localization accuracy with different quality metrics.

# 3 GPS Forecasting

## 3.1 Implementation & Sources

The diagram in Fig. 1 comprises all components of our implementation as well as all tapped data sources. A configuration file contains all necessary parameters, these are the location, time, and resolution of the measurement.

The DataGrabber is applied to gather all required information for a specific region. The main source is the web-api of Open Street Maps (OSM), which allows the possibility of selectively downloading maps in an OSM-XML format. It contains information about buildings, streets, traffic signs, places of interest, etc. In contrast to Google-Earth, OSM is an openly licensed map that is continuously updated by volunteers using local knowledge, GPS tracks, and donated sources. Due to the open data initiative there is a growing number of donations from various land registry offices and governmental institutions, especially in Europe. The high accuracy of OSM is also underpinned by the study in [24], at least for Germany. This data can be combined with SRTM files, which stands for Shuttle Radar Topography Mission and contains topological elevation information, to generate high-resolution 3D models with the help of OSM2World. The Bing web-api is applied to automatically retrieve satellite images, the minimum altitude of the terrain, and the central map coordinates. A continuously updated list of operational satellites (OPS) that describes their current state (position, velocity, inactivity) is retrieved from the Center for Space Standards & Innovation (which operates the satellite tracking web site CelesTrak).

All of this information in conjunction with the generated 3D model is used as input for our Scanner module (see also the screenshot in Fig. 2). It combines our odeViz library with the pyephem. odeViz is used to as a wrapper for ODE (Open Dynamics Engine) simulations and automatically sets up the appropriate visualization with VTK (Visualization Toolkit). pyephem is a package for performing high precision astronomy computations.

## 3.2 Line-of-Sight Analysis

Related to the physical operating principle, the availability is limited by the need for an unobstructed line-of-sight on at least four GNSS satellites. At least three satellite signals are necessary to solve the equation system related to the requested position (longitude, latitude, and altitude). An additional signal has to be evaluated due to the uncertainty of the receiver clock. For the GPS system each position is theoretically covered by up to twelve satellites simultaneously. The duration of a satellite orbit above an observer’s horizon depends on its position related to the six main orbital planes.

Due to the low transmission power of the GNSS signals a direct line-of-sight between satellite and receiver is necessary. Consequently, the number of actually available satellites is reduced by geographical conditions (mountains, canyons), vegetation or buildings. This analysis happens within the Scanner module (cf. Fig. 2), which is in fact rigid body simulation, and contains the entire 3D city model. pyephem is applied to determine the relative positions of the GPS satellites according to a certain point within the model and according to a certain point in time. By using rays, which start at a local position and extend in the direction of the satellites, it is possible to check whether a satellite is in line-of-sight or not. If there is no collision detected between the model and the ray, then the satellite should be visible from that position, otherwise not.

To demonstrate this discrepancy between the number of satellites that should be visible (above horizon) and hidden ones, we chose the the area of Manhattan already depicted in Fig. 2. In contrast to most districts in that region, it contains a free round area, buildings are not posed on a grid like structure, and furthermore the building geometries themselves are more complex than simple cubes. The screenshot in Fig. 3 shows an overlay of our model based-predictions with the results obtained from Trimble (an online GNSS planning tool) for a period of 24 hours. The black stripes indicate that a satellite is above the horizon but not in direct line-of-sight.

Our predictions matched those of Trimble for the considered GPS location and show that satellites can be out of sight for quite long periods of time, and thus not usable for position estimations, especially during satellite rise and set. Thus, most of the time, the number of visible satellites is much smaller than predicted with common methods. Space is another factor causing fluctuations. Fig. 4 contains two snapshots of a 24-hour analysis (available on our website), which depicts the difference in the number of satellites in line-of-sight in 2D for two different points in time. It is easy to see, that a deviation of one meter can cause a tremendous decrease or increase of available satellite signals.

It is primarily the proximity to one of the buildings that causes a reduction of visible satellites, but it has a further consequence (not directly visible in Fig. 4). The geometrical constellation of line-of-sight satellites is poor compared to free space, because satellite signals can only be received from limited angular positions. That causes a decrease in localization quality. Thus, the number of visible satellites cannot be directly applied as a quality metric.

In reality, some of the blocked satellite signals reach the receiver after reflections. The additional transmission length results in so called multi-path-errors and have to be excluded from further processing. Because of the shading effects, some authors have developed multi-GNSS systems that combine the satellites GPS with GLONASS or GALILEO to cope with this problem (cf. [25], [26]). Depending on the position and the level of the horizon, the improvement is significant but can be reached only with additional hardware.

## 3.3 Quality Metrics

As indicated, the number of visible satellites is a weak quality metric. It does not incorporate the magnitude of localization errors or uncertainty for a certain position, which is caused by a specific satellite constellation.

### 3.3.1 Distance Errors

For a localization based on a GNSS we evaluate the transient time between satellite and receiver in a first step. Signal propagation is disturbed by atmospheric conditions, satellite ephemeris errors, and clock drifts. The corresponding pseudo-range $\rho_i$ can be calculated by

$$\rho_i = \rho^T_{i} + c (\delta^s_i - \delta_R) \text{ for } 4 \leq i \leq n$$

where $\rho^T_{i}$ is the real range from the true position to the $i$th of $n$ satellites [15]. This distance information is overlaid by errors generated by an inaccuracy of the receiver clock ($\delta_R$) and external errors ($\delta_i$) that are related to atmospheric conditions, satellite ephemeris errors, and clock drifts. $c$ stands for the vacuum speed of light. A GNSS receiver has to solve the corresponding equation system to calculate the position $\hat{x}$ with a covariance of $C_{\hat{x}}$ according to

\begin{align}
\label{equ:C}
C_{\hat{x}}=\left(A^T C^{-1}_{\Delta\rho} A\right)^{-1}
\end{align}

$C_{\Delta\rho}$ represents the covariance matrix of the pseudo-range measurements. $A$ is an $n \times 4$ matrix of the partial derivatives of the pseudo-ranges $\rho_i$ with respect to the unknown values for position and time, while $R_{i}$ denotes the euclidean distance between the $i$th satellite and the receiver.

\begin{align}
\label{equ:A}
A =
\begin{pmatrix}
\frac{x_{1}-x}{R_{1}} & \frac{y_{1}-y}{R_{1}} & \frac{z_{1}-z}{R_{1}} & -1 \newline
\vdots & \vdots & \vdots & \vdots \newline
\frac{x_{n}-x}{R_{n}} & \frac{y_{n}-y}{R_{n}} & \frac{z_{n}-z}{R_{n}} & -1
\end{pmatrix}
\end{align}

If we assume a constant error model for all distance measurements with a particular standard deviation $\sigma_{\rho}$, we can define $C_{\hat{x}} = \sigma^2_{\rho}I$, where $I$ represents the identity matrix $n \times n$. Consequently, \eqref{equ:C} can be simplified to

\begin{align}
\label{equ:Csimple}
C_{\hat{x}}=\left(A^T A\right)^{-1} \sigma_{\rho}^2 = D \sigma_{\rho}^2
\end{align}

### 3.3.2 Dilution of Precision (DOP)

Equation \eqref{equ:C} and \eqref{equ:Csimple} show an important aspect -- the common error level is defined by the distance errors and a gain ($A$) that represents the geometrical relation of the available satellites. If all satellites are adjusted on a straight line in x-direction for instance, the rows of $A$ are linearly dependent due to $y_i, z_i = const.$ Hence, the position uncertainty based on a certain distance error $\sigma_{\rho}$ is much higher than for an orthogonal deployment of the satellites.

