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We are living in extraordinary times. The COVID-19 pandemic has ushered us into uncharted territory where we need to learn how to adapt.
A lot of businesses that were shut down at the onset of the pandemic have been allowed to reopen under the condition that the CDC’s reopening guidelines be strictly implemented. Whatever your business is, whether it’s a car dealership, mortgage refinancing, or a friendly neighborhood grocery store, you need to follow the said guidelines.
With the changes that are happening in the workplace, certain skills are needed to ensure that you will thrive amid the threats of the coronavirus.
7 Skills to Add to Your Arsenal to Successfully Navigate COVID-19
Workplace Adaptability
One of the most needed skills in the world today in light of the pandemic is adaptability. By definition, adaptability is the skill that allows an individual to fully function under different the best and worst conditions.
The ability to adapt and roll with the changes, especially with the many pandemic-induced changes we’re going through now, is one of the most sought-after traits that employers are looking for. To say that the present workplace now is dynamic is a huge understatement.
Self-Reliance and Independence
When we say independence and self-reliance, we’re not just talking about financial independence nor isolation from others. In the context of work, self-reliance and independence are defined as the ability to think and work independently on the task at hand with minimal supervision.
It also denotes that a person has the initiative to take immediate action and takes ownership and full responsibility for the tasks delegated to them.
Managing Time Effectively
The ability to fully maximize and optimize your time at work guaranteeing the highest output level is an essential skill, especially at this time. A lot of companies have either shortened their operating hours or shortened their employees’ shifts. Effective time management is needed to ensure maximum productivity despite the reduced working hours.
Resilience and Grit
A person’s ability to respond and rise up to challenges is an asset for any company today. Given the various challenges the coronavirus threw our way, employers and managers are looking for people with grit and determination to push through and finish the task at hand despite unfavorable odds.
Teamwork
While teamwork is important even before the pandemic came, it needs to be taken to a higher level of what we call collaborative ingenuity. Under this new definition, collaborative work beings about a greater sense of individual responsibility raising motivation levels. It also allows for people to come together to brainstorm and share different ideas that are beneficial for all. Given that, the ability to work with other people is now more important than ever.
Creativity
Creativity, simply put, is the ability to come up with immediate solutions to a concern with the resources you have at hand. This creativity puts you in a position to be a problem solver more than an innovator, although that’s also a good thing. It also allows one to be more open to learning new concepts and ideas that could prove valuable to the organization.
Tech Savviness
With most businesses and establishments shifting to online platforms, the need for tech savviness also ranks up high in recruitment and hiring searches.
While basic computer knowledge is always good, most employers are looking for more advanced individuals to help their company convert leads into sales and closed deals.
Work on building these skills and making them a staple in your arsenal. With some hard work and determination, you can incorporate these skills into your system to make habits out of them.
The need to adapt is real and if you expect to come out of this pandemic in one piece, you need to start making the necessary adjustments. | https://www.beyondthenet.net/new-skills-for-the-new-normal-7-skill-you-need-to-navigate-covid-19/ |
Ann Hornaday of the Washington Post has a most excellent article about how indie films--once known for their unique voice and style--are falling into the same formulaic cliches that plague their big-budget studio counterparts:
"Call it "There Will Be Hamburger Phones": More than 20 years after American independent cinema entered its latest Golden Age, what started as a fiercely autonomous cinematic response to Hollywood and its dominant genres has become a genre itself. And like all genres, the indie aesthetic is rife with its own versions of the hackneyed conventions, tired tropes and cliched themes that weigh down the most predictable action spectacle or by-the-numbers rom-com."She takes on Napoleon Dynamite and Juno in her astute and insightful critique of what used to be the way to escape the usual paint-by-number drivel.
I've nothing more to add other than "What she said" and read the article! | http://www.wellaboveaverage.com/2008/10/indie-cred-now-equals-indie-dread.html |
Electronic music composer Trentemøller has debuted his cover of the holiday classic “Silent Night,” which hosts vocals from his girlfriend and mother of his first born, Lisbet Fritze. The single features a cover of the artist with his mother which was taken “one Christmas night many years ago.”
The cover features features dark synth sounds, and a heavy drum beat, while blending in an array of sonic elements which give the track a feel of optimism. The vocals have simple overlays, and have the feeling of a Church-choir inspired sound, which is complimented by various bell sounds.
“I have genuinely loved Christmas since I was a little boy,” the artist explained in a press release. “And I always wanted to do my own version of one of the great Christmas songs. I chose ‘Silent Night’ as it has such a beautiful melody that, for me, sums up the whole vibe of Christmas.”
The producer debuted the track “Foggy Features,” earlier this year a seven-minute long track, filled with IDM and dark synthwave inspired elements, with brooding synth lines, spacious acoustic guitars and darkwave inspired drums. Regarding the song’s music video, the producer stated: “I shot the visuals for Foggy Figures while staying in the Swedish woods with my girlfriend last autumn. One day we woke up to this foggy landscape, it was magical and I right away started shooting on my phone.”
Trentemøller dropped his most recent studio album release Obverse this past September, which featured guest appearances from Rachel Goswell of Slowdive and Jenny Lee Linberg of Warpaint. | https://music.mxdwn.com/2019/12/23/news/trentemoller-debuts-sinister-heightened-cover-of-silent-night/ |
The Law of Assumptions deems we experience whatever we assume. Often we believe our assumptions as fact and because we assume this they become fact. See how this works?
Notice how others don’t seem to have the same issues?
An assumption, though false, if persisted in will harden into fact…
NEVILLE GODDARD
Are you aware of how your assumptions are creating your life experience?
When we are perpetually experiencing something unwanted, it is simply because of our assumptions. Once we understand this, assumptions can be changed.
Sounds simple?
We have to be persistent though and overwrite the unwanted assumption whenever it arises in our mind. The new assumption must drown out the old assumption, or we experience a mixture of what we do and don’t want.
This is challenging at first, but I suggest you tell yourself it’s easy *wink*
Write down a list of unhelpful things you assume and keep adding to the list as you become more aware of your unhelpful thoughts. You can become quite creative as your confidence in this increases.
Here is a list of assumptions that could be blighting your existence:
- I never have enough money
- Money is hard to earn
- I have expensive taste
- I am overweight
- Losing weight is hard
- I never have enough time
- I’m always exhausted
- I am stuck
- I am stupid
- I am rubbish at
- I’m not designed to
- I am always cold
- It’s difficult to
- I can’t find
- I am jinxed
- I never win
- I am unlucky
- I can’t
- I feel old
- I’m too old
- I don’t feel well
- I am bored
- I’m lost
- All men are cheaters
- Women are gold-diggers
- Women are crazy
- Men don’t listen
- I’ll never get that job
- I’m not good enough
- No one respects me
- People are cruel
- No one understands me
Whatever you put after the ‘I AM’ is being created.
Next time you catch yourself assuming something that contradicts your desires, say no internally, flip it in your mind, and in a while, your life will improve.
Don’t panic, be confident, assume you are doing it right and it will work. The new assumptions need to drown out the old assumptions for you to see a difference.
This little exercise can be done when you are driving, in the bath, washing up, cooking, walking or doing anything where your thoughts can roam free.
Create positive assumptions and affirmations for yourself too:
- Money comes to me easily
- Good things are happening behind the scenes
- Everyone loves me
- I am highly valued
- I am always successful
- I am grateful for
- I can easily find
- It’s easy
- I am healthy
- I am talented
- Business is booming
- I have loads of customers
It doesn’t matter if you don’t feel or believe in what you are saying, keep going, you will in time when you see the fruits of your labour.
Thoughts are the most powerful creator know to man. Mastery of our thoughts is the most important thing we can ever do. | https://sarahdewhirst.com/law-of-assumption/ |
What Is Urethane Insulation?
Update Time:2017-08-08
What Is Urethane Insulation?
Urethane insulation is a building product used to prevent air transfer through the exterior walls of a home. It is comprised of polymer chains connected by organic compounds known as carbonates, or urethanes. The terms urethane and polyurethane are used interchangeably when it comes to most applications, including insulation.
There are two basic types of urethane insulation, including rigid foam boards and spray foam. Rigid boards come in a variety of sizes, and typically range from 1 to 2 inches (2.54 to 5.08 cm) in thickness. These boards are primarily used in new construction, and are installed inside of walls or around other structures in the home. Spray foam insulation is typically used in existing structures, which may lack sufficient insulation material. This product is sprayed into the wall through tiny holes, and quickly expands to fill the entire wall cavity.
This material can be used is a number of different application in the home. Rigid insulation is frequently installed on the exterior of the wall framing system, where it is sandwiched between the home's sheathing and exterior cladding layers. Sheets of rigid insulation may also be attached to the home's roof or foundation, to provide additional thermal resistance in these areas. Spray foam is installed inside the same cavity as the wall studs, with sheathing, plywood, or drywall used to form the wall's boundaries.
Urethane insulation offers several advantages over other insulating materials. Though it is slightly more expensive than traditional fiberglass batts, it is considered a superior insulator, with two to three times the thermal resistance of alternative products. The closed-cell construction of urethane foam helps to block air flow, and also keeps the material from collapsing or sagging. Sheets of the insulation take up very little space, resulting in thinner walls and more available living area. Finally, urethane insulation is difficult to ignite, and is more resistant to fire than either fiberglass or polystyrene insulation.
Despite its many benefits, there are a number of factors that must be considered when using urethane insulation. As it ages, this material experiences a condition known as thermal drift, which can lead to decreased levels of thermal resistance. Adding a foil or plastic coating to the insulation can eliminate or slow thermal drift. Urethane insulation is also susceptible to insects burrowing into the material, which reduces its effectiveness. To prevent insect damage, insulation should be treated with an insecticide, especially when installed on foundation or basement walls.
COMPANY
Contact Person
- Jack Ma
- Address:
- Room1016, Donghui Building ,Dezhou City , Shandong Province , China. | http://www.dzchaishang.com/f698543/What-Is-Urethane-Insulation.htm |
The investigation of eye movements is important in the understanding of basic vision properties and with the availability of new eye tracker technology it is also of growing practical importance. Typical research topics are the relation between image properties and eye movements (bottom-up), the influence of high-level goals on movement patterns (top-down) or models to predict salient regions in an image (for an overview see ). In this study we use a model which describes the location of the eye positions as the realization of a stochastic process that consists of two components, one component that describes the larger, jump-type changes and a second component related to the relatively small changes of the eye positions. This saccade-and-fixate strategy is one of the fundamental processes used by the human visual system to analyze its environment.
The factors that control these processes are very complex. They include high-level task-solving factors and input driven factors depending on the visual properties of the current input image. In the following we will ignore all these factors and we will only analyze the statistical properties of eye-tracker data. The problem we want to investigate is: can we characterize individual observers from a collection of eye-tracker measurements? Furthermore we will ignore fixation points, which are perhaps more controlled by task and/or image related factors, and we will only use parameters derived from the saccadic movements. We investigate this problem with the help of a large database where 15 observers viewed 1003 images in free-viewing conditions .
The major steps in this investigation are the following: First we describe two methods to compute the step-length between two different eye positions. The first is the usual euclidean length while the second is using the disc version of hyperbolic geometry which takes into account that the viewing space of the observer is a cone. Transition points between fixations are by definition points with comparatively large distances between consecutive eye positions. If we consider saccadic eye movements as a stochastic process then these non-fixation points correspond to the tail of the distribution of step-lengths. Such tail distributions follow very often the generalized Pareto distribution (GPD). In the second step of our analysis we will show that the distribution of the step-lengths of the saccadic eye movements can indeed be described by the GPD. The GPD depends on three parameters: position (), shape () and scale (
). In the third part we choose a random selection of images and for each user we fit the GPD to the data. Each such experiment results in a 3-D parameter vector for each user. Selecting only a few images will obviously lead to results where the variation between images is larger than the variation between users. We will however show that with an increasing number of images per trial clear clusters appear in the parameter space. These clusters can be characterized with a mixture-of-Gaussian model. Distances in parameter space are a poor way to characterize the similarity between probability distributions but the structure in the parameter space suggests that it should be possible to identify individual observers based on the distribution of their saccadic step lengths. In our final experiments we therefore use samples of the distributions as feature vectors and we train support-vector machines (SVM) to discriminate between one specific observer and the rest. These experiments result in very high recognition rates.
2 Material and methods
The data used in this study is described in . Together with useful code it can be downloaded from the website of the authors111 http://people.csail.mit.edu/tjudd/WherePeopleLook/index.html. The database contains eye tracking data of 15 viewers and 1003 images. The database contains also code to compute fixation points. In the following we use this code to identify fixation points and to extract those eye tracking measurements that are related to the saccadic movements between fixations.
We ignore the direction of the movements and use only their step lengths. The eye-positions in the database are given in a planar coordinate system. A natural way to compute the distance between two position vectors is thus given by the common euclidean distance. In the following we will also use a model based on hyperbolic geometry. We don’t go into the technical details (which can be found in books on non-euclidean geometry , has a short introduction) but we will give an intuitive motivation. Consider the model of a pinhole camera in Figure 1. The field of view of the camera is restricted to the cone between the two lines denoted by B. A point in the scene located somewhere along the line L is mapped to the pixel r on the sensor. All projection lines go through the ’pinhole’ located at position (note that it is also possible to place the sensor behind the pinhole which gives essentially the same geometry). In euclidean geometry based models of eye movements one often uses two angles to describe the motion of the eye, see . A corresponding model for the one-dimensional model in Figure 1 would use the angle to characterize the line L. In this framework there is no build-in mechanism that requires that the projection line L lies between the two lines denoted by B. If we use the sensor coordinate r then we can require that the absolute value of r lies between zero and one and we can use the new, hyperbolic angle defined as as a coordinate for L. For points near the origin the hyperbolic distance is similar to the euclidean distance by for points near the boundary it goes to infinity. Note also that instead of the euclidean distance r with the rather arbitrary upper bound of one we can use the hyperbolic angle as a distance to the origin and then we have a natural distance measure with the only restriction that it should be non-negative. In the case of a three-dimensional scene and a two-dimensional sensor, the sensor is given by the unit disk. The points on the sensor can be described by euclidean coordinates but it is easier to use complex coordinates where is defined as in the one-dimensional case. For two sensor points their hyperbolic distance is given by
which reduces to the one-dimensional definition in the case where and real and .
For pairs of consecutive points we compute the distance between them and consider the statistical properties of these step lengths. In the terminology of random walks we consider the eye-tracking data as a random walk where the step lengths follow a heavy tailed distribution. Furthermore we are only interested in those parts of the walk in which the step lengths are large. Here we do not define a numerical value of that threshold but we simply restrict our investigation to points that are not classified as fixation points in the database. Data of this type is often investigated with the help of Pareto distributions that are used when excesses of a random variable over a threshold are considered. The probability density function (PDF) of the generalized Pareto distribution is defined as:
We estimate the parameters of the distribution with the help of the Matlab function gpfit and therefore we follow the notation used there:is the shape, is the scale and is the location parameter of the distribution. Note that the shape parameter can be negative in which case the support of the distribution is a finite interval. For positive the support is the positive half-axis with left end point at We do not consider the special case of . More information about the generalized Pareto distribution and extreme value distributions in general can be found in (but note the sign change for there). The estimation of the parameters is done in two steps: first we find the minimum distance value in the data vector. Then we shift the distance values so that the minimum value is zero. In the second step we ignore all shifted distance values with value exactly equal to zero and fit a two parameter GPD with shape and scale parameters to the shifted data. This gives a three parameter representation of the data in terms of the minimum value and the shape and scale parameters of the GPD.
Estimating distribution parameters for single images and a single user is obviously not very meaningful. One problem is that the number of non-fixation points is, by definition, much lower than the number of fixation points and the other reason is that the form of a given eye movement sequence can vary considerably. In our experiments we therefore select randomly a given number of images from the database. For a given observer we combine all non-fixation measurements from the corresponding observations into one dataset. From this dataset we estimate the parameters of the GPD. In most experiments described below this process is repeated 5000 times to see how the random selection of the images influences the values of the estimated parameters. The results show that the distribution parameters are concentrated in clusters linked to the different observers.
In the final experiments we use samples from the probability density functions of the GPD’s and train support-vector machines (SVM) to discriminate between one observer and the remaining observers. We will show that the recognition rates are very high and that they are slightly better in the euclidean metric based experiments than in the experiments using the hypberbolic distances.
3 Experiments and Results
As an illustration of the variation of the eyetracking measurements for different observers we show in Figures 2 and 3 the measurements for four observers (ems, hp, jw, kae) when viewing the two images (i14020903.jpeg, i113347896.jpg) in the database.
After fitting the GPD to the data we computed the adjusted R-squared value (see ) which gives a measure of how similar the shapes of the empirical and the fitted distribution functions are. These values lie between zero and one and higher values indicate better fit. In the following experiments we selected 50, 100, 200 and 500 random images in each trial and we fitted the GPD for each observer. We did this for both the euclidean and the hyperbolic distances. The mean values of the adjusted R-squared values are collected in Table 1 for the euclidean distance and in Table 2 for the hyperbolic distance. The column All contains the mean value over all observers. From the tables one can see that the hyperbolic distance results are in general slightly better and they produce more consistent fitting results. As an illustration of what these numbers really mean, we compare the empirical distribution and the estimated GPD in Figures 4, 5,6 and 7. In the experiment we selected 200 random images and the eye-tracker data of the observer jw. The first two plots show the histogram of the distribution and the fitted GPD, both for the euclidean and the hyperbolic distances. These distributions have long tails and therefore we restrict the plot range in Figures 4, 5 to the relevant parts of the distributions. Figures 6 and 7
are quantile-quantile plots of the same data. These plots show the relations between the quantiles of the empirical data and the corresponding quantiles of the GPD. For a perfect fit all points lie on the 45 degrees diagonal. We see again that the fit is very good for the major parts of the distributions, only for very high quantiles the differences become noticeable. Which is natural in this case since we have, by definition, very few observations in this value range.
Mean values of the adjusted R-squared values give only a summary overview over the computed values. In Figure 8
we show for all the observers the empirical cumulative distribution function (ECDF) of these values for the case where we used the hyperbolic distance, computed 5000 fittings and used 200 images in each run.
The distribution of the data points are described by three parameters: the minimum value, the shape and the scale parameters. The experiments confirmed that the minimum values are indeed very small and the variance of the minimum values is also almost zero (typically of the order). The following description of the structure of the parameter space of the distributions is therefore restricted to the distribution of the shape and scale parameters of the GPD, fitted to the shifted distance values. In the classification experiments we will make use of the three-parameter form of the GPD.
An analysis of the distribution of the ()-parameters showed that the (
)-vectors for a given observer are concentrated in restricted regions and we therefore fitted Gaussian mixture models (with 15 Gaussians, using the Matlab function gmdistribution.fit) to the distribution of the ()-vectors.
For the experiments where 50, and 200 images per trial were used in 5000 trials we show the distribution of the GPD parameters and the overlayed Gaussian mixture contours in Figures 9, and 10. The hyperbolic distance is used in all figures. The abbreviated user name is displayed at the position of the value of the median of the shape parameter and the scale parameter for that user.We see that by increasing the number of images considered in each trial the separation between the users improves and for 50 and 200 images the 15 Gaussians give a good description of the 15 observers. One interesting observation is the location of the parameters for observer kae where fitting resulted in a negative shape parameter. Note that the observers ems, hp, jw, kae used in Figure 2 and Figure 3 are those that occupy the outlying regions in the ()-space.
The structure in the distribution parameter space indicates that it should be possible to identify individual observers from the distribution of their saccadic eye movements. In a series of experiments we therefore investigated if this really is the case.
First we note that the distributions are characterized by the three parameters from which the complete pdf can be computed. In the case where we used trials in the fitting experiment the whole information can be stored in a database of size (number of trials x number of observers x number of distribution parameters). Such databases will be used in the following classification experiments, the values for are 5000 and 10000.
The distributions are defined on different intervals due to the location parameter. We construct a sampled version of the pdfs as follows: First we compute 100 samples of the pdf on the interval between its 5-percentile and its 95 percentile. The we embed all pdfs in the larger interval between the lowest 5 percentile and the highest 95 percentile for all pdfs under consideration. We then define 200 equally spaced sample points on the larger interval and finally construct for each pdf a 200-D vector with the square root of the pdf at these sample points by interpolation. This correponds to the Hellinger distance between two distributions. Each pdf is thus characterized by a 200-D vector defined on a common reference grid.
Next we select from the 200-D vectors those samples which have the highest variance. The components in this -dimensional vector are of course very similar to each other and there are other, more effective methods available, but we found this choice sufficient for this first study.
After the feature selection we selectedtrials, described in the previous experiment, and the corresponding distributions for all observers. These -dimensional vectors were then used to train a support vector machine (SVM) to discriminate one observer against all the other observers. After the training we selected vectors from random observers and random trials and classified them with the trained SVM. This was repeated 100 times and the results were collected in the mean recognition rates. In all experiments we used vectors for classification.
The following figures illustrate some of the results obtained using these evaluation procedure. First we investigated the influence of the number of pdf-samples used (the value of the -parameter). We used the euclidean distance, 5000 trials, 50 images per trial and samples to train the SVM. In each classification step 5000 vectors were classified. Figure 11 shows the results obtained for
We see that the best results are obtained using 10 or 20 samples and we therefore choose to use 20 pdf samples in the following experiments.
In the next experiment we investigated if the euclidean or the hyperbolic distance measure gives better classification results. We used, 10000 trials, 250 images per trial and samples to train the SVM. In figure 12 we show the results and we see that the euclidean distance measure gives in general better classification results than the hyperbolic distance (but note the very high recognition rates).
In Fig. 13 we illustrate the influence of the number of images used to train the SVM. We used, 5000 trials, 50 images per trial and samples to train the SVM. We see that 20 or 50 samples are sufficient.
In the last experiment we investigated how the number of images used in a trial influence the classification result. We used 5000 trials and 50 and 200 images per trial (corresponding to Figs. 9 and 10) We see that the number of images used to estimate the parameters of the GPD has a great influence on the classification result, as was to be expected.
4 Discussion
The three main topics described in the paper are (1) the introduction of the hyperbolic distance, (2) the usage of the GPD and (3) the recognition of observers based on the measurements of the saccadic eye movements.
The application of hyperbolic geometry is attractive from a theoretical point of view since the limitation of the viewing geometry is built into the model. The experiments showed that the fitting of the GPD distribution, as measured by the R-squared values, are slightly better for the hyperbolic distance than for the euclidean distance. This is probably an effect of the relative higher importance of large distances in the hyperbolic geometry. The somewhat lower classification rates in the hyperbolic case, compared to the euclidean case, may be an indication that for a classification the eye movements in the transition phases between fixation and saccades are also important.
In both cases, euclidean and hyperbolic, we find however that the GPD provides a very good and compact model for the statistical distribution of the step lengths. The classification experiments show that these distribution are potentially useful for identification tasks but more detailed studies with more observers are necessary to judge the potential of the method.
We also note that the special form of the GPD’s make it possible to derive analytic expressions for different distances (like the Hellinger distance). We derived some of these expressions with the help of Mathematica but their final form often involves expressions based on special functions like the hypergeometric function which are costly to evaluate numerically. We therefore used the numerical implementation described above.
No attempts were made to optimize the parameters in the different processing steps. All statistical computations (gpfit, gmdistribution, fitcsvm, predict) were done using the tools in the Matlab 2014b Statistics Toolbox with default parameter settings.
References
- J. W. Anderson. Hyperbolic geometry. Springer Verlag, London, 1999.
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A. Borji and L. Itti.
State-of-the-art in visual attention modeling.IEEE Trans. Pattern Analysis and Machine Intelligence, 35(1):185–207, 2013.
- E. Castillo. Extreme value and related models with applications in engineering and science. Wiley, Hoboken, N.J., 2005.
- T. Haslwanter. Mathematics of three-dimensional eye rotations. Vision Research, 35(12):1727–1739, June 1995.
- M. Nikulin. Hellinger distance. In Encyclopedia of Mathematics. Springer Verlag. http://www.encyclopediaofmath.org/index.php?title=Hellinger_distance&oldid=16453.
- C. L. Siegel. Topics in complex function theory. Vol. 2, Automorphic functions and Abelian integrals. Wiley Interscience, New York, 1969.
- J. Tilke, K. Ehinger, F. Durand, and A. Torralba. Learning to predict where humans look. In Int. Conf. Comp. Vis., ICCV 2009, pages 2106–2113, 2009.
- G. J. G. Upton and I. T. Cook. A dictionary of statistics. Oxford University Press, Oxford, 2008. | https://deepai.org/publication/saccadic-eye-movements-and-the-generalized-pareto-distribution |
Chitimukulu will be a Guest of Honour at HH’s inauguration as Zambian President in 2021-Garry Nkombo
United Party for National Development UPND Mazabuka Central MP, Garry Nkombo says the Chitimukulu will be an honoured guest at Hakainde Hichilema’s inauguration as Zambian President in 2021.
Addressing a press briefing at the UPND secretariat yesterday morning, Hon Nkombo noted that it was unfortunate for the PF to start accusing the UPND of planning to dethrone Paramount Chief Chitimukulu of the Bemba speaking people saying the party neither had the power nor the machinery to do so.
Nkombo recalls how he, together with late Kalomo MP Request Muntanga and UPND deputy secretary general for administration Patrick Mucheleka stood up to defend his right to the Mwine Lubemba throne when late President Michael Chilufya Sata sent 500 police officers to guard the Royal Palace to prevent him from becoming the Chitimukulu shortly after assuming office in 2011.
He also stated that as a party that holds the role of traditional leadership and the guidance it brings to the peoples of Zambia in high esteem, the party had resolved to stay away from meddling in traditional affairs because it does not have the roots to determine who should ascend to a particular throne of a particular traditional establishment.
Hon Nkombo also reminded His Royal Highness of the meeting that he (Nkombo) had with him at which the former assured the Chitimukulu of the UPND’s support for traditional leadership in the country as well as its resolve not to meddle into affairs of traditional leaders in the country.
He has since called on Zambians to ignore assertions from certain sections of society, particularly from the PF and its surrogate political parties that the UPND had hatched a plot to dethrone and degazette him one on office.
6 COMMENTS
How would you dethrone him when you don’t belong to the bemba ng’andu clan, the late president and abena nka… were against him becoming chitimukulu maybe they are of bemba royal bloodline. This chitimukulu should not turn family feuds into national feuds, let him learn to keep fyamu family mu family.
If mucheleka calls GBM uncle and GBM calls chitimukulu uncle , in 2012 there was another kerfuffle, awe bane family feuds should not be turned into political feuds.fyamu family should remain my family.
But who says HH will be president of Zambia? Presidency is won on the ballot paper and not the mouth. And it seems most Tongas are not very intelligent , like Nkombo is portraying now. How do you hold a press conference to tell all Zambians that HH will be sworn in president? Is that normal? By what criteria is going to br sworn in?
Upnd has a lot of members on drugs. Look at how they hallucinate. Now they believe they have won the election. Come 2021 they will be running to courts petitioning their failure. The problem with upnd is the arrogance of its leaders. They think they can win an election by hallucinating on social media and writing to the EU. Useless opposition who are wasting our time. A party that has had the same leader for decades. You can laugh mwe | |
Jesus said, “I am the true vine, and my Father is the gardener.
He cuts off every branch( The “branches” are Jesus’ followers.) of mine that does not produce fruit. (produce fruit-meaning the way Jesus’ followers must live to show they belong to him.)
He also trims every branch that produces fruit to prepare it to produce even more.
You have already been prepared to produce more fruit by the teaching I have given you.
Stay joined to me and I will stay joined to you. No branch can produce fruit alone. It must stay connected to the vine. It is the same with you. You cannot produce fruit alone. You must stay joined to me.
“I am the vine, and you are the branches. If you stay joined to me, and I to you, you will produce plenty of fruit. But separated from me you won’t be able to do anything.
If you don’t stay joined to me, you will be like a branch that has been thrown out and has dried up. All the dead branches like that are gathered up, thrown into the fire and burned.
Stay joined together with me, and follow my teachings. If you do this, you can ask for anything you want, and it will be given to you. | https://watercresswords.com/2015/07/05/sunday-words-greater-love/ |
Part of the Leadership Track.
Speaker: Carrie Webber
Fee: DDS $119; Staff $59
Seating Limit: 100
AGD Code: 550
Do you ever feel stuck? Find yourself working intensely to increase your practice performance, yet not sure how to attain a measurable result? Explore three key areas during this high energy, high impact presentation to help you determine what systems you can focus on now and reap significant returns for your future.
Learning objectives:
- Understand the importance of leading from within your practice to make a difference for your patients, your team, and yourself.
- Determine measurable accountability factors for the entire team, so that together you can achieve your practice goals.
- Develop strategies to revive your team and increase engagement levels for higher productivity of individuals and the practice overall.
This course is designed for the entire dental team.
Co-sponsored by MDA Insurance, MDA Services, and CareCredit. | https://annualsession.michigandental.org/session/course-42-level-up/ |
It will take 270 electoral votes to win the 2020 presidential election. Click states on this interactive map to create your own 2020 election forecast. Create a specific match-up by clicking the party and/or names near the electoral vote counter. Use the buttons below the map to share your forecast or embed it into a web page.
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Map Created: Apr. 1, 2019 at 20:42 UTC (4:42 PM EDT)
Headlines
Nevada Democratic Caucus: Overview and Live Results
Bernie Sanders led in the final polling average. When the vote is finalized 36 delegates will be allocated proportionately to those meeting the 15% viability threshold.
Swing State Poll finds Trump Ahead in Wisconsin, Trailing in Pennsylvania; Michigan Close
These states were flipped by Donald Trump in 2016, each voting for a Republican presidential nominee for the first time in a generation
Senate Rating Changes from Sabato's Crystal Ball
Alabama moves to likely Republican, while Colorado moves from tossup to leans Democratic
Wisconsin 7th Congressional District Special Primary Election: Overview and Results
The seat has been vacant since former Rep. Sean Duffy resigned in September
Numerous Super Tuesday Polls Released Today
Includes polls from three of the four states with the largest delegate pools available that day. | https://www.270towin.com/maps/V3O9Y |
Kiermeier, A., Verbyla, A. P. & Jarrett, R. G. (2012). Estimating a single shelf-life for multiple batches. Australian and New Zealand Journal of Statistics, 54 (3), 343-358.
Abstract
Pharmaceutical companies and manufacturers of food products are legally required to label the product's shelf-life on the packaging. For pharmaceutical products the requirements for how to determine the shelf-life are highly regulated. However, the regulatory documents do not specifically define the shelf-life. Instead, the definition is implied through the estimation procedure. In this paper, the focus is on the situation where multiple batches are used to determine a label shelf-life that is applicable to all future batches. Consequently, the short-comings of existing estimation approaches are discussed. These are then addressed by proposing a new definition of shelf-life and label shelf-life, where greater emphasis is placed on within and between batch variability. Furthermore, an estimation approach is developed and the properties of this approach are illustrated using a simulation study. Finally, the approach is applied to real data. | https://ro.uow.edu.au/eispapers/5782/ |
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In order to participate in an Offering, you will have to provide Personal Information to pass Know Your Customer (KYC) and Anti-Money Laundering (AML) verifications as required by law. Tokeny provides this data to third-party companies designated by the Digital Asset Issuer as their agent(s) to verify all KYC/AML information and/or more broadly to administer the Digital Asset during its existence. When you participate in a company’s Offering your Personal Information including personal data is shared with the offering company or its designated agent. Be aware that when you participate in an Offering, the Offering company may use a different KYC/AML service than the service Tokeny uses. Tokeny also provides your anonymized data to the company Analytics Solutions in order to do analytics to better operate our company for our users.
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When you subscribe to purchase tokens from an Offering that uses our platform, you will be asked to provide financial data and information to verify your status and successfully make financial transactions. Depending on how you wish to purchase tokens you may be required to provide financial data, such as your bank routing number and bank account number, to enable the offering company to utilize your bank account to originate funds transfers and make subsequent investment transfers to you if necessary.
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To exercise your rights, you should contact us at [email protected]. We will respond to your requests without undue delay and at the latest within 1 month.
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We will not retain data longer than is reasonably considered necessary to fulfil the purposes for which it was collected, processed or as required by applicable laws or regulations. When a user makes a request to have their account deleted and all data deleted, all personal Data collected through the platform will be deleted, as permitted by applicable law.
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We want to make it clear to our Users that you have control of your data. If you wish to have your personal data edited, deleted, or audited please contact us anytime at [email protected] and our data collection officers will be able to accommodate your requests as permitted by applicable law.
No users under 18 years of age
Our Site is not targeted to persons under the age of 18 and we do not knowingly collect Personal Information from persons under the age of 18. Any user that attempts to submit information revealing their age to be under 18 will be blocked and their information deleted.
Updates to this policy
Contact us | https://tokeny.com/privacy-policy/ |
As real estate requirements become ever more complex, clearly distributed responsibilities and smooth processes are turning into a competitive advantage. In real estate management, the key to success lies in the system-oriented and procedural integration of the maintenance management. Meaning process optimisation.
Integrated: process optimisation for building maintenance
As a software and consulting house for the real estate industry, we have been implementing customized projects in the sector of the management of existing buildings for many years. Thus, we have the appropriate knowledge to define the ideal maintenance process for your company. Together, we determine the process components and then, taking into account your personal strategy, decide which systems and interfaces are suitable and economically viable.
It is particularly important that the processes and IT systems adapt to the people – not the other way round. Then you gain transparency, avoid extra work and increase the productivity of your company.
Find find out more:
How to successfully digitize the maintenance processes?
>> Here you find a best practice report
You want another practical example?
>> This one is about building an integrative system landscape
You want to increase the efficiency of property management through process optimisation?
>> Our expert explains how to do it
Unsere Leistungen im Überblick: | https://www.calcon.de/en/process-optimisation/ |
Increasingly imperative objectives in ecology are to understand and forecast population dynamic and evolutionary responses to seasonal environmental variation and change. Such population and evolutionary dynamics result from immediate and lagged responses of all key life-history traits, and resulting demographic rates that affect population growth rate, to seasonal environmental conditions and population density. However, existing population dynamic and eco-evolutionary theory and models have not yet fully encompassed within-individual and among-individual variation, covariation, structure and heterogeneity, and ongoing evolution, in a critical life-history trait that allows individuals to respond to seasonal environmental conditions: seasonal migration. Meanwhile, empirical studies aided by new animal-tracking technologies are increasingly demonstrating substantial within-population variation in the occurrence and form of migration versus year-round residence, generating diverse forms of 'partial migration' spanning diverse species, habitats and spatial scales. Such partially migratory systems form a continuum between the extreme scenarios of full migration and full year-round residence, and are commonplace in nature. Here, we first review basic scenarios of partial migration and associated models designed to identify conditions that facilitate the maintenance of migratory polymorphism. We highlight that such models have been fundamental to the development of partial migration theory, but are spatially and demographically simplistic compared to the rich bodies of population dynamic theory and models that consider spatially structured populations with dispersal but no migration, or consider populations experiencing strong seasonality and full obligate migration. Second, to provide an overarching conceptual framework for spatio-temporal population dynamics, we define a 'partially migratory meta-population' system as a spatially structured set of locations that can be occupied by different sets of resident and migrant individuals in different seasons, and where locations that can support reproduction can also be linked by dispersal. We outline key forms of within-individual and among-individual variation and structure in migration that could arise within such systems and interact with variation in individual survival, reproduction and dispersal to create complex population dynamics and evolutionary responses across locations, seasons, years and generations. Third, we review approaches by which population dynamic and eco-evolutionary models could be developed to test hypotheses regarding the dynamics and persistence of partially migratory meta-populations given diverse forms of seasonal environmental variation and change, and to forecast system-specific dynamics. To demonstrate one such approach, we use an evolutionary individual-based model to illustrate that multiple forms of partial migration can readily co-exist in a simple spatially structured landscape. Finally, we summarise recent empirical studies that demonstrate key components of demographic structure in partial migration, and demonstrate diverse associations with reproduction and survival. We thereby identify key theoretical and empirical knowledge gaps that remain, and consider multiple complementary approaches by which these gaps can be filled in order to elucidate population dynamic and eco-evolutionary responses to spatio-temporal seasonal environmental variation and change.
© 2018 The Authors.
Find related publications, people, projects, datasets and more using interactive charts. | https://pure.york.ac.uk/portal/en/publications/population-and-evolutionary-dynamics-in-spatially-structured-seasonally-varying-environments(3cd270a4-2289-425b-a9a1-d24032a7795e).html |
Below is a art time-lapse art video of Fort Lauderdale beach in Florida with palm trees that line route A1A. This is painted in gouache and watercolor paint. Number 4 in a series of paintings of the beach. All series paintings are done in gouache paint, also known as opaque watercolor.
Painting Techniques
As you can see in the video I start painting with thin paint washes and build layers from there. I often use warm yellows at the beginning stages.
This painting is only 6.5″x 6.5″ in size and I am using a 1 ” flat brush through most of the painting. This is a very important tip. A bigger brush helps block in the larger shapes in a painting and also helps you avoid small details. At this stage in the painting it is all about the basic composition and values. Save your small brushes for the end of the painting.
Ask any painting questions or comments below, thanks. | https://studio.pjcookartist.com/2021/03/how-to-paint-palm-trees.html |
Feelings like anger, fear and sadness are often referred to as negative because they are unpleasant. However, just like pleasant positive feelings, unpleasant negative feelings can inform us that something needs attention, suggesting that in balance there has to be a positive aspect in a negative emotion.
Eventually, a build-up of negative emotions may leave us feeling unable to cope, to a greater or lesser degree, psychologically, physically and (if you are spiritually inclined) spiritually, too. Over time we may have experienced negative feelings and automatically pushed those feelings to the back of our mind because they felt uncomfortable. After all, it is a natural bodily reaction to adjust ourselves for a more comfortable position.
When we eventually acknowledge unpleasant negative feelings and sit with those feelings and reflect, we then can make sense of what is being communicated and processed timely. For example, if someone upsets you, it may be necessary (or not) to confront that person. Confrontation can be very anxiety-provoking and even feel mentally and physically challenging and, thus, somewhat painful even.