This effect is described by the DOP. It represents the gain of the initial distance error and can be calculated by

\begin{align}
DOP = \frac{\Delta \text{Output Location}}{\Delta \text{Measured Data}}
\end{align}

If we follow the idea of a constant non-correlated $\sigma_{\rho}$ for each satellite, the DOP is a suitable abstraction to compare relatively the GNSS localization quality for specific points and times [27]. DOP values can be calculated for different contexts. The horizontal HDOP addresses the effect of the satellite configuration on the latitude and longitude values. The VDOP focuses on the altitude quality, whereas the positional PDOP includes all three localization aspects [15]. The values can be calculated by

\begin{align}
\text{HDOP} &= \frac{\sigma_H}{\sigma_{\rho}} = \sqrt{D_{11} + D_{22}} \nonumber\newline
\text{PDOP} &= \frac{\sigma_P}{\sigma_{\rho}} = \sqrt{D_{11}+D_{22}+D_{33}}
\end{align}

In the case of an optimal satellite constellation the measurement error is notextended and consequently we receive a DOP of $1$. Worst-case scenarios generate much higher values. Tab. 1 summarizes the consequences of DOP values based on the maximum error level for $\sigma_{\rho}$. A DOP value larger than $20$ indicates that the between the GNSS measurement and the true position deviates by more than 300 m.

Tab. 1: DOP-rating for GNSS measurements [17], [27]

DOP Rating Description
1 Ideal The highest possible precision at all times.
1 − 2 Excellent Positional measurements are considered as accurate enough to meet the most sensitive applications.
2 − 5 Good Positions fulfill the uncertainty demands of inroute navigation suggestions.
5 − 10 Moderate Measurements have a limited quality. A more open view of the sky is recommended.
10−20 Fair Position measurements indicate only a very rough estimate of the location.
> 20 Poor At this level, the uncertainty reaches a nontolerable level, the measurements should be discarded.

While GNSS receivers provide run-time DOP values for specific locations, we can now pre-calculate the DOPs for a complete area. The corresponding HDOPs for the scenarios depicted in Fig. 5 are visible in the diagrams below. At most positions between the 13-floor buildings the HDOP is high (dark blue, $\text{HDOP} > 20$), a localization based on GPS measurements is thus not possible or reliable there. The situation changes at noon, where the area with a $\text{HDOP}<5$ increases, due to the fact that fewer satellites are hidden behind buildings and that they are in a better geometrical configuration. Consequently, at different points in time we can find passages between the buildings with good localization qualities.

### 3.3.3 4D-Mapping

The HDOP values in Fig. 5a and 5b were calculated at ground level and passages become visible at different points in time. Now, imagine we would lift up, the number of hidden satellites would automatically decrease, resulting in better DOPs. We can now build 3D maps where good DOPs form canyons between buildings and points with poor localization quality. In contrast to Fig. 5, the image on the first page shows a 3D representation of the PDOP values for Downtown Manhattan (a worst case environment compared to Stuyvesant Town) on 4th of February 2015. The skyline is visible, as is Battery Park in the foreground. The street canyons are overlaid by different layers representing a certain PDOP level.

Based on our GPS weather forecasts, we should be able to determine trajectories within such DOP canyons that guarantee a high localization quality at all positions. Localization information has to be combined with energy consumptions aspects. The ascent and descent to different altitudes consumes energy and extends the trajectory length for a UAV. But in contrast to most common trajectory planning algorithms we also have to deal with time as the 4th dimension. Passages with good DOPs are only open for a certain period of time. The calculated trajectories to a destination point and back can thus be completely different.

# 4. Conclusion & Outlook

In this paper we defined a new map representation for drones path planning that combines 3D building information with an expected localization quality based on GNSS measurements at each position. We described the sources and the work flow to generate the 3D environment model, the mechanisms to predict the GNSS satellite positions and the calculation of a 3D DOP map. The project achieves this using exclusively open access data and open source software.

With this paper we build the foundation for 4D-DOP based path planning for UAVs. In further steps we want to enhance the project to make it applicable for large-scale scenarios with autonomous robots and UAVs:

• At this point, the uncertainty measure evaluates only the DOP values. They just provide relative comparisons of the localization precision for different positions. We want to implement a more distance oriented error model, which provides an absolute value and considers tropospheric and atmospheric effects.

• OSM-3D includes a large number of plugins and applications for easy access on the available data. The GPS weather forecast should receive the status of an official plugin that provides the predicted uncertainty as a 2D map for driving scenarios. The 3D representation for UAVs will be available in a separate web service.

• As a part of the UAV path planning, we have to derive appropriate cost functions and to select an appropriate 4D algorithm evaluating multiple 3D GNSS maps.

# References

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[2] BBC News - Techonology, Google tests drone deliveries in Project Wing trials, Online press release, Aug. 2014. [Online]. Available: http://www.bbc.com/news/technology-28964260

[3] DHL International GmbH, DHL parcelcopter launches initial operations for research purposes, Online press release, Sep. 2014. [Online]. Available: http://www.dhl.com/en/press/releases/releases_2014/group/dhl_parcelcopter_launches_initial_operations_for_research_purposes.html

[4] U.S.News, Drones Have a Future in Deliveries, But Not for Domino’s Pizza, Online press release, Jun. 2013. [Online]. Available: http://www.usnews.com/news/articles/2013/06/06/drones-have-a-future-in-deliveries-but-not-for-dominos-pizza

[5] Huffington Post, Flying Robots Deliver Tacos To Your Location, Online press release, Mar. 2013. [Online]. Available: http://www.huffingtonpost.com/2012/03/23/tacocopter-startup-delivers-tacos-by-unmanned-drone-helicopter_n_1375842.html

[6] Techcrunch, This Ambulance Drone Can Fly Into Trouble With First Aid, Online press release, Oct. 2014. [Online]. Available: http://techcrunch.com/2014/10/31/this-ambulance-drone-can-fly-into-trouble-with-first-aid/

[7] U-T Santiago, Tijuana to use drones, Online press release, Dec. 2013. [Online]. Available: http://www.utsandiego.com/news/2013/dec/07/drones-3D-robotics-tijuana-san-diego-mayor/

[8] H. Takeda, C. Facchinetti, and J.-C. Latombe, “Planning the motions of a mobile robot in a sensory uncertainty field,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 10, pp. 1002–1017, 1994.

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[10] J. Faigl, T. Krajnık, V. Vonásek, and L. Preucil, “Surveillance planning with localization uncertainty for uavs,” in 3rd Israeli Conference on Robotics, 2010.

[11] M. Jun, A. I. Chaudhry, and R. D’Andrea, “The Navigation of Autonomous Vehicles in Uncertain Dynamic Environments: A Case Study (I),” in IEEE Conference on Decision and Control, IEEE; 1998, vol. 4, 2002, pp. 3770–3775.

[12] B. Hofmann-Wellenhof, H. Lichtenegger, and E. Wasle, GNSS – Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and more. Springer, 2007.

[13] M. Matosevic, Z. Salcic, and S. Berber, “A Comparison of Accuracy Using a GPS and a Low-Cost DGPS,” IEEE Transactions on Instrumentation and Measurement, vol. 55, no. 5, pp. 1677–1683, 2006.

[14] L. L. Arnold and P. A. Zandbergen, “Positional accuracy of the wide area augmentation system in consumer-grade GPS units,” Computers & Geo-sciences, vol. 37, no. 7, pp. 883–892, 2011.

[15] R. B. Langley, “Dilution of Precision,” GPS world, vol. 10, no. 5, pp. 52–59, 1999.

[16] L. Wang, P. D. Groves, and M. K. Ziebart, “Multi-constellation GNSS performance evaluation for urban canyons using large virtual reality city models,” Journal of Navigation, vol. 65, no. 03, pp. 459–476, 2012.

[17] D. Milbert, “Dilution of precision revisited,” Navigation, vol. 55, no. 1, pp. 67–81, 2008.

[18] P. D. Groves, Z. Jiang, L. Wang, and M. Ziebart, “Intelligent urban positioning, shadow matching and non-line-of-sight signal detection,” in 6th ESA Workshop on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (NAVITEC), 2012, pp. 1–8.

[19] L Wang, P. Groves, and M. Ziebart, “Urban Positioning on a Smartphone: Real-time Shadow Matching Using GNSS and 3D City Models,” Inside GNSS Magazine, vol. 8, no. 6, pp. 44–56, 2013.

[20] J. Meguro, T. Murata, J. Takiguchi, Y. Amano, and T. Hashizume, “GPS accuracy improvement by satellite selection using omnidirectional infrared camera,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2008, pp. 1804–1810.

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text:
# Question 0601d

Jul 20, 2017

$\text{197 g}$

#### Explanation:

If I understand your question correctly, the answer will be $\text{197 g}$.

The trick here is to realize that the combined mass of the reactants must be equal to the combined mass of the products $\to$ think the Law of conservation of mass here.

You know that you're mixing $\text{20 g}$ of aluminium metal and $\text{200 g}$ of bromine, but that the reaction only consumes

overbrace("200 g")^(color(blue)("what is available")) - overbrace("23 g")^(color(blue)("what is left behind")) = "177 g"

of bromine. This means that the combined mass of the reactants will be

$\text{20 g + 177 g = 197 g}$

Since the reaction will only produce aluminium bromide, you can say that the mass of the product must be equal to $\text{197 g}$.

CHECK THE RESULT USING MOLES

If you want, you can check the result by using the balanced chemical equation

$2 {\text{Al"_ ((s)) + 3"Br"_ (2(l)) -> "Al"_ 2"Br}}_{6 \left(a q\right)}$

Use the molar masses of the two reactants to calculate how many moles of each take part in the reaction--keep in mind that only $\text{177 g}$ of bromine react!

20 color(red)(cancel(color(black)("g"))) * "1 mole Al"/(27.0color(red)(cancel(color(black)("g")))) = "0.74 moles Al"

177 color(red)(cancel(color(black)("g"))) * "1 mole Br"_2/(159.8color(red)(cancel(color(black)("g")))) = "1.11 moles Br"_2

Aluminium reacts with bromine in a $2 : 3$ mole ratio

0.74 color(red)(cancel(color(black)("moles Al"))) * "3 moles Br"_2/(2color(red)(cancel(color(black)("moles Al")))) = "1.11 moles Al"

and produces aluminium bromide in a $2 : 1$ mole ratio

0.74 color(red)(cancel(color(black)("moles Al"))) * ("1 mole Al"_2"Br"_6)/(2color(red)(cancel(color(black)("moles Al")))) = "0.37 moles Al"_2"Br"_6

Finally, to convert this to grams, use the molar mass of aluminium bromide

0.37 color(red)(cancel(color(black)("moles Al"_2"Br"_6))) * "533.4 g"/(1color(red)(cancel(color(black)("mole Al"_2"Br"_6)))) ~~ "197 g"#

Without taking into account the number of sig figs you have for your values, you can say that th answer will once again be $\text{197 g}$ of aluminium bromide.

format: plain


text:
# ${{\boldsymbol K}^{\pm}}$ MEAN LIFE INSPIRE search

VALUE ($10^{-8}$ s) EVTS DOCUMENT ID TECN CHG  COMMENT
$\bf{ 1.2380 \pm0.0020}$ OUR FIT  Error includes scale factor of 1.8.
$\bf{ 1.2379 \pm0.0021}$ OUR AVERAGE  Error includes scale factor of 1.9.
$1.2347$ $\pm0.0030$ 15M 1
 2008
KLOE $\pm{}$ ${{\mathit \phi}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$
$1.2451$ $\pm0.0030$ 250k
 1995
CNTR ${{\mathit K}}$ at rest, U target
$1.2368$ $\pm0.0041$ 150k
 1995
CNTR ${{\mathit K}}$ at rest, Cu target
$1.2380$ $\pm0.0016$ 3M
 1971
CNTR + ${{\mathit K}}$ at rest
$1.2272$ $\pm0.0036$
 1969
CNTR + ${{\mathit K}}$ in flight
$1.2443$ $\pm0.0038$
 1965 B
CNTR + ${{\mathit K}}$ at rest
• • • We do not use the following data for averages, fits, limits, etc. • • •
$1.2415$ $\pm0.0024$ 400k 2
 1995
CNTR ${{\mathit K}}$ at rest
$1.221$ $\pm0.011$
 1967
CNTR $\pm{}$
$1.231$ $\pm0.011$
 1962
CNTR +
1  Result obtained by averaging the decay length and decay time analyses taking correlations into account.
2  KOPTEV 1995 report this weighted average of their U-target and Cu-target results, where they have weighted by 1/$\sigma$ rather than 1/$\sigma {}^{2}$.

${{\mathit K}^{\pm}}$ mean life (${10^{-8}}$s)
References:
 AMBROSINO 2008
JHEP 0801 073 Measurement of the Charged Kaon Lifetime with the KLOE Detector
 KOPTEV 1995
JETPL 61 877 Measurement of ${{\mathit \pi}^{\pm}}$ and ${{\mathit K}^{\pm}}$ Lifetime
 OTT 1971
PR D3 52 Precise Measurement of the ${{\mathit K}^{+}}$ Lifetime
 LOBKOWICZ 1969
PR 185 1676 Precise Measurement of the ${{\mathit K}^{+}}/{{\mathit K}^{-}}$ Lifetime Ratio
 FORD 1967
PRL 18 1214 Comparison of the ${{\mathit K}^{+}}$ and ${{\mathit K}^{-}}$ Decay Rates into the ${{\mathit \tau}}$ and ${{\mathit K}_{{\mu2}}}$ Modes
 FITCH 1965B
PR 140 B1088 Study of the ${{\mathit K}^{+}}$ Decay Probability
 BOYARSKI 1962
PR 128 2398 Study of the ${{\mathit K}^{+}}$ Decay
see fit info


text:
# Measure growth of small sets in $$\mathrm{SO}(3,\mathbb{R})$$

21.03.2023 15:00 - 17:00

Tran Chieu Minh (NU Singapore)

Let $$\mathrm{SO}(3,\mathbb{R})$$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $$\mu$$. Confirming a conjecture by Breuillard and Green, we show that if $$A \subseteq \mathrm{SO}(3,\mathbb{R})$$ is open and has sufficiently small measure, then $$\mu(A^2) > 3.99 \mu(A)$$. We also show a more general result for the product of two sets, which can be seen as a Brunn-Minkowski-type inequality for sets with small measure in $$\mathrm{SO}(3,\mathbb{R})$$. (Joint with Yifan Jing and Ruixiang Zhang)

Organiser:

Location:


text:
# ICHEP 2022

Jul 6 – 13, 2022
Bologna, Italy
Europe/Rome timezone

## Oscillations of Majorana neutrinos in supernova and CP violation

Jul 8, 2022, 7:05 PM
1h 25m
Bologna, Italy

#### Bologna, Italy

Palazzo della Cultura e dei Congressi
Poster Neutrino Physics

Artem Popov

### Description

Leptonic CP violation is one of the most important topics in neutrino physics. CP violation in
the neutrino sector is also strongly related to the nature of the neutrino: whether it is a Dirac
or a Majorana particle. In this contribution CP-violating effects in Majorana neutrino
oscillations in supernova media are studied. We show that resonances in neutrino-
antineutrino oscillations induced by strong magnetic fields of astrophysical objects appear in
the case of nonzero CP-violating phases. Our findings suggest a potential astrophysical setup
for studying the nature of neutrino masses and leptonic CP violation and may be important
for future neutrino experiments, for example, such as JUNO and Hyper-Kamiokande.