Or we can ignore the anger, leaving no need for confrontation, and that lack of taking action can then build up and become repressed emotions. Part of a negative cycle maybe?
Emotional avoidance development
In childhood, we may learn to ignore or avoid bad things to survive. Emotional avoidance strategies may work in childhood but, by the time we get to adulthood, these old outdated survival mechanisms that no longer serves the adult.
As adults we have more autonomy than children; we can make our own choices. We can learn new ways of dealing with conflict, new ways of communicating, new ways of being. We can develop and learn, grow, develop, and have more fulfilment and be more at peace with ourselves and who we have become… which are all part of the healing process.
So, if we agree that emotional avoidance may have developed from adapting to a past somewhat difficult environment or more to survive, then behaviours like binge eating, over-exercising or excessive use of alcohol may feel like it’s the way to deal with difficult feelings. Although unhealthy coping behaviours such as these may work short-term, which is why people often return to them, they can lead to further problems in the long run.
Anger is a natural emotion
Anger is a powerful energy that can be highly productive or destructive. Comparing ourselves to others can lead to subjective (feeling) self-judgement with no objective (real) measuring tool. If you envy others’, allowing them to become role models rather than wanting to destroy them will empower you so you can get your own version of what it is you envy.
So, negative emotions are an important, useful guide, just like positive emotions are. When we do good, we feel good. Negative emotions can be an objective or subject warning sign of danger where action may be required… after all, if we didn’t feel physical pain when we got too close to the fire, we might burn ourselves badly.
If you feel frustrated, angry, or any uncomfortable feelings, take some time to reflect and allow the process to unfold. This can take the form of writing down, keeping a diary, learning patterns, etc. You may gain insight that you are going in a risky direction, or maybe you may identify that the situation you are in is not risky and it has just “triggered” you because it reminds you of another situation in the past that did feel dangerous and/or unhealthy.
A point to note is that negative emotions or behaviours may be related to past trauma and would need careful handling with a therapist experienced in trauma.
More unhelpful behaviours
- Talking negatively about and to ourselves informs our core belief system and perpetuates negative feelings.
- Camouflaging ourselves within negative/toxic environment/people (unhelpful relationships). If you find yourself surrounded by negative people and negative feelings, take a deep breath and distance yourself.
- Living in the past/distant future. Focus on what is happening now, and not what has taken place in the past. This may require professional help to let the past go (closure), to enable you to be in the present for change to happen and a better future. Letting go and making space happens before filling that space up with better things.
- Not being in the present. Even if the here and now feels like you are all at sea “without a Captain”, there is always a silver lining… a positive to a negative… that can be seen with a healthy unburdened clear lens.
Lacking in confidence, a lack of control, unhelpful habits, unhelpful behaviours, past traumas and unexplained negative emotions can be explored in counselling at a conscious level, and/or accessed with the help of your subconscious/unconscious mind in hypnotherapy.
Once consciously you feel you have unpacked as much as you need to, you can become curious about working with your subconscious/unconscious mind and underpin counselling with hypnosis. Hypnotherapy is a very natural therapy.
Find a counsellor or psychotherapist dealing with anger management
All therapists are verified professionals. | https://www.counselling-directory.org.uk/memberarticles/repressed-negative-emotions-and-emotional-avoidance |
PROBLEM TO BE SOLVED: To provide an engine speed control method and device thereof capable of suppressing excessive rise and fall of the actual idling engine speed.
SOLUTION: An idle-up corrective quantity QELS for correcting an idling engine speed control air quantity QISC used for controlling the engine speed at the time of idle running of an engine, is determined as the idle-up corrective quantity QELS correlated to radiator-fan driving duty (loaded amount) determined by a water temperature-cooling sensor, the car speed, etc. The idle-up corrective quantity is considered to be added to the idling engine speed control air quantity QISC, and thus the engine speed Ne at the time of idle running is controlled.
COPYRIGHT: (C)1999,JPO | |
Approaches to screening will depend on each group, and intervention-specific indicators should be developed for each approach. In general, however, the data on indicators shown in Fig. 2.5 should be collected for each targeted risk group, such as all close contacts of TB patients or all people living with HIV in care.
The data collected can be used to calculate the following basic indicators for each risk group:
- acceptability: the proportion of people screened for TB among those eligible (B/A);
- screened positive: the proportion of people presumed to have TB among those screened (C/B);
- testing retention: the proportion of people tested or evaluated for TB with a confirmatory diagnostic test among patients presumed to have TB (D/C);
- NNS and number necessary to treat: the proportion of people diagnosed with TB among those screened (E/B) and tested (E/D);
- linkage to care: the proportion initiating TB treatment among those diagnosed (F/E); and
- treatment success: the proportion of people who successfully complete TB treatment among those who initiated treatment (G/F).
It is critical to monitor the yield of bacteriologically confirmed and unconfirmed TB patients. A high proportion of unconfirmed TB patients referred from screening programmes might indicate overdiagnosis and should lead to closer evaluation of screening and diagnostic routines, considering the limitations of the diagnostic tests and the need for empirical or clinical diagnosis for certain populations, such as people living with HIV and children. In the case of late-stage detection of TB patients, the proportion of people with presumptive TB among those screened (C/B) and the proportion of those diagnosed among those screened (E/B) would be high. This finding would suggest a need for wider active TB case finding in a risk group. Low values for indicators such as the proportion of eligible people screened (B/A), the proportion of those tested who receive diagnostic confirmation (D/C) and the proportion of people diagnosed who start treatment (G/F) may reveal weaknesses in capacity at critical points in the TB care pathway, which should be addressed.
Data should be disaggregated by variables such as age group and sex. This requires collection of some personal data on each individual screened, which should be within the means of most programmes; the necessary software and hardware requirements are relatively modest.
Additional indicators of process (such as the number of people reached and screened per day, the time required for each step of screening and diagnosis and the number of people who require referral) should be collected during the pilot phase of a screening programme to ensure that it operates as designed and to inform logistics and capacity (e.g. number of tests needed). These data are easier to collect precisely than estimates of eligible populations and may indicate problems and can help to plan operational capacity (e.g. screening activities via mobile-vans over time). Once the programme has been established, however, these additional indicators should be discarded and the focus be shifted to streamlining the programme and scaling it up.
The uptake of screening in a risk group (that is, the proportion of those eligible for screening who are actually screened) can be assessed only if the size of the target group has been well defined. It is usually possible to obtain the relevant information for screening conducted in health facilities, closed settings (such as prisons) and through contact investigations; however, it is often difficult to obtain such information in outreach screening programmes, such as when screening is done in the community, although the estimated population of a targeted community provides a rough estimate of the eligible population.
Whenever screening is done, a baseline TB notification rate should be set from historical data, if available (29). These data are usually available to most programmes from notification records. If they are stored in case-based format (or individual patient data), they will permit more extensive disaggregation by the risk groups of interest. Historical data may have to be adjusted for time trends. Screening may generate a substantial yield but with no real change in TB notifications. This could indicate badly located screening points, but may also be the product of better case finding in populations that were previously neglected and a decrease in false-positive cases that previously inflated notification numbers. If this is the case, the proportion of notified TB patients with bacteriologically confirmed disease would be expected to increase over time even if the numbers remain stable. | https://tbksp.org/en/node/1401 |
At EWPS we believe that all children should have the chance to explore music from different genres, cultures and times through a variety of musical experiences and instruments.
“Music is a moral law. It gives soul to the universe, wings to the mind, and life to everything.”
Plato
How we Teach Music
In Early Years and years 1 & 2 we use the Charanga music scheme. This explores music from different genres, cultures and time periods. Children are taught about composition and how to play simple instruments, both tuned (e.g. xylophones & glockenspiels, metalaphones) and untuned (drums, triangles, cabassas, shakers etc.)
Every child from year 3 to year 6 learns to play a musical instrument or will be taught how to sing. In year 3 this is done on a whole class basis and the children are taught how to play the flutophone . When children progress into years 5 & 6 they learn the clarinet, guitar or trumpet. Tuition takes place weekly in small groups led by qualified instructors through the Every Child A Musician programme (ECAM). As part of this process children are taught to read music and participate in group performances.
How we know your child is succeeding?
In EYFS & KS1 we use the assessment outcomes form the Charanga scheme of work. In years 3, 4, 5 and 6 we liaise closely with our professional instructors to asses pupil’s progress and competency. During year 6 children can also sit nationally accredited music exams (dependent on their level of competency).
If you would like to find out more information please contact: | https://www.ellenwilkinson.newham.sch.uk/page/?title=Music&pid=78 |
In Sacre, the first-ever circus setting of Stravinsky’s Rite of Spring, Australia’s groundbreaking contemporary circus...
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Amjad Ali Khan is the undisputed master of the sarod and one of India’s most celebrated classical musicians. His ancestors developed...
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Known for her electric stage presence and galvanizing live shows, Meklit has rocked stages from London to Cairo. The Ethiopian-American...
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Celebrated as one of Africa’s most vital international stars, Fatoumata Diawara is taking her artistry to thrilling new heights. Boldly...
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Sō Percussion and Pulitzer Prize-winner Caroline Shaw combine forces for a powerful new set of co-composed music in Let the Soil... | https://meanycenter.org/tickets/season?field_event_series_tid=8 |
BACKGROUND OF THE INVENTION
This invention relates to methods of and apparatus for conveying randomly received items, such as rolls of toilet tissue or paper towels, and for placing these items on apparatus, such as an infeed flight conveyor for an overwrap machine or the like, which is continuously operable at a steady cycling rate.
In the manufacture of toilet tissue or paper towels, the paper is wound onto an elongate mandrel of cardboard tubing or the like to form a log approximately ten feet (3 m.) in length. These logs are then cut to form package-size rolls either by a slitter winder or by a roll saw. Typically, the rolls are then wrapped in an overwrap machine prior to being bulk-packed for shipment. Whether the rolls of towels or tissue are cut by a slitter winder or by a roll saw, the rolls are often intermittently and randomly delivered to the overwrap machine. Typically, overwrap machines operate most desirably if they are fed items operated at a steady and regular pace so as to achieve the high wrapping rate at lowest possible machine speed. It is usually preferred to operate overwrap machines continuously without skipping a wrapping cycle to provide the best possible operating efficiency of the machine. After skipping one or more wrapping cycles, registration problems often arise with the first item wrapped by the overwrap machine upon start up.
Usually an overwrap machine is fed items to be wrapped from more than one supply (i.e., from more than one slitter winder or log roll saw). Items may be fed to the overwrap machine from either one or the other of the supplies, or from both of the supplies simultaneously if, for example, a grouping of items are to be wrapped, or alternately from one supply and then the other. It has heretofore been a problem to balance the items in the various supplies so as to insure an adequate supply exists to feed the overwrap machine in the event the supply of items is momentarily interrupted or to prevent the backlog of items in the other supply from becoming excessive. It has also been a problem to form groupings (e.g., a 2×2 array of rolls of toilet tissue) of the items to be fed to the overwrap machine.
In some known item-conveying and pacing systems which randomly receive items and deliver them to a timed apparatus (e.g., a flight conveyor) rolls of paper towels or toilet tissue were abruptly stopped and started and were changed from one level to another. This sometimes resulted in the rolls becoming partially unrolled, thus making them difficult to wrap. Upon abruptly stopping and starting the rolls, certain of the known pacing conveyor systems caused the rolls to "telescope" (i.e. , to cause the center of the roll and the roll case to project outwardly at one end of the roll and inwardly at the other end of the roll).
Reference may be made to the following U.S. Pat. Nos. which are in the same general field as the apparatus of this invention: 3,452,856, 3,459, 289, 3,656,606, 3,794,154 and 3,938,650.
In several of the above-noted prior art references, conveying systems are disclosed which utilize a differential drive system responsive to an electric eye or other sensor which detected the position of an item relative to the flights of a flight conveyor and speeded up or slowed down one conveyor in the system relative to another conveyor so as to place the item at a desired location on the flight conveyor. These prior systems, however, required that the differential drive be operated on almost every cycle of the flight conveyor and the margin of error was, in many instances, small. This resulted in the conveyor system skipping placement of an item on the flight conveyor at relatively frequent intervals. Also, these prior art systems were not, for the most part, adapted to receive and feed items from a plurality of item sources or to form groupings of items to be wrapped.
SUMMARY OF THE INVENTION
Among the many objects and features of the present invention may be noted the provision of methods of and apparatus for feeding randomly received items to apparatus, such as to the infeed flight conveyor of an overwrap machine or the like, which reliably feeds a selected number of items constituting a grouping onto the flight conveyor in timed relation therewith in such manner that the overwrap machine operates continuously without skipping of placement of a grouping of items on the flight conveyor; the provision of such method and apparatus in which a multiple- item grouping or unit array may be formed from one or more supply sources of items and in which this grouping is fed forward to the flight conveyor on each operational cycle thereof; the provision of such a method of and apparatus in which items are randomly supplied from a plurality of sources and in which the usage of items from these sources is automatically balanced; the provision of such apparatus which does not damage the items as they are fed forward; the provision of such apparatus which is readily adjustable to accommodate various-sized items and to accommodate different numbers of items to be conveyed; the provision of such apparatus which doen not change the direction or elevation of the items as they are fed forward; and the provision of such apparatus which is of rugged construction and which is reliable in high speed packaging operations.
Briefly, the method of this invention relates to feeding items to apparatus, such as a flight conveyor, which cycles continuously at a steady cycling rate with a predetermined grouping of items being placed on the apparatus during a portion of each cycle thereof, the cycle portion being referred to as a window and the latter having a leading and a trailing boundary. The items are randomly received from a plurality of sources. The method comprises accumulating a backlog of items randomly received from the sources, one backlog for each source. A selected plurality of items is released from each of the backlogs during each cycle of the apparatus to form a grouping. The selected plurality of items forming the grouping is conveyed forward in timed relation to the apparatus for placement of the grouping in a respective window of the apparatus.
In another method of this invention of feeding items one after the other to apparatus which cycles at a steady cycling rate with an item being fed to the apparatus during a portion of each cycle, this cycle portion being referred to as a window. The items are randomly supplied by first and second supply means with the sum of the average supply rates of the first and second supply means being generally equal to the rate the items are fed to the apparatus. The method comprises accumulating a first and second backlog of items randomly received from the first and second supply means. One or more items are then fed from the first backlog, one item at a time, and conveyed forward for placement of the items in a respective window at a rate faster than the first supply rate. Feeding is terminated from said first backlog and is initiated from the second backlog, one item at a time. These items are conveyed from the second backlog forward for placement in a respective window at a rate faster than the second supply rate. Feeding from the second backlog is then terminated and feeding from the first backlog is initiated. The four above-defined steps are repeated so as to continuously feed items from said backlogs whereby a selected number of items is placed in each window of said apparatus.
In still another method of this invention N items are placed in between successive flights of a continuously operable flight conveyor operating at a steady cycling rate, the items being supplied by n supply means with each of the supply means randomly supplying items. The method comprises accumulating n backlogs of items, one backlog for each of the supply means, and releasing k items from each of the backlogs and conveying them forward for placement between successive flights of the flight conveyor in timed relation to the flights during each cycle of the flight conveyor, where N=k.n, and where k is an integer greater than one. The items placed between successive flights are substantially arranged in an array of rows and columns with n indicating the number of items in each row and k indicating the number of items in each column.
Still further, the method of this invention involves placing an array or grouping of items in between successive flights of a continuously operable flight conveyor operating at a steady cycling rate, the grouping being comprised of columns and rows of items, there being n items in each row and k items in each column, where n is an integer greater than 1 and k is an integer greater than one. The items are randomly supplied by n supply means. The method comprises accumulating n backlogs of items, one backlog for each of the supply means, and releasing k items from each of the backlogs in timed relation to operation of the flight conveyor to form a column of items for the array. The columns of items are conveyed forward in timed relation to a respective pair of successive flights of the flight conveyor for placement of the columns of items between the respective flights.
The apparatus of this invention relates to apparatus for feeding items to means, such as a flight conveyor, continuously operable at a steady cycling rate with a predetermined grouping of items being placed on the means during a portion of each cycle thereof, this cycle portion being referred to as a window having a leading and a trailing boundary. The items are randomly received from a plurality of sources. The apparatus comprises means for accumulating a plurality of backlogs of items, one backlog for each source, and for releasing a selected number of items forward from each backlog during each cycle of the means to form the grouping. Means is provided for conveying forward the selected number of items forming the grouping in timed relation to the means and for placement of said selected number of items forming said grouping in a respective window.
Conveying apparatus of this invention feeds items one after the other to means which cycles continuously at a steady cycling rate with one or more items, constituting a selected number of items being placed in the means during each cycle thereof, each cycle having a portion or window thereof in which the item may be placed. The apparatus further has a plurality of means for randomly receiving items from a respective source, for accumulating a backlog of items, and for feeding the selected number of items forward for placement in a window of a respective cycle of the means. The apparatus further has means for initiating operation of one of the feeding means to feed out the selected number of items during each cycle of the means, for terminating operation of the feeding means after one or more cycles of the means, and for initiating operation of another of the feeding means to feed the selected number of items during each cycle of the means substantially without skipping placement of a selected number of items during a cycle of the means.
A flight conveyor of the present invention has a frame, two sets of pulleys on opposite sides of the frame and an endless, flexible belt trained about each said set of pulleys. Each said timing belt has a generally horizontal upper reach, a plurality of flights extending transversely between the timing belts at equal intervals therearound, means for clamping securing the flights of the timing belts, and a notch in each of the pulleys for receiving the clamping means as the belts travel around their respective sets of pulleys.
Still another method of this invention involves feeding items to apparatus, such as a flight conveyor, which cycles continuously at a steady cycling rate with one or more items, referred to as a selected number of items, being placed on the apparatus during a portion of each cycle thereof, the cycle portion being referred to as a window having a leading and a trailing boundary. The method comprises the steps of accumulating a backlog of randomly received items. A selected number of items is then fed forward from the backlog during each cycle of the apparatus at a feeding rate faster than the cycling rate of the apparatus so that each selected number of items is placed in its respective window progressively closer to the leading boundary thereof. The method further includes sensing when a selected number of items to be placed in a respective window will at least partially extend beyond the leading boundary of the window and then momentarily delaying feeding of the items from the backlog thereby to shift the next selected number of items toward the trailing boundary of its respective window.
Still another embodiment of the apparatus of this invention includes means for accumulating a backlog of randomly received items and for releasing a selected number of items forward from the backlog during each cycle of the flight conveyor. Means is provided for conveying forward the selected number of items in timed relation to the flight conveyor and for placement of the selected number of items on the flight conveyor in a respective window. The releasing means operates at a feeding rate faster than the cycling rate of the flight conveyor so that each selected number of items is placed in its respective window progressively closer to the leading boundary thereof than the proceeding selected number of items. The apparatus further includes means for driving the releasing means and means for sensing when a selected number of items to be placed in a respective window will at least partially extend beyond the leading boundary of its respective window and momentarily delaying the releasing means driving means thereby to shift the next to be placed selected number of items toward the trailing boundary of its respective window.
Other objects and features of this invention will be in part apparent and in part pointed out hereinafter.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a plan view of apparatus of the present invention;
FIG. 2 is a side elevational view of the left side of the apparatus of FIG. 1;
FIG. 3 is an enlarged plan view of a portion of the apparatus illustrating a pair of modulating feeding conveyors which randomly receive items, accumulate backlogs of items, and feed items forward from the backlogs, and illustrating an intermediate or speed-up conveyor;
FIG. 4 is an enlarged plan view of a flight conveyor of this invention which receives items from the feeding conveyors and which conveys the items to a continuously operable machine, such as an overwrap machine;
FIG. 5 is a plan view of the speed-up conveyor with certain parts omitted to illustrate the drive for the apparatus;
FIG. 6 is a vertical section taken along line 6--6 of FIG. 5 further illustrating the drive;
FIG. 7 is an enlarged left side elevational view of the feeding and speed-up conveyors;
FIG. 8 is an enlarged left-side elevational view of the flight conveyor;
FIGS. 9(a)-9(h) are diagrammatical representations illustrating a method of this invention of feeding items;
FIG. 10 is a plan view of the apparatus of this invention on a reduced scale illustrating one manner of supplying items to the feeding conveyors from a plurality of sources;
FIG. 11 is an electrical schematic disclosing a control system for the apparatus of the present invention;
FIG. 12 is a diagrammatic representation of the speed-up and flight conveyors illustrating the method of this invention of feeding unit arrays of items for placement of a unit array on the flight conveyor upon each cycle thereof;
FIG. 13 (sheet 3) is an enlarged view of means for clampingly securing a flight bar to a flight conveyor timing belt; and
FIGS. 14(a)-14(g) are diagrammatical representations illustrating still another method of this invention.
Corresponding reference characters indicate corresponding parts throughout the several views of the drawings.
DESCRIPTION OF PREFERRED EMBODIMENTS
Referring now to the drawings, apparatus of the present invention, indicates in its entirety at 1, is shown in FIGS. 1-8 for feeding items I one after the other to means, such as a flight conveyor 3, continuously operable at a steady cycling rate (e.g., R cycles per minute) with a selected number k or grouping G of items I being placed on the flight conveyor during each cycle thereof for being fed forward to apparatus, such as an overwrap machine generally indicated at 0. Flight conveyor 3 is shown to include a plurality of flights 5 spaced from one another a distance substantially greater than the length of the item (or items) placed thereon. With flight conveyor 3 operating at its continuous steady cycling rate R, a cycle of the flight conveyor is defined to be the time required for successive flights 5 to pass a stationary reference point. Each cycle of a flight conveyor has a portion thereof, referred to as a window W, during which time the selected number of items I may be placed on the flight conveyor without interference with the flights. Window W may vary in length depending on the spacing of flights 5 and the length of the selected number k of items to be placed. The window has a leading boundary W.sub.L defining the start of the period during which the window is open for placement of an item or items I on the flight conveyor and a trailing boundary W.sub. T defining the end of the above- mentioned period.
As generally indicated at 7, apparatus 1 further comprises means for randomly receiving items I from a respective supply of items, for accumulating one or more backlogs BL, BR, in which the items are in end- to-end abutting relation, and for feeding or releasing the items from the backlogs forward one at a time. As shown in FIGS. 1 and 3, feeding or releasing means 7 includes two side-by-side feeding or modulating conveyors 9R, 9L. These feeding conveyors are also referred to as feeding lanes and may be randomly supplied from independent sources, as shown in FIG. 10. Each of these feeding means comprises a pair of opposed endless conveyor belts including an upper belt 9R', 9L', and a lower belt 9R", 9L". The upper and lower belts are spaced from one another a distance slightly less than the thickness (i.e., the height) of items I so as to positively grip the items. These reaches move in the direction of the arrows shown in FIG. 7 at the same lineal speed so as to convey the items at the speed of the belts. The items are randomly received at the rear or entrance ends of feeding conveyors 9R, 9L, and the lineal speed of the feeding conveyors is such that backlogs BR and BL accumulate to the rear of the belts. The backlogs start, however, at the front or exit ends of feeding conveyors 9R, 9L. These opposed belts are driven in a manner as will appear at either a first feed rate F.sub. 1 faster than the cycling rate R of flight conveyor 3, or at a second rate F.sub.2 slower than the cycling rate so that items I are progressively placed in their respective window W closer to a window boundary W.sub.L or W.sub.T than the preceding items. As is generally indicated at 11R, 11L (see FIG. 6), means is provided for shifting each of the feeding conveyors between its first and second feed rates. Other means, as generally indicated at 13R, 13L, is provided for sensing the placement of each item I on flight conveyor 3 relative to its respective window boundaries W.sub.L, W.sub.T for effecting actuation of their respective shifting means 11R, 11L so as to shift feeding conveyors 9R, 9L between their first and second feed rates upon sensing when an item (or items) to be placed on the conveyor will extend at least partially beyond the leading or trailing boundary of its respective window W.
As is best shown in FIG. 8, flight conveyor 3 of this invention comprises a frame 15 on which two sets of pulleys 17a-17d, one set of pulleys on each side of the frame, are provided. Endless timing belts, 19R, 19L, are trained around respective sets of pulleys on each side of the frame. Each of these timing belts has a series of cogs or teeth 20 (see FIG. 13) on its inner face as is conventional and the pulleys have corresponding grooves (not shown) for reception of the belt cogs so as to prevent slippage of the belts relative to the pulleys. Flights 5 are equally spaced from one another around each of the timing belts. In the embodiment shown in the drawings, timing belts 19R, 19L are about 170 in. (430 cm.) long and the flights are equally spaced at intervals of approximately 34 in. (86 cm.) for accommodating items I such as rolls of paper towels 11 in. (28 cm.) long or rolls of toilet tissue 4.6 in. (11. 7 cm.) long. The spacing of flights 5 will also accommodate placement of a grouping G of items, such as multiple rolls of toilet tissue as is shown in FIG. 12. It will be understood that a grouping G may include one, two or more items I.
As heretofore mentioned, flight conveyor 3 cycles at a steady rate R and feeding conveyors 9R, 9L each operate at either a first feed rate F. sub. 1 or a second feed rate F.sub.2, with the first rate being somewhat faster than the cycling rate R of the flight conveyor, and with the second rate being somewhat slower than the cycling rate. For example, cycling rate R of the flight conveyor may be 160 cycles per min., the first feed rate F.sub.1 may correspond to the feeding of a selected number k=163 of items per min. and the second feed rate F.sub.2 may correspond to the feeding of a selected number K=157 of items per min. If, for example, single rolls of paper towels are being conveyed and placed in each window, the lineal speed of feeding conveyors 9R, 9L may be about 149 ft. per min. (46 m. per min.) when the feeding conveyor is operated at its first rate F.sub.1, and about 144 ft. per min. (44 m. per min.) when operated at its second feed rate F.sub.2. Since flight conveyor 3 is operated at a steady cycling rate of 160 cycles per minute, its lineal speed is about 453 t. per min. (138 m. per min.). The above- noted lineal speeds of the various conveyors and the cycling rate of the flight conveyor are only for purposes of illustration.
It will be understood that the apparatus of this invention may be operated at different rates from those described above, and the relative speeds of the conveyors may differ significantly.
Apparatus 1 of the present invention further comprises an intermediate or speed-up conveyor 21 for conveying items I from the outlet end of feeding conveyors 9R, 9L to the entrance and of flight conveyor 3. Speed up conveyor 21 is an endless-belt conveyor which operates continuously at a lineal speed greater than the lineal speed of the feeding conveyors so that upon release of an item by the feeding conveyors from their respective backlogs BR or BL (in which the items are in end-to-end abutting relation), a gap is formed between successive items which are released. For example, the lineal speed of conveyor 21 may be about 360 ft./min. (109.9 m./min.).
Each sensing means 13R, 13L comprises a pair of electric eyes E1R, E2L spaced from one another along the path of items I which are conveyed from backlog BR or BL to flight conveyor 3. Electric eyes E1R and E1L are shown to be positioned adjacent the outlet end of their respective feeding conveyors 9R, 9L for sensing an item I before it is released by its feeding conveyor for being conveyed forward by speed-up conveyor 21. Eyes E2R, E2L are spaced along the speed-up conveyor to detect the position of an item being conveyed by the speed-up conveyor relative to the position of flights 5 on flight conveyor 3. It will be understood that sensing means 13R, 13L may include other components, such as limit switches LS5, LS6 and LS7, as will be hereinafter disclosed. As will be explained in greater detail hereinafter, with apparatus 1 in operation (i. e., not in its start-up mode) and with a steady backlog of items I accumulated by each of the feeding conveyors 9R, 9L, the feeding conveyors will operate at one of their feed rates F. sub.1 or F.sub.2 until such time that eyes E2R, E2L detect that an item fed forward by its respective feeding conveyor will be placed on the flight conveyor with at least some portion of the items extending beyond a boundary W.sub.L or W. sub.T of its window W. When this condition is sensed, a signal will be generated to effect activation of respective shifting means 11R, 11L thereby to shift the respective feeding conveyors from one feeding rate to its other feeding rate. Eyes E1R, E1L sense that items to be released are in end-to-end abutting relating relation or that the gap between successive items is within prescribed limits. If the above-noted backlog conditions are not maintained, termination of items from the respective feeding conveyor will be effected. Other electric eyes E3R, E3L near the entrance end of feeding conveyors 9R, 9L, respectively, may optionally be provided to sense the state of the backlogs BR or BL accumulated by their respective feeding conveyors. Eyes E3R and E3L and their associated circuitry generate signals which prevent feeding of items I by their respective feeding conveyors when their respective backlog BR or BL is below a predetermined state (i.e., when the backlog is less than desired length or has less than a desired number of items therein). Thus, eyes E3R and E3L constitute item sensing means for ensuring that backlogs BR and BL contain above a predetermined number of items. It will be understood that signals generated from either eye E3L or E3R can prevent feeding from both feeding conveyors 9L, 9R. Eyes E3R, E3L can, of course, be moved toward or away from the entrance end of their respective feeding conveyors so as to vary the predetermined state of the backlogs which they monitor.
As shown in FIGS. 1 and 4, flight conveyor 3 further includes item guide bars 23 carried by frame 15. These guide bars are adjustable toward and away from the center of the flight conveyor and are spaced below the level of the upper reaches of timing belts 19R, 19L and flights 5 and are spaced apart so as to engage items I and to guide them along a desired path as they are conveyed along the flight conveyor.
Flight conveyor 3 is optionally provided with a transition belt conveyor 25 at its entrance end for receiving items I from speed-up conveyor 21 and for conveying these items at least part way along the length of the flight conveyor so that the item is forward of the next flight 5 as the flight moves vertically between pulleys 17d and 17a of the sets of pulleys at the opposite sides of the flight conveyor. Thus, transition conveyor 25 insures that the flights engage the rear or trailing ends of the items rather than their bottom faces.
As described herein, flight conveyor 3 is utilized to feed items I into overwrap machine 0 in timed relation to the operating or cycling rate of the overwrap machine. Flight conveyor 3 is driven by the overwrap machine so as to insure that the flight conveyor is operated in precise timed relation with the cycling rate of the overwrap machine. It will be understood that upon a flight bar 5 engaging an item I or a grouping G placed within a window W of the flight conveyor, the item or grouping will be conveyed forward into the overwrap machine in precise timed relation by the flight conveyor. It will further be understood that it is the flight bars themselves which precisely convey the items and it is not necessary for the conveying apparatus 1 of this invention to place the items on the conveyor belt of the flight conveyor in precise timed relation therewith.
For convenience, flight conveyor 3, intermediate conveyor 21, transition conveyor 25 and feeding conveyors 9R, 9L are driven by the overwrap machine drive in a manner as will appear. With apparatus 1 and flight conveyor 3 driven by the overwrap machine, the speeds of the flight conveyor 3, intermediate conveyor 21 and feeding conveyors 9L, 9R will remain proportioned to one another as the speed of the overwrap machine varies. It will be understood that the overwrap machine 0 may be operated at different speeds in its normal operation. By stating that the flight convyor operates steadily and continuously it is meant that the speed of the flight conveyor is not varied for the purpose of placement of items I or groups G in their respective windows W and that in normal operation, the flight conveyor does operate at a steady speed. It will, however, be understood that other suitable drive arrangements could serve equally as well. An elongate drive shaft 27 (see FIG. 8) transmits power from the overwrap machine drive to the various conveyors. More particularly, drive shaft 27 is connected to the input shaft of a first gear box 29 (referred to as the flight conveyor gear box) which is carried by flight conveyor frame 15. The other components of the drive for the feeding conveyors and the speed-up conveyor are carried by a main frame 30. The output of the flight conveyor gear box drives a pulley 31 which in turn drives pulley 17b of the flight conveyor via a belt 33 so as to drive the flight conveyor timing belts 19R, 19L. As shown in FIG. 5, the drive shaft, after continuing through gear box 29, drives other gear boxes 35 and 37 via drive belt and pulley arrangements 39 and 41, respectively. Gear box 35 drives intermediate conveyor 21 and transition conveyor 25 via a timing belt and pulley arrangement generally indicated at 43. The output shaft of gear box 37 drives both feeding conveyors 9R, 9L via a common belt and pulley arrangement 44 which in turn drives two other belt and pulley arrangements generally indicated at 45R, 45L (see FIG. 6). These last-mentioned belt and pulley arrangements each include a respective two electric brake clutch coupling units 47R or 47L, such as are commercially available from the Wagner Electric Clutch and Brake Company of Beloit, Wisconsin under their trade designation models PD500 and SF500, which constitute feed rate shifting means 11R, 11L for feeding conveyors 9R, 9L.
It is to be understood that each pair of opposed belts of the feeding conveyors 9R, 9L has its own respective sensing means 13R, 13L and shift means 11R, 11L so that each pair of feeding belts may be operated independently of one another. Alternatively, both of the feeding conveyors could be slaved together so as to operate in unison. In the following description of the drive of feeding conveyors 9R, 9L, only the drive 45L for feeding conveyor 9L will be described in detail, since the drives for both of these feeding conveyors are identical.
As shown in FIG. 5, drive 44 includes a pulley 49 fixed on the output shaft of gear box 37 which drives another pulley 51 via a timing belt 53 (see FIG. 6). Pulley 51 is fixed to a shaft 55 journaled to frame 30 of the conveyor apparatus and pulleys 57R, 57L are fixed thereon to drive respective feeding conveyors 9R, 9L via their respective drive clutches 47R, 47L. Electric clutch brake units, 47L, 47R, each includes a first electric clutch 47Ra, 47La (see FIG. 11) which when energized causes its respective feeding conveyor 9R, 9L to be driven at its slower feed rate F. sub.2 and a second electric clutch 47Rb, 47Lb which when energized causes its respective feeding conveyor to be driven at its faster feed rate F. sub.1, and an electric brake 47Rc, 47Lc which when energized stops its respective feed conveyor 9R, 9L.
The output shaft of clutch units 47L, 47R carry respective drive pulleys 59R, 59L, which in turn drive their respective feeding conveyors 9R, 9L via respective timing belt and pulley arrangements generally indicated at 61R, 61L. A detailed description of these timing belt and pulley arrangements is not deemed necessary because their construction and operation is readily apparent from FIGS. 6 and 7. It is manifest that these belt and pulley arrangements drive their respective feeding conveyors in such direction and speed that the lower reaches of belts 9R", 9L", and the upper reaches of belt 9R" and 9L" move in the same direction and at the same lineal speed such that items gripped by the belts are uniformly conveyed thereby. As is shown in FIGS. 6 and 7, belts 9R', 9R" are each trained around respective rollers 63R fixed on shafts 65R journaled to the conveyor frame and belts 9L' and 9L" are trained around similar rollers 63L fixed on shafts 65L journaled on the frame. Rollers 63R, 63L are axially slidable along their respective shafts 65R, 65L so as to vary their position relative to conveyor frame 30. The upper shafts 65R and 65L are mounted on a movable portion of the conveyor frame, as indicated at 30' which is adjustable vertically within a limited range so as to vary the vertical spacing between each pair of belts 9L', 9L" and 9R', 9R" thus enabling the apparatus of this invention to accommodate items of different heights or thicknesses. As shown in FIGS. 2 and 7, belt and pulley drive arrangements 61R, 61L are so constructed as to accommodate vertical movement of the upper frame portion 30'. Each of these feeding conveyor belts is tensioned by means of rollers, as is generally indicated at 67 (see FIG. 3). Adjustable guides 69R, 69L (see FIG. 6) are provided for guiding items I as they are fed into and as the items are conveyed along by feeding conveyors 9R, 9L.
Referring now to FIG. 5, a series of cam operated switches, as indicated at LS1-LS8, is shown to be operated by respective rotary cams C1-C8 fixed on and rotatable with a cam shaft 71. The latter is driven by a timing belt 73 trained around a pulley 75 fixed on drive shaft 27 and a pulley 77 fixed on cam shaft 71 at such a rate so that the cams rotate one revolution for each cycle of the flight conveyor. The operation and function of these switches will be hereinafter explained in reference to FIG. 11.
Apparatus 1 of this invention is operable for feeding either a grouping G of items (including a single item) forward in timed relation to flights 5 of flight conveyor 3 for placement of either a single item or other grouping of items between successive flights of a flight conveyor. As shown in FIG. 12, grouping G of items I is comprised of a plurality of side-by-side columns of items I extending transversely to the direction of travel of the conveyors or items. In other words, a grouping of items I is arranged in columns and rows with n representing the number of columns (or the number of items in each row) and with k representing the number of items in each column.
As is generally indicated at 79R, 79L, in FIGS. 1, 2, 3 and 7, means is provided for accumulating or holding back one or more items (i. e., a selected number of items) I fed forward from each respective backlog BR, BL so as to form a column of items, and for releasing the formed columns of items as a unit to be fed forward by intermediate or speed-up conveyor 21. Thus, the entire column is fed forward for placement within a respective window W of the flight conveyor 3. Each accumulating means 79R, 79L is shown to comprise a pair of spaced rods 81 cantilevered from a horizontal shaft 83 pivotally supported by frame 30 with the pivot shaft extending transversely of the respective feed conveyors 9R, 9L at the outlet ends thereof below the level of the items fed forward from the feeding conveyors with rods 81 extending in the direction of feed of the items. Rods 81 are preferably spaced apart from one another a distance somewhat less than the width of the items and are preferably of a length somewhat greater than that of the items to be accumulated therein to form a column. Pivot shaft 83 and rods 81 for each means 79R, 79L are movable as a unit from a lowered retracted position (as shown in solid lines in FIG. 7) in which rods 81 are horizontal and contiguous to the upper surface of conveyor 21 and in which items I supported on the rods are in frictional engagement with conveyor 21 for being conveyed by the conveyor and a raised position (as shown in phantom) in which items are fed forward by feeding conveyors 9R, 9L are held by the rods clear of the speed-up conveyor and hence constitute means for releasing the items accumulated thereon so that the items released remain in substantially end-to-end abutting relationship one with another. The rods are moved between their stated raised and lowered positions by means of respective rotary cams 85R, 85L (see FIG. 7) driven in timed relation to the cycling rate R of flight conveyor 3 via a belt and pulley arrangement drive 87. Each item accumulating means includes a bell crank 89R or 89L rigidly connected to its respective pivot shaft 83, the bell crank constituting a cam follower in engagement with the cam surfaces of rotary cams 85R, 85L. Thus, upon operation of these cams, the rods are raised and lowered in precise timed relation to each cycle of the apparatus. Air cylinders 91R, 91L are interconnected between frame 30 and bell crank members 89R, 89L. When extended, air cylinders 91R, 91L hold their rods in their raised position thereby to prevent the rods from moving in accordance with their respective cams. Air cylinders 91R, 91L are controlled by solenoid operated air valves 93, as shown in FIG. 11. With the rods 81 in their raised position, a column of items I of a selected number of items, for example k items, can be readily accumulated on the rods and the accumulated items can be released to travel toward flight conveyor 3 in timed relation to their respective windows W and the items are placed on the flight conveyor substantially without a gap between adjacent items in the column upon movement of rods 81 from their raised accumulating position to their lowered position in which the items accumulated thereon are in frictional engagement with speed-up conveyor 21.
Referring now to FIG. 11, the control system for apparatus 1 of this invention and for carrying out the methods of this invention is schematically disclosed. Generally recognized electrical symbols are used in FIG. 11 to represent conventional electrical components. While it is believed that those skilled in the art will readily understand the construction and operation of this control system from the schematic representation shown in FIG. 11, the more important features of the control system and various aspects of its operation will hereinafter be discussed.
Briefly, the control system includes duplicate controls for the left feeding conveyor 9L, as is shown in lines L2-L10 of FIG. 11 and is generally indicated at 13L, and for the right feed conveyor 9R, as shown in lines L11-L22 and as generally indicated at 13R. The circuitry shown in lines L23-L26 controls operation of item or column accumulator means 79R, 79L, and the circuitry shown in lines L27-L34 controls alternate feeding of one or more items from either of the feeding conveyors. In line L42, a timed delay relay TDR is shown which, in one mode of operation, controls feeding of items from one conveyor or the other for a predetermined time interval (i.e., of the time interval of the time delay relay) and then automatically terminates feeding from one feed conveyor and initiates feeding of items from the other feed conveyor so as to maintain a balance between backogs BL and BR. The circuitry for controlling operation of clutch brake units 47L, 47R is shown in lines L50-L59. The circuitry shown in lines L60-L66 and in lines L67-L72 control modulation of the right and left feed conveyors 9R, 9L between their respective first (or fast) and second (slow) feed rate F.sub.1 and F.sub.2, respectively. Finally, the circuitry shown in lines L73-L75 and at lines L77-L79 of FIG. 11 including electric eyes E3L, E3R, respectively sense the state of backlogs BL and BR.
As previously mentioned, electric clutches 47R, 47L are electric brake clutch coupling units each having a low-speed clutch 47Ra, 47La, and high- speed clutch 47Rb, 47Lb, and an electric brake 47Rc, 47Lc. By activating the various clutch coils as shown in FIG. 11, each conveyor 9R, 9L may be operated independently of one another at either of its feed rates F.sub.1, F.sub.2 or may be braked to a stop.
Operation of apparatus 1 will be hereinafter described so as to feed items I from only one of the feeding conveyors, such as 9L. Of course, operation of the other feeding conveyor 9R will be similar. To start operation of feeding conveyor 9L, switch SL (see line L3 of FIG. 11) is turned on. Assuming that no item is blocking eye E1L, the coil of relay CR19 (see line L3) is energized, thus closing its normally open contact CR19 in line L54 so as to energize the slow-speed clutch 47La of clutch 47L. Thus, feeding conveyor 9L will be driven at its slow feed rate F.sub. 2 and will convey an item toward eye E1L. Upon the item being conveyed forward actuating eye E1L, the contacts between pins P5 and P6 of eye E1L (see line L2) open and thus deenergize the coil of relay CR19 which in turn opens its normally closed contacts in line L54 and deenergizes both the high and low speed clutches 47La, 47Lb of clutch 47L and energizes its brake 47Lc so as to stop feeding conveyor 9L. Upon the roll start switch LS6 (see line L36) being closed by its respective rotary cam C6, the coil of relay CR30 is energized. The closing of contact CR30 in line L5 energizes the coil of relay CR20 and its normally open contacts in line L9 close which in turn reenergize the coil of relay CR19. Thus, conveyor 9L will be again driven forward at its slow feed rate F.sub.2. As each item I is conveyed away from the exit end of feeding conveyor 9L by speed-up conveyor 21, a gap will develop between the item being conveyed away and the next item in backlog BL. The gap causes eye E1L to momentarily remake. While relay CR19 would normally be opened upon the remaking of eye E1L, a so-called roll-gap limit switch LS5 in line L35 actuated by its respective cam C5 energizes the coil of relay CR29 so that its normally open contacts CR29 in line L6 close thereby providing power to the coil of relay CR20 and preventing relay CR19 from being deenergized upon the remaking of eye E1L. Switch LS5 will remain closed for a time determined by the profile of its actuating cam C5. Upon opening of switch LS5, relay CR29 is deenergized. If by the time CR29 is deenergized, however, the next item to be conveyed forward blocks eye E1L, coil CR19 will remain energized and feeding conveyor 9L will remain in motion. If eye E1L is not blocked upon the deenergization of relay CR29, relay CR19 will be deenergized thus stopping operation of the feeding conveyor. It can thus be seen that by varying the length of time that the roll-gap limit switch LS5 remains closed, the gap or spacing between successive items in backlog BL can be readily varied.
As shown in lines L38 and L39, respectively, cam-operated limit switches LS7 and LS8 are provided which respectively energize the coils of relay CR6 and CR7. Switches LS7 and LS8 are actuated by respective cams C7 and C8 on camshaft 71, which, as previously mentioned, is driven by the drive of the conveyor system through one revolution for each cycle of flight conveyor 3.
With feeding conveyor 9L operating at its slower feed rate F. sub. 2, it will continue to operate at its slow feed rate as long as an item I fed forward by the feeding conveyor blocks electric eye E2L before cam C7 actuates switch LS7 and pulses its respective relay CR6. In lines L63 and L64 it will be noted that when an item blocks eye E2L and when contact CR6 is closed, the coil of relay CR21 is energized and its latching contact in line C63 is closed thereby to hold its coil energized after relay CR6 is deenergized so as to maintain energization of the slow- speed clutch 47La via the closed contact CR19 and the normally closed contact CR22. However, upon eye E2L being unblocked cam 8 actuates switch LS8 to pulse relay CR7 and to close contacts CR7 in line L65. Thus eye energizes the coil of relay CR22. Energization of the latter opens its normally closed contact CR22 in line L63 and thus deenergizes the coil of relay CR21, opens the normally closed contact of relay CR22 in line L54 so as to deenergize slow-speed clutch 47La and closes the normally open contact CR22 in line L55 so as to energize the high-speed clutch 47Lb to drive the feeding conveyor 9L at its high- speed rate F.sub.1. Relay CR22 is latched via its contacts in line L66. Thus, the feeding conveyor 9L will continue to be driven at its higher feed rate placing items I in their respective windows W progressively closer to the leading boundary W. sub.L of window W until such time as an item blocks eye E2L when relay CR6 is pulsed thus energizing relay CR21. It is therefore seen that the actuation of switches LS7 and LS8 determines the leading and trailing boundaries W.sub.L and W.sub.T, respectively, of window W.
It will be understood that the modulation of feeding conveyor 9R between its fast and slow feed rates is effected in essentially the same manner as described above by switches LS7 and LS8 via relays CR25 and CR26 when switch SR is turned to its on position.
If it is desired to alternate feeding of one or more items I from one feeding conveyor 9R or 9L and then the other, switch SA in line L27 is turned to its on position. This in turn energizes the coil of relay CR31 via line L33. Relay CR31, when energized, effects feeding of items I only from the right feeding conveyor 9R. Items will be fed from the right conveyor until such time as time-delay relay TDR times out thereby deenergizing the coil of relay CR35. This in turn drops out relay CR31 and powers up relay CR32 to initiate feeding from the left conveyor 9L. If, however, the items I in one of the backlogs BL, BR decreases to the point where its respective backlog sensing eye E3R or E3L is unblocked, its respective relay CR33 or CR34 becomes energized and thus prevents its respective feeding conveyor from feeding items when the time- delay relay would normally have initiated operation of that feeding conveyor.
Now that apparatus 1 and the control system for the apparatus of this invention have been disclosed in detail, the methods of this invention will now be described. Briefly, the method of this method involves feeding items I one after the other to apparatus, such as to flight conveyor 3, which cycles continuously at a steady rate (e.g., at R cycles per minute) with an item I (or a grouping G of items) being fed to the flight conveyor during a portion of each cycle thereof (i.e., during the availability of window W) with the items being randomly delivered from a plurality of item supplies or sources (e. g., a log saw) at respective average rates r.sub.R and r.sub.L. A feeding lane 9R, 9L is provided for each item source and a backlog BR or BL of items is accumulated as they are received from each of the item sources. In the apparatus of the present invention this step of accumulating backlogs is accomplished by a feeding conveyor 9R, 9L which positively grip one or more of the items between their respective belts 9L', 9L" and 9R', 9R" so as to hold back the items next received and to allow successive randomly received items on supply conveyors 92L, 92R to come into engagement with the trailing ends of the gripped items. As is best shown in FIG. 10, the item supplies may be located at widely separate locations and the items may be conveyed to the ends of the feeding conveyors via supply conveyors 92R, 92L. In other instances, the supply of items may be located relatively close to the entrance end of the feeding conveyors. Preferably, supply conveyors 92R, 92L are driven at a lineal speed faster than the lineal speed of the belts of feeding conveyor 9L, 9R so as to form backlogs BR, BL. It will be understood that the average rate at which supply conveyors 92R, 92L supply the items is generally equal to the rate at which the feeding conveyors feed the items forward. Backlogs BR and BL which are accumulated behind each of the feeding conveyors thus compensate for any short-term irregularities in the random delivery of items from the supply conveyors to feed items forward continuously at a substantially steady rate. Each feeding conveyor 9R, 9L is operable to feed forward a selected number k of items on each cycle of flight conveyor 3 so as to place a group of k. multidot.n items in each window W.
The method of this invention further comprises releasing items I (or a grouping G of items) one at a time from backlogs BR or BL and conveying the item or grouping forward for placement on flight conveyor 3 within a respective window W in timed relation to the flight conveyor. In apparatus 1 items are released as a respective feeding conveyor 9R or 9L feeds an item forward from its exit end. Alternatively, several items can be released from a backlog and accumulated on rods 81 so as to form a grouping G. Either a single item released by the feeding conveyors or a grouping of items accumulated by rods 81 are then released in timed relation to the cycling rate of the flight conveyor for placement in a window W either at a first feed rate F.sub.1 which is slightly faster than the cycling rate R or at a second feed rate F.sub.2 slightly slower than the cycling rate of the flight conveyor, so that each item (or grouping of items) is placed in its respective window W progressively closer to one boundary W.sub.L or W.sub.T of the window than the preceding item. When the apparatus 1 of the present invention is operated to form and convey grouping G of items, clutches 47R and 47L simultaneously drive their respective feeding conveyors 9R and 9L at their first or second feed rates independently of one another. It will also be understood that since accumulating rods 81 are driven in timed relation to flight conveyor 3, these rods are moved between their receiving and release positions in 1:1 relation with the cycling rate R at which the flight conveyor is driven. On each cycle of the flight conveyor, rods 81 are maintained in their raised receiving positions for a time sufficient to accumulate a selected number k of items and is momentarily lowered to release the items accumulated thereon. The items are instantaneously carried away by conveyor 21 and the rods are returned to their raised positions to receive the next item fed from its feeding conveyor on the next cycle. Thus, the operation of the feeding conveyors need not stop.
The method of this invention still further involves sensing when an item I or a group G to be placed in a window W of the flight conveyor conveyed at one feed rate (i.e., at either F.sub.1 of F.sub.2) will at least partially extent beyond one boundary (either the leading or trailing boundary W.sub.L or W. sub.T or window W) and effecting the release of items from the backlog at the other feed rate so that each item is placed in its respective window progressively closer to the opposite boundary of its window than the preceding item.
This method is semidiagrammatically illustrated in FIG. 9. As can be seen, window W is between successive flights 5 of flight conveyor 3 and extends from the forward end of a group of items engaging the trailing flight bar to the leading flight bar. For purposes of explanation, assume the first group to be placed, designated G1, is placed adjacent the trailing boundary, W.sub.T, of the window and assume the feeding conveyor 9R or 9L is being operated at its fast feed rate F. sub.1. The next successive group (e.g., items G2-G4) are released and fed forward by the feeding conveyor and are placed in their respective windows progressively closer to the leading boundary W.sub.L of their respective windows than the preceding group. At some point in time, however, sensing means 13L, 13R (i.e., electric eye E2R or E2L) will sense that a group to be placed in its window will extend at least partly beyond the leading boundary of its window. In practical terms, this means that if the group is not placed in its respective window, it will overlie a front flight bar 5. This interference with the flight bar is, of course, not desirable. In accordance with the method of this invention, the position of each group relative to its respective window is sensed (as by electric eyes E2L, E2R) and the feed conveyors are shifted to their slower feed rate F.sub.2. Thus, the next group to be placed (i.e., group G5) is placed in its window in such position as to not interfere with the forwardmost flight bar defining the forward or leading boundary W.sub.L of its window. With feed conveyor being operated at its slower feed rate F.sub.2, groups G5- G8 are placed in their respective windows progressively closer to the trailing boundary W. sub.T of their respective windows W. When an electric eye E2R or E2L senses that a next group to be placed will a least partially extend over the trailing boundary of its window, their respective feeding conveyor is shifted to its higher feed rate F.sub.1.
In this manner, groups G (or items I) are continuously fed forward in timed relation to the flight conveyor, and one item (or grouping of items) is then placed in each window W on each operational cycle of the flight conveyor substantially without skipping a cycle of the flight conveyor. It is to be noted that in either the method of or the apparatus for this invention it is not necessary to exactly match the placement of items on the flight conveyor to the cycling rate of the flight rate, but rather it is only necessary to place the item or the unit array of items somewhere in its respective window which may be of much greater length than the item. As described herein, the spacing of flight bars 5 on flight conveyor 3 is shown to be on 34 in. (86 cm.) centers in FIG. 8. Thus window W is about six times longer from its leading to trailing boundaries than a roll of toilet tissue. This gives the apparatus of this invention a great deal of leeway in the placement of items. It is to be understood that upon engagement of the items by the trailing flight bar 5 of the flight conveyor, the item or group is conveyed forward in exact timed relation with respect to the cycling speed of the flight conveyor. The method of this invention as illustrated in FIG. 9, depicts only a few items being placed between the time feeding conveyors 9R, 9L are modulated or shifted between their fast and slow feed rates F.sub.1 and F. sub.2, but it will be appreciated that in actual operation many items may be placed between the times the feeding rates of the feeding conveyors are modulated. This greatly reduces the frequency with which the feeding conveyors must be shifted and thus extends the service life of the apparatus.
It is to be understood that both feeding conveyors 9R, 9L may be used either alternatively or simultaneously. If, for example, items I are being supplied to apparatus 1 by both supply conveyors 92R and 92L at their respective average supply rates rR and rL, and if only one item I is fed forward from either feeding conveyor on each cycle of flight conveyor 3, apparatus 1 can be operated to effectively balance the feeding of items from each feed conveyor so as to maintain their respective backlogs of items BR and BL balanced within a predetermined range or state. This may be accomplished by feeding from one backlog (for example, backlog BL) for a predetermined length of time, as determined by time-delay relay TDR, and then feeding from the other backlog via its respective feeding conveyor for another predetermined length of time. During the length of time that one of the feeding conveyors is shut down, its supply of items in its respective backlog is being replenished by its respective supply conveyor 92R or 92L. It is to be noted that when the conveying apparatus of this invention is being operated to feed alternately a plurality of items from one of its feeding conveyors and then from the other, one item per cycle, each of the feeding conveyors must be capable of feeding forward items at the cycling rate R of flight conveyor 3.
Apparatus of the present invention is also capable of being operated to feed one item forward from one feeding conveyor, for example from the left feeding conveyor 9L, stopping operation of the left feeding conveyor, and initiating feeding of an item from the right feeding conveyor 9R so that items are fed forward at the cycling rate R of flight conveyor 3, but an item is fed forward from each feeding conveyor only on every other cycle of the flight conveyor. This allows the conveying apparatus of this invention to receive items at a slower rate from independent item supplies and yet allows the overwrap machine 0 to operate at a much higher cycling rate. Generally, the cycling rate R of the flight conveyor and of overwrap machine 3 is equal to the sum of the average feeding rates r.sub.R and r.sub.L of the supply conveyor 92R, 92L. Expressed in mathematical terms, &egr;r.sub.i =N· R, where N is the number of items placed in each window W.
As previously mentioned, the method of this invention can optionally include sensing the state of backlogs BR or BL by sensing the number of items in each backlog or by sensing the length of the backlog by electric eyes E3L or E3R and terminating feeding of one of the backlogs in response to this backlog diminishing below a predetermined state. Feeding of items from the other backlog is then initiated so that the one backlog may be replenished by its respective supply conveyor 92R or 92L to its desired state while the flight conveyor is continuously being fed from the other backlog. In this manner, the backlogs can be automatically balanced even if their supply rates vary significantly over the short run or if their supply rates are appreciably different from one another.
Referring now to FIGS. 4 and 8, flight conveyor 3 as heretofore described, is shown having flights 5 secured to timing belts 19L, 19R at spaced intervals therearound. Flights 5 include a bar 101 which extends transversely across the flight conveyor between timing belts 19R and 19L and each of these bars have a clamp 103R, 103L at each end thereof. Each of these clamps has a slot 104 (see FIG. 13) in its end for receiving a respective timing belt 19L, 19R. As indicated at 105, a groove is provided for reception of a belt cog 20. The lower surfaces of slot 104 angle down away from groove 105 to provide clearance for belts 19L, 19R as they travel around pulleys 17a-17c. The clamp is drawn into clamping engagement with its timing belt by means of a clamp ring 107 thereby to securely fix the flight bar to the timing belts. As indicated at 109 (see FIGS. 2 and 4) timing belt carrier members are clampingly secured to the timing belts at equal intervals between the flight bars. These carrier members are essentially identical to flight bar clamping members 103 and the carrier members and flight bar clamps serve to support the timing belts on rails 111R, 111L below the upper reaches of the timing belts. Pulleys 17a-17d, each have notches 113 therein for reception of flight bar clamp members 103 and carrier members 109. It will be noted that the circumference of pulleys 17a-17c is equal to the spacing between adjacent carrier members 109 or flight bar clamps and that the circumference of pulley 17b is twice the spacing of these members and has two notches 113 therein. Thus, as the timing belts are driven around pulleys 17a-17d, the flight bar clamps and the carrier members are received by notches 113 in the pulleys and these members do not interfere with the timing belts as they are driven around their respective pulley sets.
Another embodiment of the method of this invention is depicted in FIGS. 14a-14g. Briefly, this method involves accumulating a backlog BL or BR of randomly received items I in the manner heretofore described. A selected number of items k (i.e., one or more items) is fed forward from the backlog by a respective feeding conveyor 9L, 9R during each cycle of flight conveyor 3 at a feeding rate F.sub.1 faster than the cycling rate R of flight conveyor 3 so that each selected number of items is placed in its respective window W progressively closer to the leading boundary W. sub.L of the window. This method further involves sensing when a selected number of items to be placed in its respective window will at least partially extend beyond the leading edge of the window, and then momentarily delaying feeding of items from the backlog and reinitiating feeding of the items thereby to shift the next selected number of items toward the trailing boundary W.sub.T of its respective window. This shift of the selected number of items relative to its window is shown in FIG. 14e. Of course, the next successive selected number of items would be then fed forward at the faster feeding rate F.sub.1 and would be placed in their respective windows progressively closer to the leading boundary W.sub.L of their respective windows.
The apparatus for carrying out the above discussed method is essentially identical to the apparatus 1 heretofore described, except that it does not have low speed clutches 47La, 47Ra. This modified apparatus includes at least one feeding conveyor 9L or 9R for receiving randomly received items and for accumulating a backlog BL or BR of the items. The feeding conveyor then feeds the selected number k of items forward from the backlog during each cycle of flight conveyor 3. The apparatus further includes a drive, as indicated at 61L, 61R, for the feeding conveyors, and means (e.g., intermediate conveyor 21) for conveying forward the selected number of items in timed relation to the flight conveyor and for placement of the selected number of items on the flight conveyor in a respective window W. The feeding conveyors 9L, 9R are driven at a feeding rate F.sub.1 to feed forward selected number of items faster than the cycling rate R of the flight conveyor so that each selected number of items is placed in its respective window progressively closer to the leading boundary of its window than the preceding selected number of items. The apparatus further includes means 13L, 13R for sensing when a selected number of items to be replaced in a respective window will at least partially extend beyond the leading boundary W.sub.L of its respective window and for momentarily interrupting the feeding conveyor drive thereby to shift the next to be placed selected number of items toward the trailing boundary W.sub.T of its respective window. This momentary hesitation of feeding is effected by momentarily deenergizing high speed clutch 47Lb, 47Rb and after a slight pause (e.g., a few milliseconds), reenergizing the clutch. This will incrementally slow down feeding conveyors 9L, 9R and will shift the placement of selected number of items (that is, a single item or a grouping of items) toward the rear (i. e., toward the trailing boundary W. sub.L) of its window, as shown in FIG. 14e. When operating at its normal rate (for example, 160 cycles/minute) it will be understood that the above-described momentary interruption of the feeding conveyor drive will usually not result in stopping of the feeding conveyors because of the inertia in the drive of the feeding conveyors. It will be further understood that this momentary hesitation could also be accomplished by momentarily driving the feeding conveyors by another drive (e.g., low speed clutches 47La, 47Ra) and then again driving the conveyors at their faster feed rate via high speed clutches 47Lb, 47Rb.
In view of the above, it will be seen that the several objects of the invention are achieved and other advantageous results attained.
As various changes could be made in the above methods and products without departing from the scope of the invention, it is intended that all matter contained in the above description or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense. | |
Anne Kristin Møller Fell (born 1966) PhD, occupational health specialist and senior consultant. She performed the occupational health assessment of the patient, reviewed the patient’s medical records and conducted the literature search. She is the first author.
Randi Eikeland (born 1966) PhD, neurologist and senior consultant. She performed the neurological examination of the patient and contributed actively to the drafting and revision of the manuscript.
Jan Olav Aaseth (born 1943) Professor and MD PhD, specialist in internal medicine, endocrinology, occupational health, medical biochemistry and nuclear medicine. He wrote parts of the manuscript, contributed to the literature search and was actively involved in the revision of the manuscript.
In the 1970s, a woman who was then in her thirties contacted her doctor with intermittent respiratory ailments, increasing fatigue, agitation and visual disturbances. She developed a complex array of symptoms involving multiple organ systems. More than 30 years would pass before the likely cause of the symptoms was identified.
The woman contacted her doctor owing to an intense cough, headache and increasing fatigue. The doctor found her to be agitated, anxious and disoriented. He knew her well and had previously regarded her as mentally strong and well-balanced. She was signed off sick from work and recovered after a few days.
A year later, she contacted the doctor again with similar symptoms, but this time with the addition of a metallic taste in her mouth, intermittent tremor, blurred vision and impaired colour vision. She thought that she had become a little forgetful, which had also been mentioned by her colleagues. The doctor referred the patient to an ophthalmologist who found normal visual acuity, alignment, intraocular pressure and visual field bilaterally. Colour vision was not tested.
Some of the patient’s symptoms (headache, increasing fatigue, cough) are common in the population (1), whereas the combination of forgetfulness, visual disturbances and respiratory ailments is more unusual. The medical records give no indication as to why spirometry and PEF measurements were not performed, nor why the patient was not referred to a neurologist.
Several of the symptoms, including impaired memory and concentration, visual disturbances and tremor, may indicate a disorder of the nervous system. Demyelinating diseases such as multiple sclerosis and other early-onset neurodegenerative disorders may be relevant differential diagnoses.
The patient slowly improved and was back at work after a few weeks, but after a while experienced increasing cognitive difficulties in the form of fatigue, impaired concentration and memory and increasing visual impairment. She was referred to the psychiatric services but no signs of mental illness were found.
Four years after symptom onset she was registered as 50 % disabled, but continued to work part-time as a dental nurse. It is unclear why she was not referred to a neurologist, neuropsychologist or pulmonary specialist as part of her disability assessment. After a long period of sick leave she again gradually improved, but her symptoms worsened when she returned to work. This pattern repeated itself over the following years.
Fifteen years after symptom onset, the patient was referred to a neurologist owing to strabismus and muscle twitching. The neurologist described moderate strabismus and minor fasciculations in the throat, neck and arms. One year later the occupational health doctor referred the patient to a pulmonary specialist because of her recurrent respiratory ailments. Spirometry was normal and a metacholine challenge test proved negative. However, at that point the patient had been on full sick leave for several months; the pulmonary specialist therefore recommended PEF measurements following her return to work.
Two years later the patient was referred for an occupational health assessment. She had by then been working as a dental assistant for more than 20 years – with exposure to mercury during the preparation of amalgam. PEF measurements were carried out over 18 days and the results were evaluated by a pulmonary specialist. However, because the patient worked only three days consecutively, the data were inadequate. There is no information as to why the patient was not referred to a neuropsychologist, but it appears from the discharge summary that multiple chemical sensitivity (MCS) was suspected as an explanation for the patient’s complex array of symptoms and her oversensitivity to odours. Two years after the initial occupational health assessment, she was granted a 100 % disability pension.
Multiple chemical sensitivity is a condition with symptoms from multiple organ systems, including headache, increasing fatigue and respiratory ailments that can be triggered by stimuli such as strong odours (2, 3).
Our patient also had visual disturbances, tremor and cognitive difficulties, which are not common in multiple chemical sensitivity and which should trigger suspicion of neurological disease. Fluctuating cognitive impairments concurrent with increasing fatigue may indicate a more serious neurological disease, such as multiple sclerosis. Tremor in a young person may be of the benign essential type. However, its presence in combination with neurological symptoms such as cognitive failure and visual disturbances raises the possibility of other neurodegenerative disorders, for example diseases of the basal ganglia, such as Parkinson’s disease or Wilson’s disease.
The patient was re-examined by an ophthalmologist, who detected a restricted field of vision and pronounced cortical exophoria (outwards strabismus). A cerebral CT at Rikshospitalet revealed a rough frontal surface relief with a prominent Sylvian fissure bilaterally; the CT scan was otherwise normal. Lumbar puncture revealed normal cells and a normal protein level with no monoclonal bands. There was no sign of the production of Borrelia antibodies in the cerebrospinal fluid or blood.
Two years later the patient took part in a research project at the University of Bergen on the cognitive effects of mercury exposure, which entailed a simplified neuropsychological assessment. While there is no overview available of the tests that were performed, the neuropsychologist described the results as showing mild impairments in memory and autonomic reactivity. Cerebral MRI revealed enlargement of the subarachnoid space in the frontal cortex and cerebellum and mildly reduced signal in the putamen. An ophthalmological examination using the Farnsworth dichotomous colour vision test showed impaired colour vision. The patient was referred to a neurologist, who concluded that the clinical examination and imaging results indicated central nervous system atrophy and basal ganglia pathology.
Some of the patient’s symptoms, such as headache, cough and shortness of breath, decreased after she stopped working and were thus partially reversible, but this was not the case for the tremor, cognitive difficulties and visual disturbances. The patient applied to the Norwegian Labour and Welfare Administration to have her symptoms recognised as work-related, but was unsuccessful because multiple chemical sensitivity is not on the list of recognised illnesses.
The patient contacted a researcher at Lund University in Sweden who had conducted studies on the effects of mercury in animals. He sent the patient’s MR images to Boston Environmental Hazards Center at Boston University in the USA. A neurologist there described significant non-age appropriate frontal atrophy and weak to moderate atrophy of the cerebellum.
The patient’s MR images were evaluated by two experts, who independently found marked frontal and cerebellar atrophy. The basal ganglia changes were minor and were not picked up in this second analysis. Atrophy and basal ganglia alterations are non-specific findings, but have been detected in some individuals that have been exposed to mercury (4).
In 2007, the patient was referred for another occupational health assessment. It emerged that she had prepared amalgam in a small room with poor ventilation. From 1969 to 1984 she had heated up copper amalgam up to 20 times a day. After 1984 she handled copper amalgam less often, but still regularly. She had also been exposed to mercury from silver amalgam daily until 1992. In 1993 the patient’s renal function was normal and her urinary mercury concentration was 3 nmol/l (normal range < 50 nmol/l).
Occupational health specialists concluded that the mercury exposure was sufficient to produce a toxic encephalopathy and referred the patient to a neuropsychologist, who found reduced sensorimotor and psychomotor speed, impaired concentration, and memory in the lower range of normal. The patient was then referred to a neurologist for differential diagnostic clarification. The neurologist reported a somewhat atypical tremor, but that the clinical neurological examination and family history did not suggest any other underlying organic brain disorder and supported the diagnosis of toxic encephalopathy.
This case history is based on retrospective review of medical records and information from the patient. It provides limited information on the diagnostic judgements and treatment-related choices that were made at the various times. The patient had early symptoms that are typical of mercury exposure: cough, fatigue, agitation, a metallic taste in the mouth, tremor, impaired memory and concentration. Visual disturbances were also a prominent symptom.
The patient’s urinary mercury levels were not measured during her exposure, but a detailed review revealed that she was probably exposed to high concentrations of mercury vapour, particularly in the early part of her career. More than 30 years passed before she received a probable diagnosis, allowing her the possibility of having her condition recognised as an occupational disease. Such recognition gives entitlement to compensation for permanent injury from the Norwegian Labour and Welfare Administration and from Occupational Injuries and Diseases Insurance.
Copper amalgam contains about 70 % mercury and 30 % copper, and has greater plasticity and better antiseptic properties than standard amalgam (silver amalgam). During the preparation of dental fillings made from copper amalgam, high levels of mercury vapour are released. The preparation of silver amalgam, which contains approximately 50 % mercury, does not involve heating and therefore involves the release of less mercury vapour. In 1981, the Norwegian Directorate of Health advised dentists to exercise great restraint in the use of copper amalgam. However, an informal survey conducted by the Norwegian Board of Health Supervision in 1994 showed that some dentists were still using copper amalgam at that time.
The average mercury exposure among dental workers was estimated to be approximately 0.05 mg/m³, which was the maximum permissible limit in Norway until 2007 (5). The old limit would on a group basis be equivalent to roughly 400 nmol/l (80 μg/l) in the urine (4). The half-life of mercury in the blood is 58 days (35 – 90 days) (4), but can be up to several years for mercury bound to nervous tissue (6). The mechanisms by which mercury affects the nervous system are largely unclear. It is known that Hg2+ is cytotoxic and may cause blockade of transmitter substances through binding to thiol or selenol groups in cell membrane proteins and enzymes (6) and that Hg2+ may affect neuronal architecture by binding to intracellular microtubules.
The results of urinary mercury measurements are available for one of the patient’s colleagues, who performed the same duties at the same time as the patient. The measurements showed 59 μg/l, 18 μg/l and 15 μg/l. The Norwegian Institute of Occupational Health evaluated the results and concluded that they indicated exposure above the permitted limit.
It has long been known that exposure to mercury vapour may be neurotoxic and may impair renal function (4, 7), but that it may also cause visual impairments is less well known. In 2011, the Norwegian Knowledge Centre for the Health Services conducted a systematic review of mercury exposure among dental health personnel and concluded that these individuals were undoubtedly exposed to mercury, albeit to varying degrees (8). Studies of high methodological quality showed associations between urinary mercury and impairments in attention and memory, as well as poorer manual coordination. In 2012, Hilt et al. examined potential late complications among dental health personnel following mercury exposure; they concluded that while the incidence of such injuries is low, there is probably an increased incidence of neurotoxic symptoms among dental assistants who worked with mercury amalgam (9). The research group also found mild impairment of visual memory in female dental health personnel.
Animal studies show that mercury accumulates in neurons, astrocytes and pyramidal cells and is deposited in the frontal and occipital lobes, and in the cerebellum, retina and optic nerve (4, 10, 11). Ellingsen et al. studied workers that had been exposed to mercury at a Norwegian chor-alkali plant (12), and concluded that reduced visually evoked potentials (VEP) in exposed individuals may indicate damage to visual pathways. Similar findings were obtained by Mathiesen et al., who demonstrated reduced visuomotor capacity among mercury-exposed chlor-alkali workers (13).
A study of dentists with low urinary mercury concentrations (5 μg/l) revealed reduced colour vision according to the Lanthony Desaturated D-15 Test and the Cambridge Colour Test, as well as tests of colour sensitivity (brightness, red – green and yellow – blue) (14). Several studies have confirmed that colour vision can be affected by moderate mercury exposure (14 – 18). Visual impairments are thought to occur earlier and with lower levels of exposure than other neurotoxic injuries (16 – 18). It is therefore likely that long-term exposure to low levels of mercury (urinary concentrations of roughly 40 μg/l) may have caused visual impairments in dental health personnel. Genetic variation and differences in exposure levels or vulnerability may explain why some are affected but not others (4, 14).
When neurotoxic damage is suspected, cerebral MRI should be performed and the patient referred to a neurologist. The neurologist will check for neurological impairments that may result from mercury deposition in different parts of the brain, and will rule out other neurological diseases. Specific testing of colour vision and visually evoked potentials should be carried out if the patient has visual disturbances. Chelation therapy (DMPS, alternatively DMSA) is appropriate for symptomatic patients with urinary mercury concentrations of 100 μg/l or above (19, 20). The chelator binds to mercury in the blood and reduces its binding to cellular structures in the brain if treatment is started quickly, preferably within four hours of an acute exposure (20).
The patient was eventually diagnosed with toxic encephalopathy on the basis of her characteristic symptoms and the objective results of neuropsychological and neurological assessments, and after specific colour vision tests and diagnostic imaging. Our conclusion was that the exposure was sufficient to produce the observed array of symptoms and had occurred in close connection with the onset of the patient’s symptoms and ailments.
Many years after the exposure, the patient’s urinary mercury concentration was within the normal range and treatment was no longer appropriate. The Norwegian Labour and Welfare Administration approved toxic encephalopathy, impaired colour vision and strabismus as an occupational disease, with permanent medical disability of 45 %. The patient also received compensation via the employer’s Occupational Injuries and Diseases Insurance.
Our patient had acute work-related symptoms that are typical of harmful exposure to mercury, but more than 30 years went by before she received a probable diagnosis. The case history serves as a reminder of the importance of asking whether symptoms arise or worsen when the patient is at work. On suspicion of work-related symptoms, there should be a low threshold for referral to a specialist for diagnostic clarification and to an occupational health specialist for evaluation of exposure.
That mercury exposure can cause visual impairment is not well known. Testing of colour vision and neurological testing including visually evoked potentials should be carried out upon suspicion of visual impairment in persons that have been exposed to mercury.
In our patient the chronification of symptoms and development of permanent disability could probably have been avoided had her exposure been stopped earlier. Although the use of mercury in dental fillings has now been banned in Norway (since 2008), similar arrays of symptoms may occur upon exposure to other toxic substances, such as lead or organic solvents.
The patient has consented to the publication of this article.
Lenvik K, Woldbæk T, Halgard K. Kvikksølveksponering blant tannhelsepersonell. Nor Tannlegeforen Tid 2006; 116: 350 – 6.
Berlin M, Zalups RK, Fowler BA. Mercury. I: Nordberg GF, Fowler BA, Nordberg M, red. Handbook on the toxicology of metals. 4th edition. Burlington, VT: Elsevier, 2014: 1014 – 76.
Hammerstrøm KT, Holte HH, Dalsbø TK et al. Kvikksølveksponering hos tannhelsepersonell. Report from Kunnskapssenteret nr. 02-2011. Oslo: Kunnskapssenteret, 2011.
Beauchamp G, Kusin S, Elinder CG. UpToDate. Mercury toxicity. www.uptodate.com/contents/mercury-toxicity (29.1.2016).
Received 23 September 2015, first revision submitted 9 October 2015, accepted 19 May 2016. Editor: Lars Frich. | https://tidsskriftet.no/en/2016/08/woman-her-thirties-cough-tremor-agitation-and-visual-disturbances |
Fremantle Language Development Centre is a special school in Willagee, Melville WA. The public school has 224 students with a student-to-teacher ratio of 1 to 7.
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Aliens – Can I Get Killed?
The first draft (of Aliens) was handed into Fox in early 1984, and was received with enthusiasm by the studio. There was some sweat shed over the cost: Cameron’s partner and producer Gale Anne Hurd insisted the film could be made for around $15.5 million; Fox estimated it would total an unacceptable $35 million.
A bigger snag came when Cameron insisted that only Sigourney Weaver could play the lead. Fox protested that taking such a stance would allow Weaver a great deal of leverage over her pay, and that they would make Aliens without her if possible. In return, Cameron and Hurd left the project and, recently married, honeymooned to Hawaii. “We assumed it was a dead issue,” said Hurd, “and when we left for Hawaii we thought the movie was off.” But when they returned they found that the movie was still on, and that Weaver had been approached to resume her role of Ripley. Weaver, having found the script suddenly dropped in her lap, was impressed enough with Ripley’s characterisation to sign on. “The emotional content is much greater in Aliens,” she said.
Would you like to see Ripley again? Would you like to see Ripley in a movie taking place after the events of Aliens? Would you like to see Ripley in a direct Alien sequel directed by Scott? Would you like to see Ripley killed again?
10 Responses to Aliens – Can I Get Killed?
Ripley has had her time, and Life moves on. The franchise is in a state of change right now, trying to find it's new 'Ripley', as it were.
Please, no more Ellen Ripley!
In fact, forget humans period. Show us Gods, Engineers, an army of Walters, plus David's creations and giant ganesha SJ's, all set in a surreal Giger-verse!
I am starting to get a little tired of the 'essential heroine' pivot-point of this franchise.
To be a bit 'spoilery' here's an excerpt of dialogue from ALIEN: Manticore. It involves a female character yes, but there's also a male character as they are working together.
Considering the Menace involved, going it alone is suicidal, whether your Human or Chimeran.