Based on: A.Popov, A.Studenikin, “Manifestations of nonzero Majorana CP-violating phases
in oscillations of supernova neutrinos”, Phys.Rev.D 103 (2021) 11, 115027.

In-person participation No


text:
# Using \columncolor outside column definition

I'm trying to create a command, which calls on an environment, which defines a custom column definition with one argument using \newcolumntype, and defines a second command, containing parameters for a table column. Then, still within the first command, I begin the tabular environment and call the custom column, with the second command as its argument. I always get an undefined control sequence error, though. Any thoughts on why that might be?

I'm sure that was clear as mud, but I've cut it down to a minimal example, which follows:

\documentclass{article}
\usepackage{array}
\usepackage[table,dvipsnames]{xcolor}
\usepackage{colortbl}

\newenvironment{tablecolor}{
\newcommand{\tlight}{\columncolor{SkyBlue!30}\color{black}}
\newcolumntype{C}[1]{>{##1}c}
}{}

\newcommand{\tablegoals}{
\begin{tablecolor}
\begin{tabular}{l|C{\tlight}}
Goal 7  & \\
\end{tabular}
\end{tablecolor}
}

\begin{document}
\tablegoals
\end{document}

Here's my second attempt. Different approach, same undefined control sequence error with \columncolor.

\documentclass{article}
\usepackage{array}
\usepackage[table,dvipsnames]{xcolor}
\usepackage{colortbl}
\usepackage{ifthen}

\newenvironment{tablecolor}{
\newcolumntype{C}[1]{>{
\ifthenelse{\equal{dark}{##1}}{
\columncolor{Cyan}\color{white}
}{
\columncolor{SkyBlue!30}\color{black}
}
}c}
}{
}

\newcommand{\tablegoals}[1]{
\begin{tablecolor}
\begin{tabular}{l|C{dark}}
Goal 7  &   #1
\end{tabular}
\end{tablecolor}
}

\begin{document}
\tablegoals{Check mark}
\end{document}

-
Welcome to TeX.SE. Thanks for providing a MWE, that really helps. However, in this case I am not clear what you want, even after reading the comments in the other answer. I would suggest you edit this question to do a table that has ONE of the types that you want and then document that your would like to be able to reuse the same table and just change the specific colors. Then we could help you with the \newcommand –  Peter Grill Nov 13 '11 at 4:33
I would like to be able to quickly generate the same table, but I'd also like to be able to generate tables with different layouts employing the same color scheme, as well as a greyscale color scheme. –  volfied Nov 13 '11 at 5:18

Wow, you've got a bit of a mess on your hands there! First off, you don't need that new environment, or at least you don't need those commands in it. Try this:

\documentclass{article}
\usepackage{array}
\usepackage[table,dvipsnames]{xcolor}
\usepackage{colortbl}
\newenvironment{tablegoals}[1]
{\begin{tabular}{l|>{\columncolor{#1}}c}}
{\end{tabular}}

\begin{document}
\begin{tablegoals}{SkyBlue!30}
Goal 7  & Something \\
\end{tablegoals}
\end{document}

The \newwnvironment here takes the name of your new environment as the first argument, the stuff that comes before the content of your environment as the second, and the stuff at the end your environment as the third. Thus the second argument contains the \begin{tabular}{...} including the colour definition of the second column. The third argument simply ends the table. There is no need to define any new commands, since this new environment allows you to type all this only once anyway.

If you find yourself defining a lot of tables with different column colours, then perhaps the following newcommand can help:

\newcommand{\newcolumnwithcolor}[2]{\newcolumntype{#1}{>{\columncolor{#2}}c}}

Now in your preamble you can define lots of column types quickly and readably

\newcolumnwithcolor{C}{SkyBlue!30}

which you may use in a tabular environment.

\begin{tabular}{l|C}
Goal 7  & Something \\
\end{tabular}

Note that is seems that these may only be named with a single character.

Finally, you can define a special column that takes arguments. If you use the same column definition and only change the colour for example, then the following works:

\newcolumntype{D}[1]{>{\columncolor{#1}}c}

\begin{tabular}{|l|D{SkyBlue!10}|}
Goal 7  & Something \\
\end{tabular}

The number of additional arguments may be more than one, so you can add alignment as an option etc.

-
Well, I did simplify to the extent that some of it makes no real sense. I'm trying to save myself the trouble of repeating things over and over. I need the ability to make a color table and a B&W table. The environment would redefine the parameters for the dark and light colors (blue, light blue; gray, white), along with arrayrulecolors. I'm not sure my method is the most efficient way to do it, but I'd like the flexibility to start an environment that changes the table to color or B&W, then start tabular and not have to muck around with lots of > commands. Am I making sense? –  volfied Nov 13 '11 at 3:50
I'm not sure I'm following. Why not define two tables like the above, one with colour and one without? Or you could add additional parameters to take a colour as part of the environment. –  qubyte Nov 13 '11 at 3:56
And if I want to use \newcolumntype, I just need to define separate ones for dark and light, lcr of each? I guess I was hoping to avoid that, since the column names can only be one character and they start getting completely arbitrary. Entirely possible I'm not making my life any less complicated this way, of course. –  volfied Nov 13 '11 at 3:56
Well, you don't need to define a new column type at all if you're using custom environments. You'll have to write it all once of course. I have the feeling that we're not on the same page though... –  qubyte Nov 13 '11 at 3:59
While I'm sure you're not inclined to wade through a ton of code, here's the full context, if you've interested and have the time. I appreciate whatever help you can offer. dl.dropbox.com/u/1716280/indiana.sty dl.dropbox.com/u/1716280/8-brumback.tex –  volfied Nov 13 '11 at 4:02

I asked David Carlisle, who wrote both the colortbl package and \newcolumntype in the array package about this problem. Here's his response:

the error message is telling you that \columncolor is not defined and in fact looking at colortbl.sty there is no definition for such a command, it is just used as a token in

\expandafter\CT@extract\the\toks\@tempcnta\columncolor!\@nil

which means essentially that after the normal array package table processing has done its stuff, colortbl goes in finds that token as a marker and inserts some extra stuff.

I can't remember why it was written that way (the log says it's 15 years ago:-) but probably there was a reason at the time.

The result (which is probably not documented too well) is that if you hide \columncolor in a macro it won't work, it has to be there explicitly at the top level. You could use newcolumntypes defining one color or the other, or taking a parameter taking the colour, as they are expanded out by array.sty before colortbl starts looking for its tokens.

-


text:
Why will my led not light up & why does my input voltage fluctuate when I add a wire?

This design is suppose to light up the LED When the sensor is under light, and turn off the LED when the sensor is covered.