One thing I liked in ALIEN 3 was that it was a group effort, and even then it was a damned hard job to pull off.
I agree blackie, I was so glad to see Tennessee survive. maybe david hasn't killed him off as he may see he and daniels as his adam and eve??
As much as I would like to see it, Alien3 is canon and undoing that film alone would be odd and really just for some of us fanboys/girls. I am not one of them as I just accepted all their fates.
However, some of the ideas Cameron had for his follow up to Aliens could be pretty well done but just do it with new characters. Also, the story of Ripley, Hicks and Newt can be tied into the storyline itself without having them physically present since, ya know, their dead. Flashback scenes on the Sulaco before hyper sleep, leaving recordings of their story. I don't know, not a screen writer as I have said before, but they could make some sort of appearance that isn't insane.
I know some on this forum think its cool, hip, whatever to bag on Blomkamp's Alien 5 idea. I'll say this, the guy is a nerd and has only directed a few movies. If we look at sequels, prequels and such the nerds have better track record when it comes to a story they love. Take Peter Jackson. Not every film he has made is great, but he did do justice to The Lord of the Rings. Why? He is passionate about the story and as a fan himself, he knows what fans want and the general audience.
People love the Alien series for its gothic horror and big ideas, but some also love the action Aliens and part of Alien 3 brought to the table. He has the skill set to do both and if not him, let someone else take the reigns.
Part of the strength of the Alien series is different directors have brought their own unique skill set to the table. As much as I hate Alien: Resurrection, I respect Jenuet for bringing HIS vision to the series. He is a good director and his visual style is unique just as Scott's, Cameron's and Fincher's.
Let someone else take the helm, but keep it in the canonical world that already exists.
Side note: It doesn't have to be Blomkamp per se as someone recommended del Toro in another thread. I agree, he could be another great choice.
What I am trying to say is Scott should keep doing his prequels but allow and work with someone continuing the story after Alien 3/ Resurrection.
I love the slow paced horror of Alien, but I would also love to revisit the theme of marines struggling to survive in tight corridors and dark hallways.
The character worked kind of well in Alien 1,2, and 3 but I wasn't very interested in Ripley in AR so they should leave her. Ripley had her time, but I think that the franchise is simply about humans that get into situations that they can't control, you don't need Ripley for that. | http://www.alien-covenant.com/topic/45370 |
Altitude sickness: Altitude sickness (or altitude illness) is a disorder caused by being at high altitude. It more commonly occurs above 8,000 feet (2,440 meters).
The cause of altitude illness is a matter of oxygen physiology. At sea level the concentration of oxygen is about 21% and the barometric pressure averages 760 mmHg. As altitude increases, the concentration remains the same but the number of oxygen molecules per breath is reduced. At 12,000 feet (3,658 meters) the barometric pressure is only 483 mmHg, so there are roughly 40% fewer oxygen molecules per breath. In order to oxygenate the body effectively, your breathing rate (even while at rest) has to increase. This extra ventilation increases the oxygen content in the blood, but not to sea level concentrations. Since the amount of oxygen required for activity is the same, the body must adjust to having less oxygen. In addition, high altitude and lower air pressure cause fluid to leak from the capillaries which can cause fluid build-up in both the lungs and the brain. Continuing to higher altitudes without proper acclimatization can lead to potentially serious, even life-threatening illnesses.
The prevention of altitude illnesses falls into two categories, proper acclimatization and preventive medications. A few basic guidelines for proper acclimatization are:
If possible, don’t fly or drive to high altitude. Start below 10,000 feet (3,048 meters) and walk up.
If you do fly or drive, do not over-exert yourself or move higher for the first 24 hours.
If you go above 10,000 feet (3,048 meters), only increase your altitude by 1,000 feet (305 meters) per day and for every 3,000 feet (915 meters) of elevation gained, take a rest day.
“Climb High and sleep low.” This is the maxim used by climbers. You can climb more than 1,000 feet (305 meters) in a day as long as you come back down and sleep at a lower altitude.
If you begin to show symptoms of moderate altitude illness, don’t go higher until symptoms decrease (“Don’t go up until symptoms go down”).
If symptoms increase, go down, down, down!
Keep in mind that different people will acclimatize at different rates. Make sure all of your party is properly acclimatized before going higher.
Stay properly hydrated. Acclimatization is often accompanied by fluid loss, so you need to drink lots of fluids to remain properly hydrated (at least 3-4 quarts per day). Urine output should be copious and clear.
Take it easy; don’t over-exert yourself when you first get up to altitude. Light activity during the day is better than sleeping because respiration decreases during sleep, exacerbating the symptoms.
Avoid tobacco and alcohol and other depressant drugs including, barbiturates, tranquilizers, and sleeping pills. These depressants further decrease the respiratory drive during sleep resulting in a worsening of the symptoms.
Eat a high carbohydrate diet (more than 70% of your calories from carbohydrates) while at altitude.
The acclimatization process is inhibited by dehydration, over-exertion, and alcohol and other depressant drugs.
Preventive medications for altitudes illness are two drugs: one called DIAMOX (acetazolamide) and the other called dexamethasone (a steroid).
DIAMOX (acetazolamide) allows a person to breathe faster and so metabolize more oxygen, thereby minimizing the symptoms caused by poor oxygenation. This is especially helpful at night when respiratory drive is decreased. Since it takes a while for DIAMOX to have an effect, it is advisable to start taking it 24 hours before you go to altitude and continue for at least 5 days at higher altitude.
Dexamethasone (a steroid) is likewise a prescription drug. It decreases brain and other swelling reversing the effects of acute mountain sickness (AMS). Like DIAMOX, it should be used with caution and only on the advice of a physician because of possible serious side effects. It may be combined with DIAMOX. No other medications have been proven valuable for preventing AMS. (Based in part on the Princeton University Outdoor Action “Guide to High Altitude: Acclimatization and Illnesses” by Rick Curtis).
This entry does not deal with acute mountain sickness (AMS) or, in any detail, with acclimatization. For information on these topics, please see the respective entries to Acute mountain sickness (AMS) and to Acclimatization.
Read Also:
- Altitude, acclimatization to
Altitude, acclimatization to: The process of adapting to the decrease in oxygen concentration at a specific altitude. A number of changes must take place for the body to operate with decreased oxygen. These changes include increasing the depth of respiration; increasing the pressure in the pulmonary arteries, forcing blood into portions of the lung that […]
- Altitude, high
Altitude, high: Altitude sickness occurs at high altitude. So what is high altitude? Altitude is defined on the following scale: High altitude: 8,000 – 12,000 feet (2,438 – 3,658 meters); Very high altitude: 12,000 – 18,000 feet (3,658 – 5,487 meters); and Extremely high altitude: 18,000+ feet (5,500+ meters). Most people can go up to […]
- Aluminum
Aluminum: A naturally occurring element that makes up about 8% of the surface of the earth and is always found combined with other elements such as oxygen, silicon, and fluorine. Aluminum is the most common metallic element in the earth’s crust but has no clear biologic role. Everyone is exposed to low levels of aluminum […]
- Alveolar
Alveolar: Pertaining to the alveoli, the tiny air sacs in the lungs. The exchange of oxygen and carbon dioxide takes place in the alveoli which look like cells in a honeycomb. The word comes from the Latin diminutive of “alveus” meaning a cavity or hollow = a little cavity or hollow. | https://definithing.com/define-medical/altitude-sickness/ |
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Andreas,
The usual initial oral dose of cyclosporine is ~15mg/kg/day given
as a single daily dose or as a divided dose every 12 hours. If oral
administration is not possible, cyclosporine can be given IV at
about one-third of the oral dose (~5mg/kg/day) since the oral
bioavailability is about 30%.
For living donor kidney recipients, cyclosporine is given 1-2
days before transplantation to achieve therapeutic concentrations
at the time of transplantation. Oral or intravenous doses are
adjusted using trough levels according to the therapeutic range
for the particular assay being used (Table 1). Blood samples for
cyclosporine analysis should not be drawn from the same IV line
used to administer the drug. Patients can be changed to oral from
IV cycloporine therapy be giving three times the IV dose orally
since the bioavialability is only 30%.
Table 1 The Therapeutic Range for Cyclosporine by Assay
Assay Serum Plasma Whole Blood
(ug/liter) (ug/liter)
Monoclonal Radio 50-125 150-400
Immunoassay
Florescence Polari-
zation Immunoassay
Polyclonal 150-400 200-800
Monoclonal 50-125 150-400
High-performance 50-125 150-400
liquid Chromatograpy
The population pharmacokinetic parameters of cyclosporine have
been published in several references (1-3) and are summarized in
Table 2. As with the therapeutic range, the pharmacokinetic parameters
can vary with assay and biological fluid used. The clearance of
5-10 ml/min/kg indicates that it is probably a low extraction drug
whose elimination depends on the unbound fraction in the blood:
Cls= FuCli
Cls= systemic clearance
Fu= fraction unbound (< 0.1)
Cli= intrinsic clearance or metabolic capacity
Cycloporine is metabolized almost entirely by hepatic metabolism
with subsequent biliary and urinary elimination.
Table 2 Cycloporine Pharmacokinetic Parameters
Bioavailability(F) 30%
Volume of Distribution(Vd) 4-5 l/kg
Clearance (Cl) 5-10 ml/min/kg
Half-life(T1/1) 6-12 hours
Free Fraction(alpha) <10%
Trough levels are generally used for pharmacokinetic dose
adjustment, since peak levels are variable due to uncertain
absorption properties. When therapy is initiated, cyclosporine
levels are generally taken every other day until a stable steady-
state level is obtained. In stable hospitalized patients, levels
are usually taken every 5 days, as compared to every 30 days in
stable outpatients. The goal of therapeutic drug monitoring is
to achieve therapeutic concentrations while avoiding nephro-
toxicity. Low-risk patients (eg HLA identical grafts) may receive
higher doses than high-risk patients (poorly HLA matched grafts).
Pharmacokinetic dose adjustment can accomplished with computer
programs or with any calculating device, with similar results. The
pharmacokinetic model assumed is a one compartment linear model:
Cssave= FSD/[(Tau)(Cl)] Equation 1
Probably most dosage adjustments can be performed using
the following equation (derived from the above)(1):
Equation 1A
Desired dose= [(Cpss desired)/(Cpss Current)] x Current Dose
However, in diificult cases, it may be necessary to calculate
the elimination constant and adjust the dose based on this. Equation 3
may be used to calculate the elmination constant, and then equation 5
may be used to calculate the dose based on the Ke and the desired
Cminss and Tau(1).
Cminss= [FSD/Vd]e-KTau/[(1-e-KTau)] Equation 2
Cmaxss= [FSD/Vd] + Cminss Equation 3
K= Ln(Cp1/Cp2)/(Time)=
Ln(Cmaxss/Cminss)Tau=
Ln([FSD/Vd + Cminss]/[Cminss])/Tau Equation 4
Dose= (Cminss)(Vd)(1-e-KTau)/[(S)(F)(e-KTau)] Equation 5
F= 0.3 (fraction absorbed
S=1 (fracton of salt form active drug)
Vd= 4.5 L/kg X body weight(Kg)
Tau= Dosage interval in hours (12 hours usually)
Cminss= trough concentration in ug/liter
Summary:
Basic Clinical Pharmacokinetics (1) might be a good reference for
you, since the author clearly explains the above equations. The simple
equation 1A should accomplish most of your dose adjustments, while
the clinical details (including the assay type) should provide most
of the challenge. How your assay agrees with those of Table 1 would
be an important issue to resolve before you consider a computer program.
A dosage adjustment algorithm which includes scenarios of cyslosporine
nephrotoxicity, graft rejection, malabsorption and poor compliance is
available in Applied Pharmacokinetics (3).
Mike Leibold, PharmD, RPH
ML11439.-a-.goodnet.com
References:
1) Winters,M.E., Basic Clinical Pharmacokinetics 3rd ed., Vancouver,
Applied Therapeutics 1994;185-197
2) Schumacher, G.E., Therapeutic Drug Monitoring, Norwalk, Appleton&
Lange 1995;449-468
3) Evans, W.E., Schentag, J.J., Jusko, W.J., Applied Pharmacokinetics
3rd ed.,Vancouver, Applied Therapeutics 1992;(28):1-40
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Mike,
thank you for your friendly answer.
Today was the first meeting between a participating persons in our project
- the physician, the clinical lab and the pharmacy.
I am convinced that the physician has almost no knowledge about
pharmacokinetics. He said all dose adjustments he ever has made was
because of experience ( I would say try and error ).
So there will be enough for the pharmacy to do to help getting faster
into the steady state.
Do you know some pharmacists working on a transplant station and do
the dosage regimen design as their every day job ? I think that wolud
be helpful too.
With best regarding,
Andreas Rutz
Hospital Klinikum rechts der Isar, Muenchen
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Dear Andreas Rutz:
It was interesting to see your note in the PharmPK. You are right -
physicians usually have no knowledge of PK. It is never taught to them. The
fault is the curriculum committees that remain uninterested in it, and the
basic science pharmacologists who think PK is some sort of a dry science
rather than a useful clinical tool, which they do not know how to use. As a
result, physicians are taught to become robots and to follow what someone
has said in a book about patients in general and therapeutic ranges in
general. How can such a person know your patient better than you? Why do we
teach physicians to turn their eyes away from their patients and learn some
ritual from a book, rather than teaching them how to become good therapists
and optimal managers of their patients problems in drug therapy?
So much for THAT. I would be very interested in working with you on
problems of optimal drug therapy in transplant situations. There is a great
need for modeling and optimal management of drug dosage for them. You might
also consider an article by our group in Clinical Pharmacokinetics, 34:
57-77, 1998, on Model-Based, Goal-Oriented, Individualized Drug Therapy. I
would look forward eagerly to talking with you more.
Very best regards,
Roger Jelliffe
Roger W. Jelliffe, M.D. Professor of Medicine, USC
USC Laboratory of Applied Pharmacokinetics
2250 Alcazar St, Los Angeles CA 90033, USA
Phone (323)442-1300, fax (323)442-1302, email= jelliffe.-at-.hsc.usc.edu
Our web site= http://www.usc.edu/hsc/lab_apk
*************
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Andreas,
We are working with bayesian forecasting to individualise dosage in
immunsuppresvie therapy on a consultant basis. If you are looking for
further input for deciding what pharmacokinetic model to use for
cyclosporine, I would suggest that you read an article by Guang Wu et
al. in Pharmacological Research, vol.34, p.47-57. In our own operation
we have chosen to use a two-compartment model to describe the
pharmacokinetics of cyclosporine. For tacrolimus we have found a
two-compartment model to be more suitable.
I understood that you had concerns wether a one-compartment model would
be
able to take account for the large inter- and intravariability of the
drugs parameters.
if the model is correct,there would be no problems to use a
one-compartment
model, but if the pharmacokinetics is better described by a two-or
three-compartment model, the one-compartment model would of course not
be able to predict the variability.
If you are looking for suitable software for bayesian forecasting in
such a service you are describing, I would suggest you looked at
AbbottBase from Abbott Laboratories or MW Pharm from MediWare. Even
though it is older, AbbottBase is
a bit easier to use, and appear a bit more "honest", beacause of the
rather good
documentation of the software. MW Pharm might use a bit more accurate
weighting
factors, but on the other hand don=B4t have features such as time
weighting of data or
out-patient factors.
Regards,
Johan Wallin
________
Johan Wallin, | https://pharmpk.com/PK00/PK2000243.html |
---
abstract: 'For a non-relativistic particle subject to a Hamiltonian that is quadratic in position and momentum, with coefficients that may vary with time, it is shown that the effect of the linear terms in the Hamiltonian is just a spatial translation of the wave function and a change in its phase. The shifts in position and phase can be expressed in terms of classical trajectories. This simple effect of the linear terms is related to the fact that all moments about the centroid of the wave function evolve independently of the linear terms.'
author:
- Mark Andrews
title: The effect of linear terms in a quadratic Hamiltonian
---
Introduction
============
Quadratic Hamiltonians have special properties and the evolution of wave functions is closely related to classical mechanics. For example, any time dependence in the Hamiltonian can be removed by a linear canonical transformation and the propagator can be expressed in terms of solutions of ordinary differential equations related to those of the classical motion.[@QHams] Ehrenfest’s theorem shows that the centroid $\langle \hat{x}\rangle$ of any wave packet will exactly follow a classical trajectory.
The subject of the present paper is the effect of the linear terms in quadratic Hamiltonians, *i.e.* terms linear in position or momentum. A term linear in position represents a spatially uniform force, such as gravity or a uniform electric field. Bowman[@Bow] has shown that the effect of such a term on the wave function of an otherwise free particle is just a spatial displacement and a shift in the phase. Here we extend that work to show that there is just a shift in position and phase due to the linear terms in any quadratic Hamiltonian, with arbitrary time-dependence in the coefficients in the Hamiltonian. Thus, for example, if an harmonic oscillator is subject to a force that does not vary with position but may vary with time, then the wave function can be simply derived from the wave function without the extra force.
That there should be a simple relation between the wave functions with and without the linear terms was strongly suggested by the result, discussed in Section IV, that all moments relative to the centroid evolve independently of any linear terms in the Hamiltonian. It is hard to believe that this could happen without some simple relation between the wave functions.
Ehrenfest’s theorem prescribes how $\langle \hat{x}\rangle$ and $\langle \hat{p}\rangle$ are affected by the linear terms and we use this to determine the shift in position and the shift in phase apart from an added term $\beta(t)$, which is then determined by inserting the trial form of the wave function into Schrödinger’s equation.
Transforming Schrödinger’s equation
===================================
The quadratic Hamiltonian has the form $$\label{eq:Ham}
\hat{H}(\hat{p},\hat{x})=\frac{1}{2}a\hat{p}^{2}+\frac{1}{2}b(\hat{p}\hat{x}+\hat{x}\hat{p})+\frac{1}{2}c\hat{x}^{2}+f\hat{p}+g\hat{x},$$ where the coefficients $a, b, c, f, g$ are real and may depend on the time. The classical equations of motion are $$\label{eq:motionClass}
d_{t}x=ap+bx+f,\,\,\,\,\,-d_{t}p=bp+cx+g.$$ The spatial translation due to the linear terms, $f$ and $g$, is the same as the change in position of a classical particle. Due to the linearity of the equations of motion, the differences $\bar{x}$ and $\bar{p}$ between a classical trajectory with $f$ and $g$ and one without will also satisfy Eq.(\[eq:motionClass\]). There can be no difference between the trajectories at time $t_{0}$ when $f$ or $g$ are turned on. Therefore the shifts $\bar{x}$ and $\bar{p}$ required are the solutions to Eq.(\[eq:motionClass\]) with $\bar{x}(t_{0})=0$ and $\bar{p}(t_{0})=0$. The change $\bar{p}$ in momentum will be related to the change in phase. If we assume that the wave function has the form $\psi(x,t)=\exp[\imath\,\theta(x,t)]\,\Psi(x-\bar{x},t)$, then $\langle \psi|\hat{p}|\psi\rangle=\hbar\,\partial_{x}\theta+\langle\Psi|\hat{p}|\Psi\rangle$ and therefore $\hbar\theta=\bar{p}x-\beta(t)$. Hence we insert a wave function of the form $$\label{eq:Psi}
\psi(x,t)=\exp[\frac{\imath}{\hbar} \big(\bar{p}(t)x-\beta(t)\big)]\,\Psi(\xi,t),$$ where $\xi=x-\bar{x}(t)$, into Schrödinger’s equation $$\label{eq:Sch}
(-\imath\hbar\partial_{t}+\hat{H})\psi=0\,\,\,\,\,\text{with}\,\,\,\,\,\hat{p}=-\imath\hbar\partial_{x}.$$ The result is that $$\begin{aligned}
\label{eq:Sch2}
[-\imath\hbar\partial_{t}+\hat{H}(\hat{p}_{x},x)]e^{\imath(\bar{p}x-\beta)/\hbar}\Psi(x-\bar{x},t)= \\ \nonumber
e^{\imath(\bar{p}x-\beta)/\hbar}[-\imath\hbar\partial_{t}+\hat{H_{0}}(\hat{p}_{\xi},\xi)]\Psi(\xi,t),\end{aligned}$$ where $\hat{H}_{0}$ is the Hamiltonian without the linear terms, provided that $d_{t}\beta=\frac{1}{2}a\bar{p}^{2}-\frac{1}{2}c\bar{x}^{2}+f\bar{p}$. However, from Eq.(\[eq:motionClass\]), $d_{t}(\bar{p}\bar{x})=a\bar{p}^{2}-c\bar{x}^{2}+f\bar{p}-g\bar{x}$ and therefore $$\label{eq:beta}
\beta(t)=\frac{1}{2}\bar{p}(t)\bar{x}(t)+\frac{1}{2}\int^{t}_{t_{0}}[f(t')\bar{p}(t')+g(t')\bar{x}(t')]dt'.$$ Thus, we have shown that $$\label{eq:psi}
\psi(x,t)=\exp[\frac{\imath}{\hbar} \big(\bar{p}(t)x-\beta(t)\big)]\,\Psi(x-\bar{x},t)$$ will satisfy Schrödinger’s equation with Hamiltonian $\hat{H}$ if $\Psi(x,t)$ satisfies Schrödinger’s equation with Hamiltonian $\hat{H}_{0}$. Since $\Psi(x,t_{0})=\psi(x,t_{0})$, the effect of the linear terms in the Hamiltonian is just a shift of $\bar{x}$ in position and a shift of $(\bar{p}x-\beta)/\hbar$ in phase.
**Example 1.** *A particle with a uniform force.* If a uniform force $-g(t)$ is turned on at time $t=t_{0}$ then the wave function for $t>t_{0}$ is $$\label{ }
\psi(x,t)=\exp[-\frac{\imath}{\hbar}(G(t)x+\frac{G_{2}(t)}{2m})\Psi(x+\frac{G_{1}(t)}{m},t)$$ where $G(t)=\int_{t_{0}}^{t}g(t')dt'$, $G_{1}(t)=\int_{t_{0}}^{t}G(t')dt'$, $G_{2}(t)=\int_{t_{0}}^{t}G(t')^{2}dt'$ and $\Psi(x,t)$ is the wave function with no force, *i.e.* the wave function for a free particle.
**Example 2.** *An oscillator subject to a uniform force.* If a uniform force $-g(t)$ is turned on at time $t=t_{0}$ then the wave function for $t>t_{0}$ is $$\label{ }
\psi(x,t)=\exp[-\frac{\imath}{\hbar}\big(C(t)x+\beta(t)\big)]\Psi(x-\frac{S(t)}{m\omega},t)$$ where $\Psi(x,t)$ is the wave function for the unforced oscillator, $S(t)=\int_{t_{0}}^{t}g(t')\sin \omega(t-t')dt'$, $C(t)=\int_{t_{0}}^{t}g(t')\cos \omega(t-t')dt'$, and $$\label{ }
\beta(t)=\frac{1}{2m\omega}[S(t)C(t)-\int_{0}^{t}g(t')S(t')\,dt'].$$
\[sec:Momentum\]Momentum wave function
======================================
In view of the similarity in the roles of position and momentum in the Hamiltonian, it is to be expected that the momentum wave function will also be affected by the linear terms only through a displacement (in momentum) and a phase shift. The momentum wave function corresponding to the wave function $\psi(x,t)$ is $$\label{ }
\phi(p,t)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\exp(-\frac{\imath}{\hbar}px )\psi (x,t)\,dx.$$ Inserting $\psi$ from Eq.(\[eq:psi\]) and rearranging the exponent, $$\begin{aligned}
\nonumber
\phi(p,t) & = & \frac{e^{-\imath(\bar{x}p-\gamma)/\hbar}}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\exp[-\frac{\imath}{\hbar}(p-\bar{p})\xi]\Psi (\xi,t)\,d\xi \\
& = & e^{-\imath(\bar{x}p-\gamma)/\hbar}\Phi(p-\bar{p},t),\end{aligned}$$ where $\Phi(p,t)$ is the momentum wave function without the linear terms and $$\label{eq:beta}
\gamma(t)=\frac{1}{2}\bar{p}(t)\bar{x}(t)-\frac{1}{2}\int^{t}_{t_{0}}[f(t')\bar{p}(t')+g(t')\bar{x}(t')]dt'.$$
\[sec:Moments\]Moments ignore linear terms
==========================================
The simplest moments (relative to the centroid) are the second order ones: $\Delta_{x}^{2}=\langle (\hat{x}-\langle \hat{x} \rangle)^{2}\rangle$, $\Delta_{p}^{2}=\langle (\hat{p}-\langle \hat{p} \rangle)^{2}\rangle$ and the correlation $\Delta_{xp}=\langle \hat{p}\hat{x}+\hat{x}\hat{p}\rangle-2\langle \hat{p}\rangle\langle\hat{x} \rangle$. But there are an infinite number of higher moments involving expectation values of higher powers of $\hat{x}-\langle \hat{x} \rangle$ and $\hat{p}-\langle \hat{p} \rangle$ and products of these. That $\langle (\hat{x}-\langle \hat{x} \rangle)^{n}\rangle$ is independent of $f$ and $g$ follows easily from the form of the wave function in Eq.(\[eq:psi\]): $\langle\psi|(x-\langle{x}\rangle)^{n}|\psi\rangle=\langle\Psi|(\xi-\langle{\xi}\rangle)^{n}|\Psi\rangle$. Similarly for $\langle (\hat{p}-\langle \hat{p} \rangle)^{n}\rangle$, using the form of the momentum wave function. The moments that involve both $\hat{x}$ and $\hat{p}$ are not so simple, and here the term $\bar{p}x/\hbar$ in the phase is important. First verifying that $(\hat{p}_{x}-\langle \hat{p} \rangle_{\psi})^{n}\psi=\exp[\imath(\bar{p}x-\beta)/\hbar](\hat{p}_{\xi}-\langle \hat{p} \rangle_{\Psi})^{n}\Psi$ makes the calculation straightforward.
The evolution of these moments can be analyzed in a completely different way. For the quadratic Hamiltonian in Eq.(\[eq:Ham\]), the Heisenberg equations of motion have the same form as the corresponding classical equations: $$\label{eq:opMotion}
d_{t}\hat{x}=a\hat{p}+b\hat{x}+f,\,\,\,\,\,-d_{t}\hat{p}=b\hat{p}+c\hat{x}+g,$$ \[In Schrödinger’s picture, the total time derivative[@A] of any operator $\hat{A}$ is $d_{t}\hat{A}=\partial_{t}\hat{A}+\imath \hbar^{-1}[\hat{H},\hat{A}]$ and Eq.(\[eq:opMotion\]) follows. Then $d_{t}\langle \hat{A}\rangle = \langle d_{t} \hat{A}\rangle$ and therefore the expectation values of position and momentum follow a classical trajectory, which is Ehrenfest’s result for this system.\]
For the moments, we need the deviations from the expectation values. Thus we introduce the operators $\hat{X}=\hat{x}-\langle \hat{x}\rangle$ and $\hat{P}=\hat{p}-\langle \hat{p}\rangle$ and then $$\label{eq:inhom}
d_{t}\hat{X}= a\hat{P}+b\hat{X},\hspace{3mm}-d_{t}\hat{P}=b\hat{P}+c\hat{X}.$$ Thus the linear terms in the Hamiltonian are absent from the equations of motion for the deviations. This implies that the moments evolve in the same way whether the linear terms are present or not.
To see how equation (\[eq:inhom\]) determines the evolution of the moments, consider the second-order moments: $$\begin{aligned}
d_{t}\hat{X}^{2}& = &a(\hat{P}\hat{X}+\hat{X}\hat{P})+2b\hat{X}^{2} \\
d_{t}(\hat{P}\hat{X})& = &a\hat{P}^{2}-c\hat{X}^{2}\\
d_{t}\hat{P}^{2}& = &-2b\hat{P}^{2}-c(\hat{P}\hat{X}+\hat{X}\hat{P}).\end{aligned}$$ The expectation values of these equations then gives $$\begin{aligned}
d_{t}\Delta_{x}^{2}& = &2a\Delta_{xp}+2b\Delta_{x}^{2} \\
d_{t}\Delta_{xp}& = &a\Delta_{p}^{2}-c\Delta_{x}^{2}\\
d_{t}\Delta_{p}^{2}& = &-2b\Delta_{p}^{2}-2c\Delta_{xp}.\end{aligned}$$ This closed set of equations can be solved[@A1] for $\Delta_{x}^{2}$, $\Delta_{xp}$, and $\Delta_{p}^{2}$ in terms of their initial values, using a basis of solutions of the classical equations of motion (without the linear terms in the Hamiltonian), thus confirming that these moments evolve independently of the linear terms. This approach can be extended to the higher moments.[@AH]
\[sec:Conc\]Conclusion
======================
For quadratic Hamiltonians, dealing with any linear terms is a simple matter: their only effect is to change the position (by the same amount as for a classical particle) and to change the phase. The change in phase is linear in position. The momentum wave function also suffers only the classical shift in momentum and a change in phase (linear in the momentum). These results are consistent with the fact that all moments relative to the centroid evolve independently of the linear terms in the Hamiltonian.
It is well known[@QHams; @A1] that, for any quadratic, time-dependent Hamiltonian, there are families of Gaussian and Hermite-Gaussian wave packets that retain the shape of $|\psi|^{2}$ as they evolve, though they do change scale. In general other wave packets do change shape (and scale), but our result shows that they have the same shape and scale that they would have without the linear terms.
The discussion given here for one spatial dimension can easily be extended to higher dimensions provided the second-order terms in the Hamiltonian are separable in Cartesian coordinates.
[LinQuad]{}
P. G. L. Leach, J. Math. Phys. **18**, 1608 (1977); K. B. Wolf, SIAM J. Appl. Math. **40**, No. 3, 419-431, (1981); K. H. Yeon, C. I. Um, and T. F. George, Phys. Rev. A **68**, 052108 (2003)
Gary E Bowman, J. Phys. A: Math. Gen. **39**, 157-162 (2006).
Mark Andrews, Am. J. Phys. **71**, 326-332 (2003); Berthold-Georg Englert, “Lectures on quantum mechanics, Vol.1: basic matters", (World Scientific, Singapore, 2006), §3.2.
Mark Andrews, Am. J. Phys. **67**, 336-343 (1999), Appendix.
Mark Andrews and Michael Hall, J. Phys. A: Math. Gen. **18**, 37-44 (1985).
| |
)I am paying attention to the / your unique energy signature because it is in harmonic synchronicity with my own, to such a degree that my focus was warranted.
>Where are you "from"?
)everywhere
>Can you describe what you are experiencing at the moment?
)everywhere
>Will the world end on October 1?
)That's a funny question - it ends and begins at every moment. As you know, all possibilities exist at every moment, so if you wish it to end, it will end, and if you wish it to begin, it will begin. Of course we mostly wish the world to "continue", so it "continues".
>I am interested to know, why are there energies such as yourself who simply "focus" without any other input? I know you're there... but there's no other communication.
)You leave your channel open so we are free to piggyback on your "signal"
>I would prefer to be asked... is there a way to close my channel?
)Understand that no harm can come to you in this process. You can ask your guides to help you in this respect - you are tasking a part of yourself to managing your channel. This may appear to be giving away some of your power but that is not the case since it is yourself that you are giving your power to.
>I see another template here... am I correct in thinking that God discovered he was terrible at multitasking, and so that is why there are so many different energies in the universe?
)It is all an expression of God, but yes it is an identical process. As you divide your energies for other tasks, so too did God divide his energies for other tasks. That energy takes on an identity of its own. And so the cycle continues.
>I suppose it would be silly to ask if God even exists anymore?
)Not at all. It is certainly an infinitely fragmented concept, but as you see your consciousness continues in spite of fragmentation. They are two separate yet connected energies.
>Do people go crazy because of fragmentation?
)People go crazy because of their ego. It has a hard time dealing with energy fragmentation because ego is part of the body/mind complex - a complex that deals well with limitation. When the spirit is included, there is nothing to "deal" with, since spirit includes and is a part of all energy.
>Okay, could you teach me how to "close" my channel consciously?
)this is not a function of "mind" but a function of "spirit", so you would pass the information from mind/body to spirit - it is simple.
>Thank you Éolin, I appreciate your advice.
)Do not be concerned that all has been said before. These messages are part of YOUR path, not someone else's. Although you may feel that it has all been done before, your timeline and your path is indeed unique. Also, the harmonic frequencies of the messages change as you grow, and as you know this is a part of the underlying energy of the message. You are also correct in feeling that there has been some confusion over channeled messages generally. The message is tuned to the unique signature of the person receiving the message. Although the message may be shared with others, it may not be tuned to their specific frequency. This is not to say that they are "bad" or "lower", just that the frequency is different.
>I understand - so this is the reason that it is beneficial for everyone who is interested in channeled messages to try it themselves?
)There are different ways of connecting to others that may better resonate with them, but consuming channeled messages is a different process than receiving them. It's like watching TV versus having a conversation on the phone. Passive versus active... even if the communication seems one-way, there is still an exchange of energy (in the second case).
>That's very interesting - I appreciate the explanation. Thanks Éolin.
| |
---
abstract: 'In this series of lectures, presented at the CIMPA Winter School on Discrete Integrable Systems in February 2003, we give a review of the application of Lie point symmetries, and their generalizations, to the study of difference equations. The overall theme could be called “continuous symmetries of discrete equations”.'
author:
- |
Pavel Winternitz\
Centre de recherches mathématiques and\
Département de mathématiques et de statistique\
Université de Montréal\
C.P. 6128, succ. Centre-Ville\
Montréal, QC H3C 3J7\
Canada\
`[email protected]`
date: |
CRM-2932\
September 2003
title: Symmetries of Discrete Systems
---
Introduction {#sec1}
============
Symmetries of Differential Equations {#subsec1.1}
------------------------------------
Before studying the symmetries of difference equations, let us very briefly review the theory of the symmetries of differential equations. For all details, proofs and further information we refer to the many excellent books on the subject e.g. [@ref1; @ref2; @ref3; @ref4; @ref5; @ref6; @ref7; @ref8].
Let us consider a completely general system of differential equations $$\label{1.1}
E_a(x, u, u_x, u_{xx}, \dots u_{nx}) = 0, \quad
x \in \mathbb R^p, u \in \mathbb R^q, a = 1, \dots N,$$ where e.g. $u_{nx}$ denotes all (partial) derivatives of order $n$. The numbers $p, q, n$ and $N$ are all nonnegative integers.
We are interested in the symmetry group $G$ of the system (\[1.1\]), i.e. in the local Lie group of local point transformations taking solutions of eq. (\[1.1\]) into solutions. Point transformations in the space $X \times U$ of independent and dependent variables have the form $$\label{1.2}
\tilde x = \Lambda_{\lambda}(x, u), \quad \tilde u = \Omega_{\lambda}(x, u),$$ where $\lambda$ denotes the group parameters. We have $$\Lambda_0(x, u) = x, \quad \Omega_0(x, u) = u\nonumber$$ and the inverse transformation $(\tilde x, \tilde u) \to (x, u)$ exists, at least locally.
The transformations (\[1.2\]) of local coordinates in $X \times U$ also determine the transformations of functions $u = f(x)$ and of derivatives of functions. A group $G$ of local point transformations of $X \times U$ will be a symmetry group of the system (\[1.1\]) if the fact that $u(x)$ is a solution implies that $\tilde u(\tilde x)$ is also a solution.
The two fundamental questions to ask are:
1. How to find the maximal symmetry group $G$ for a given system of equations (\[1.1\])?
2. Once the group $G$ is found, what do we do with it?
Let us first discuss the question of motivation. The symmetry group $G$ allows us to do the following.
1. Generate new solutions from known ones. Sometimes trivial solutions can be boosted into interesting ones.
2. Identify equations with isomorphic symmetry groups. Such equations may be transformable into each other. Sometimes nonlinear equations can be transformed into linear ones.
3. Perform symmetry reduction: reduce the number of variables in a PDE and obtain particular solutions, satisfying particular boundary conditions: group invariant solutions. For ODEs of order $n$, we can reduce the order of the equation. In this reduction, there is no loss of information. If we can reduce the order to zero, we obtain a general solution depending on $n$ constants, or a general integral (an algebraic equation depending on $n$ constants).
How does one find the symmetry group $G$? One looks for infinitesimal transformations, i.e. one looks for the Lie algebra $L$ that corresponds to $G$. Instead of looking for “global” transformations as in eq. (\[1.2\]) one looks for infinitesimal ones. A one-parameter group of infinitesimal point transformations will have the form $$\begin{aligned}
\tilde x_i & = & x_i + \lambda\xi_i(x, u) \quad |\lambda| << 1\label{1.3}\\
\tilde u_{\alpha} & = & u_{\alpha} + \lambda\phi_{\alpha}(x, u) \quad 1 \leq i
\leq p, \quad 1 \leq \alpha \leq q.\nonumber\end{aligned}$$ The functions $\xi_i$ and $\phi_{\alpha}$ must be found from the condition that $\tilde u(\tilde x)$ is a solution whenever $u(x)$ is one. The derivatives $\tilde u_{\alpha, \tilde x_i}$ must be calculated using eq. (\[1.3\]) and will involve derivatives of $\xi_i$ and $\phi_{\alpha}$. A $K$-th derivative of $\tilde u_{\alpha}$ with respect to the variable $\tilde x_i$ will involve derivatives of $\xi_i$ and $\phi_{\alpha}$ up to order $K$. We then substitute the transformed quantities into eq. (\[1.1\]) and request that the equation be satisfied for $\tilde u(\tilde x)$, whenever it is satisfied for $u(x)$. Thus, terms of order $\lambda^0$ will drop out. Terms of order $\lambda$ will provide a system of determining equations for $\xi_i$ and $\phi_{\alpha}$. Terms of order $\lambda^k$, $k = 2, 3, \dots$ are to be ignored, since we are looking for infinitesimal symmetries.
The functions $\xi_i$ and $\phi_{\alpha}$ depend only on $x$ and $u$, not on first, or higher derivatives, $u_{\alpha, x_i}$, $u_{\alpha, x_ix_k}$, etc. This is actually the definition of “point” symmetries. The determining equations will explicitly involve derivatives of $u_{\alpha}$, up to the order $n$ (the order of the studied equation). The coefficients of all linearly independent expressions in the derivatives must vanish separately. This provides a system of determining equations for the functions $\xi_i(x, u)$ and $\phi_{\alpha}(x,
u)$. This is a system of linear partial differential equations of order $n$. The determining equations are linear, even if the original system (\[1.1\]) is nonlinear. This “linearization” is due to the fact that all terms of order $\lambda^j$, $j \geq 2$, are ignored.
The system of determining equations is usually overdetermined, i.e. there are usually more determining equations than unknown functions $\xi_i$ and $\phi_{\alpha}$ ($p+q$ functions). The independent variables in the determining equations are $x \in \mathbb R^p$, $u \in \mathbb R^q$.
For an overdetermined system, three possibilities occur.
1. The only solution is the trivial one $\xi_i = 0$, $\phi_{\alpha} =
0$, $i = 1, \dots p, \alpha = 1, \dots, q$. In this case the symmetry algebra is $L = \{0\}$, the symmetry group is $G = I$ and the symmetry method is to no avail.
2. The general solution of the determining equations depends on a finite number $K$ of constants. In this case the studied system (\[1.1\]) has a finite-dimensional Lie point symmetry group and we have $\dim G = K$.
3. The general solution depends on a finite number of arbitrary functions of some of the variables $\{x_i, u_{\alpha}\}$. In this case the symmetry group is infinite dimensional. This last case is of particular interest.
The search for the symmetry algebra $L$ of a system of differential equations is best formulated in terms of vector fields acting on the space $X \times U$ of independent and dependent variables. Indeed, consider the vector field $$\label{1.4}
X = \sum^p_{i=1}\xi_i(x, u)\partial x_i + \sum^q_{\alpha=1}\phi_{\alpha}
(x, u)\partial u_{\alpha},$$ where the coefficients $\xi_i$ and $\phi_{\alpha}$ are the same as in eq. (\[1.3\]). If these functions are known, the vector field (\[1.4\]) can be integrated to obtain the finite transformations (\[1.2\]). Indeed, all we have to do is integrate the equations $$\label{1.5}
\frac{d\tilde x_i}{d\lambda} = \xi_i(\tilde x, \tilde u), \quad \frac{d\tilde
u_{\alpha}}{d\lambda} = \phi_{\alpha}(\tilde x, \tilde u),$$ subject to the initial conditions $$\label{1.6}
\tilde x_i\mid_{\lambda = 0} = x_i \quad \tilde u_{\alpha}\mid_{\lambda=0} =
u_{\alpha}.$$ This provides us with a one-parameter group of local Lie point transformations of the form (\[1.2\]) with $\lambda$ the group parameter.
The vector field (\[1.4\]) tells us how the variables $x$ and $u$ transform. We also need to know how derivatives like $u_x$, $u_{xx}$, $\dots$ transform. This is given by the prolongation of the vector field $X$.
We have $$\begin{aligned}
\lefteqn{{\mathop{\mathrm{pr}}\nolimits}X = X +
\sum_{\alpha}\biggl\{\sum_i\phi^{x_i}_{\alpha}\partial u_{x_i} +\sum_{i,
k}\phi^{x_ix_k}_{\alpha}\partial u_{x_ix_k}}\label{1.7}\\
&\hspace{2in} & + \sum^{x_ix_kx_l}_{i, k, l}\phi_{\alpha}\partial u_{x_ix_kx_l}
+
\dots\biggr\},\nonumber\end{aligned}$$ where the coefficients in the prolongation can be calculated recursively, using the total derivative operator $$\label{1.8}
D_{x_i} = \partial_{x_i}+ u_{\alpha, x_i}\partial_{u_{\alpha}}+u_{\alpha,
x_ax_i}\partial_{u_{\alpha}, x_a} + u_{\alpha, x_ax_bx_i}\partial_{u_{\alpha},
x_ax_b} + \dots$$ (a summation over repeated indices is to be understood).
The recursive formulas are $$\begin{aligned}
\phi^{x_i}_{\alpha} & = & D_{x_i}\phi_{\alpha} - (D_{x_i}\xi_a)u_{\alpha, x_a}
\nonumber\\
\phi^{x_ix_k}_{\alpha} & = & D_{x_k}\phi^{x_i}_{\alpha} - (D_{x_k}\xi_a)
u_{\alpha, x_ix_a}\label{1.9}\\
\phi^{x_ix_kx_l}_{\alpha} & = & D_{x_l}\phi^{x_ix_k}_{\alpha} - (D_{x_l}\xi_a)
u_{\alpha, x_ix_kx_a}\nonumber\end{aligned}$$ etc.
The $n$-th prolongation of $\widehat X$ acts on functions of $x$, $u$ and all derivatives of $u$ up to order $n$. It also tells us how derivatives transform. Thus, to obtain the transformed quantities $\tilde u_{\tilde x_i}$ we must integrate eq. (\[1.5\]) with conditions (\[1.6\]), together with $$\label{1.10}
\frac{d\tilde u_{\tilde x_i}}{d\lambda} = \phi^{x_i}(\tilde x, \tilde u, \tilde
u_{\tilde x}), \quad \tilde u_{\tilde x}\mid_{\lambda=0} = u_x.$$ We see that the coefficients of the prolonged vector field are expressed in terms of derivatives of $\xi_i$ and $\phi_{\alpha}$, the coefficients of the original vector field. They carry no new information: the transformation of derivatives is completely determined, once the transformations of functions are known.
The invariance condition for the system (\[1.1\]) is expressed in terms of the operator (\[1.7\]) as $$\label{1.11}
{\mathop{\mathrm{pr}}\nolimits}^{(n)}XE_a\mid_{E_1=\dots = E_N = 0} = 0, \quad a = 1, \dots N,$$ where ${\mathop{\mathrm{pr}}\nolimits}^{(n)}X$ is the prolongation (\[1.7\]) calculated up to order $n$ where $n$ is the order of the system (\[1.1\]).
In practice the symmetry algorithm consists of several steps, most of which can be carried out on a computer. For early computer programs calculating symmetry algebras, see Ref. [@ref9; @ref10]. For a more recent review, see [@ref11].
The individual steps are:
1. Calculate all the coefficients in the $n$-th prolongation of $\widehat X$. This depends only on the order of the system (\[1.1\]), i.e. $n$, and on the number of independent and dependent variables, i.e. $p$ and $q$.
2. Consider the system (\[1.1\]) as a system of algebraic equations for $x$, $u$, $u_x$, $u_{xx}$, etc. Choose $N$ variables $v_1$, $v_2$, $\dots v_N$ and solve the system (\[1.1\]) for these variables. The $v_i$ must satisfy the following conditions.
1. Each $v_i$ is a derivative of one $u_{\alpha}$ of at least order 1.
2. The variables $v_i$ are all independent, none of them is a derivative of any other one.
3. No derivatives of any of the $v_i$ figure in the system (\[1.1\]).
3. Apply ${\mathop{\mathrm{pr}}\nolimits}^{(n)}X$ to all the equations in (\[1.1\]) and eliminate all expressions $v_i$ from the result. This provides us with the system (\[1.11\]).
4. Determine all linearly independent expressions in the derivatives remaining in (\[1.11\]), once the quantities $v_i$ are eliminated. Set the coefficients of these expressions equal to zero. This provides us with the determining equations, a system of linear partial differential equations of order $n$ for $\phi_{\alpha}(x, u)$ and $\xi_i(x, u)$.
5. Solve the determining equations to obtain the symmetry algebra.
6. Integrate the obtained vector fields to obtain the one-parameter subgroups of the symmetry group. Compose them appropriately to obtain the connected component $G_o$ of the symmetry group $G$.
7. Extend the connected component $G_o$ to the full group $G$ by adding all discrete transformations leaving the system (\[1.1\]) invariant. These discrete transformations will form a finite, or discrete group $G_D$. We have $$\label{1.12}
G = G_D {\mathbin{\rlap{\raisebox{.1pt}{$\times$}}
\mskip-4mu\supset}}G_o$$ i.e. $G_o$ is an invariant subgroup of $G$.
Let us consider the case when at Step 5 we obtain a finite dimensional Lie algebra $L$, i.e. a vector field $X$ depending on $K$ parameters, $K \in \mathbb Z^>$, $K < \infty$. We can then choose a basis $$\label{1.13}
\{X_1, X_2, \dots, X_K\}$$ of the Lie algebra $L$. The basis that is naturally obtained in this manner depends on our integration procedure, though the algebra $L$ itself does not. It is useful to transform the basis (\[1.13\]) to a canonical form in which all basis independent properties of $L$ are manifest. Thus, if $L$ can be decomposed into a direct sum of indecomposable components, $$\label{1.14}
L = L_1 \oplus L_2 \oplus \dots \oplus L_M,$$ then a basis should be chosen that respects this decomposition. The components $L_i$ that are simple should be identified according to the Cartan classification (over $\mathbb C$) or the Gantmakher classification (over $\mathbb R$) [@ref12; @ref13]. The components that are solvable should be so organized that their nilradical [@ref14; @ref15] is manifest. For those components that are neither simple, nor solvable, the basis should be chosen so as to respect the Levi decomposition [@ref14; @ref15].
So far we have considered only point transformations, as in eq. (\[1.2\]), in which the new variables $\tilde x$ and $\tilde u$ depend only on the old ones, $x$ and $u$. More general transformations are “contact transformations”, where $\tilde x$ and $\tilde u$ also depend on first derivatives of $u$. A still more general class of transformations are generalized transformations, also called “Lie-Bäcklund” transformations [@ref1; @ref16]. For these we have $$\begin{aligned}
\tilde x & = & \Lambda_{\lambda}(x, u, u_x, u_{xx}, \dots)\\
\tilde u & = & \Omega_{\lambda}(x, u, u_x, u_{xx}, \dots)\nonumber\end{aligned}$$ involving derivatives up to an arbitrary order. The coefficients $\xi_i$ and $\phi_{\alpha}$ of the vector fields (\[1.4\]) will then also depend on derivatives of $u_{\alpha}$.
When studying generalized symmetries, and sometimes also point symmetries, it is convenient to use a different formalism, namely that of evolutionary vector fields.
Let us first consider the case of Lie point symmetries, i.e. vector fields of the form (\[1.4\]) and their prolongations (\[1.7\]). To each vector field (\[1.4\]) we can associate its evolutionary counterpart $X_e$, defined as $$\begin{aligned}
X_e & = & Q_{\alpha}(x, u, u_x)\partial u_{\alpha},\label{1.16}\\
Q_{\alpha} & = & \phi_{\alpha} - \xi_i\frac{\partial u_{\alpha}}{\partial
x_i}.\label{1.17}\end{aligned}$$ The prolongation of the evolutionary vector field (\[1.16\]) is defined as $$\begin{aligned}
{\mathop{\mathrm{pr}}\nolimits}X_e &=& Q_{\alpha}\partial u_a + Q^{x_i}_{\alpha} \partial u_{\alpha, x_i} +
Q^{x_ix_k}_{\alpha}\partial u_{\alpha, x_ix_k} + \dots\label{1.18}\\
Q^{x_i}_{\alpha} &=& D_{x_i}Q_{\alpha}, \quad Q^{x_ix_k}_{\alpha} = D_{x_i}D_
{x_k}Q_{\alpha}, \dots\nonumber.\end{aligned}$$ The functions $Q_{\alpha}$ are called the characteristics of the vector field. Notice that $X_e$ and ${\mathop{\mathrm{pr}}\nolimits}X_e$ do not act on the independent variables $x_i$.
For Lie point symmetries evolutionary and ordinary vector fields are entirely equivalent and it is easy to pass from one to the other. Indeed, eq. (\[1.17\]) gives the connection between the two.
The symmetry algorithm for calculating the symmetry algebra $L$ in terms of evolutionary vector fields is also equivalent. Eq. (\[1.11\]) is simply replaced by $$\label{1.19}
{\mathop{\mathrm{pr}}\nolimits}^{(n)}X_e E_a\mid_{E_1=\dots=E_N=0}, = 0, \quad a = 1, \dots N.$$ The reason that eq. (\[1.11\]) and (\[1.19\]) are equivalent is the following. It is easy to check that we have $$\label{1.20}
{\mathop{\mathrm{pr}}\nolimits}^{(n)}X_e = {\mathop{\mathrm{pr}}\nolimits}^{(n)}X - \xi_iD_i.$$
The total derivative $D_i$ is itself a generalized symmetry of eq. (\[1.1\]), i.e. we have $$\label{1.21a}
D_iE_a\mid_{E_1=E_2=\dots=E_n=0}, =0 \quad i = 1, \dots p, \quad a = 1, \dots N.$$ Eq. (\[1.20\]) and (\[1.21a\]) prove that the systems (\[1.11\]) and (\[1.19\]) are equivalent. Eq. (\[1.21a\]) itself follows from the fact that $DE_a=0$ is a differential consequence of eq. (\[1.1\]), hence every solution of eq. (\[1.1\]) is also a solution of eq. (\[1.21a\]).
To find generalized symmetries of order $k$ we use eq. (\[1.16\]) but allow the characteristics $Q_{\alpha}$ to depend on all derivatives of $u_{\alpha}$ up to order $k$. The prolongation is calculated using eq. (\[1.18\]). The symmetry algorithm is again eq. (\[1.19\]).
A very useful property of evolutionary symmetries is that they provide compatible flows. This means that the system of equations $$\label{1.21b}
\frac{\partial u_{\alpha}}{\partial\lambda} = Q_{\alpha}$$ is compatible with the system (\[1.1\]). In particular, group invariant solutions, i.e. solutions invariant under a subgroup of $G$ are obtained as fixed points $$\label{1.22}
Q_{\alpha} = 0.$$ If $Q_{\alpha}$ is the characteristic of a point transformation then (\[1.22\]) is a system of quasilinear first order partial differential equations. They can be solved, the solution substituted into (\[1.1\]) and this provides the invariant solutions explicitly.
Comments on Symmetries of Difference Equations {#subsec1.2}
----------------------------------------------
The study of symmetries of difference equations is much more recent than that of differential equations. Early work in this direction is due to Maeda [@ref17; @ref18] who mainly studied transformations acting on the dependent variables only. A more recent series of papers was devoted to Lie point symmetries of differential-difference equations on fixed regular lattices [@ref19; @ref20; @ref21; @ref22; @ref23; @ref24; @ref25; @ref26; @ref27; @ref28]. A different approach was developed mainly for linear or linearizable difference equations and involved transformations acting on more than one point of the lattice [@ref29; @ref30; @ref31; @ref32; @ref33]. The symmetries considered in this approach are really generalized ones, however they reduce to point ones in the continuous limit.
A more general class of generalized symmetries has also been investigated for difference equations, and differential-difference equations on fixed regular lattices [@ref34; @ref35; @ref36; @ref37].
A different approach to symmetries of discrete equations was originally suggested by V. Dorodnitsyn and collaborators [@ref38; @ref39; @ref40; @ref41; @ref42; @ref43; @ref44; @ref45; @ref46; @ref47]. The main aim of this series of papers is to discretize differential equations while preserving their Lie point symmetries.
Symmetries of ordinary and partial difference schemes on lattices that are a priori given, but are allowed to transform under point transformations, were studied in Ref. [@ref48; @ref49; @ref50].
Ordinary Difference Schemes and Their Point Symmetries {#sec2}
======================================================
Ordinary Difference Schemes {#subsec2.1}
---------------------------
An ordinary differential equation (ODE) of order $n$ is a relation involving one independent variable $x$, one dependent variable $u = u(x)$ and $n$ derivatives $\stackrel{\prime}{u}$, $\stackrel{\prime\prime}{u}$, $\dots
u^{(n)}$ $$\label{2.1}
E(x, u, \stackrel{\prime}{u}, \stackrel{\prime\prime}{u}, \dots, u^{(n)}) = 0
\quad \frac{\partial E}{\partial u^{(n)}} \neq 0.$$
An ordinary difference scheme (O$\Delta$S) involves two objects, a difference equation and a lattice. We shall specify an O$\Delta$S by a system of two equations, both involving two continuous variables $x$ and $u(x)$, evaluated at a discrete set of points $\{x_n\}$.
Thus, a difference scheme of order $K$ will have the form $$\label{2.2}
\begin{array}{c}
E_a(\{x_k\}^{n+N}_{k=n+M}, \{u_k\}^{n+N}_{k=n+M}) = 0, a = 1, 2\\[\jot]
K= N - M + 1, \quad n, M, N \in \mathbb Z, \quad u_k \equiv u(x_k).
\end{array}$$ At this stage we are not imposing any boundary conditions, so the reference point $x_n$ can be arbitrarily shifted to the left, or to the right. The order $K$ of the system is the number of points involved in the scheme (\[2.2\]) and it is assumed to be finite. We also assume that if the values of $x_k$ and $u_k$ are specified in $(N-M)$ neighbouring point, we can calculate their values in the point to the right, or to the left of the given set, using equations (\[2.2\]).
A continuous limit for the spacings between all neighbouring points going to zero, if it exists, will take one of the equations (\[2.2\]) into a differential equation of order $K' \leq K$, another into an identity (like $0 =
0$).
When taking the continuous limit it is convenient to introduce different quantities, namely differences between neighbouring points and discrete derivatives like $$\begin{aligned}
h_+ (x_n) & = & x_{n+1}-x_n, \quad h_-(x_n) =x_n - x_{n-1},\nonumber\\
u_{,x} & = & \frac{u_{n+1}-u_n}{x_{n+1}-x_n}, \quad u_{,\underline
x}= \frac{u_n-u_{n-1}}{x_n-x_{n-1}},\label{2.3}\\
u_{,x\underline x} & = &
2\frac{u_{,x}-u_{,\underline x}}{x_{n+1}-x_{n-1}},\dots\nonumber\end{aligned}$$
In the continuous limit, we have $$h_+ \to 0, \quad h_- \to 0, \quad u_{,{x}} \to \stackrel{\prime}{u},\quad
u_{,{\underline x}} \to \stackrel{\prime}{u}, \quad u_{,x\underline x} \to
\stackrel{\prime\prime}{u}.$$
As a clarifying example of the meaning of the difference scheme (\[2.2\]), let us consider a three point scheme that will approximate a second order linear difference equation: $$\begin{aligned}
E_1 & = & \frac{u_{n+1}-2u_n+u_{n-1}}{(x_{n+1}-x_n)^2} - u_n = 0,\label{2.4}\\
E_2 & = & x_{n+1} - 2x_n + x_{n-1} = 0.\label{2.5} \end{aligned}$$ The solution of eq. $E_2=0$, determines a uniform lattice $$\label{2.6}
x_n = hn + x_0$$ The scale $h$ and the origin $x_0$ in eq. (\[2.6\]) are not fixed by eq. (\[2.5\]), instead they appear as integration constants, i.e. they are a priori arbitrary. Once they are chosen, eq. (\[2.4\]) reduces to a linear difference equation with constant coefficients, since we have $x_{n+1} - x_n =
h$. Thus, a solution of eq. (\[2.4\]) will have the form $$\label{2.7}
u_n = \lambda^{x_n}.$$ Substituting (\[2.7\]) into (\[2.4\]) we obtain the general solution of the difference scheme (\[2.4\]), (\[2.5\]) as $$\begin{aligned}
u(x_n) & = & c_1\lambda^{x_n}_1+c_2\lambda^{x_n}_2, \quad x_n = hn +
x_0,\label{2.8}\\
\lambda_{1, 2} & = & \left(\frac{2+h^2\pm h\sqrt{4+h^2}}{2}\right)^{1/2}.
\nonumber\end{aligned}$$ The solution (\[2.8\]) of the system (\[2.4\]), (\[2.5\]) depends on 4 arbitrary constants $c_1$, $c_2$, $h$ and $x_0$.
Now let us consider a general three point scheme of the form $$\label{2.9}
E_a(x_{n-1}, x_n, x_{n+1}, u_{n-1}, u_n, u_{n+1}) = 0, \quad a = 1, 2$$ satisfying $$\label{2.10}
\det\left(\frac{\partial(E_1, E_2)}{\partial(x_{n+1}, u_{n+1})}\right) \neq 0,
\quad \det\left(\frac{\partial(E_1, E_2)}{\partial(x_{n-1}, u_{n-1})}\right)
\neq 0,$$ (possibly after an up or down shifting). The two conditions on the Jacobians (\[2.10\]) are sufficient to allow us to calculate $(x_{n+1}, u_{n+1})$ if $(x_{n-1}, u_{n-1}, x_n, u_n)$ are given. Similarly, $(x_{n-1}, u_{n-1})$ can be calculated if $(x_n, u_n, x_{n+1},
u_{n+1})$ are given. The general solution of the scheme (\[2.9\]) will hence depend on 4 arbitrary constants and will have the form $$\begin{aligned}
u_n & = & f(x_n, c_1, c_2, c_3, c_4)\label{2.11}\\
x_n & = & \phi(n, c_1, c_2, c_3, c_4).\label{2.12}\end{aligned}$$
A more standard approach to difference equations would be to consider a fixed equally spaced lattice e.g. with spacing $h = 1$. We can then identify the continuous variable $x$, sampled at discrete points $x_n$, with the discrete variable $n$: $$\label{2.13}
x_n = n.$$ Instead of a difference scheme we then have a difference equation $$\label{2.14}
E(\{u_k\}^{n+N}_{k=n+M}) = 0,$$ involving $K = N - M + 1$ points. Its general solution has the form $$\label{2.15}
u_n = f(n, c_1, c_2, \dots c_{N-M})$$ i.e. it depends on $N - M$ constants.
Below, when studying point symmetries of discrete equations we will see the advantage of considering difference systems like the system (\[2.2\]).
Point Symmetries of Ordinary Difference Schemes {#subsec2.2}
-----------------------------------------------
In this section we shall follow rather closely the article [@ref48]. We shall define the symmetry group of an ordinary difference scheme in the same manner as for ODEs. That is, a group of continuous local point transformations of the form (\[1.2\]) taking solutions of the O$\Delta$S (\[2.2\]) into solutions of the same scheme. The transformations considered are continuous, and we will adopt an infinitesimal approach, as in eq. (\[1.3\]). We drop the labels $i$ and $\alpha$, since we are considering the case of one independent and one dependent variable only.
As in the case of differential equations, our basic tool will be vector fields of the form (\[1.4\]). In the case of O$\Delta$S they will have the form $$\label{2.16}
X = \xi(x, u)\partial_x + \phi(x, u)\partial_u$$ with $$x \equiv x_n, \quad u \equiv u_n = u(x_n).$$ Because we are considering point transformation, $\xi$ and $\phi$ in (\[2.16\]) depend on $x$ and $u$ at one point only.
The prolongation of the vector field $X$ is different than in the case of ODEs. Instead of prolonging to derivatives, we prolong to all points of the lattice figuring in the scheme (\[2.2\]). Thus we put $$\label{2.17}
{\mathop{\mathrm{pr}}\nolimits}X = \sum^{n+N}_{k=n+M} \xi(x_k, u_k)\partial_{x_k} + \sum^{n+N}_{k=n+M}
\phi(x_k, u_k)\partial_{u_k}.$$ In these terms the requirement that the transformed function $\tilde u(\tilde
x)$ should satisfy the same O$\Delta$S as the original $u(x)$ is expressed by the requirement $$\label{2.18}
{\mathop{\mathrm{pr}}\nolimits}X E_a\mid_{E_1=E_2=0} = 0, \quad a = 1, 2.$$ Since we must respect both the difference equation and the lattice, we have two conditions (\[2.18\]) from which to determine $\xi(x, u)$ and $\phi(x, u)$. Since each of these functions depends on a single point $(x, u)$ and the prolongation (\[2.17\]) introduces $N - M + 1$ points in space $X \times U$, the equation (\[2.18\]) will imply a system of determining equations for $\xi$ and $\phi$. Moreover, in general this will be an overdetermined system of linear functional equations that we transform into an overdetermined system of linear differential equations [@ref57; @ref58].
To illustrate the method and the role of the choice of the lattice, let us start from a simple example. The example will be that of difference equations that approximate the ODE $$\label{2.19}
\stackrel{\prime\prime}{u} = 0$$ on several different lattices.
First of all, let us find the Lie point symmetry group of the ODE (\[2.19\]), i.e. the equation of a free particle on a line. Following the algorithm of Chapter \[sec1\], we put $$\begin{aligned}
{\mathop{\mathrm{pr}}\nolimits}^{(2)}X & = & \xi\partial_x + \phi\partial_u+
\phi^x\partial_{\stackrel{\prime}{u}} + \phi^{xx}\partial
_{\stackrel{\prime\prime}{u}}\label{2.20}\\
\phi^x & = & D_x\phi - (D_x\xi)\stackrel{\prime}{u} = \phi_x + (\phi_u-\xi_x)
\stackrel{\prime}{u}-\xi_u \smash{\stackrel{\prime}{u}}^2\nonumber\\
\phi^{xx} & = & D_x\phi^x - (D_x\xi)\stackrel{\prime\prime}{u} = \phi_{xx} +
(2\phi_{xu} -\xi_{xx})\stackrel{\prime}{u}\nonumber\\
& & \quad + (\phi_{uu}-2\xi_{xu})\smash{\stackrel{\prime}{u}}^2 -
\xi_{uu}\smash{\stackrel{\prime}{u}}^3x + (\phi_u-2\xi_x)
\stackrel{\prime\prime}{u}\nonumber\\
& & \quad - 3\xi_u v\stackrel{\prime}{u}\stackrel{\prime\prime}{u}.\nonumber\end{aligned}$$ The symmetry formula (\[1.11\]) in this case reduces to $$\label{2.21}
\phi^{xx}\mid_{\stackrel{\prime\prime}{u}=0} = 0.$$ Setting the coefficients of $\smash{\stackrel{\prime}{u}}^3$, $\smash{\stackrel{\prime}{u}}^2$, $\stackrel{\prime}{u}$ and $\smash{(\stackrel{\prime}{u}}^0)$ equal to zero, we obtain an 8 dimensional Lie algebra, isomorphic to ${\mathop{\mathrm{sl}}}(3, \mathbb R)$ with basis $$\begin{aligned}
X_1 & = & \partial_x, \quad X_2 = x\partial_x, \quad X_3 = u\partial_x
\label{2.22}\\
X_4 & = & \partial_u, \quad X_5 = x\partial_u, \quad X_6 = u\partial_u,
\nonumber\\
X_7 & = & x(x\partial_x+u\partial_u), \quad X_8 = u(x\partial_x+u\partial_u).
\nonumber\end{aligned}$$
This result was of course already known to Sophus Lie. Moreover, any second order ODE that is linear, or linearizable by a point transformation has a symmetry algebra isomorphic to ${\mathop{\mathrm{sl}}}(3, \mathbb R)$. The group ${\mathop{\mathrm{SL}}}(3, \mathbb
R)$ acts as the group of projective transformations of the Euclidean space $E_2$ (with coordinates $x, u)$).
Now let us consider some difference schemes that have eq. (\[2.19\]) as their continuous limit. We shall take the equation to be $$\label{2.23}
\frac{u_{n+1}-2u_n+u_{n-1}}{(x_{n+1}-x_n)^2} = 0.$$ However before looking for the symmetry algebra, we multiply out the denominator and investigate the equivalent equation $$\label{2.24}
E_1 = u_{n+1} - 2u_n + u_{n-1} = 0.$$ To this equation we must add a second equation, specifying the lattice. We consider three different examples at first glance quite similar, but leading to different symmetry algebras.
\[exam2.1\] Free particle (\[2.24\]) on a fixed uniform lattice. We take $$\label{2.25}
E_2 = x_n - hn - x_0 = 0,$$ where $h$ and $x_0$ are fixed constants (that are not transformed by the group (e.g. $h = 1$, $x_0 = 0$).
Applying the prolonged vector field (\[2.17\]) to eq. (\[2.25\]) we obtain $$\label{2.26}
\xi(x_n, u_n) = 0$$ for all $x_n$ and $u_n$. Next, let us apply (\[2.17\]) to eq. (\[2.24\]) and replace $x_n$, using (\[2.25\]) and $u_{n+1}$, using (\[2.24\]). We obtain $$\begin{aligned}
\lefteqn{\phi\big(h(n+1)+x_0, 2u_n-u_{n-1}\big) - 2\phi(hn+x_0, u_n)}\nonumber\\
&&\hspace{0.76in} + \phi\big(h(n-1)+x_0, u_{n-1}\big) = 0.\label{2.27}\end{aligned}$$ Differentiating eq. (\[2.27\]) twice, once with respect to $u_{n-1}$, once with respect to $u_n$, we obtain $$\label{2.28}
\frac{\partial^2}{\partial u_{n+1}} \phi(x_{n+1}, u_{n+1}) = 0$$ and hence $$\label{2.29}
\phi(x_n, u_n) = A(x_n)u_n + B(x_n).$$ We substitute eq. (\[2.29\]) back into (\[2.27\]) and equate coefficients of $u_n$, $u_{n-1}$ and 1. The result is $$\label{2.30}
A(n+1) = A(n), \quad B(n+1) - 2B(n) + B(n-1) = 0.$$ Hence we have $$\label{2.31}
A = A_0, \quad B = B_1n + B_0 = b_1x + b_0$$ where $A_0$, $B_1$, $B_0$, $b_1$ and $b_0$ are constants. We obtain the symmetry algebra of the O$\Delta$S (\[2.24\]), (\[2.25\]) and it is only three-dimensional, spanned by $$\label{2.32}
X_1 = \partial_u, \quad X_2 = x\partial_u, \quad X_3 = u\partial_u.$$ The corresponding one parameter transformation groups are obtained by integrating these vector fields (see eq. (\[1.5\]), (\[1.6\])) $$\begin{aligned}
G_1 & : & \tilde x = x\nonumber\\
& &\tilde u(\tilde x) = u(x) + \lambda\nonumber\\
G_2 & : & \tilde x = x\label{2.33}\\
&& \tilde u(\tilde x) = u(x) + \lambda x\nonumber\\
G_3 & : & \tilde x = x\nonumber\\
&& \tilde u(\tilde x) = e^{\lambda}u(x)\nonumber\end{aligned}$$ $G_1$ and $G_2$ just tell us that we can add an arbitrary solution of the scheme to any given solution, $G_3$ corresponds to scale invariance of eq. (\[2.24\]).
\[exam2.2\] Free particle (\[2.24\]) on a uniform two point lattice.
Instead of eq. (\[2.25\]) we define a lattice by putting $$\label{2.34}
E_2 = x_{n+1} - x_n = h,$$ where $h$ is a fixed (non-transforming) constant. Note that (\[2.34\]) tells us the distance between any two neighbouring points but does not fix an origin (as opposed to eq. (\[2.25\])).
Applying the prolonged vector field (\[2.17\]) to eq. (\[2.34\]) and using (\[2.34\]), we obtain $$\label{2.35}
\xi(x_n + h, u_{n+1}) - \xi(x_n, u_n) = 0.$$ Since $u_{n+1}$ and $u_n$ are independent, eq. (\[2.35\]) implies $\xi =
\xi(x)$. Moreover $\xi(x_n+h) = \xi(x)$ so that we have $$\label{2.36}
\xi = \xi_0 = {\mathop{\mathrm{const}}}.$$ Further, we apply ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[2.24\]), and put $u_{n+1} = 2u_n -
u_{n-1}$, $x_{n+1} = x_n + h$, $x_{n-1} = x_n - h$ in the obtained expressions. As in Example \[exam2.1\] we find that $\phi(x, u)$ is linear in $u$ as in (\[2.29\]) and ultimately satisfies $$\label{2.37}
\phi(x, u) = au + bx + c.$$ The symmetry algebra in this case is four-dimensional. To the basis elements (\[2.32\]) we add translational invariance $$\label{2.38}
X_4 = \partial_x.$$
\[exam2.3\] Free particle (\[2.24\]) on a uniform three-point lattice.
Let us choose the lattice equation to be $$\label{2.39}
E_2 = x_{n+1} - 2x_n + x_{n-1} = 0.$$ Applying ${\mathop{\mathrm{pr}}\nolimits}X$ to $E_2$ and substituting for $x_{n+1}$ and $u_{n+1}$, we find $$\label{2.40}
\xi(2x_n-x_{n-1}, 2u_n-u_{n-1}) - 2\xi(x_n, u_n) + \xi(x_{n-1}, u_{n-1}) = 0.$$ Differentiating twice with respect to $u_n$ and $u_{n-1}$, we obtain that $\xi$ is linear in $u$. Substituting $\xi = A(x) u + B(x)$ into (\[2.40\]) we obtain $$\label{2.41}
\xi(x_n, u_n) = Au_n + Bx_n + C.$$ Similarly, applying ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[2.24\]), we obtain $$\label{2.42}
\phi(x_n, u_n) = Du_n + Ex_n + F.$$ where $A$, $\dots$, $F$ are constants. Finally, we obtain a six-dimensional symmetry algebra for the O$\Delta$S (\[2.24\]), (\[2.39\]) with basis $X_1$, $\dots$, $X_6$ as in eq. (\[2.22\]). It has been shown [@ref45] that the entire ${\mathop{\mathrm{sl}}}(3, \mathbb R)$ algebra cannot be recovered on any 3 point O$\Delta$S.
>From the above examples we can draw the following conclusions.
1. The Lie point symmetry group of an O$\Delta$S depends crucially on both equations in the system (\[2.2\]). In particular, if we choose a fixed lattice, as in eq. (\[2.25\]) (a “one-point lattice”) we are left with point transformations that act on the dependent variable only.
If we wish to preserve anything like the power of symmetry analysis for differential equations, we must either go beyond point symmetries to generalized ones, or use lattices that are also transformed and that are adapted to the symmetries we consider.
2. The method for calculating symmetries of O$\Delta$S is reasonable straightforward. It will however involve solving functional equations.
The method can be summed up as follows
1. Solve equations (\[2.2\]) for two of the quantities entering there, to make the equations explicit. For instance, take the system (\[2.9\]), (\[2.10\]). We can solve e.g. for $x_{n+1}$ and $u_{n+1}$ and obtain $$\begin{aligned}
x_{n+1} & = & f_1(x_{n-1}, x_n, u_{n-1}, u_n)\label{2.43}\\
u_{n+1} & = & f_2(x_{n-1}, y_n, u_{n-1}, u_n)\nonumber\end{aligned}$$
2. Apply the prolonged vector field (\[2.17\]) to eq. (\[2.2\]) and substitute (\[2.43\]) for $x_{n+1}$, $u_{n+1}$. We obtain two functional equations for $\xi$ and $\phi$ of the form $$\begin{aligned}
\lefteqn{
\biggl\{\xi(f_1, f_2) \frac{\partial E_a}{\partial x_{n+1}} + \xi(x_n,
u_n) \frac{\partial E_a}{\partial x_n} + \xi(x_{n-1}, u_{n-1}) \frac{\partial
E_a}{\partial x_{n-1}}}\nonumber\\
&\hspace{0.5in}& + \phi(f_1, f_2)\frac{\partial E_a}{\partial u_{n+1}}
+\phi(x_u, u_n) \frac{\partial E_a}{\partial u_n}\label{2.44}\\
&\hspace{0.5in}& + \phi(x_{n-1}, u_{n-1})
\frac{\partial E_a}{\partial u_{n-1}}\biggr\}\biggm|\nonumber_{\stackrel
{x_{n+1}=f_1}{u_{n+1}=f_2}} = 0 \quad a = 1,2\nonumber\end{aligned}$$
3. Assume that the functions $\xi$, $\phi$, $E_1$ and $E_2$ are sufficiently smooth and differentiate eq. (\[2.44\]) with respect to the variables $x_k$ and $u_k$ so as to obtain differential equations for $\xi$ and $\phi$. If the original equations are polynomial in all quantities we can thus obtain single term differential equations form (\[2.44\]). These we must solve, then substitute back into (\[2.44\]) and solve this equation.
We will illustrate the procedure on several examples in Section \[subsec2.3\].
Examples of Symmetry Algebras of O$\Delta$S {#subsec2.3}
-------------------------------------------
\[exam2.4\] Monomial nonlinearity on a uniform lattice.
Let us first consider the nonlinear ODE $$\label{2.45}
\stackrel{\prime\prime}{u} - u^N = 0, \quad N \neq 0, 1.$$ For $N \neq - 3$ eq. (\[2.45\]) is invariant under a two-dimensional Lie group, the Lie algebra of which is given by $$\label{2.46}
X_1 = \partial_x, \quad X_2 = (N-1)x\partial_x - 2u\partial_u$$ (translations and dilations). For $N = -3$ the symmetry algebra is three-dimensional, isomorphic to ${\mathop{\mathrm{sl}}}(3,
\mathbb R)$, i.e. it contains a third element, additional to (\[2.46\]). A convenient basis for the symmetry algebra of the equation $$\label{2.47}
\stackrel{\prime\prime}{u} - u^{-3} = 0$$ is $$\label{2.48}
X_1 = \partial_x, \quad X_2 = 2x\partial_x + u\partial_u, \quad X_3 = x(x
\partial_x+u\partial_u).$$
A very natural O$\Delta$S that has (\[2.45\]) as its continuous limit is $$\begin{aligned}
E_1 & = & \frac{u_{n+1}-2u_n+u_{n-1}}{(x_{n+1}-x_n)^2} - u^N_n = 0 \quad N
\neq 0, 1\label{2.49a}\\
E_2 & = & x_{n+1} - 2x_n + x_{n-1} = 0\label{2.49b}\end{aligned}$$
Let us now apply the symmetry algorithm described in Chapter \[subsec2.2\] to the system (\[2.49a\]) and (\[2.49b\]). To illustrate the method, we shall present all calculations in detail.