I noticed two problems:

1. My LED won't light up.

2. When I measure the voltage input difference (expecting +15- V) without the wire connecting the second prong of my 50k potentiometer (25th node) to the second prong of the op-amp, I measure 15 volts. But when I insert that wire how it is in the picture, my same input voltage measurement starts fluctuating. I assume this is a problem

Of course, I tried changing the values of the potent. and sensor but had no luck. I also tested my op-amp and it seems to be working correctly. The LED is also working correctly because it lit up in a different scenario. & Yes, I am also aware there is no voltage input in the photo. These were taken after I unplugged The input.

Will you please look over my design and help me locate the problem?

EDIT: New circuit based off new understanding

EDIT2: Just talked to my instructor. I indeed originally wired the purple wire wrong, but it should be fixed in the last two photos. I also confirmed that my resistor is not shorted and the photo is just confusing everyone, so I will upload a new one once I get home.

EDIT3: Less confusing photo (made sure you could tell the led is not shorted)

SOLUTION: Thank you @jonk and @TonyStewart!

My circuit was wrong in that I did not realize my potentiometer should not connect to the op-amp's 3pin (basically I didn't understand the schematic).

Later on, @jonk helped me come to the realization that there was something wrong with my op-amp, so I used a new one and it now works.

• Probably not the cause, but where is your decoupling capacitor? Mar 1 '17 at 6:11
• @winny I haven't learned about / used decoupling capacitors. This lab should work without one. Mar 1 '17 at 6:14
• Pot is pointing at 15V try <1V Mar 1 '17 at 6:34
• I can see the arrow is pointing to the pin connected to V+. Get a voltmeter and report every pin with pot set to 0V. Mar 1 '17 at 6:48
• Look carefully at your LED connection. You have both LED pins on the same horizontal row, which means they are connected together, and nothing connecting to them. You need to rotate the LED 90 degrees, then connect the resistor and comparator properly. Mar 1 '17 at 12:42

I think you may have wired it up wrong. It looks to me like you did this:

simulate this circuit – Schematic created using CircuitLab

I think you want this:

simulate this circuit

Pull the end of the purple jumper off of pin 3 and move it to your $+15\:\textrm{V}$ rail.

Below is a piece of your schematic and some information I've added. The arrow points to a "cross-over" and NOT to a connection. Just in case that helps.

Your newly added picture does appear to show a shorted LED. However, your first picture does not appear to show a shorted LED. I wouldn't have been able to tell from when I was reading your post, because your last picture wasn't added by then. Now, looking at it it does appear that it could be shorted out. Did you move it between pictures???

Anyway, that is another detail to look at and double-check. I think you already know (knew) that all five holes along a row are all connected together. That's a detail that everyone needs to know just to use a protoboard. So I assume you are aware of that. But anyone can accidentally get a lead in the wrong hole, too. So it's worth checking out just to make sure.

Best wishes here.

• Comments are not for extended discussion; this conversation has been moved to chat. Mar 2 '17 at 2:05

@jonk was correct in identifying the purple wire was wrong.

@WhatRoughBeast was also correct , your LED is shorted out and R is connected wrong to LED. FIX IT You must use a a voltmeter to debug your connections and learn to say "pin one" and write as p1 or U1-1 or pin1.

Compare my expectations below and record your results.

Pin    Function  Voltage
---    -------  --------
1     (out)   15V  V+ with LED off     0V with LED on
2      (-)     0 to 15V pot V . Set to middle pin3 range
3      (+)     0V in dark  3V in bright light
4       V+      Supply voltage

11      V-      Supply return or ground ( 0V reference )

The LED must never see reversed voltages or inserted backwards as -15V will damage LED. (-5V max) The Series R limits the current from the output switched from 0 to V+ and the voltage across RED LEDs rises to 2V when lit.

The pin 2 & 3 connections are reversed on schematic to match your description. Ambient light +V in makes output + and LED goes off when connected to LED -ve (cathode)

As is Pin 2 (-) is your reference level input from pot and if room is lit, Pin 3 (+) rises and if more than (-) then output goes high. ( We think by saying if + is greater than - then output is +true (high))

• It's starting to make sense now but I won't be home for a little while to do some more work. I'll check on it again, but I'm pretty sure the LED isn't shorted, it's just bad picture quality (Should be the same as it is in picture 1). Mar 1 '17 at 15:22
• By shorted , I mean in the same row of pins meaning connected together. and the resistor goes between them without shorting but to the wrong pins. Try to layout the leads more logically like the schematic. I think this caused your confusion. They were also shorted in picture 1 but @WhatRoughbeast had better eyes on this than I did. Mar 1 '17 at 15:30
• I will look at it. Are you sure they are shorted in the very first photo? Because I cannot, for the life of me, see how they aren't in series. Mar 1 '17 at 18:05
• Sorry 1st photo is now correct Mar 1 '17 at 18:15
• Haha! Alright. Was worried! I just started learning about op-amps today so once I get home I will double check the input voltages. Mar 1 '17 at 18:35


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions. (English) Zbl 1203.65225
Summary: We propose a tailored-finite-point method for a type of linear singular perturbation problem in two dimensions. Our finite point method has been tailored to some particular properties of the problem. Therefore, our new method can achieve very high accuracy with very coarse mesh even for very small $ϵ$, i.e. the boundary layers and interior layers do not need to be resolved numerically. In our numerical implementation, we study the classification of all the singular points for the corresponding degenerate first order linear dynamic system. We also study some cases with nonlinear coefficients. Our tailored finite point method is very efficient in both linear and nonlinear coefficients cases.
##### MSC:
 65N08 Finite volume methods (BVP of PDE)
##### References:
 [1] Berger, A.E., Han, H.D., Kellogg, R.B.: A priori estimates and analysis of a numerical method for a turning point problem. Math. Comput. 42, 465–492 (1984) · doi:10.1090/S0025-5718-1984-0736447-2 [2] Brayanov, I., Dimitrova, I.: Uniformly convergent high-order schemes for a 2D elliptic reaction-diffusion problem with anisotropic coefficients. Lect. Notes Comput. Sci. 2542, 395–402 (2003) [3] Cheng, M., Liu, G.R.: A novel finite point method for flow simulation. Int. J. Numer. Methods Fluids 39, 1161–1178 (2002) · Zbl 1053.76056 · doi:10.1002/fld.365 [4] Ge, L., Zhang, J.: High accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids. J. Comput. Phys. 171, 560–578 (2001) · Zbl 0990.65117 · doi:10.1006/jcph.2001.6794 [5] Han, H.D.: The artificial boundary method–numerical solutions of partial differential equations on unbounded domains. In: Li, T., Zhang, P. (eds.) Frontiers and Prospects of Contemporary Applied Mathematics. Higher Education Press/World Scientific, Singapore (2005) [6] Han, H., Huang, Z.: A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous medium. J. Comput. Math. 26, 728–739 (2008) [7] Han, H., Huang, Z., Kellogg, B.: A Tailored finite point method for a singular perturbation problem on an unbounded domain. J. Sci. Comput. 36, 243–261 (2008) · Zbl 1203.65221 · doi:10.1007/s10915-008-9187-7 [8] Huang, Z.: Tailored finite point method for the interface problem. Netw. Heterogeneous Media 4, 91–106 (2009) · Zbl 1187.65128 · doi:10.3934/nhm.2009.4.91 [9] Hemker, P.W.: A singularly perturbed model problem for numerical computation. J. Comput. Appl. Math. 76, 277–285 (1996) · Zbl 0870.35020 · doi:10.1016/S0377-0427(96)00113-6 [10] Il’in, A.M.: Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes 6, 596–602 (1969) [11] Iooss, G., Joseph, D.D.: Elementary Stability and Bifurcation Theory. Springer, New York (1980) [12] Li, J., Chen, Y.: Uniform convergence analysis for singularly perturbed elliptic problems with parabolic layers. Numer. Math. Theor. Methods Appl. 1, 138–149 (2008) [13] Li, J., Navon, I.M.: Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: convection-diffusion. Comput. Methods Appl. Mech. Eng. 162, 49–78 (1998) · Zbl 0936.65134 · doi:10.1016/S0045-7825(97)00329-0 [14] Lin, H., Atluri, S.N.: The Meshless Local Petrov-Galerkin (MLPG) method for solving incompressible Navier-Stokes equations. CMES 2, 117–142 (2001) [15] Mendeza, B., Velazquez, A.: Finite point solver for the simulation of 2-D laminar incompressible unsteady flows. Comput. Methods Appl. Mech. Eng. 193, 825–848 (2004) · Zbl 1106.76423 · doi:10.1016/j.cma.2003.11.010 [16] Miller, J.J.H.: On the convergence, uniformly in $ϵ$, of difference schemes for a two-point boundary singular perturbation problem. In: Hernker, P.W., Miller, J.J.H. (eds.) Numerical Analysis of Singular Perturbation Problems, pp. 467–474. Academic Press, San Diego (1979) [17] Morton, K.W.: Numerical Solution of Converction-Diffusion Problems. Applied Mathematics and Mathematical Computation, vol. 12. Chapman and Hall, London (1996) [18] Morton, K.W., Stynes, M., Süli, E.: Analysis of a cell-vertex finite volume method for convection-diffusion problems. Math. Comput. 66, 1389–1406 (1997) · Zbl 0885.65121 · doi:10.1090/S0025-5718-97-00886-7 [19] Oñate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L.: A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39, 3839–3866 (1996) · doi:10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R [20] Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, New York (1996) [21] Shishkin, G.I.: A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation. Numer. Math. Theor. Methods Appl. 1, 214–234 (2008) [22] Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005) · Zbl 1115.65108 · doi:10.1017/S0962492904000261 [23] Wesseling, P.: Uniform convergence of discretization error for a singular perturbation problem. Numer. Methods Partial Differ. Equ. 12, 657–671 (1996) · doi:10.1002/(SICI)1098-2426(199611)12:6<657::AID-NUM2>3.0.CO;2-R