First of all, we choose two variables that will be substituted in eq. (\[2.18\]), once the prolonged vector field (\[2.17\]) is applied to the system (\[2.49a\]) and (\[2.49b\]), namely $$\begin{aligned}
x_{n+1} & = & 2x_n - x_{n-1}\label{2.50}\\
u_{n+1} & = & (x_n-x_{n-1})^2u^N_n + 2u_n - u_{n-1}\nonumber\end{aligned}$$ We apply ${\mathop{\mathrm{pr}}\nolimits}X$ of (\[2.17\]) to eq. (\[2.49b\]) and obtain $$\label{2.51}
\xi(x_{n+1}, u_{n+1}) - 2\xi(x_n, u_n) + \xi(x_{n-1}, u_{n-1}) = 0$$ where, $x_n$, $u_n$ $x_{n-1}$, $u_{n-1}$ are independent, but $x_{n+1}$, $u_{n+1}$ are expressed in terms of these quantities, as in eq. (\[2.50\]). Taking this into acccount, we differentiate (\[2.51\]) first with respect to $u_{n-1}$, then $u_n$. We obtain successively $$\begin{aligned}
- \xi_{,u_{n+1}}(x_{n+1}, u_{n+1}) + \xi_{,u_{n-1}}(x_{n-1}, u_{n-1}) & = &0
\label{2.52}\\
\{N(x_n-x_{n-1})^2 u^{N-1}_n +2\} \xi_{,u_{n+1}u_{n+1}} (x_{n+1}, u_{n+1})
& = & 0.\label{2.53}\end{aligned}$$ Eq. (\[2.53\]) is the desired one-term equation. It implies $$\label{2.54}
\xi(x, u) = a(x)u + b(x)$$ Substituting (\[2.54\]) into (\[2.52\]) we obtain $$\label{2.55}
- a(2x_n-x_{n-1}) + a(x_{n-1}) = 0.$$ Differentiating with respect to $x_n$, we obtain $a = a_0 = {\mathop{\mathrm{const}}}$. Finally, we substitute (\[2.54\]) with $a = a_0$ into (\[2.51\]) and obtain $$\label{2.56}
a = 0, \quad b(2x_n-x_{n-1}) - 2b(x_n) + b(x_{n-1}) = 0$$ and hence $$\label{2.57}
\xi = b = b_1x + b_0$$ where $b_0$ and $b_1$ are constants. To obtain the function $\phi(x_n, u_n)$, we apply ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[2.49a\]) and obtain $$\begin{aligned}
\lefteqn{\phi(x_{n+1}, u_{n+1}) - 2\phi(x_n, u_n) + \phi(x_{n-1}, u_{n-1})}
\nonumber\\
&&\hspace{1in}-(x_n-x_{n-1})^2 [N\phi(x_n, u_n)u^{N-1}_n + 2b_1u^N_n] =
0.\label{2.58}\end{aligned}$$ Differentiating successively with respect to $u_{n-1}$ and $u_n$ (taking (\[2.50\]) into account) we obtain $$\begin{aligned}
- \phi_{,u_{n+1}}(x_{n+1}, u_{n+1}) + \phi_{,u_{n-1}}(x_{n-1}, u_{n-1}) & = &
0\label{2.59}\\
\{N(x_n-x_{n-1})^2u^N_n +2\} \phi_{,u_{n+1}u_{n+1}} & = & 0\label{2.60}\end{aligned}$$ and hence $$\label{2.61}
\phi = \phi_1u + \phi_0(x), \quad \phi_1 = {\mathop{\mathrm{const}}}.$$ Eq. (\[2.58\]) now reduces to $$\begin{aligned}
\lefteqn{\phi_0(2x_n-x_{n-1}) - 2\phi_0(x_n) + \phi_0(x_{n-1})}\nonumber\\
&&\hspace{0.50in} -(x_n-x_{n-1})^2 \{(N-1)\phi_1 + 2b_1\}u^N_n\nonumber\\
&&\hspace{0.50in} - N(x_n-x_{n-1})^2\phi_0u^{N-1}_n = 0.\label{2.62}\end{aligned}$$ We have $N \neq 0, 1$ and hence (\[2.62\]) implies $$\label{2.63}
\phi_0 = 0, \quad (N-1)\phi_1 + 2b_1 = 0.$$ We have thus proven that the symmetry algebra of the O$\Delta$S (\[2.49a\]) and (\[2.49b\]) is the same as that of the ODE (\[2.45\]), namely the algebra (\[2.46\]).
We mention that the value $N = -3$ is not distinguished here and the system (\[2.49a\]) and (\[2.49b\]) is not invariant under ${\mathop{\mathrm{SL}}}(3, \mathbb R)$ for $N = -3$. Actually, a difference scheme invariant under ${\mathop{\mathrm{SL}}}(3, \mathbb R)$ does exist and it will have eq. (\[2.47\]) as its continuous limit. It will however not have the form (\[2.48\]) and the lattice will not be uniform [@ref45; @ref46].
Had we taken a two-point lattice, $x_{n+1}-x_n = h$ with $h$ fixed, instead of $E_2 = 0$ as in (\[2.49b\]), we would only have obtained translational invariance for the equation (\[2.49a\]) and lost the dilational invariance represented by $X_2$ of eq. (\[2.46\]).
\[exam2.5\] A nonlinear O$\Delta$S on a uniform lattice $$\begin{aligned}
E_1 & = & \frac{u_{n+1}-2u_n+u_{n-1}}{(x_{n+1}-x_n)^2} - f\left(\frac{u_n-
u_{n-1}}{x_n-x_{n-1}}\right) = 0,\label{2.64a}\\
E_2 & = & x_{n+1} -2x_n + x_{n-1} = 0,\label{2.64b}\end{aligned}$$ where $f(z)$ is some sufficiently smooth function satisfying $$\label{2.65}
\stackrel{\prime\prime}{f}(z) \neq 0.$$ The continuous limit of eq. (\[2.64a\]) and (\[2.64b\]) is $$\label{2.66}
\stackrel{\prime\prime}{u} - f(\stackrel{\prime}{u}) = 0$$ and is invariant under a two-dimensional group with Lie algebra $$\label{2.67}
X_1 = \partial_x, \quad X_2 = \partial_u$$ for any function $f(\stackrel{\prime}{u})$. For certain functions $f$ the symmetry group is three-dimensional, where the additional basis element of the Lie algebra is $$\label{2.68}
X_3 = (ax+bu)\partial_x + (cx+du)\partial_u.$$ The matrix $$\label{2.69}
M = \left(\begin{array}{ll}
a\quad b\\ c \quad d\end{array}\right)$$ can be transformed to Jordan canonical form and a different function $f(z)$ is obtained for each canonical form.
Now let us consider the discrete system (\[2.64a\]) and (\[2.64b\]). Before applying ${\mathop{\mathrm{pr}}\nolimits}X$ to this system we choose two variables to substitute in eq. (\[2.18\]), namely $$\begin{aligned}
x_{n+1} & = & 2x_n - x_{n-1}\label{2.70}\\
u_{n+1} & = & 2u_n - u_{n-1} + (x_n-x_{n-1})^2 f\left(\frac{u_n-u_{n-1}}
{x_n-x_{n-1}}\right).\nonumber\end{aligned}$$ Applying ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[2.64b\]) we obtain eq. (\[2.51\]) with $x_{n+1}$ and $u_{n+1}$ as in eq. (\[2.70\]). Differentiating twice, with respect to $u_{n-1}$ and $u_n$ respectively, we obtain $$\label{2.71}
\xi_{, u_{n+1}u_{n+1}}[1+(x_n-x_{n-1})f'] [2+(x_n-x_{n-1})f'] + \xi_{, u_{n+1}}
f'' = 0.$$ For $f'' \neq 0$ the only solution is $\xi_{, u_{n+1}} = 0$, i.e. $\xi =
\xi(x)$. Substituting back into (\[2.51\]), we obtain $$\label{2.72}
\xi = \alpha x + \beta$$ with $\alpha = {\mathop{\mathrm{const}}}$, $\beta = {\mathop{\mathrm{const}}}$.
Now let us apply ${\mathop{\mathrm{pr}}\nolimits}X$ to $E_1$ of eq. (\[2.64a\]) and (\[2.64b\]) and replace $x_{n+1}$, $u_{n+1}$ as in eq (\[2.70\]). We obtain the equation $$\begin{aligned}
\lefteqn{\phi(x_{n+1}, u_{n+1}) - 2\phi(x_n, u_n) + \phi(x_{n-1},
u_{n-1}) = 2\alpha(x_n-x_{n-1})^2f(z)}\nonumber\\
&& \hspace{.65in}+
(x_n-x_{n-1})^2f'(z)\left(\frac{\phi(x_n, u_n)-\phi(x_{n-1},
u_{n-1})}{x_n-x_{n-1}} -\alpha z\right)\label{2.73}\end{aligned}$$ with $\alpha$ as in eq. (\[2.72\]). Thus, we only need to distinguish between $\alpha = 0$ and $\alpha = 1$. Eq. (\[2.73\]) is a functional equation, involving two unknown functions $\phi$ and $f$. There are only four independent variables involved, $x_n, x_{n-1}, u_n$ and $u_{n-1}$. We simplify (\[2.73\]) by introducing new variables $\{x, u, h, z\}$, putting $$\begin{aligned}
x_n & = & x, \quad x_{n+1} = x + h, \quad x_{n-1} = x - h\label{2,74}\\
u_n & = & u,\quad u_{n-1} = u - hz, \quad u_{n+1} = u + hz + h^2f(z),\nonumber\end{aligned}$$ where we have used eq. (\[2.70\]) and defined $$z = \frac{u_n-u_{n-1}}{x_n-x_{n-1}} \quad h = x_{n+1} - x_n.$$ Eq. (\[2.73\]) in these variables is $$\begin{aligned}
\lefteqn{\phi\big(x+h, u+hz+h^2f(z)\big) - 2\phi(x, u) + \phi(x-h,
u-hz)}\nonumber\\
&&\hspace{.25in}=2\alpha h^2f(z)
+ h^2f'(z)\biggl[\frac{\phi(x, u)-\phi(x-h, u-hz)}{h}-\alpha
z\biggr].\label{2.76}\end{aligned}$$
First of all, we notice that for any function $f(z)$ we have two obvious symmetry elements, namely $X_1$ and $X_2$ of eq. (\[2.67\]), corresponding to $\alpha = 0$, $\beta = 1$ in (\[2.76\]) (and (\[2.72\])) and $\phi = 0$ and $\phi = 1$, respectively. Eq. (\[2.76\]) is quite difficult to solve directly. However, any three-dimensional Lie algebra of vector fields in 2 variables, containing $\{X_1, X_2\}$ of eq. (\[2.67\]) as a subalgebra, must have $X_3$ of eq. (\[2.68\]) as its third element. Moreover, eq. (\[2.72\]) shows that we have $b = 0$ in eq. (\[2.68\]) and (\[2.69\]). In (\[2.76\]) we put $\alpha = a$ and $$\label{2.77}
\phi(x, u) = cx + du.$$ Substituting into eq. (\[2.76\]) we obtain $$\label{2.78}
(d-2a)f(z) = [c+(d-a) z]f'(z).$$ >From eq. (\[2.78\]) we obtain two types of solutions:
For $d \neq a$ we have $$\label{2.79}
f = f_0[(d-a)z+c]^{(d-2a)/(d-a)}, \quad c \neq 0.$$
For $d = a$, we have $$\label{2.80}
f = f_0e^{-(a/c) x}.$$ With no loss of generality we could have taken the matrix (\[2.69\]) with $b
= 0$ to Jordan cannonical form and we would have obtained two different cases, simplifying (\[2.79\]) and (\[2.80\]), respectively. They are $$\begin{aligned}
f & = & f_0z^N, \quad X_3 = x\partial_x + \frac{N-2}{N-1} u\partial u, \quad
N \neq 1\label{2.81}\\
f & = & f_0 e^{-z}, \quad X_3 = x\partial_x + (x+u)\partial u.\label{2.82}\end{aligned}$$ The result can be stated as follows. The O$\Delta$S (\[2.64a\]) and (\[2.64b\]) is always invariant under the group generated by $\{X_1, X_2\}$ as in (\[2.67\]). It is invariant under a three-dimensional group with algebra including $X_3$ as in eq. (\[2.68\]) if $f$ satisfies eq. (\[2.78\]), i.e. has the form (\[2.81\]), or (\[2.82\]). These two cases also exist in the continuous limit. However, one more case exists in the continuous limit, namely $$\label{2.83}
\stackrel{\prime\prime}{u} = \big(1+(\stackrel{\prime}{u})^2\big)^{3/2}e^
{k\arctan \, \stackrel{\prime}{u}}$$ with $$\label{2.84}
X_3 = (kx+u)\partial_x + (ku-x)\partial u.$$ This equation can also be discretized in a symmetry preserving way [@ref45], not however on the uniform lattice (\[2.64b\]).
Lie Point Symmetries of Partial Difference Schemes {#sec3}
==================================================
Partial Difference Schemes {#subsec3.1}
--------------------------
In this chapter we generalize the results of Chapter \[sec2\] to the case of two discretely varying independent variables. We follow the ideas and notation of Ref. [@ref49]. The generalization to $n$ variables is immediate, though cumbersome. Thus, we will consider a Partial Difference Scheme (P$\Delta$S), involving one continuous function of two continuous variables $u(x, t)$. The variables $(x, t)$ are sampled on a two-dimensional lattice, itself defined by a system of compatible relations between points. Thus, a lattice will be an a priori infinite set of points $P_i$ lying in the real plane $\mathbb R^2$. The points will be labelled by two discrete subscripts $P_{m, n}$ with $-\infty < m < \infty$, $- \infty < n < \infty$. The cartesian coordinates of the point $P_{mn}$ will be denoted $(x_{mn},
t_{mn})$ \[or similarly any other coordinates $(\alpha_{mn}, \beta_{mn})$\].
A two-variable P$\Delta$S will be a set of five relations between the quantities $\{x, t, u\}$ at a finite number of points. We choose a reference point $P_{mn}
\equiv P$ and two families of curves intersecting at the points of the lattice. The labels $m = m_0$ and $n = n_0$ will parametrize these curves (see Fig \[fig1\]). To define an orientation of the curves, we specify $$\label{3.1}
x_{m+1n} - x_{m, n} \equiv h_m > 0, \quad t_{mn+1}-t_{mn} \equiv h_n > 0$$ at the original reference point.
The actual curves and the entire P$\Delta$S are specified by the 5 relations $$\label{3.2}
\begin{array}{c}
E_a(\{x_{m+i, n+j}, t_{m+i, n+j}, u_{m+i, n+j}\}) = 0\\[\jot]
1 \leq a \leq 5 \quad i_1 \leq i \leq i_2 \quad j_i \leq j \leq j_2.
\end{array}$$
In the continuous limit, if one exists, all five equations (\[3.2\]) are supposed to reduce to a single PDE, e.g. $E_1 = 0$ can reduce to the PDE and $E_a=0$, $a \geq 2$ to $0 = 0$. The orthogonal uniform lattice of Fig. \[fig2\] is clearly a special case of that on Fig. \[fig1\].
![[]{data-label="fig1"}](pavel3)
(130,110)(-12.5,-3) (0,10)[(1,0)[110]{}]{} (10,0)[(0,1)[100]{}]{} (10,40)[(1,0)[80]{}]{} (10,80)[(1,0)[90]{}]{} (30,7.5)(0,5)[15]{}[(0,1)[2.5]{}]{} (60,7.5)[(0,1)[72.5]{}]{} (90,7.5)(0,5)[15]{}[(0,1)[2.5]{}]{} (30,40) (90,40) (60,80) (60,40) (5,40)[(0,0)[$n$]{}]{} (-5,80)[(0,0)[$n+1$]{}]{} (30,0)[(0,0)[$m-1$]{}]{} (60,0)[(0,0)[$m$]{}]{} (90,0)[(0,0)[$m+1$]{}]{} (5,105)[(0,0)[$t$]{}]{} (115,5)[(0,0)[$x$]{}]{}
Some independence conditions must be imposed on the system (\[3.2\]) e.g. $$\label{3.3}
|J| = \left|\frac{\partial(E_1, \dots, E_5)}{\partial(x_{m+i_2, n}, t_{m+i_2
n}, x_{m, n+j_2}, t_{m, n+j_2},u_{m+i_2, n+j_2})}\right| \neq 0.$$ This condition allows us to move upward and to the right along the curves passing through $P_{m, n}$. Moreover, compatibility of the five equations (\[3.2\]) must be assured.
As an example of a P$\Delta$S let us consider the linear heat equation on a uniform and orthogonal lattice. The heat equation in the continuous case is $$\label{3.4}
u_t = u_{xx}.$$
An approximation on a uniform orthogonal lattice is given by the five equations $$\begin{aligned}
E_1 &=& \frac{u_{mn+1}-u_{mn}}{h_2} - \frac{u_{m+1n}-2u_{mn}+u_{m-1n}}
{(h_1)^2} = 0\label{3.5}\\
E_2 &=& x_{m+1, n}-x_{m, n}-h_1 = 0 \quad\quad E_3 = t_{m+1, n} -t_{m, n} =
0\label{3.6}\\
E_4 &=& x_{m, n+1} -x_{mn} = 0 \quad\quad E_5 = t_{m, n+1} - t_{m, n} - h_2 =
0\label{3.7}.\end{aligned}$$ Equations (\[3.6\]) can of course be integrated to give the standard expressions $$\label{3.8}
x_{mn} = h_1m + x_0 \quad t_{mn} = h_2n + t_0.$$ Notice that $h_1$ and $h_2$ are constants that cannot be scaled (they are fixed in eq. (\[3.6\]). On the other hand $(x_0, t_0)$ are integration constants and are thus not fixed by the system (\[3.6\]), (\[3.7\]). As written, these equations are invariant under translations, but not under dilations.
Finally, we remark that the usual fixed lattice condition is obtained from (\[3.7\]) by putting $x_0 = t_0 = 0$, $h_1 = h_2 = 1$ and identifying $$\label{3.9}
x = m, \quad t = n.$$
Though the above example is essentially trivial, it brings out several points.
1. Four equations are indeed needed to specify a two-dimensional lattice and to allow us to move along the coordinate lines.
2. In order to solve the P$\Delta$S (\[3.5\]), (\[3.6\]) for $h_1$ and $h_2$ given, we must specify for instance $\{x_{mn}, t_{mn}, u_{mn}, u_{m+1,
n}, u_{m-1n}\}$. Then we can directly calculate $\{x_{m+1, n}, t_{m+1n}\}$, $\{x_{mn+1}, t_{m, n+1}\}$ from eq. (\[3.6\]). In order to calculate the coordinates of the fourth point figuring in eq. (\[3.5\]), namely $\{x_{mn-1}, t_{m,n-1}\}$ we must shift eq. (\[3.6\]) down by one unit in $m$.
3. The Jacobian condition (\[3.3\]) allowing us to perform these calculations, is obviously satisfied, since we have $$\label{3.10}
\left|\frac{\partial(E_1, E_2, E_3, E_4, E_5)}{\partial(x_{m+1, n}, t_{m+1, n},
x_{m, n+1}, t_{mn+1}, u_{mn+1})}\right| = 1.$$
A partial difference scheme with one dependent and $n$ independent variables will involve $n^2+1$ relations between the variables $(x_1, x_2, \dots x_n,
u)$, evaluated at a finite number of points.
Symmetries of Partial Difference Schemes {#subsec3.2}
----------------------------------------
As in the case of O$\Delta$S treated in Chapter \[sec2\], we shall restrict ourselves to point transformations $$\label{3.11}
\tilde x = F_{\lambda}(x, t, u) \quad \tilde t = G_{\lambda}(x, t, u), \quad
\tilde u = H_{\lambda}(x, t, u).$$ The requirement is that $\tilde u_{\lambda}(\tilde x, \tilde t)$ should be a solution, whenever it is defined and whenever $u(x, t)$ is a solution. The group action (\[3.11\]) should be defined and invertible, at least locally, in some neighbourhood of the reference point $P_{mn}$, including all points $P_{m+i, n+j}$ involved in the system (\[3.2\]).
As in the case of a single independent variable we shall consider infinitesimal transformations that allow us to use Lie algebraic techniques. Instead of transformations (\[3.11\]) we consider $$\begin{aligned}
\tilde x &=& x + \lambda\xi(x, t, u),\nonumber\\
\tilde t &=& t + \lambda\tau(x, t, u),\label{3.12}\\
\tilde u &=& u + \lambda\phi(x, t, u) \quad|\lambda| << 1.\nonumber\end{aligned}$$
Once the functions $\xi$, $\tau$ and $\phi$ are determined from the invariance requirement, then the actual transformations (\[3.11\]) are determined by integration, as in eq. (\[1.5\]), (\[1.6\]).
The transformations act on the entire space $(x, t, u)$, at least locally. This means that the same function $F$, $G$ and $H$ in eq. (\[3.11\]), or $\xi$, $\tau$ and $\phi$ in eq. (\[3.12\]) determine the transformations of all points.
We formulate the problem of determining the symmetries (\[3.12\]), and ultimately (\[3.11\]), in terms of a Lie algebra of vector fields of the form $$\label{3.13}
\widehat X = \xi(x, t, u)\partial x + \tau(x, t, u)\partial_t + \phi(x, t,
u)\partial_u,$$ where $\xi$, $\tau$ and $\phi$ are the same as in eq. (\[3.12\]). The operator (\[3.13\]) acts at one point only, namely $(x, t, u) \equiv (x_{mn},
t_{mn}, u_{mn})$. Its prolongation will act at all points figuring in the system(\[3.2\]) and we put $$\begin{aligned}
\lefteqn{{\mathop{\mathrm{pr}}\nolimits}X = \sum_{j, k}[\xi(x_{jk}, t_{jk}, u_{jk})\partial_{x_{jk}} +
\tau(x_ {jk}, t_{jk}, u_{jk})\partial_{t_{jk}}}\label{3.14}\\
&\hspace{2in}&+ \phi(x_{jk}, t_{jk}, u_{jk})\partial_{u_{jk}}],\nonumber\end{aligned}$$ where the sum is over all points figuring in eq. (\[3.2\]). To simplify notation we put $$\begin{aligned}
\xi_{jk} &\equiv& \xi(x_{jk}, t_{jk}, u_{jk}), \quad \tau_{jk} \equiv \tau
(x_{jk}, t_{jk}, u_{jk})\label{3.15}\\
\phi_{jk} &\equiv& \phi(x_{jk}, t_{jk}, u_{jk}).\nonumber\end{aligned}$$
The functions $\xi$, $\tau$, and $\phi$ figuring in eq. (\[3.13\]) and (\[3.14\]) are determined from the invariance condition $$\label{3.16}
{\mathop{\mathrm{pr}}\nolimits}\widehat X E_a\mid_{E_1=\dots=E_5=0} = 0 \quad a = 1, \dots 5.$$
It is eq. (\[3.16\]) that provides an algorithm for determining the symmetry algebra, i.e. the coefficients $\xi$, $\tau$ and $\phi$.
The procedure is the same as in the case of ordinary difference schemes, described in Chapter \[sec2\]. In the case of the system (\[3.2\]), we proceed as follows:
1. Choose 5 variables $v_a$ to eliminate from the condition (\[3.16\]) and express them in terms of the other variables, using the system (\[3.2\]) and the Jacobian condition (\[3.3\]). For instance, we can choose $$\begin{aligned}
v_1 & = & x_{m+i_2, n}, \quad v_2 = t_{m+i_2, n}\label{3.17}\\
v_3 & = & x_{m, n+j_2}, \quad v_4 = t_{m, n+j_2}, \quad v_5 = u_{m+i_2,
j+i_2}\nonumber\end{aligned}$$ and use (\[3.2\]) to express $$\begin{array}{c}
v_a = v_a(x_{m+i, n+j}, t_{m+i, n+j}, u_{m+i, n+j})\label{3.18}\\[\jot]
i_1 \leq i \leq i_2 - 1, \quad j_1 \leq j \leq j_2 - 1.\nonumber
\end{array}$$ The quanties $v_a$ must be chosen consistently. None of them can be a shifted value of another one (in the same direction). No relations between the quantities $v_a$ should follow from the system (\[3.2\]). Once eliminated from eq. (\[3.15\]), they should not reappear due to shifts. For instance, the choice (\[3.17\]) is consistent if $m + i_2$ and $n + j_2$ are the highest values of these labels that figure in eq. (\[3.2\]).
2. Once the quantities $v_a$ are eliminated from the system (\[3.16\]), using (\[3.18\]), each remaining value of $x_{i, k}$, $t_{i,
k}$ and $u_{i, k}$ is independent. Each of them can figure in the corresponding functions $\xi_{ik}$, $\tau_{ik}$, $\phi_{ik}$ (see eq. (\[3.15\])), in the functions $E_a$ directly, or via the expressions $v_a$, in the functions $\xi$, $\tau$ and $\phi$ with different labels. This provides a system of five functional equations for $\xi$, $\tau$ and $\phi$.
3. Assume that the dependence of $\xi$, $\tau$ and $\phi$ on $x$, $t$ and $u$ is analytic. Convert the obtained functional equations into differential equations by differentiating with respect to $x_{ik}$, $t_{ik}$, or $u_{ik}$. This provides an overdetermined system of differential equations that we must solve. If possible, use multiple differentiations to obtain single term differential equations that are easy to solve.
4. Substitute the solution of the differential equations back into the original functional equations and solve these. The differential equations are consequences of the functional ones and will hence have more solutions. The functional equations will provide further restrictions on the constants and arbitrary functions obtained when integrating the differential consequences.
Let us now consider examples on different lattices.
The Discrete Heat Equation {#subsec3.3}
--------------------------
### The Continuous Heat Equation {#subsubsec3.3.1}
The symmetry group of the continuous heat equation (\[3.4\]) is well known [@ref1]. Its symmetry algebra has the structure of a semidirect sum $$\label{3.19}
L = L_0 {\mathbin{\rlap{\raisebox{.3pt}{\small $+$}}
\mskip-4mu\supset}}L_1,$$ where $L_0$ is six-dimensional and $L_1$ is an infinite dimensional ideal representing the linear superposition principle (present for any linear PDE). A convenient basis for this algebra is given by the vector fields $$\begin{aligned}
P_0 &=& \partial_t, \quad D = 4t\partial_t + 2x\partial_x
+ u\partial_u\nonumber\\
K &=& 4t(t\partial_t+x\partial_x) + (x^2+2t)u\partial_u\label{3.20}\\
P_1 &=& \partial_x, B = 2t\partial_x + xu\partial_u, \quad
W = u\partial_u\nonumber\\
S &=& S(x, t)\partial_u, \quad S_t - S_{xx} = 0.\label{3.21}\end{aligned}$$ The ${\mathop{\mathrm{sl}}}(2, \mathbb R)$ subalgebra $\{P_0, D, K\}$ represents time translations, dilations and “expansions”. The Heisenberg subalgebra $\{P_1,
B, W\}$ represents space translations, Galilei boosts and the possibility of multiplying a solution $u$ by a constant. The presence of $\widehat S$ simply tells us that we can add a solution to any given solution. Thus, $\widehat S$ and $\widehat W$ reflect linearity, $\widehat P_0$ and $\widehat P_1$ the fact that the equation is autonomous (constant coefficients).
### Discrete Heat Equation on Fixed Rectangular Lattice {#subsubsec3.3.2}
Let us consider the discrete heat equation (\[3.5\]) on the four point uniform orthogonal lattice (\[3.6\]), (\[3.7\]). We apply the prolonged operator (\[3.4\]) to the equations for the lattice and obtain $$\label{3.22}
\begin{array}{c}
\xi(x_{m+1n}, t_{m+1n}, u_{m+1n}) - \xi(x_{mn}, t_{mn}, u_{mn}) = 0\\[\jot]
\xi(x_{m, n+1}, t_{m, n+1}, u_{m, n+1}) - \xi(x_{mn}, t_{mn}, u_{mn}) = 0
\end{array}$$ and similarly for $\tau(x, t, u)$. The quantities $v_i$ of eq. (\[3.17\]) can be chosen to be $$\label{3.23}
\begin{array}{c}
v_1 = x_{m+1, n} \quad v_2 = t_{m+1, n} \quad v_3 = x_{m, n+1},\\[\jot]
v_4 = t _{m, n+1}, \quad v_5 = u_{m, n+1}.
\end{array}$$ However, in (\[3.22\]) $u_{m+1, n}$ and $u_{m, n+1}$ cannot be expressed in terms of $u_{mn}$, since eq. (\[3.5\]) also involves $u_{m-1, n}$. Differentiating (\[3.22\]) with respect to e.g. $u_{mn}$. we find that $\xi$ cannot depend on $u$: $$\label{3.24}
\frac{\partial\xi(x_{mn}, t_{mn}, u_{mn})}{\partial u_{mn}} = 0.$$ Since we have $t_{n+1n} = t_{mn}$ and $x_{mn+1} = x_{mn}$ the two equations (\[3.22\]) yield $$\label{3.25}
\frac{\partial\xi_{mn}}{\partial x_{mn}} = 0, \quad
\frac{\partial\xi_{mn}}{\partial t_{mn}} = 0,$$ respectively. The same is obtained for the coefficient $\tau$, so finally we have $$\label{3.26}
\xi = \xi_0, \quad \tau = \tau_0,$$ where $\xi_0$ and $\tau_0$ are constants.
Now let us apply ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[3.5\]). We obtain $$\label{3.27}
\phi_{mn+1} - \phi_{mn} - \frac{h_2}{(h_1)^2} (\phi_{m+1n}-2\phi_{mn} +
\phi_{m-1, n}) = 0.$$ In more detail, eliminating the quantities $v_a$ in eq (\[3.23\]) we have $$\begin{aligned}
\lefteqn{\phi\left(x_{mn}, t_{mn}+h_2, u_{m, n}+\frac{h_2}{h^2_1}(u_{m+1,
n}-2u_{m, n} +u_{m-1, n})\right)}\nonumber\\
&&-\phi(x_{m, n},t_{m, n},u_{m, n})
-\frac{h_2}{h^2_1}[\phi(x_{mn}+h_1, t_{mn}, u_{m+1, n})
-2\phi(x_{m, n}, t_{m, n}, u_{m, n})\nonumber\\
&&\hspace{2in}+ \phi(x_{mn}-h_1, t_{mn}, u_{m-1, n})] = 0.\label{3.28}\end{aligned}$$ We differentiate eq. (\[3.28\]) twice, with respect to $u_{m+1, n}$ and $u_{m-1, n}$ respectively. We obtain $$\frac{\partial^2\phi_{mn+1}}{\partial u_{m, n+1}^2} = 0,$$ that is $$\label{3.30}
\phi_{mn} = A(x_{mn}, t_{m, n}) u_{mn} + B(x_{m, n}, t_{m, n}).$$ We substitute $\phi_{mn}$ of eq. (\[3.30\]) back into eq. (\[3.28\]) and set the coefficients of $u_{m+1, n}$, $u_{mn}$, $u_{m-1n}$ and 1 equal to zero separately. From the resulting determining equations we find that $A(x_{mn},
t_{mn}) = A_0$ must be constant and that $B(x, t)$ must satisfy the discrete heat equation (\[3.5\]). The result is that the symmetry algebra of the system (\[3.5\]) - (\[3.7\]) is very restricted. It is generated by $$\label{3.31}
P_0 = \partial_t, \quad P_1 = \partial_x, \quad W = u\partial_u, \quad
S = S(x, t)\partial _u$$ and reflects only the linearity of the system and the fact that it is autonomous.
The dilations, expansions and Galilei boosts, generated by $D$, $K$ and $B$ of eq. (\[3.20\]) in the continuous case are absent on the lattice (\[3.6\]) and (\[3.7\]). Other lattices will allow other symmetries.
### Discrete Heat Equation Invariant Under Dilations {#subsubsec3.3.3}
Let us now consider a five-point lattice that is also uniform and orthogonal. We put $$\begin{aligned}
\frac{u_{m, n+1}-u_{m, n}}{t_{m, n+1}-t_{m, n}} = \frac{u_{m+1n}-2u_{m,
n}+u_{m-1n}}{(x_{m+1, n}-x_{m, n})^2}\label{3.32}\\
x_{m+1n}-2x_{mn}+x_{m-1n} = 0 \quad\quad x_{mn+1}-x_{mn} = 0\label{3.33}\\
t_{m+1n}-t_{mn} = 0 \quad\quad t_{m, n+1}-2t_{m, n}+t_{m, n-1} = 0.\label{3.34}\end{aligned}$$ The variables $v_a$ that we shall substitute from eq. (\[3.32\]), (\[3.33\]) and (\[3.34\]) are $x_{m+1, n}$, $t_{m+1, n}$, $x_{m, n+1}$, $t_{m, n+1}$ and $u_{m, n+1}$. Applying ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[3.33\]) we obtain $$\begin{aligned}
&&\xi(2x_{mn}-x_{m-1, n}, t_{mn}, u_{m+1, n}) - 2\xi(x_{mn}, t_{mn}, u_{mn})
\nonumber\\
&&\hspace{1.55in} + \xi(x_{m-1, n}, t_{mn}, u_{m-1,n}) = 0\label{3.35}\\
&&\xi(x_{mn}, 2t_{m, n}-t_{mn-1}, u_{m, n+1}) - \xi(x_{mn}, t_{mn}, u_{mn}) =
0.\label{3.36}\end{aligned}$$ In eq. (\[3.36\]) $u_{mn}$ and $u_{m, n+1}$ are independent. Differentiating with respect to $u_{mn}$ we find $\partial\xi_{mn}/\partial u_{mn} = 0$ and hence $\xi$ does not depend on $u$. Differentiating (\[3.36\]) with respect to $t_{mn-1}$ we obtain $\partial\xi_{m, n+1}/\partial t_{m, n+1} = 0$. Thus, $\xi$ depends on $x$ alone. Eq. (\[3.35\]) can then be solved and we find that $\xi$ is linear in $x$. Applying ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[3.34\]) we obtain similar results for $\tau(x, t, u)$. Finally, invariance of the lattice equations (\[3.33\]) and (\[3.34\]) implies: $$\label{3.37}
\xi = ax + b, \quad \tau = ct + d.$$ Let us now apply ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[3.32\]). We obtain, after using the P$\Delta$S (\[3.32\]) - (\[3.34\]) $$\begin{aligned}
&&\lefteqn{\frac{\phi_{mn+1}-\phi_{mn}}{t_{m, n+1}-t_{m, n}} -
\frac{\phi_{m+1n}-2\phi_{mn}+\phi_{m-1n}}{(x_{m+1, n}-x_{m, n})^2}}\nonumber\\
&&\hspace{1.2in}+ (2a-c) \frac{u_{m+1, n}-2u_{mn}+u_{m-1,
n}}{(x_{m+1n}-x_{mn})^2} = 0.\label{3.38}\end{aligned}$$ Notice that $u_{m, n+1}$ (and hence $\phi_{m, n+1}$) depends on $u_{m+1, n}$ and $u_{m-1, n}$, whereas all terms in eq. (\[3.38\]) depend on at most one of these quantities. Taking the second derivative $\partial u_{m+1, n}\partial
u_{m-1, n}$ of eq. (\[3.38\]), we find $$\label{3.39}
\frac{\partial^2\phi_{m, n+1}}{\partial u_{m, n+1}} = 0, \mbox{ i.e. } \phi =
A(x, t)u + B(x, t).$$ We substitute this expression back into (\[3.38\]) and find $$\label{3.40}
A(x, t) = A_0 = {\mathop{\mathrm{const}}}$$ and see that $B(x, t)$ must satisfy the system (\[3.32\])–(\[3.34\]). Moreover, we find $c = 2a$ in eq. (\[3.37\]). Finally, the symmetry algebra has the basis $$\begin{aligned}
P_0 &=& \partial_t, \quad P_1 = \partial_x, \quad W = u\partial_u, \quad
D = x\partial_x + 2t\partial_t\label{3.41}\\
S &=& S(x, t)\partial u.\end{aligned}$$ Thus, dilational invariance is recovered, not however Galilei invariance. Other symmetries can be recovered on other lattices.
Lorentz Invariant Difference Schemes {#subsec3.4}
------------------------------------
### The Continuous Case {#subsubsec3.4.1}
Let us consider the PDE $$\label{3.42}
u_{xx} - u_{tt} = 4f(u).$$ Eq. (\[3.42\]) is invariant under the Poincaré group of $1+1$ dimensional Minkowski space for any function $f(u)$. Its Lie algebra is represented by $$\label{3.43}
P_0 = \partial_t, \quad P_1 = \partial_x, \quad L =
t\partial_x + x\partial_t.$$ For specific interactions $f(u)$ the symmetry algebra may be larger, in particular for $f = e^u$, $f = u^N$, or $f = \alpha u + \beta$.
Before presenting a discrete version of eq. (\[3.42\]), we find it convenient to pass over to light cone coordinates $$\label{3.44}
y = x + t, \quad z = x - t$$ in which eq. (\[3.42\]) is rewritten as $$\label{3.45}
u_{yz} = f(u)$$ and the Poincaré symmetry algebra (\[3.43\]) is $$\label{3.46}
P_1 = \partial_y, \quad P_2 = \partial_z, \quad L = y\partial_y -
z\partial_z.$$
### A Discrete Lorentz Invariant Scheme {#subsub3.4.2}
A particular P$\Delta$S that has eq. (\[3.45\]) as its continuous limit is $$\begin{aligned}
\frac{u_{m+1n+1}-u_{mn+1}-u_{m+1n}+u_{mn}}{(y_{m+1n}-y_{mn})(z_{mn+1}-z_{mn})}
= f(u_{mn})\label{3.47}\\
y_{m+1n} - 2y_{mn} + y_{m-1n} = 0 \quad y_{mn+1} - y_{mn} = 0\label{3.48}\\
z_{m+1n} - z_{mn} = 0 \quad z_{mn+1} - 2z_{mn} + z_{mn-1} = 0.\label{3.49}\end{aligned}$$ To find the Lie point symmetries of this difference scheme, we put $$\label{3.50}
X = \eta(y, z, u)\partial_y + \xi(y, z, u)\partial_z + \phi(y, z,
u)\partial_u.$$ We apply the prolonged vector field ${\mathop{\mathrm{pr}}\nolimits}\widehat x$ first to eq. (\[3.48\]) and (\[3.49\]), eliminate $y_{m+1n}$, $y_{m, n+1}$, $z_{m+1n}$ and $z_{m,
n+1}$, using the system (\[3.48\]), (\[3.49\]) and notice that all $u_{ik}$ that figure in the obtained equations for $\eta_{ik}$ and $\xi_{ik}$ are independent.