text:
?

Free Version
Difficult

# Electrical Suspension

APPH12-Y7EXM5

What is the distance between two electrons if the electric force between them is equal to the weight of an electron at the Earth's surface?

A

Approximately 0 m, but the electric force would never quite be as large as the gravitational force.

B

5.08 m

C

25.8 m

D

$3.2\times {10}^8$ m


text:
## Experimental Mathematics

### Artin's Conjecture, Turing's Method, and the Riemann Hypothesis

Andrew R. Booker

#### Abstract

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (nonmonomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to $S_5$ and $A_5$ representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of nonabelian extensions of $\Q$.

In addition, we discuss two methods for locating zeros of arbitrary $L$-functions. The first uses the explicit formula and techniques developed in A. R. Booker and A. Strömbergsson,“Numerical Computations with the Trace Formula and the Selberg Eigenvalue Conjecture,” for computing with trace formulas. The second method generalizes that of Turing for verifying the Riemann hypothesis. In order to apply it we develop a rigorous algorithm for computing general $L$-functions on the critical line via the fast Fourier transform.

Finally, we present some numerical results testing Artin's conjecture for $S_5$ representations, and the Riemann hypothesis for Dedekind zeta functions of $S_5$ and $A_5$ fields.

#### Article information

Source
Experiment. Math., Volume 15, Issue 4 (2006), 385-408.

Dates
First available in Project Euclid: 5 April 2007


text:
What does shear mean?

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both computationally and geometrically), but what does shear mean in this context? Is it related to the shear mapping that is used in Linear Algebra? What does it really mean and how do we use it?

Consider a little block in the fluid with edge lengths $a,b,c$. Velocity at the origin is $\vec{v}=(u,v,w)$. With help of a Taylor series expansion we can approximately write the velocity at $\vec{r} = (a,b,c)$ as: $$\vec{v}_P = \vec{v}(\vec{r}) = \vec{v}(a,b,c) \approx \vec{v} + \frac{\partial \vec{v}}{\partial x}a + \frac{\partial \vec{v}}{\partial y}b + \frac{\partial \vec{v}}{\partial z}c \\ = \vec{v} + \left(\vec{\nabla} \vec{v}\right)^T\vec{r} \quad \mbox{with} \quad \vec{r} = \begin{bmatrix}a \\ b \\ c\end{bmatrix}$$ and $$\left(\vec{\nabla} \vec{v}\right) = \large \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x} \\ \frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} \\ \frac{\partial u}{\partial z} & \frac{\partial v}{\partial z} & \frac{\partial w}{\partial z} \end{bmatrix}$$ The gradient of the velocities $\left(\vec{\nabla} \vec{v}\right)$ can be split into a diagonal part, a symmetric part (with zero trace) and an anti-symmetric part: $$\left(\vec{\nabla} \vec{v}\right) = \large \begin{bmatrix} \frac{\partial u}{\partial x} & 0 & 0 \\ 0 & \frac{\partial v}{\partial y} & 0 \\ 0 & 0 & \frac{\partial w}{\partial z} \end{bmatrix} \\ + \large \begin{bmatrix} 0 & \frac{1}{2}\left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right) & \frac{1}{2}\left(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}\right)\\ \frac{1}{2}\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right) & 0 & \frac{1}{2}\left(\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}\right) \\ \frac{1}{2}\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right) & \frac{1}{2}\left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}\right) & 0 \end{bmatrix} \\ + \large \begin{bmatrix} 0 & \frac{1}{2}\left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) & \frac{1}{2}\left(\frac{\partial w}{\partial x} - \frac{\partial u}{\partial z}\right) \\ \frac{1}{2}\left(\frac{\partial u}{\partial y} - \frac{\partial v}{\partial x}\right) & 0 & \frac{1}{2}\left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}\right) \\ \frac{1}{2}\left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}\right) & \frac{1}{2}\left(\frac{\partial v}{\partial z} - \frac{\partial w}{\partial y}\right) & 0 \end{bmatrix}$$ The divergence (or dilatation) is recognized in the first matrix, the trace of that matrix to be precise. The rotation (curl) is recognized in the third matrix. For the sake of completeness, put: $$\vec{\omega} = \begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix} = \large \begin{bmatrix} \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \\ \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \\ \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \end{bmatrix}$$ then the vector representing rotation becomes: $$\frac{1}{2} \begin{bmatrix} 0 & \omega_z & -\omega_y \\ -\omega_z & 0 & \omega_x \\ \omega_y & - \omega_x & 0 \end{bmatrix}^T \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \frac{1}{2} \begin{bmatrix} \omega_y c - \omega_z b \\ \omega_z a - \omega_x c \\ \omega_x b - \omega_y a \end{bmatrix} = \frac{1}{2} \vec{\omega} \times \vec{r}$$ So the angular velocity of the fluid's rotation is half $\times\, \vec{\omega}$ .
But let's concentrate now on the shear (or distortion) part, as requested by the OP. Consider one of the terms, take $\large \frac{1}{2}\left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right)$ , in two dimensions for simplicity.

During a time $dt$ a (square) fluid particle is transported in the plane. Then $\large \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right)$ is the velocity of the change of the two angles, as shown in the picture. (Also notice that the angles, say $\phi$ , are very small, so $\,\tan(\phi) \approx \phi$ ) The fluid particle is deformed herewith. Therefore the symmetric (traceless) matrix is also known as a deformation tensor. The deformation tensor contains half the velocities of the deformation of a fluid particle.

EDIT. At least that's what an old reader has teached us. But I'm not satisfied with the last picture. There must be a simpler way to look at it. Let's get rid of the partial derivatives in the first place,
by employing Green's theorem : $$\iint \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dx\,dy = \oint \left( L\,dx + M\,dy \right)$$ When specified for our three special cases, it reads: $$\begin{cases} \mbox{divergence :} & \large \iint \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) dx\,dy = \oint \left( -v \,dx + u \,dy \right) \\ \mbox{curl :} & \large \iint \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) dx\,dy = \oint \left( u \,dx + v \,dy \right) \\ \mbox{shear : } & \large \iint \left( \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right) dx\,dy = \oint \left( - u \,dx + v \,dy \right) \end{cases}$$ A picture says more than a thousand words:

Working out the shear part separately - performing vector addition $\,(\pm u,\pm v)\,dt\,$ - results in:

BONUS. Still let's have another look at it in some other, more "advanced" way : Lie series.
A square (2-D) fluid particle is changed a little bit into an arbitrary quadrilateral, due to velocity gradients, in general with a transformation $dT$ of the following sort: $$dT = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} a & b \\ c & d \end{bmatrix} dt$$ If we assume that this transformation is occurring repeatedly, within a timespan $t$ , say $N$ times. Then $dt = t/N$ and: $$dT^N = \left(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} a & b \\ c & d \end{bmatrix} \frac{t}{N}\right)^N$$ The limit of this transformation for $N \to \infty$ is: $$T = \lim_{N\to\infty} dT^N = \Large e^{t \begin{bmatrix} a & b \\ c & d \end{bmatrix}}$$ In the main part of the current answer, we have split the general (infinitesimal) transformation $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ into three parts, like this: $$\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a & 0\\ 0 & d \end{bmatrix} + \begin{bmatrix} 0 & (b+c)/2 \\ (b+c)/2 & 0 \end{bmatrix} + \begin{bmatrix} 0 & (b-c)/2 \\ -(b-c)/2 & 0 \end{bmatrix}$$ Let us consider the accumulated effect of each such transformations occuring repeatedly and separately. Let $p=(b+c)/2$ and $\omega=(b-c)/2$ , both constant, then: $$\begin{cases} D = \Large e^{t \begin{bmatrix} a & 0 \\ 0 & d \end{bmatrix}} & \mbox{: divergence} \\ S = \Large e^{t \begin{bmatrix} 0 & p \\ p & 0 \end{bmatrix}} & \mbox{: shear} \\ \Omega = \Large e^{t \begin{bmatrix} 0 & \omega \\ -\omega & 0 \end{bmatrix}} & \mbox{: curl} \end{cases}$$ The well-known series expansion of $e^x$ will be employed for our purpose: $$D = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + t \begin{bmatrix} a & 0 \\ 0 & d \end{bmatrix} + \frac{1}{2} t^2 \begin{bmatrix} a^2 & 0 \\ 0 & d^2 \end{bmatrix} + \cdots = \begin{bmatrix} e^{ta} & 0 \\ 0 & e^{td} \end{bmatrix}$$ The result is a scaling transformation. The edges of the fluid square particle are enlarged with factors $e^{ta}$ and $e^{td}$ in x- and y- direction respectively. The total volume change is the determinant $e^{t(a+d)}$ , with $\;a+d = \partial u/\partial x+\partial v/\partial y$ . $$\Omega = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + (\omega t) \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} + \frac{1}{2} (\omega t)^2 \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} + \frac{1}{3!} (\omega t)^3 \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} + \cdots \\ = \begin{bmatrix} 1 - (\omega t)^2/2! + \cdots & (\omega t) - (\omega t)^3/3! + \cdots \\ - (\omega t) + (\omega t)^3/3! + \cdots & 1 - (\omega t)^2/2! + \cdots \end{bmatrix} = \begin{bmatrix} \cos(\omega t) & \sin(\omega t) \\ -\sin(\omega t) & \cos(\omega t) \end{bmatrix}$$ The result is a continuing rotation, with angular velocity $\omega$. Last but not least: shear. $$S = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + (p t) \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} + \frac{1}{2} (p t)^2 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \frac{1}{3!} (p t)^3 \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} + \cdots \\ = \mbox{in very much the same way} = \begin{bmatrix} \cosh(p t) & \sinh(p t) \\ \sinh(p t) & \cosh(p t) \end{bmatrix}$$ The result is a quadrilateral which has been (de)formed as follows: $$\begin{bmatrix} 1 \\ 0 \end{bmatrix} \to \begin{bmatrix} \cosh(p t) \\ \sinh(p t) \end{bmatrix} \quad ; \quad \begin{bmatrix} 0 \\ 1 \end{bmatrix} \to \begin{bmatrix} \sinh(p t) \\ \cosh(p t) \end{bmatrix}$$ Make a simple sketch to see that this is indeed a deformation. The volume of the fluid particle is invariant under this transformation because of $$\begin{vmatrix} \cosh(p t) & \sinh(p t) \\ \sinh(p t) & \cosh(p t) \end{vmatrix} = \cosh^2(p t) - \sinh^2(p t) = 1$$

The decomposition is:

$\nabla_i v_j=\frac 12\varepsilon_{ijk}(\nabla\times v)_k+\frac 1n(\nabla\cdot v)\delta_{ij}+\left(\frac{\nabla_i v_j+\nabla_j v_i}{2}-\frac 1n(\nabla\cdot v)\delta_{ij}\right)$

The last term is the one we care about. It is symmetric and trace-free. Consider a basis in which it is diagonal. For example it could be $\mathop{diag}(-3,1,2)$. Imagine $v$ is the velocity field of a fluid. This term would represent fluid being compressed along the x-direction and spreading out in the y and z-directions (twice as much in the z-direction).

An example of a field, $v$, with no curl or divergence but with this last term being $\mathop{diag}(-3,1,2)$ would be $v=(-3x_1,x_2,2x_3)$.


text:
Recently, I read Tektronix's Probe Bandwidth Calculations. It point out that Tektronix and Agilent have different probing philosophy:

There are two schools of thought on what an oscilloscope probe should actually display on an oscilloscope. Tektronix subscribes to the philosophy that a probe should measure the unloaded, or original, signal. Agilent subscribes to a different philosophy that says a probe should measure the loaded signal.

Because I've never used Agilent's scope, I want to know if you've noticed this. And what about other scope vendors such as LeCroy, RIGOL, etc, which philosophy they subscribe to.

It would appear that LeCroy follow Agilent/Keysight in this respect (or, at least, Tektronix's presentation of what Agilent's method is). This can be seen from their probe manuals, for example for the ZS4000 active (single-ended) probe. They provide the probe impedance as a function of frequency and advocate that the user corrects for it when interpreting the measurement, using the formula:

$$V_\mathrm{out} = Z_\mathrm{probe}/(Z_\mathrm{probe}+Z_\mathrm{source}) \times V_\mathrm{in}$$

I avoid quoting further from their manual to avoid potential copyright issues (because it would require the whole section to be quoted to reproduce it properly here), but if you follow the link and read the manual, you will find that everything is quite clearly stated.