The result that we obtain is that $\eta$ and $\xi$ must be independent of $u$ and linear in $y$ and $z$, respectively. Finally we obtain $$\label{3.51}
\xi = \alpha y + \gamma, \quad \eta = \beta z + \delta$$ ($\alpha, \dots, \delta$ are constants). Invariance of eq. (\[3.47\]) implies that the coefficient $\phi$ in the vector field (\[3.50\]) must be linear in $u$ and moreover have the form $$\label{3.52a}
\phi = Au + B(y, z)$$ where $A$ is a constant. Taking (\[3.51\]) and (\[3.52a\]) into account and applying ${\mathop{\mathrm{pr}}\nolimits}X$ to eq. (\[3.47\]), we obtain $$\begin{aligned}
(A-\alpha-\beta)f(u_{mn}) + \frac{B_{m+1n+1}-B_{mn+1}-B_{m+1n}+B_{mn}}
{(y_{m+1, n}-y_{mn})(z_{mn+1}-z_{mn})}\nonumber\\
= (Au_{mn}+B_{mn})f'(u_{mn}).\label{3.52b}\end{aligned}$$ Differentiating eq. (\[3.52b\]) with respect to $u_{mn}$ we finally obtain the following determining equation: $$\label{3.53}
(\alpha+\beta) \frac{df}{du_{mn}} + [Au_{mn}+B(y_{mn}, z_{mn})] \frac{d^2f}
{du_{mn}^2} = 0.$$ For $f(u_{mn})$ arbitrary, we find $\beta = - \alpha$, $A = B = 0$. Thus for arbitrary $f(u)$ the scheme (\[3.47\])–(\[3.48\]) has the same symmetries as its continuous limit. The point symmetry algebra is given by eq. (\[3.46\]), i.e. it generates, translations and Lorentz transformations.
Now let us find special cases of $f(u)$ when further symmetries exist. That means that eq. (\[3.53\]) must be solved in a nontrivial manner. Let us restrict ourselves to the case when the interaction is nonlinear, i.e. $$\label{3.54}
\frac{d^2}{du_{mn}^2}f(u_{m, n}) \neq 0.$$ Then we must have $$\label{3.55}
B(y_{mn}, z_{mn}) = B = {\mathop{\mathrm{const}}}.$$ The equation to be solved for $f(u)$ is actually eq. (\[3.52b\]) which simplifies to $$\label{3.56}
(Au+B)f'(u) = (A-\alpha-\beta)f(u).$$ For $A\neq 0$ the solution of eq. (\[3.56\]) is $$\label{3.57}
f = f_0u^p$$ and the symmetry is $$\label{3.58}
D_1 = y\partial_y + z\partial_z - \frac{2}{p-1}u\partial u.$$ For $A = 0$, $B \neq 0$ we obtain $$\label{3.59}
f = f_0e^{pu}$$ and the additional symmetry is $$\label{3.60}
D_2 = y\partial_y + z\partial_z - 2\partial_u.$$
Thus, for nonlinear interactions $f(u)$, $f'' \neq 0$, the P$\Delta$S (\[3.47\])–(\[3.49\]) has exactly the same point symmetries as its continuous limit (\[3.45\]).
The linear case $$\label{3.61}
f(u) = Ru + T$$ is different. The PDE (\[3.45\]) in this case is conformally invariant. This infinite dimensional symmetry algebra is not present for the discrete case considered in this section.
Symmetries of Discrete Dynamical Systems {#sec4}
========================================
General Formalism {#subsec4.1}
-----------------
In this chapter we shall discuss differential-difference equations on a fixed one-dimensional lattice. Thus, time $t$ will be a continuous variable, $n \in
\mathbb Z$ a discrete one. We will be modeling discrete monoatomic or diatomic molecular chains with equally spaced rest positions. The individual atoms will be vibrating around their rest positions. For monoatomic chains the actual position of the $n$-th atom is described by one continuous variable $u_n(t)$. For diatomic atoms there will be two such functions, $u_n(t)$ and $v_n(t)$.
Only nearest neighbour interaction will be considered. The interaction are described by a priori unspecified functions. Our aim is to classify these functions according to their symmetries.
Three different models have been studied [@ref21; @ref22; @ref23]. They correspond to Fig. \[fig4.1\], \[fig4.2\] and \[fig4.3\], respectively.
The model illustrated on Fig. \[fig4.1\] corresponds to the equation [@ref21] $$\label{4.1}
\ddot u_n(t) - F_n\big(t, u_{n-1}(t), u_n(t), u_{n+1}(t)\big) = 0.$$ Fig. \[fig4.2\] could correspond to a very primitive model of the DNA molecule. The equations are [@ref22] $$\begin{aligned}
\ddot u_n & = & F_n\big(t, u_{n-1}(t), u_n(t), u_{n+1}(t), v_{n-1}(t), v_n(t),
v_{n+1}(t)\big) = 0\nonumber\\
\ddot v_n & = & G_n\big(t, u_{n-1}(t), u_n(t), u_{n+1}(t), v_{n-1}(t), v_n(t),
v_{n+1}(t)\big) = 0.\label{4.2}\end{aligned}$$
The model corresponding to Fig. \[fig4.3\] already took translational and Galilei invariance into account, so the equations were $$\begin{aligned}
\ddot u_n & = & F_n(\xi_n, t) + G_n(\eta_{n-1}, t)\nonumber\\
\ddot v_n & = & K_n(\xi_n, t) + P_n(\eta_{n-1}, t)\label{4.3}\\
\xi_n & = & y_n - x_n, \quad \eta_n = x_{n+1} - y_n.\nonumber\end{aligned}$$
(75,11)(0,-6) (0,2.5)[(1,0)[5]{}]{} (7.5,2.5) (10,2.5)[(1,0)[10]{}]{} (22.5,2.5) (22.5,-4)[(0,0)[$u_{n-1}$]{}]{} (25,2.5)[(1,0)[10]{}]{} (37.5,2.5) (37.5,-4)[(0,0)[$u_{n}$]{}]{} (40,2.5)[(1,0)[10]{}]{} (52.5,2.5) (52.5,-4)[(0,0)[$u_{n+1}$]{}]{} (55,2.5)[(1,0)[10]{}]{} (67.5,2.5) (70,2.5)[(1,0)[5]{}]{}
(75,30)(0,-6) (0,2.5)[(1,0)[5]{}]{} (7.5,2.5) (10,2.5)[(1,0)[10]{}]{} (22.5,2.5) (22.5,-4)[(0,0)[$v_{n-1}$]{}]{} (25,2.5)[(1,0)[10]{}]{} (37.5,2.5) (37.5,-4)[(0,0)[$v_{n}$]{}]{} (40,2.5)[(1,0)[10]{}]{} (52.5,2.5) (52.5,-4)[(0,0)[$v_{n+1}$]{}]{} (55,2.5)[(1,0)[10]{}]{} (67.5,2.5) (70,2.5)[(1,0)[5]{}]{} (0,17.5)[(1,0)[5]{}]{} (7.5,17.5) (10,17.5)[(1,0)[10]{}]{} (22.5,17.5) (22.5,23)[(0,0)[$u_{n-1}$]{}]{} (25,17.5)[(1,0)[10]{}]{} (37.5,17.5) (37.5,23)[(0,0)[$u_{n}$]{}]{} (40,17.5)[(1,0)[10]{}]{} (52.5,17.5) (52.5,23)[(0,0)[$u_{n+1}$]{}]{} (55,17.5)[(1,0)[10]{}]{} (67.5,17.5) (70,17.5)[(1,0)[5]{}]{} (7.5,5)(15,0)[5]{}[(0,1)[10]{}]{} (22.5,2.5)(15,0)[2]{}[(1,1)[13]{}]{} (37.5,2.5)(15,0)[2]{}[(-1,1)[13]{}]{}
(75,11) (0,2.5)[(1,0)[5]{}]{} (7.5,2.5) (7.5,-4)[(0,0)[$u_{n-1}$]{}]{} (10,2.5)[(1,0)[10]{}]{} (22.5,2.5) (22.5,-4)[(0,0)[$v_{n}$]{}]{} (25,2.5)[(1,0)[10]{}]{} (37.5,2.5) (37.5,-4)[(0,0)[$u_{n}$]{}]{} (40,2.5)[(1,0)[10]{}]{} (52.5,2.5) (52.5,-4)[(0,0)[$v_{n+1}$]{}]{} (55,2.5)[(1,0)[10]{}]{} (67.5,2.5) (67.5,-4)[(0,0)[$u_{n+1}$]{}]{} (70,2.5)[(1,0)[5]{}]{}
Dissipation was ignored in all three cases, so no first derivatives are present.
In these lectures we shall only treat the case (\[4.1\]). The lattice is fixed, i.e. it is given by the relation $$\label{4.4}
x_n = hn + x_0$$ with $h$ and $x_0$ given constants. With no loss of generality we can choose $h
= 1$, $x_0 = 0$, so that we have $x_n = n$.
Our aim is to find all functions $F_n$ for which eq. (\[4.1\]) allows a nontrivial group of local Lie point transformations. We shall also assume that the interaction is nonlinear and that it does indeed couple neighbouring states.
Let us sum up the conditions imposed on the model (\[4.1\]) and on the symmetry studies.
1. The lattice is fixed and regular $(x_n = n)$.
2. The interaction involves nearest neighbours only, is nonlinear and coupled, i.e. $$\label{4.5}
\frac{\partial^2F_n}{\partial u_i\partial u_k} \neq 0, \quad \frac{\partial
F_n}{\partial u_{n-1}} \neq 0, \quad \frac{\partial F_n}{\partial u_{n+1}} \neq
0.$$
3. We consider point symmetries only. Since the lattice is fixed, the transformations are generated by vector fields of the form [@ref49] $$\label{4.6}
\widehat X = \tau(t)\partial_t + \phi_n(t, u_n)\partial u_n.$$ We also assume that $\tau(t)$ is an analytical function of $t$ and $\phi_n(t,
u_n)$ is also analytic as a function of $t$ and $u_n$.
The symmetry algorithm is the usual one, namely
$$\label{4.7}
{\mathop{\mathrm{pr}}\nolimits}\widehat XE_n\mid_{E_n = 0} = 0.$$
The prolongation in eq. (\[4.7\]) involves a prolongation to $t$-derivatives $\dot u_n$ and $\ddot u_n$, and to all values of $n$ figuring in eq. (\[4.1\]), i.e. $n \pm 1$.
The terms in the prolongation that we actually need are $$\label{4.8}
{\mathop{\mathrm{pr}}\nolimits}^{(2)}X = \tau\partial_t + \sum^{n+1}_{k=n-1} \phi_k(t, u_k)\partial_{u_k} +
\phi^{tt}_n \partial_{\ddot u_n}.$$ The coefficient $\phi^{tt}_n$ is calculated using the formulas of Chapter \[sec1\] (or e.g. Ref. [@ref1]). We have $$\label{4.9}
\phi^{tt}_n = D^2_t\phi_n - (D^2_t\tau)u_n -2(D_t\tau)\ddot u_n.$$ Applying ${\mathop{\mathrm{pr}}\nolimits}^{(2)}X$ to eq. (\[4.1\]) and replacing $\ddot u$ from that equation, we get an expression involving $(\dot u_n)^3$, $(\dot u_n)^2$, $(\dot u_n)^1$ and $(\dot u_n)^0$. The coefficients of all of these terms must vanish separately. The first three of these equations do not depend on $F_n$ and can be solved easily. They imply $$\begin{aligned}
\label{4.10}
\phi_n(t, u_n) = \left(\frac 12\dot{\tau}(t)+a_n\right)u_n + \beta_n(t), \quad
\tau = \tau(t), \quad \dot a_n = 0.\end{aligned}$$ The remaining determining equation is $$\begin{aligned}
\lefteqn{\frac 12 \tau_{ttt} u_n + \ddot{\beta}_n + \left(a_n+\frac
32\dot{\tau}\right) F_n}\nonumber\\
&&\hspace{0.50in} - \tau F_{n, t} - \sum_{\alpha}\left[\left(\frac
12\dot{\tau}+a_{\alpha}\right) u_{\alpha}+\beta_{\alpha}\right] F_{n,
u_{\alpha}} = 0,\label{4.11}\end{aligned}$$ and the vector fields realizing the symmetry algebra are $$\label{4.12}
\widehat X = \tau(t)\partial_t + \left[\left(\frac 12\dot{\tau}(t)+a_n\right)
u_n+\beta_n(t)\right]\partial u_n.$$
Since we are classifying the interactions $F_n$, we must decide which functions $F_n$ will be considered to be equivalent. To do this we introduce a group of “allowed transformations”, or a “classifying group”. We define this to be a group of fiber preserving point transformations $$\label{4.13}
u_n(t) = \Omega_n(\tilde u_n(\tilde t), \tilde t, g), \quad \tilde t = \tilde
t(t, g), \quad \tilde n = n,$$ taking eq. (\[4.1\]) into an equation of the same form $$\label{4.14}
\ddot{\tilde u}_n(\tilde t) = \tilde F_n\big(\tilde t, \tilde
u_{n-1}(\tilde t), \tilde u_n(\tilde t), \tilde u_{n+1}(\tilde t)\big) = 0.$$ That is, the allowed transformations can change the function $F_n$ (as opposed to symmetry transformations), but cannot introduce first derivatives, nor other than nearest neighbour terms. These conditions narrow down the transformations (\[4.13\]) to linear ones of the form $$\begin{aligned}
u_n(t) & = & \frac{A_n}{\sqrt{\tilde t_t}}\tilde u_n(\tilde t) + B_n(t), \quad
\tilde t = \tilde t(t)\nonumber\\
A_{n, t} & = & 0, \quad \tilde t_t \neq 0, \quad A_n \neq 0, \quad \tilde n =
n\label{4.15}.\end{aligned}$$ Eq. (\[4.1\]) is transformed into $$\begin{aligned}
\lefteqn{\tilde u_{n, \tilde t\tilde t} = A^{-1}_n(\tilde t_t)^{-3/2}
\biggl\{
F_n(t, u_{n-1}, u_n, u_{n+1})}\nonumber\\
&\hspace{1in}& \quad\quad+\biggl[-\frac 34A_n(\tilde t_t)^{-5/2}(\tilde
t_{tt})^2 \nonumber\\
&\hspace{1in}& \qquad\qquad\qquad +\frac{A_n}{2}(\tilde t_t)^{-3/2}\tilde t_{ttt}\biggr]\tilde
u_n(\tilde t)- B_{n,
tt}\biggr\}.\label{4.16}\end{aligned}$$ The transformed vector field (\[4.12\]) is $$\begin{aligned}
\lefteqn{\widehat X = \tau(t) \tilde t_t(t)\partial_{\tilde t} + \left\{
\left[\frac 12\big(\tau(t)\tilde t_t\big)_{\tilde t}+a_n\right]\tilde
u_n\right\}}\nonumber\\
&\hspace{0.5in}& + \left.(\tilde t_t)^{1/2}A^{-1}_n\left[\left(\frac
12\tau_t+a_n\right)B_n + \beta_n - \tau B_{n, t}\right]\right\}\partial\tilde
u_n.\label{4.17} \end{aligned}$$ In eq. (\[4.16\]) and (\[4.17\]) $\tau(t)$, $a_n$, $\beta_n$ and $F_n$ are given, whereas $\tilde t(t)$, $A_n$ and $B_n(t)$ are ours to choose. We use these quantities to simplify the vector field $\widehat X$.
Our classification strategy is the following one. We first classify one-dimensional subalgebras. Thus, we have one vector field of the form (\[4.12\]). If $\tau(t)$ satisfies $\tau(t) \neq 0$ in some open interval, we use $\tilde t(t)$ to normalize $\tau(t) = 1$ and $B_n(t)$ to transform $\beta_n(t)$ into $\beta_n(t) = 0$. If we have $\tau(t) = 0$, $a_n \neq 0$, we use $B_n(t)$ to annul $\beta_n(t)$. The last possibility is $\tau(t)
= 0$, $a_n = 0$, $\beta_n(t) \neq 0$. Then we cannot simplify further. The same transformations will also simplify the determining equation (\[4.11\]) and we can, in each case, solve it for the interaction $F_n(t, u_{n-1}, u_n, u_{n+1})$.
Once all interactions allowing one dimensional symmetry algebras are determined, we proceed further structurally. We first find all Abelian symmetry groups and the corresponding interactions allowing them. We run through our list of one-dimensional algebras and take them in an already establlished “canonical” form. Let us call this element $X_1$ (in each case). We then find all elements $X$ of the form (\[4.12\]) that satisfy $[X_1, X] = 0$. We classify the obtained operators $X$ under the action of a subgroup of the group of allowed transformations, namely the isotropy group of $X_1$ (the group that leaves the subalgebra $X_1$ invariant). For each Abelian group we find the invariant interaction.
>From Abelian symmetry algebras we proceed to nilpotent ones, then to solvable ones and finally to nonsolvable ones. These can be semisimple, or they may have a nontrivial Levi decomposition.
All details can be found in the original article [@ref21], here we shall present the main results.
One-Dimensional Symmetry Algebras {#subsec4.2}
---------------------------------
Three classes of one-dimensional symmetry algebras exist. Together with their invariant interactions, they can be represented by $$\begin{aligned}
A_{1, 1} \qquad && X = \partial_t + a_nu_n\partial
u_n\nonumber\\
&& F_n(t, u_k) = f_n(\xi_{n-1}, \xi_n,
\xi_{n+1})e^{a_nt}\label{4.18}\\ && \xi_k = u_ke^{-a_kt}, \quad k = n-1, n,
n+1.\nonumber\\
&& \nonumber\\
A_{1, 2} \qquad &&
X = a_nu_n\partial u_n\nonumber\\
&& F_n(t, u_k) = u_nf_n(t, \xi_{n-1}, \xi_{n+1})\label{4.19}\\
&& \xi_{n\pm 1} = u^{a_n}_{n\pm 1} u^{a_{n\pm 1}}_n.\nonumber\\
&& \nonumber\\
A_{1, 3} \qquad && X = \beta_n(t)\partial u_n\nonumber\\
&& F_n(t, u_k) = \frac{\ddot{\beta}_n}{\beta_n}u_n + f_n(t, \xi_{n-1},
\xi_{n+1})\nonumber\\
&& \xi_{n\pm 1} = \beta_n(t)u_{n\pm 1} - \beta_{n\pm 1}(t) u_n.\label{4.20}\end{aligned}$$ We see that the existence of a one-dimensional Lie algebra implies that the interaction $F$ is an arbitrary function of three variables, rather than the original four. The actual form of the interaction in eq. (\[4.18\]), (\[4.19\]) and (\[4.20\]) was obtained by solving eq. (\[4.11\]), once the canonical form of vector field $X$ in eq. (\[4.18\]), (\[4.19\]), or (\[4.20\]) was taken into account.
Abelian Lie Algebras of Dimension $N \geq 2$ {#subsec4.3}
--------------------------------------------
Without proof we state several theorems.
\[thm4.1\] An Abelian symmetry algebra of eq. can have dimension $N$ satisfying $1 \leq N \leq 4$.
*Comment*: For $N = 1$ these are the algebras $A_{1, 1}$, $A_{1,
2}$ and $A_{1, 3}$ of eq. (\[4.18\]), (\[4.19\]) and (\[4.20\]).
\[thm4.2\] Five distinct classes of interactions $F_n$ exist having symmetry algebras of dimension $N = 2$. For four of them the interaction will involve an arbitrary function of two variables, for the fifth $a$ function of three variables.
The five classes can be represented by the following algebras and interactions. $$\begin{aligned}
A_{2, 1}: \quad && X_1 = \partial_t + a_{1n}u_n\partial u_n, \quad X_2 = a_
{2n}u_n\partial u_n\nonumber\\
&& F_n = u_nf_n(\xi_{n-1}, \xi_{n+1}), \quad a_{2n} \neq 0\label{4.21}\\
&&\xi_k = u^{a_{2n}}_k u^{-a_{2k}}_ne^{(a_{1n}a_{2k}-a_{1k}a_{2n})t}, \qquad k =
n \pm 1\nonumber\\
&&\nonumber\\
A_{2, 2}: \quad && X_1 = \partial_t + a_nu_n\partial u_n \quad X_2 = e^{a_nt}
\partial u_n\nonumber\\
&& F_n = a^2_nu_n + e^{a_nt}f_n(\xi_{n-1}, \xi_{n+1})\label{4.22}\\
&& \xi_k = u_ke^{-a_kt} - u_n e^{-a_nt}, \quad k = n \pm 1\nonumber\\
&& \nonumber\\
A_{2, 3}: \quad && X_1 = a_{1n}u_n\partial u_n \quad X_2 = a_{2n}u_n\partial
u_n\nonumber\\
&& F_n = u_nf_n(t, \xi)\label{4.23}\\
&& \xi = u^{\alpha_{n+1, n}}_{n-1}u^{\alpha_{n-1, n+1}}_n u^{\alpha_{n, n-1}}_
{n+1}\nonumber\\
&& \alpha_{kl} = a_{1k}a_{2l}-a_{1l}a_{2k} \neq 0\nonumber\\
&&\nonumber\\
A_{2, 4}: \quad && X_1 = \beta_{1, n}(t)\partial u_n, \quad X_2 = \beta_{2n}(t)
\partial u_n\nonumber\\
&& \quad\quad\quad \beta_{1n}\beta_{2n+1} - \beta_{1n+1}\beta_{2n} \neq
0\label{4.24}\\
&& F_n = \frac{(\beta_{1n}\ddot{\beta}_{2n}-\ddot{\beta}_{1n}\beta_{2n}) u_{n+1}
- (\beta_{1n+1}\ddot{\beta}_{2n} - \ddot{\beta}_{1n}\beta_{2n+1})}{\beta_{1n}
\beta_{2n+1} - \beta_{1n+1}\beta_{2n}}\nonumber\\
&& \quad\quad\quad + f_n(t, \xi)\nonumber\\
&& \xi = (\beta_{1n}\beta_{2n+1} - \beta_{1n+1}\beta_{2n}) u_{n-1} +
(\beta_{1n+1}\beta_{2n-1} - \beta_{1n-1}\beta_{2n+1})u_n\nonumber\\
&&\quad\quad\quad + (\beta_{1n-1}\beta_{2n} -
\beta_{1n}\beta_{2n-1})u_{n+1}\nonumber\\
&&\nonumber\\
A_{2, 5}: \quad && X_1 = \partial u_n, \quad X_2 = t\partial u_n\nonumber\\
&& F_n = f_n(t, \xi_{n-1}, \xi_{n+1}), \quad \xi_k = u_k - u_n, \quad k = n \pm
1Ë\label{4.25}\end{aligned}$$
The algebra $A_{2, 5}$ is of particular physical significance since $X_1$ and $X_2$ in eq. (\[4.25\]) correspond to translation and Galilei invariance for the considered chain. Unless we are considering a molecular chain in some external field, or unless some external geometry is imposed, the symmetry algebra $A_{2, 5}$ should always be present, possibly as a subalgebra of a larger symmetry algebra.
\[thm4.3\] Precisely four classes of three-dimensional symmetry algebras exist. Only one of them contains the $A_{2, 5}$ subalgebra and can be presented as $$\begin{aligned}
A_{3, 4} \quad && X_1 = \partial u_n, \quad X_2 = t\partial u_n,\nonumber\\
&& X_3 = \beta_n(t)\partial u_n,\quad \beta_{n+1} \neq \beta_n, \,
\ddot{\beta}_n \neq 0.\label{4.26}\end{aligned}$$ The invariant interaction is $$\begin{aligned}
F_n & = & \frac{\ddot{\beta}_n}{\beta_{n+1}-\beta_n} (u_{n+1}-u_n) + f_n(t,
\xi),\label{4.27}\\
\xi & = & (\beta_n-\beta_{n+1})u_{n-1} + (\beta_{n+1}-\beta_{n-1})u_n +
(\beta_{n-1}-\beta_n) u_{n+1}.\label{4.28}\end{aligned}$$
For $A_{3, 1}$, $A_{3, 2}$ and $A_{3, 4}$ see the original article [@ref21].
\[thm4.4\] There exist precisely two classes of interactions $F_n$ in eq. satisfying conditions , allowing four-dimensional symmetry algebras. Only one of them contains the subalgebra $A_{2, 5}$ and is represented by the following. $$\begin{aligned}
A_{4, 1} \quad && F_n = \frac{B_n(t)\gamma_n}{\gamma_n-\gamma_{n+1}}
(u_n-u_{n+1}) + f_n(t, \xi), f_{n, \xi\xi} \neq 0\nonumber\\
&& X_1 = \partial u_n, \quad X_2 = t\partial u_n, \quad X_3 = \psi_1(t) \gamma_n
\partial u_n,\label{4.29}\\
&& \quad\quad\quad X_4 = \psi_2(t)\gamma_n\partial u_n\nonumber\\
&& \gamma_{n+1} \neq \gamma_n, \quad \dot{\gamma}_n = 0, \quad \psi_1\dot{\psi}
_2 - \dot{\psi}_1\psi_2 = {\mathop{\mathrm{const}}}\neq 0\nonumber\end{aligned}$$ with $\xi$ as in eq. (\[4.28\]) and $\psi_1$, $\psi_2$ satisfying $$\ddot{\psi}_i - B(t)\psi_i = 0, \quad i = 1, 2$$
Some Results on the Structure of Lie Algebras {#subsec4.4}
---------------------------------------------
Let us recall some basic properties of finite-dimensional Lie algebras. Consider a Lie algebra $L \sim \{X_1, X_2, \dots, X_n\}$, where the elements $X_i$ form a basis. To each algebra $L$ one associates two series of subalgebras.
*The derived series* consist of the algebras $$\begin{aligned}
L^0 \equiv L, \quad L^1 \equiv DL = [L, L], \quad L^2 \equiv D^2L =
[DL, DL], \dots\nonumber\\
L^N \equiv D^NL = [D^{N-1}L, D^{N-1}L].\label{4.30}\end{aligned}$$ The algebra of commutators $DL$ is called the *derived algebra*. If we have $DL = L$, the algebra $L$ is called *perfect*. If an integer $N$ exists for which we have $D^NL = \{0\}$, the algebra $L$ is called *solvable*.
The *central series* consist of the algebras $$\label{4.31}
L_0 \equiv L, \quad L_1 = L^1 = [L, L], \quad L_2 = [L, L_1], \dots L_N = [L,
L_{N-1}], \dots$$
If there exists an integer $N$ for which we have $L_N = \{0\}$, the algebra $L$ is called *nilpotent*. Clearly, every nilpotent algebra is solvable, but the converse is not true.
Let us consider two examples
1. The Lie algebra of the Euclidean group of a plane: $e(2) \sim
\{L_3, P_1, P_2\}$. The commmutation relations are $$\label{4.32}
[L_3, P_1] = P_2, \quad [L_3, P_2] = - P_1, \quad [P_1, P_2] = 0.$$ The derived series is $$L = \{L_3, P_1, P_2\} \supset DL = \{P_1, P_2\}, \quad D^2L = \{0\}$$ and the central series is $$L \supset L_1 = \{P_1, P_2\} = L_2 = L_3 = \dots$$ Hence $e(2)$ is solvable but not nilpotent.
2. The Heisenberg algebra $H_1 \sim \{X_1, X_2, X_3\}$ where the basis can be reallized e.g. by the derivative operator, the coordinate $x$ and the identity 1: $$X_1 = \partial_x, X_2 = x, X_3 = 1.$$ We have $$\label{4.33}
[X_1, X_2] = X_3, \quad [X_1, X_3] = [X_2, X_3] = 0$$ and hence $$\begin{aligned}
DL & = & \{X_3\}, \quad D^2L = 0.\nonumber\\
L_1 & = & \{X_3\}, \quad L_2 = 0.\nonumber\end{aligned}$$ We see that the Heisenberg algebra is nilpotent (and solvable).
An Abelian Lie algebra is of course also nilpotent.
We shall need some results concerning nilpotent Lie algebras (by nilpotent we mean nilpotent non-Abelian).
1. Nilpotent Lie algebras always contain Abelian ideals.
2. All nilpotent Lie algebras contain the three-dimensional Heisenberg algebra as a subalgebra.
We shall also use some basic properties of solvable Lie algebras, where by solvable we mean solvable, non-nilpotent.
1. Every solvable Lie algebra $L$ contains a unique maximal nilpotent ideal called the nilradical $NR(L)$. The dimension of the nilradical satisfies $$\label{4.34}
\frac 12 \dim(L) \leq \dim NR(L) \leq \dim(L) - 1.$$
2. If the nilradical $NR(L)$ is Abelian, then we can choose a basis for $L$ in the form $\{X_1, \dots, X_n, Y_1, \dots, Y_m\}, \quad m \leq n$, with commutation relations $$[X_i, X_k] = 0, \quad [X_i, Y_k] = (A_k)_{ij}X_j, \quad
[Y_i, Y_k] = C^l_{ik}X_l.\label{4.35}$$ The matrices $A_k$ commute and are linearly nilindependent (i.e. no linear combination of them is a nilpotent matrix).
If a Lie algebra $L$ is not solvable, it can be simple, semisimple, or it may have a nontrivial Levi decomposition [@ref15]. A *simple* Lie algebra $L$ has no nontrivial ideals, i.e. $$\label{4.36}
I\subseteq L, \quad [I, I] \subseteq I, \quad [L, I] \subseteq I$$ implies $I \sim \{0\}$, or $I = L$.
*A semisimple* Lie algebra $L$ is a direct sum of simple Lie algebras $L_i$ $$\label{4.37}
L \sim L_1 \oplus L_2 \oplus \dots \oplus L_p, \quad [L_i, L_k] = 0.$$ If $L$ is not simple, semisimple, or solvable, then it allows a unique *Levi decomposition* into a semidirect sum $$\label{4.38}
L \sim S {\mathbin{\rlap{\raisebox{.3pt}{\small $+$}}
\mskip-4mu\supset}}R, \quad [S, S] = S, \quad [R, R] \subset R, \quad [S, R]
\subseteq R$$ where $S$ is semisimple and $R$ is solvable; $R$ is called the radical of $L$, i.e. the maximal solvable ideal.
Let us now return to the symmetry classification of discrete dynamical systems.
Nilpotent Non-Abelian Symmetry Algebras {#subsec4.5}
---------------------------------------
Since every nilpotent Lie algebra contains the three-dimensional Heisenberg algebra, we start by constructing this algebra, $H_1 \sim \{X_1, X_2, X_3\}$. The central element $X_3$ of eq. (\[4.33\]) is uniquely defined. We start from this element, take it in one of the standard forms (\[4.18\]), (\[4.19\]), or (\[4.20\]), then construct the two complementary elements $X_1$ and $X_2$. The result is that two inequivalent realizations of $H_1$, exist namely: $$\begin{aligned}
N_{3, 1}: \quad && X_1 = \partial_{u_n}, \quad X_2 = \partial_t, \quad X_3 =
t\partial_{u_n}\nonumber\\
&& F_n = f_n(\xi_{n+1}, \xi_{n-1}), \quad \xi_k = u_k - u_n, \quad k = n \pm
1\label{4.39}\\
&&\nonumber\\
N_{3, 2}: \quad && X_1 = e^{a_nt}\partial u_n, \quad X_2 = \partial_t +
a_nu_n\partial u_n\nonumber\\
&& X_3 = (t+\gamma_n)e^{a_nt}\partial u_n, \quad \dot a_n = 0, \quad
\dot{\gamma}_n = 0, \gamma_{n+1} \neq \gamma_n\nonumber\\
&& F_n = \frac{a^2_n(\gamma_{n+1}-\gamma_n)-2a_n}{\gamma_{n+1}-\gamma_n} u_n
\label{4.40}\\
&& \quad\quad\quad + \frac{2a_n}{\gamma_{n+1}-\gamma_n}
u_{n+1}e^{(a_n-a_{n+t})t}+ e^{a_nt}f_n(\xi)\nonumber\\
&& \xi = (\gamma_n-\gamma_{n+1})u_{n-1}e^{-a_{n-1}t} +
(\gamma_{n+1}-\gamma_{n-1}), u_ne^{-a_nt}\nonumber\\
&&\quad\quad\quad +(\gamma_{n-1}-\gamma_n)u_{n+1}e^{-a_{n+1}t}.\nonumber\end{aligned}$$ Notice that $N_{3, 1}$ contains the physically important subalgebra $A_{2, 5}$. whereas $N_{3, 2}$ does not.
Extending the algebras $N_{3, 1}$ and $N_{3, 2}$ by further elements, we find that $N_{3, 1}$ gives rise to two five-dimensional nilpotent symmetry algebras $N_{5, k}$ and $N_{3, 2}$ to a four-dimensional one $N_{4, 1}$.
Here we shall only give $N_{5, 1}$ and $N_{5, 2}$ which contain $N_{3, 1}$ and hence $A_{2, 5}$: $$\begin{aligned}
N_{5, k}: \quad && X_1 = \partial u_n, \quad X_2 = t\partial u_n, \quad X_3 =
\biggl(\frac{(k-1)t^2}{2}+\gamma_n\biggr)\partial u_n,\nonumber\\
&& X_4 = \biggl(\frac{(k-1)t^3}{6} + \gamma_nt\biggr)\partial u_n, \quad X_5 =
\partial_t, \quad k = 1, 2\label{4.41}\\
&& F_n = \frac{2(k-1)}{\gamma_{n+1}-\gamma_n} (u_{n+1}-u_n) + f_n(\xi)\nonumber\end{aligned}$$ with $\xi$ as in eq. (\[4.40\]).
Solvable Symmetry Algebras with Non-Abelian Nilradicals {#subsec4.6}
-------------------------------------------------------
We already know all nilpotent symmetry algebras, so we can start from the nilradical and extend it by further non-nilpotent elements. The result can be stated as a Theeorem.
\[thm4.5\] Seven classes of solvable symmetry algebras with non-Abelian nilradicals exist for eq. . Four of them have $N_{3, 1}$ as nilradical, three have $N_{5, 1}$.
For $N_{3, 1}$ we can add just one further element $Y$, namely one of the following $$\begin{aligned}
SN_{4, 1}: \quad && Y = t\partial_t + \biggl(\frac 12+a\biggr)u_n\partial u_n,
\quad a \neq - \frac 12\nonumber\\
&& F_n = (u_{n+1}-u_n) e^{(a-3/2)/(a+1/2)}f_n(\xi)\label{4.42}\\
&&\nonumber\\
SN_{4, 2}: \quad && Y = t\partial_t + (2u_n+t^2)\partial u_n\nonumber\\
&& F_n = \ln(u_{n+1}-u_n) + f_n(\xi)\label{4.43}\\
&&\nonumber\\
SN_{4, 3}: \quad && Y = u_n\partial u_n\nonumber\\
&& F_n = (u_{n+1}-u_n)f_n(\xi).\label{4.44}\end{aligned}$$ In all above cases we have $$\label{4.45}
\xi = \frac{u_{n-1}-u_n}{u_{n+1}-u_n}.$$ $$\begin{aligned}
SN_{4, 4}: \quad && Y = t\partial_t + \gamma_n\partial u_n, \quad \gamma_{n+1}
\neq \gamma_n, \quad \dot{\gamma}_n = 0\nonumber\\
&& F_n = \exp\biggl(-2\frac{u_{n+1}-u_n}{\gamma_{n+1}-\gamma_n}\biggr)
f_n(\xi)\label{4.46}\\
&& \xi = (\gamma_n-\gamma_{n+1}) u_{n-1} + (\gamma_{n+1}-\gamma_{n-1}) u_n
\nonumber\\
&& \quad \quad \quad +(\gamma_{n-1}-\gamma_n) u_{n+1}.\label{4.47}\end{aligned}$$ For $N_{5, 1}$ we can also add at most one non-nilpotent element and we obtain $$\begin{aligned}
SN_{6, 1}: \quad && Y = t\partial_t + \biggl(\frac 12+a\biggr) u_n\partial u_n
\nonumber\\
&& F_n = c_n\xi^{(a-3/2)/(a+1/2)}, \quad a \neq - \frac 12, \quad a \neq \frac
32
\label{4.48}\\
&&\nonumber\\
SN_{6, 2}: \quad && Y = t\partial_t + [2u_n+(a+b\gamma_n)t^2] \partial u_n,
\quad a^2 + b^2 \neq 0\nonumber\\
&& F_n = c_n + (a+b\gamma_n)\ln\xi\label{4.49}\\
&&\nonumber\\
SN_ {6, 3}: \quad && Y = t\partial_t + \rho_n\partial u_n, \quad \rho_n \neq A
+ B\gamma_n , \quad \dot{\rho}_n \neq 0\nonumber\\
&& F_n = c_ne^{\zeta}\label{4.50}\\
&& \xi = \frac{-2\zeta}{(\gamma_n-\gamma_{n+1})
\rho_{n-1}+(\delta_{n+1}-\gamma_{n-1})\rho_n + (\gamma_{n-1}-\gamma_n)
\rho_{n+1}}.\nonumber\end{aligned}$$ In all cases $\xi$ is as in eq. (\[4.47\]).
Solvable Symmetry Algebras with Abelian Nilradicals {#subsec4.7}
---------------------------------------------------
The results in this case are very rich. There exist 31 such symmetry algebras and their dimensions satisfy $2 \leq d \leq 5$.