For the differential probes operating in the 10 GHz range (for example, the WaveLink D1030), their approach is slightly different to either of the ones presented in the Tektronix technical brief. The probes measure the loaded signal, as per Agilent, but they provide equalization software (Virtual Probe) to recover the unloaded signal. One models the circuit impedances and indicates the type and location of the probe, and the de-embedding is done accordingly. They summarize it as follows (quoting from the WaveLink probe manual):

Teledyne LeCroy probes are calibrated at the factory using a Vector Network Analyzer (VNA) to measure a system (probe plus test fixture) frequency response. The test fixture is de-embedded from the measurement using Teledyne LeCroy's Eye Doctor tools so the remaining frequency response is due to the combination of the test signal and the probe loading on the test circuit. The system frequency response is then calculated for these remaining circuit elements.

If you wish to de-embed the effect of probe loading on your circuit, you can use the appropriate equivalent circuit model ... and Teledyne LeCroy's Eye Doctor tools to accomplish this.

You can also use Teledyne LeCroy’s Virtual Probe option. This option allows you to select the probe tip from a list of supported tips. Your selection applies a corresponding s-parameter file that is derived from the equivalent circuit model of the tip.

However, I haven't actually used these probes, so I can't comment on how good the software is.

I am a user of Agilent MSO oscilloscopes. I had never heard about anything like "probing philosophy" before you showed me this document. But I have heard much about "signal measurement techniques".

In short, this document is about marketing rubbish, as you (probably) know that Tektronix and Agilent compete for the market and Tek comes after Agilent here. I will not advocate any of them, they both provide good, state-of-the-art products, but the use of a "compare to the leader" method in a market competition is generally used by the one following the leader, not by the leader itself.

Look for yourself, the document discusses the "probe bandwidth", saying nothing about the components from which that bandwidth results, i.e. capacitance (I found no mention of the "pF" unit in the whole doc) in the first place. There are no physics, only claims in the doc.

The Agilent o-scopes I'm familiar with, have internal switches to select between 10 Megs and 50 Ohms input resistance while input capacitance is mostly a property of a passive probe. There are also active probes to make HF/VHF measurements better/possible.

What to use depends on the appropriate signal measuring technique which, in turn, depends on the situation you are facing. There is no instrumentation task that can be done with Tek while not with Agilent, and vice versa, of course.

Therefore, such "philosophy" is about marketing [rubbish], not electronics [truth :-].

P.S. Also, there is Agilent's answer for Tektronix's question on the philosophy.

• I think this is somewhat arguable. Given that Tek is saying that their method of calibration assumes 50 Ohm sources, I would personally prefer the Agilent approach when one can expect to encounter variable source impedance. As such, I'd say it's not bullshit, but important distinction that the user needs to be aware of. Whether Agilent/Keysight agree about Tek's characterization of their philosophy is another matter. Apr 4, 2015 at 17:28
• @OleksandrR. does Tek not implement such a 50R/10M switch at all?
– alex
Apr 4, 2015 at 17:34
• I have no idea. I use LeCroy oscilloscopes. But I also think it's somewhat beside the point, because Tek are talking here about high frequencies for which nobody is going to use passive probes. Even when using active probes, you still have to account for the effect of the probe loading somehow (even if it's much less than 1pF). This is where the calibration of the probe frequency response comes in. Apr 4, 2015 at 17:45
• @OleksandrR. if the switch exists then i see no diff between Tek and Agi. i also know no docs where Agi calims its "philosophy" in all and "50R mandatory" in particular. generally, it discusses various approaches depending on what you want to measure, what you face with, and what you have.
– alex
Apr 4, 2015 at 17:46
• @alex: I have read the link you gave. It conforms that Agilent does have such a philosophy as Tek mentioned. Since they have state their philosophy clearly, so l do not think it's just a 'marketing'. Unless they can make a perfect probe without loading to the circuits under test, the two philosophyies should always result different products or different methods to specify their products. Apr 7, 2015 at 13:32

Informative only:

• Agilent probes catalog

• LeCroy probes catalog

• Rigol probes catalog

Probes & Accessories [HTML]

I think theirs philosophy (if any) is simple: "we provide solution for any your problem".

@diverger,

From Agilent's Side-by-Side Comparison of Agilent and Tektronix Probing Measurements on High-Speed Signals referenced by me earlier:

# Conclusion

Regardless of whether you use Tektronix’ or Agilent’s measurement performance standards for characterizing the performance of high-bandwidth active probes, Agilent’s probes outperform Tektronix probes on all counts as documented in this application note. Agilent’s 1134A InfiniiMax active probes using various probe heads (browser, solder-in) demonstrate less probe loading and more accurate reproduction of the signal applied to the probe tips. Even if you apply a Tektronix standard where you ignore probe loading and compare measured signals to unloaded/unprobed signals, Agilent’s probes still out-perform Tektronix probes when you compare the same probing configurations (Tektronix browser connection versus Agilent browser connection, Tektronix solder-in connection versus Agilent solder-in connection) and when you use identical sampling techniques (real-time), as shown in this application note.

With the introduction of the new 1130 Series InfiniiMax differential active probes, Agilent adopted a new probe architecture/topology where the probe amplifier is physically displaced from the probe head using precision RF transmission line technology for high-impedance connections. This new probe technology enhances usability and measurement performance for high-bandwidth applications. In fact, Agilent’s new 1130 Series InfiniiMax active probing system was recently selected as the 2002 Test & Measurement Product-of-The-Year award sponsored by EDN magazine. To our knowledge, this is the first time EDN has selected an “accessory” for this award in the Test & Measurement category.

IMO, I made a mistake in my first answer having suspected only Tektronix in marketing rubbish. It was be necessary to suspect both of them, remembering however that Agi was the defensing side. (And we need to recognize that Agi won this PR boxing round.)

I think neither Tek nor Agi but the engineer selects the method to measure and the corresponding background to interpret the result. Both Tek and Agi give the instruments for both approaches.

From the both documents I understand the one technical thing: they both beaten capacitance issues so well, they both beaten attenuation issues so well, but they beaten inductance issues not so, one did it better than another. I also assume that one patented some on this earlier and wider than another. It's very probable the philosophy (as PR base) itself and its difference (as PR task) were grown up on that.

As my conclusion, what do you prefer to buy from Tek, Agi, LeCroy, etc: a good thing you need or a puffing story about why other sucks? Tek/Agi drops its shit to the fan directed to the competitor, calling this "philosophy". Let's they taste their shit self, without us.

Good luck.

P.S. Both documents look like a WWF wrestling battle there each word and action are set up long before the battle begins :-]

• I think this is a little off topic as an answer. OP wants to know if different scope manufacturers adhere to different "probing philosophies". Instead of making another answer you should add this to the one you already have, or just delete this as it doesn't add anything to your previous answer. "There is not probing philosophy, it is all marketing BS" gets your point across without all that text. Apr 7, 2015 at 18:43
• @alex: The two docs do have some "marketing" purpose. And I think the "philosophy" is different to "performance". Even with equal performance the two philosophy will give us different results. I think they both can produce good probes by subscribing to their own philosophies. This is not only a "marketing" problem, because their philosophy does have some impact on the users experience, that is, what you will get when you using their probes, "what was" or "what is". It's sure if they can produce perfect probes, then the two philosophies will give our same results. But for now, it won't. Apr 8, 2015 at 1:45
• @alex: You've put Agi on the defence side. But in the "Probe Bandwidth Calculations", at page 12. it sates the source "Side-by-Side Comparison of Agilent and Tektronix Probing Measurements on High-Speed Signals". So if we can think it's Agi start the "war"? Apr 8, 2015 at 2:04