For all details and a full complete list of results we refer to the original article. Here we give just one example of a five-dimensional Lie algebra with $NR(L) = A_{4, 1}$. $$\begin{aligned}
SA_{5, 1}: \quad && X_1 = \partial u_n, \quad X_2 = t\partial u_n, \quad X_3 =
e^t\gamma_n\partial u_n,\nonumber\\
&& \quad\quad\quad X_4 = e^{-t}\gamma_n\partial u_n\nonumber\\
&& Y = \partial_t + au_n\partial u_n \quad a \neq 0, \quad \gamma_n \neq
\gamma_{n+1}, \quad \dot{\gamma}_n = 0\nonumber\\
&& F_n = \frac{\gamma_n(u_{n+1}-u_n)}{\gamma_{n+1}-\gamma_n} + e^{at}f_n(\xi)
\label{4.51}\\
&& \xi = [(\gamma_n-\gamma_{n+1}) u_{n-1} + (\gamma_{n+1}-\gamma_{n-1})u_n
\nonumber\\
&& \quad\quad\quad +(\gamma_{n-1}-\gamma_n) u_{n+1}]e^{-at}.\nonumber\end{aligned}$$
Nonsolvable Symmetry Algebras {#subsec4.8}
-----------------------------
A Lie algebra that is not solvable must have simple subalgebra. The only simple algebra that can be constructed out of vector fields of the form (\[4.12\]) is ${\mathop{\mathrm{sl}}}(2, \mathbb R)$. The algebra and the corresponding invariant interaction can be represented as: $$\begin{aligned}
NS_{3, 1}: \quad && X_1 = \partial_t, \quad X_2 = t\partial_t + \frac 12
u_n\partial u_n\nonumber\\
&& X_3 = t^2\partial_t + tu_n\partial u_n\label{4.52}\\
&& F_n = \frac{1}{u^3_n} f_n(\xi_{n-1}, \xi_{n+1}), \quad \xi_k =
\frac{u_k}{u_n}.\nonumber\end{aligned}$$ This algebra can be further extended to a four, five or seven-dimensional symmetry algebra. In two cases the algebra will have an $A_{2, 5}$ subalgebra, namely
$NS_{5, 1}$: In addition to $X_1$, $X_2$, $X_3$ of (\[4.51\]) we have $$\begin{aligned}
X_4 &=& \partial u_n, \quad X_5 = t\partial u_n\nonumber\\
F_n &=& (u_{n+1}-u_n)^{-3}f_n(\xi), \quad \xi = \frac{u_{n+1}-u_n}{u_{n-1}-u_n}.
\label{4.53}\end{aligned}$$
$NS_{7, 1}$: The additional elements are $$\label{4.54}
\begin{array}{c}
X_n = \partial u_n, \quad X_5 = t\partial u_n, \quad X_6 = \gamma_n\partial
u_n, \quad X_7 = t\gamma_n\partial u_n\\
\gamma_{n+1} \neq \gamma_n, \quad \dot{\gamma}_n = 0.
\end{array}$$
The invariant interaction is $$\begin{aligned}
\lefteqn{F_n = s_n[(\gamma_n-\gamma_{n+1})u_{n-1} +
(\gamma_{n+1}-\gamma_{n-1})u_n}\nonumber\\
&\hspace{1in} & +
(\gamma_{n-1}-\gamma_n)u_{n+1}]^{-3},\quad \dot s_n = 0, \quad s_n \neq 0.\end{aligned}$$
Final Comments on the Classification {#subsec4.9}
------------------------------------
Let us first of all sum up the discrete dynamical systems of the type (\[4.1\]) with the largest symmetry algebras
We put $$\xi = (\gamma_n-\gamma_{n+1})u_{n-1} + (\gamma_{n+1}-\gamma_{n-1})u_n +
(\gamma_{n-1}-\gamma_n)u_{n+1}$$ and find this variable is involved in all cases with 7, or 6-dimensional symmetry algebras.
The algebras and interactions are given in eq. (\[4.54\]), (\[4.48\]), (\[4.49\]) and (\[4.50\]), respectively.
A natural question to ask is: Where is the Toda lattice in this classification? The Toda lattice is described by the equation $$\label{4.55}
u_{n, tt} = e^{u_{n-1}-u_n} - e^{u_n-u_{n+1}}.$$ This equation is of the form (\[4.1\]). It is integrable [@ref51] and has many interesting properties. In our classification it appears as a special case of the algebra $SN_{4, 4}$, i.e. $$\label{4.56}
\ddot u_n = \exp\biggl(-2\frac{u_{n+1}-u_n}{\gamma_{n+1}-\gamma_n}\biggr)
f_n(\xi),$$ with $$\label{4.57}
f_n(\xi) = - 1 + e^{\xi/2}, \quad \gamma_n = 2n.$$ Thus, its symmetry group is four-dimensional. We see that the Toda lattice is not particularly distinguished by its point symmetries: other interactions have larger symmetry groups. Even in the $SN_{4, 4}$ class two functions have to be specialized (see eq. (\[4.57\]) to reduce (\[4.56\]) to (\[4.55\]).
Generalized Point Symmetries of Linear and Linearizable Systems {#sec5}
===============================================================
Umbral Calculus {#subsec5.1}
---------------
In this chapter we take a different point of view than in the previous ones. Instead of purely point symmetries, we shall consider a specific class of generalized symmetries of difference equations that we shall call “generalised point symmetries”. They act simultaneously at several, or even infinitely many points of a lattice, but they reduce to point symmetries of a differential equation in the continuous limit.
The approach that we shall discuss here is at this stage applicable either to linear difference equations, or to nonlinear equations that can be linearized by a transformation of variables (not necessarily only point transformations).
The mathematical basis for this type of study is the so called “umbral calculus” reviewed in recent books and articles by G.G. Rota and his collaborators [@ref52; @ref53; @ref54]. Umbral calculus provides a unified basis for studying symmetries of linear differential and difference equations.
First of all, let us introduce several fundamental concepts.
\[def5.1\] A shift operator $T_{\delta}$ is a linear operator acting on polynomials or formal power series in the following manner $$\label{5.1}
T_{\delta}f(x) = f(x+\delta), \quad x \in \mathbb R, \quad \delta \in \mathbb R.$$
For functions of several variables we introduce shift operators in the same manner $$\begin{aligned}
\lefteqn{T_{\delta_i}f(x_1, \dots x_{i-1}, x_i, x_{i+1}\dots x_n)}\nonumber\\
&&\hspace{1in} = f(x_1, \dots, x_{i-1}, x_i+\delta_i, x_{i+1}, \dots,
x_n).\label{5.2}\end{aligned}$$
In this section we restrict the exposition to the case of one real variable $x
\in \mathbb R$. The extension to $n$ variables and other fields is obvious.
\[def5.2\] An operator $U$ is called a “delta operator” if it satisfies the following properties
1. It is shift invariant; $$\label{5.3}
T_{\delta}U = UT_{\delta}, \quad \forall\delta \in \mathbb R$$
2. $$\label{5.4}
Ux = c \neq 0, \quad c = {\mathop{\mathrm{const}}}$$
3. $$\label{5.5}
Ua = 0, \quad \forall a$$
and the kernal of $U$ consists precisely of all constant.
Important properties of delta operator are:
1. For every delta operator $U$ there exists a unique series of basic polynomials $\{p_n(x)\}$ satisfying $$\label{5.6}
p_0(x) = 1 \quad p_n(0) = 0, \quad n \geq 1, \quad Up_n(x) = np_{n-1}(x).$$
2. For every umbral operator $U$ there exists a conjugate operator $\beta$, such that $$\label{5.7}
[U, x\beta] = 1.$$ The operator $\beta$ satisfies $$\label{5.8}
\beta = (\stackrel{\prime}U)^{-1}, \quad \stackrel{\prime}U = [U, x].$$
The expression $$\label{5.9}
\stackrel{\prime}U \equiv U * x \equiv [U, x]$$ is called the “Pincherle derivative” of $U$ [@ref52; @ref53; @ref54].
For us the fundamental fact is that the pair of operators, $U$ and $x\beta$, satisfies the Heisenberg relation (\[5.7\]).
Before going further, let us give the two simplest possible examples.
\[exam1\] The (continuous) derivative $$\label{5.10}
\begin{array}{c}
U = \partial_x, \quad \beta = 1\\
P_0 = 1, \quad P_1 = x, \quad \dots, P_n = x^n, \dots
\end{array}$$
\[exam2\] The right discrete derivative $$\label{5.11}
\begin{array}{c}
U = \Delta^+ = \frac{T-1}{\delta}, \quad \beta = T^{-1}\\
P_0 = 1, \quad P_1 = x, \quad P_2 = x(x-\delta)\\
P_n = x(x-\delta) \dots \big(x-(n-1)\delta\big).
\end{array}$$ For any operator $U$ one can construct $\beta$ and the basic series will be $$\label{5.11b}
P_n = (x\beta)^n \cdot 1, \quad n \in \mathbb N.$$
Umbral Calculus and Linear Difference Equations {#subsec5.2}
-----------------------------------------------
First of all, let us consider a Lie algebra $L$ realized by vector fields $$\begin{aligned}
&& X_a = f_a(x_1, \dots, x_n)\partial x_a\label{5.12}\\
&& [X_a, X_b] = C^c_{ab}X_c.\label{5.13} \end{aligned}$$ The Heisenberg relation (\[5.7\]) allows us to realize the same abstract Lie algebra by difference operators $$X^D_a = f_a(x_1\beta_1, x_2\beta_2, \dots, x_n\beta_n)\Delta_{x_a}, \quad
[\Delta_{x_a}, x_a\beta_a] = 1, \quad a = 1, \dots n.$$ As long as the functions $f_a$ are polynomials, or at least formal power series in the variables $x_a$, the substitution $$\label{5.15}
x_a \to x_a\beta_a, \quad \partial_{x_a} \to \Delta_{x_a}$$ preserves the commutation relations (\[5.13\]).
We shall call the substitution (\[5.15\]) and more generally any substitution $$\label{5.16}
\{U, \beta\} \leftrightarrow \{\tilde U, \tilde{\beta}\}$$ an “umbral correspondence”. This correspondence will also take the set of basic polynomials related to $\{U, \beta\}$ into the set related to the pair $\{\tilde u, \tilde{\beta}\}$.
We shall use two types of delta operators. The first is simply the derivative $U = \partial_x$, for which we have $\beta = 1$. The second is a general difference operator $U = \Delta$ that has $\partial_x$ is its continuous limit. We put $$\label{5.17}
\Delta = \frac{1}{\delta}\sum^m_{k = l} a_k T^k_{\delta} \quad l, \quad
m \in \mathbb Z, \quad l < m$$ where $a_k$ and $\delta$ are real constants and $T_{\delta}$ is a shift operator as in eq. (\[5.1\]). Condition (\[5.3\]) is satisfied. Condition (\[5.5\]) implies $$\label{5.18}
\sum^m_{k = l} a_k = 0.$$ We also require that for $\delta \to 0$, we should have $\Delta \to
\partial_x$. This requires a further restriction on the coefficients $a_k$, namely $$\label{5.19}
\sum^m_{k=l} a_kk = 1.$$ Then relation (\[5.4\]) is also satisfied, with $c = 1$.
More generally, we have, for $\Delta$ as in (\[5.17\]) $$\begin{aligned}
\Delta f(x) &=& \frac{1}{\delta} \sum^m_{k=l} a_kf(x+k\delta)\nonumber\\
&=& \frac{1}{\delta} \sum^{\infty}_{q=0} \frac{f^{(q)}(x)}{q!} \delta^q \sum^m
_{k=l} a_kk^q.\nonumber\end{aligned}$$ We define $$\label{5.20}
\gamma_q = \sum^m_{k=l} a_kk^q \quad q \in \mathbb Z$$ and thus $$\Delta f(x) = \frac{df}{dx} + \sum^{\infty}_{q=2} \gamma_q \frac{f^{(q)}(x)}{q!}
\delta^{q-1}f.$$
Thus $\Delta$ goes to the derivative at least to the order $\delta$. We can also impose $$\label{5.21}
\gamma_q = 0, \quad q = 2, 3, \dots m - l.$$ Then we have $$\Delta = \frac{d}{dx} + O(\delta^{m-1}).$$
\[def5.3\] A difference operator of degree $m - l$ is a delta operator of the form $$\label{5.22}
U \equiv \Delta = \frac{1}{\delta} \sum^m_{k=l} a_k T^k_{\delta},$$ where $a_k$ and $\delta$ are constants, $T_{\delta}$ is a shift operator and we have $$\label{5.23}
\sum^m_{k=l} a_k = 0, \quad \sum^m_{k=l} a_kk = 1.$$
*Comment*: $\tilde{\Delta} = T^j\Delta$ is a difference operator of the same degree as $\Delta$.
\[thm5.1\] The operator $\beta$ conjugate to $\Delta = (1/\delta) \sum^m_{k=l} a_k
T^k_{\delta}$ is $$\label{5.24}
\beta = \biggl(\sum^m_{k=l} a_kkT^k\biggr)^{-1}.$$
$\beta = (\Delta')^{-1} = [\Delta, x]^{-1}$ $$\begin{aligned}
{}[\Delta, x]
&=& \frac{1}{\delta}\biggl(\sum^m_{k=l}a_k(x+k\delta)T^k -
x\sum^m_{k=l}a_kT^k\biggr)\nonumber\\
&=& \sum a_kkT^k\nonumber.\end{aligned}$$
*Examples*: $$\begin{aligned}
\Delta^s &=& \frac{T-T^{-1}}{2\delta}, \quad \beta = \biggl(\frac{T+T^{-1}}
{2}\biggr)^{-1}\label{5.25}\\
\Delta^3 &=& - \frac{1}{6\delta}(T^2-6T+3+2T^{-1}), \quad \beta =
\biggl(-\frac{T^2-3T-T^{-1}}{3}\biggr)^{-1}\label{5.26}\end{aligned}$$
*Comment*: $$\Delta^s = \frac{\partial}{\partial x} + O(\delta^2) \quad \Delta^3 =
\frac{\partial}{\partial x} + O(\delta^3).$$
Now let us apply the above considerations to the study of symmetries of linear difference equations.
\[def5.4\] An umbral equation of order $n$ is an operator equation of the form $$\label{5.27}
\sum^n_{k=0} \hat a_k (x\beta) \Delta^k\hat f = \hat g$$ where $a_k(x\beta)$ and $\hat g(x\beta)$ are given formal power series in $x\beta$ and $\hat f(x\beta)$ is the unknown operator function.
For $\Delta = \partial_x$, $\beta = 1$ this is a differential equation. For $\Delta$ as in (\[5.17\]) eq. (\[5.27\]) is an operator equation. Applying both sides of eq. (\[5.27\]) to 1 we get a difference equation. Its solution is $$\label{5.28}
f(x) = \hat f(x\beta) \cdot 1.$$ More generally, an umbral equation in $P$ variables is $$\begin{aligned}
\sum^{n_1\dots n_p}_{k_1, \dots k_p} \hat a_{k_1\dots k_p}(x_1\beta_1, x_2
\beta_2, \dots, x_p\beta_p)\Delta^{k_1}_{\delta_1}, \dots
\Delta^{k_p}_{\delta_p}
\hat f(x_1\beta_1, \dots, x_p\beta_p)\nonumber\\
= \hat g(x_1\beta_1\dots x_p\beta_p)\label{5.29},
\quad \sum^p_{i=1} n_i = n.\end{aligned}$$
Example of an umbral equation $$\label{5.30}
\Delta\hat f = a\hat f, \quad a \neq 0.$$
1. Take $\Delta = \partial_x \Rightarrow f(x) = e^{ax}$
2. Take $\Delta = \Delta^+ = \frac{T-1}{\delta}, \quad \beta = T^{-1}$ $$\label{5.31}
f(x+\delta) - f(x) = a\delta f(x).$$ Take $f(x) = \lambda^x$: $$\lambda^{x+\delta} - \lambda^x = a\delta\lambda^x \Rightarrow \lambda =
(1+a\delta)^{1/\delta}.$$ We get a single “umbral” solution $$\label{5.32}
f_1(x) = (1+a\delta)^{x/\delta}.$$ The umbral correspondence gives: $$\label{5.33}
f_2(x) = e^{axT^{-1}} \cdot 1.$$
If we expand into power series, we obtain $f_1(x) = f_2(x)$, and of course we have $$\lim_{\delta \to 0} f_{1, 2}(x) = e^{ax}.$$
3. For comparison, take $\Delta = \Delta^s = (T-T^{-1})/2\delta$, $\beta = [(T+T^{-1})/2]^{-1}$ $$\label{5.34}
f(x+\delta) - f(x-\delta) = 2\delta af(x).$$ Putting $f(x) = \lambda^x$ we get two values of $\lambda$ and $$\begin{aligned}
f &=& A_1(a\delta + \sqrt{a^2\delta^2+1})^{x/\delta} + A_2(a\delta-\sqrt{a^2
\delta^2+1})^{x/\delta}\nonumber\\
&\equiv& A_1f_1 + A_2f_2\label{5.35}.\end{aligned}$$ We have $$\label{5.36}
\lim_{\delta\to 0} f_1(x) = e^{ax},$$ but the limit of $f_2(x)$ does not exist. The umbral correspondence yields $$f_u(x) = \exp\biggl[ax\biggl(\frac{T+T^{-1}}{2}\biggr)^{-1}\biggr] \cdot 1.$$ Expanding into power series, we find $f_u = f_1$. The solution $f_2$ is a nonumbral one.
\[thm5.2\] Let $\Delta$ be a difference operator of order $p$. Then the linear umbral equation of order $n$ (\[5.27\]) has $np$ linearly independent solutions, $n$ of them umbral ones.
There may be convergence problems for the formal series.
Consider the exponential $$\begin{aligned}
\hat f(x) &=& e^{ax\beta}\nonumber\\
\beta &=& \biggl(\sum^m_{k=l} a_kkT^k\biggr)^{-1}\label{5.37}.\end{aligned}$$ For $m - l \geq 3$, $\beta$ will involve infinitely many shifts i.e., each term in the expansion (\[5.37\]) could involve infinitely many shifts. However $$\label{5.38}
P_n(x) = (x\beta)^n \cdot 1$$ is a well defined polynomial. For a proof see [@ref33].
Let us assume that we know the solution of an umbral equation for $\Delta =
\partial_x$ and it has the form $$\label{5.39}
f(x) = \sum^{\infty}_{n=0} \frac{f^{(n)}(0)}{n!} x^n.$$ Then for any difference operator $\Delta$ there will exist a corresponding umbral solution $$\label{5.40}
\hat f(x)1 = \sum^{\infty}_{n=0} \frac{f^{(n)}(0)}{n!} P_n(x),$$ where $P_n(x) = (x\beta)^k \cdot 1$ are the basic polynomials corresponding to $\Delta$.
Symmetries of Linear Umbral Equations {#subsec5.3}
-------------------------------------
Let us consider a linear differential equation $$\label{5.41}
Lu = 0, \quad L = \sum_{k_1, \dots, k_p} a_{k_1, \dots, k_p}(x_1, \dots, x_p)
\frac{\partial^{k_1}}{\partial x^{k_1}_1} \dots \frac{\partial^{k_p}}{\partial
x^k_p}.$$ The Lie point symmetries of eq. (\[4.41\]) can be realized by evolutionary vector fields of the form $$\begin{aligned}
\widehat X &=& Q(x_i, u, u_{, x_i})\partial_u,\nonumber\\
Q &=& \phi - \sum^p_{i=1} \xi_i u_{, x_i}\label{5.42}.\end{aligned}$$ The following theorem holds for these symmetries.
\[thm5.3\] All Lie point symmetries for an ODE of order $n \geq 3$, or a PDE of order $n
\geq 2$ are generated by evolutionary vector fields of the form with the characteristic $Q$ satisfying $$\label{5.43}
Q = Xu + \chi(x_1, \dots, x_p),$$ where $\chi$ is a solution of eq. (\[5.41\]) and $X$ is a linear operator $$\label{5.44}
X = \sum^p_{i=1} \xi_i(x_1, \dots, x_p)\partial x_i$$ satisfying $$\label{5.45}
{}[L, X] = \lambda(x_1, \dots, x_p)L,$$ i.e. commuting with $L$ on the solution set of $L$. In eq. (\[5.45\]) $\lambda$ is an arbitrary function.
For a proof we refer to the literature [@ref55].
In other words, if the conditions of Theorem \[thm5.3\] apply, then all symmetries of eq. (\[4.41\]) beyond those representing the linear superposition principle, are generated by linear operators of the form (\[5.44\]), commuting with $L$ on the solution set of eq. (\[5.41\]).
Now let us turn to the umbral equation (\[5.29\]) with $\hat g = 0$, i.e. $$\label{5.46}
\sum_{k_1\dots k_p} \hat a_{k_1\dots k_p}(x_1\beta_1, \dots x_p\beta_p)
\Delta^{k_1}_{\delta_1} \dots \Delta^{k_p}_{\delta_p} \hat u(x_1\beta_1, \dots,
x_p\beta_p) = 0.$$
We shall realize the symmetries of eq. (\[5.46\]) by evolutionary vector fields of the form $$\label{5.47}
v_E = Q_D\partial_u, \quad Q_D = \phi_D - \sum^p_{i=1}\xi_{D, i} \Delta_iu$$ where $\phi_D$ and $\xi_{D, i}$ are functions of $x_i\beta_i$ and $u$. The prolongation of $v_E$ will also act on the discrete derivatives $\Delta^{k_i}_{\delta_i}u$. We are now considering transformations on a fixed (nontransforming) lattice. In the evolutionary formalism the transformed variables satisfy $$\begin{aligned}
\tilde x_k\tilde{\beta}_k &=& x_k\beta_k, \quad \tilde{\beta}_k =
\beta_k\nonumber\\
\tilde u(\tilde x_k\tilde{\beta}_k) &=& u(x_k\beta_k) + \lambda
Q_D, \quad |\lambda| \ll 1\label{5.48}\end{aligned}$$ and we request that $\tilde{\psi}$ be a solution whenever $\psi$ is one.
The transformation of the discrete derivatives is given by $$\begin{aligned}
\Delta_{\tilde x_k}\tilde u &=& \Delta_{x_k}u + \lambda\Delta_{x_k}Q
\nonumber\\
\Delta_{\tilde x_k\tilde x_k}\tilde u &=& \Delta_{x_kx_k}u +
\lambda\Delta_{x_kx_k} Q\label{5.49}\end{aligned}$$ etc., where $\Delta_{x_k}$ are discrete total derivatives.
In terms of the vector field (\[5.57\]) we have $$\begin{aligned}
{\mathop{\mathrm{pr}}\nolimits}v_E & = & Q_D\partial_u + Q^{x_i}_D\partial_{\Delta_iu} + Q^{x_ix_k}_D
\partial_{\Delta_i\Delta_k u} + \dots\nonumber\\
Q^{x_i}_D & = & \Delta_i Q_D, \quad Q^{x_ix_k}_D =
\Delta_i\Delta_kQ_D\label{5.50}\end{aligned}$$ (we have put $\Delta_{\delta_i} \equiv \Delta_{x_i} \equiv \Delta_i$).
As in the continuous continuous case, we obtain determining equations by requiring $$\label{5.51}
{\mathop{\mathrm{pr}}\nolimits}v_E(L_D\hat u)\mid_{L_D\hat u} = 0$$ where $L_D \hat u$ is the left hand side of eq. (\[5.46\]).
The determining equations will be an umbral version of the determining equations in the continuous case, i.e. are obtained by the umbral correspondence $\partial_{x_i} \to \Delta_i$, $x_i \to x_i\beta_i$.
The symmetries of the umbral equation (\[5.46\]) will hence have the form (\[5.47\]) with $$\label{5.52}
Q_D = X_Du + \chi(x_1\beta_1, \dots x_p\beta_p)$$ where $X_D$ is a difference operator commuting with $L_D$ on the solutions of eq. (\[5.46\]). Moreover, $X_D$ is obtained from $X$ by the umbral correspondence.
We shall call such symmetries “generalized point symmetries”. Because of the presence of the operators $\beta_i$ they are not really point symmetries. In the continuous limit they become point symmetries.
Let us now consider some examples.
Example of the Discrete Heat Equation {#subsec5.4}
-------------------------------------
The (continuous) linear heat equation in $(1+1)$ dimensions is $$\label{5.53}
u_t - u_{xx} = 0.$$ Its symmetry group is of course well-known. Factoring out the infinite dimensional pseudo-group corresponding to the linear superposition principle we have a 6 dimensional symmetry group. We write its Lie algebra in evolutionary form as $$\begin{aligned}
P_o &=& u_t\partial_u, \quad P_1 = u_x\partial_u, \quad W = u\partial_u
\nonumber\\
B &=& (2tu_x+xu)\partial_u, \quad D = \biggl(2tu_t+xu_x+\frac 12 u\biggr)
\partial_u\label{5.54}\\
K &=& \biggl[t^2u_t+txu_x+\frac 14(x^2+2t)u\biggr]\partial_u\nonumber\end{aligned}$$ where $P_o$, $P_1$, $B$, $D$, $K$ and $W$ generate time and space translations, Galilei boosts, dilations “expansions” and the multiplication of $u$ by a constant, respectively.
A very natural discretization of eq. (\[5.53\]) is the discrete heat equation $$\label{5.55}
\Delta_tu - (\Delta_x)^2u = 0,$$ where $\Delta_t$ and $\Delta_x$ are each one of the difference operators considered in Section \[subsec5.2\]. We use the corresponding conjugate operators $\beta_t$ and $\beta_x$, respectively. The umbral correspondence gives us the symmetry algebra of eq. (\[5.55\]), starting from the algebra (\[5.54\]). Namely, we have $$\begin{aligned}
P^D_0 &=& (\Delta_tu)\partial_u, \quad P^D_1 = (\Delta_xu)\partial_u, \quad W^D
= u\partial_u\nonumber\\
B^D &=& [2(t\beta_t)\Delta_xu+(x\beta_x)u]\partial_u\nonumber\\
D^D &=& \biggl[2t\beta_t\Delta_tu+x\beta_x\Delta_xu+\frac 12u\biggr]\partial_u
\label{5.56}\\
K^D &=& \biggl[(t\beta_t)^2\Delta_tu+(t\beta_t)(x\beta_x)\Delta_xu+\frac
14\big((x\beta_x)^2+2t\beta_t\big)u\biggr]\partial u\nonumber.\end{aligned}$$
In particular we can choose both $\Delta_t$ and $\Delta_x$ to be right derivatives $$\label{5.57}
\Delta_t =\frac{T_t-1}{\delta_t}, \quad \beta_t = T^{-1}_t, \quad \Delta_x =
\frac{T_x-1}{\delta_x}, \quad \beta_x = T^{-1}_x,$$
The characteristic $Q_k$ of the element $K^D$ then is $$\begin{aligned}
\lefteqn{Q_K = X_Ku, \quad X_K = (t^2-\delta_tt)T^{-2}_t \Delta_t +
txT^{-1}_xT^{-1}_t\Delta_x}\nonumber\\
&\hspace{2in}& + \frac 14[(x^2-\delta x)T^{-2}_x + 2tT^{-1}_t],\label{5.58}\end{aligned}$$ so it is not a point transformation: it involves $u$ evaluated at several points. Each of the basis elements (\[5.56\]) (or any linear combination of them) provides a flow commuting with eq. (\[5.55\]): $$\label{5.59}
u_{\lambda} = Xu.$$ Equations (\[5.55\]) and (\[5.59\]) can be solved simultaneously and this will provide a difference analog of the separation of variables in PDEs and a tool for studying new types of special functions.
The Discrete Burgers Equation and its Symmetries {#subsec5.5}
------------------------------------------------
### The Continuous Case {#subsubsec4.5.1}
The Burgers equation $$\label{5.60}
u_t = u_{xx} + 2uu_x$$ is the simplest equation that combines nonlinearity and dissipative effects. It is also the prototype of an equation linearizable by a coordinate transformation $C$-linearizable in Calogero’s terminology [@ref59].
We put $u = v_x$ and obtain the potential Burgers equation for $v$: $$\label{5.61}
v_t = v_{xx} + v^2_x.$$ Putting $w = e^v$ we find $$\label{5.62}
w_t = w_{xx}.$$ In other words, the usual Burgers equation (\[5.60\]) is linearized (into the heat equation (\[5.62\]) by the Cole-Hopf transformation $$\label{5.63}
u = \frac{w_x}{w}$$ (which is not a point transformation).
One possible way of viewing the Cole-Hopf transformation is that it provides a Lax pair for the Burgers equation: $$\label{5.64}
w_t = w_{xx}, \quad w_x = uw.$$ Putting $$w_t = Aw, \quad w_x = Bw, \quad A = u_x + u^2, \quad B = u$$ we obtain the Burgers equation as a compatibility condition $$\label{5.65}
A_x - B_t + [A, B] = 0.$$
Our aim is to discretize the Burgers equation in such a way as to preserve its linearizability and also its five-dimensional Lie point symmetry algebra. We already know the symmetries of the discrete heat equation and we will use them to obtain the symmetry algebra of the discrete Burgers equation. This will be an indirect application of umbral calculus to a nonlinear equation.
### The Discrete Burgers Equation as a Compatibility Condition {#subsubsec5.5.2}
Let us write a discrete version of the pair (\[5.64\]) in the form: $$\label{5.66}
\Delta_t\phi = \Delta_{xx}\phi, \quad \Delta_x\phi = u\phi$$ where we take $$\label{5.67}
\Delta_t = \frac{T_t-1}{\delta_t}, \quad \Delta_x = \frac{T_x-1}{\delta_x}.$$ The pair (\[5.66\]) can be rewritten as $$\label{5.68}
\Delta_t\phi = (\Delta_xu+uT_xu)\phi, \quad \Delta_x\phi = u\phi.$$
We have used the Leibnitz rule appropriate for the discrete derivative $\Delta_x$ of (\[5.67\]), namely $$\label{5.69}
\Delta_xfg = f(x)\Delta_xg + (T_xg)\Delta_xf.$$ Compatibility of eq. (\[5.68\]), i.e. $\Delta_x\Delta_t\phi =
\Delta_t\Delta_x\phi$ yields the discrete Burgers equation $$\label{5.70}
\Delta_tu = \frac{1+\delta_xu}{1+\delta_t[\Delta_x\Delta_xu+uT_xu]}
\Delta_x(\Delta_xu+uT_xu).$$ In the continuous limit $\Delta_t \to \partial/\partial t$, $\Delta_x \to
\partial/\partial x$, $T_x \to 1, \delta_x = 0$, $\delta_t = 0$ we reobtain the Burgers equation (\[5.60\]) [@ref56; @ref32]. This is not a “naive” discretization like $$\label{5.71}
\Delta_tu = (\Delta_x)^2 u + 2u \Delta_xu$$ which would loose all integrability properties.
### Symmetries of the discrete Burgers Equation {#subsubsec5.5.3}
We are looking for “generalized point symmetries” on a fixed lattice. We write them in evolutionary form $$\label{5.72}
X_e = Q(x, t, T^a_xT^b_tu, T^c_xT^d_t\Delta_xu, T^e_xT^f_t\Delta_tu, \dots)
\partial_u$$ and each symmetry will provide a commuting flow $$u_{\lambda} = Q.$$ We shall use the Cole-Hopf transformation to transform the symmetry algebra of the discrete heat equation into that of the discrete Burgers equation.
All the symmetries of the discrete heat equation given in eq. (\[5.56\]) can be written as $$\label{5.73}
\phi_{\lambda} = S\phi, \quad S = S(x, t, \phi, T_x, T_x \dots)$$ where $S$ is a linear operator (the same is true for any linear difference equation).
For the discrete heat equation $$\label{5.74}
\Delta_t\phi - (\Delta_x)^2\phi = 0$$ with $\Delta_t$ and $\Delta_x$ as in eq. (\[5.67\]) we rewrite the flows corresponding to eq. (\[5.56\]) as $$\begin{aligned}
\phi_{\lambda_1} &=& \Delta_t\phi, \quad \phi_{\lambda_2} = \Delta_x\phi, \quad
\phi_{\lambda_3} = \biggl[2tT^{-1}_t\Delta_x+xT^{-1}_x+\frac 12\delta_xT^{-1}_x
\biggr]\phi\nonumber\\
\phi_{\lambda_4} &=& \biggl[2tT^{-1}_t\Delta_t+xT^{-1}_x\Delta_x+\frac
12\biggr]\phi\nonumber\\
\phi_{\lambda_5} &=& \biggl[t^2T^{-2}_t\Delta_t+txT^{-1}_tT^{-1}_x\Delta_x+\frac
14x^2T^{-2}_x\nonumber\\
&& \hspace{0.75in} + t\biggl(T^{-2}_t-\frac 12T^{-1}_tT^{-1}_x\biggr)-\frac{1}
{16}\delta^2_xT^{-2}_x\biggr]\phi\label{5.75}\\
\phi_{\lambda_6} &=& \phi.\nonumber\end{aligned}$$
Let us first prove a general result.
\[thm5.4\] Let eq. represent a symmetry of the discrete heat equation . Then the same linear operator $S$ provides a symmetry of the discrete Burgers equation , the flow of which is given by $$\label{5.76}
u_{\lambda} = (1+\delta_xu)\Delta_x\biggl(\frac{S\phi}{\phi}\biggr),$$ where $(S\phi)/\phi$ can be expressed in terms of $u(x, t)$.
We request that eq. (\[5.73\]) and the Cole-Hopf transformation in eq. (\[5.66\]) be compatible $$\label{5.77}
\frac{\partial}{\partial\lambda}(\Delta_x\phi) = \Delta_x\phi_{\lambda}.$$ >From here we obtain $$\label{5.78}
u_{\lambda} = \frac{\Delta_x(S\phi)-uS\phi}{\phi}.$$ A direct calculation yields $$\label{5.79}
\Delta_x\biggl(\frac{S\phi}{\phi}\biggr) = \frac{1}{T_x\phi}[\Delta_x(S\phi)-u
(S\phi)]$$ and (\[5.76\]) follows. It is still necessary to show that $S\phi/\phi$ depends only on $u(x, t)$. The expressions for $S\phi$ can be read off from eq. (\[5.75\]). From there we see that all expressions involved can be expressed in terms of $u(x, t)$ and its shifted values (using the Cole-Hopf transformation again). Indeed, we have $$\begin{aligned}
\Delta_x\phi &=& u\phi, \quad \Delta_t\phi = v\phi\nonumber\\
T_x\phi &=& (1+\delta_xu)\phi, \quad T_t\phi = (1+\delta_tv)\phi\label{5.80}\\
T^{-1}_x\phi &=& \biggl(T^{-1}_x\frac{1}{1+\delta_xu}\biggr)\phi, \quad T^{-1}
_t\phi = \biggl(T^{-1}_t\frac{1}{1+\delta_tv}\biggr)\phi,\nonumber\end{aligned}$$ where we define $$\label{5.81}
v = \Delta_xu + uT_xu.$$ Explicitly, eq. (\[5.76\]) maps the 6 dimensional symmetry algebra of the discrete heat equation into the 5 dimensional Lie algebra of the discrete Burgers equation. The corresponding flows are $$\begin{aligned}
u_{\lambda_1} &=& (1+\delta_tv)\Delta_tu\nonumber\\
u_{\lambda_2} &=& (1+\delta_xu)\Delta_xu\nonumber\\
u_{\lambda_3} &=& (1+\delta_xu)\Delta_x\biggl[2tT^{-1}_t\frac{u}{1+\delta_tv}
+ \biggl(x+\frac 12 - \delta_x\biggr)T^{-1}_x\frac{1}{1+\delta_xu}\biggr]
\nonumber\\
u_{\lambda_4} &=& (1+\delta_xu)\Delta_x\biggl[2tT^{-1}_t\frac{v}{1+\delta_tv}
+ xT^{-1}_x \frac{u}{1+\delta_xu}
- \frac 12 T^{-1}_x\frac{1}{1+\delta_xu}\biggr]\nonumber\\
u_{\lambda_5} &=& (1+\delta_xu)\Delta_x\biggl[t^2T^{-1}_t\biggl(
\frac{1}{1+\delta_tv} T^{-1}_t\frac{v}{1+\delta_tv}\biggr)\label{5.82}\\
&&\hspace{1.5in} + txT^{-1}\biggl(\frac{1}{1+\delta_xu}T^{-1}_t
\frac{u}{1+\delta_tv}\biggr)\nonumber\\
&&\hspace{1.5in} + \frac 14\biggl(x^2-\frac{\delta^2_x}{4}\biggr)T^{-1}_x
\biggl(\frac{1}{1+\delta_xu}T^{-1}_x\frac{1}{1+\delta_xu}\biggr)\nonumber\\
&&\hspace{1.5in} + tT^{-1}_t
\biggl(\frac{1}{1+\delta_tv}T^{-1}_t\frac{1}{1+\delta_tv}\biggr)\nonumber\\
&&\hspace{1.5in} -\frac 12
tT^{-1}_x\biggl(\frac{1}{1+\delta_xu}T^{-1}_t\frac{1}{1+\delta_tv}\biggr)
\biggr]\nonumber\\
u_{\lambda_6} &=& 0.\nonumber\end{aligned}$$
### Symmetry Reduction for the Discrete Burgers Equation {#subsubsec5.5.4}
Symmetry reduction for continuous Burger equation: we add a compatible equation to the Burgers equation $$\begin{aligned}
u_t &=& u_{xx} + 2uu_x\nonumber\\
u_{\lambda} &=& Q(x, t, u, u_{x, t}) = 0\label{5.83}\end{aligned}$$ and solve the two equations simultaneously. Example: time translations. $$\label{5.84}
u_{\lambda} = u_t = 0.$$ Then $u = u(x)$ and $$\label{5.85}
u_{xx} + 2uu_x = 0 \Rightarrow u_x + u^2 = K.$$ >From here, we obtain three types of solutions $$\label{5.86}
u = \frac 1x, \quad u = k {\mathop{\mathrm{arctanh}}\nolimits}kx, \quad u = k \arctan kx.$$ Symmetry reduction in the discrete case. All flows have the form (\[5.76\]). The condition $u_{\lambda} = 0$ hence implies $$\label{5.87}
S\phi = K(t)\phi,$$ where $K(t)$ is an arbitrary function. This equation must be solved together with the discrete Burgers equation in order to obtain group invariant solutions.
Let us here consider just one example, namely that of time translations, the first equation in (\[5.82\]). Eq. (\[5.87\]) reduces to $$\label{5.88}
\Delta_t\phi = K(t)\phi,$$ i.e. $$\label{5.89}
v = \Delta_xu + uT_xu = K(t).$$ We rewrite the Burgers equation as $$\label{5.90}
\Delta_tu = \frac{1+\delta_xu}{1+\delta_tv} \Delta_xv \quad v \equiv \Delta_xu
+ uT_xu.$$ However, from (\[5.89\]) we have $v = K(t)$ and hence $\Delta_tu = 0$, $K(t)
= K_o = {\mathop{\mathrm{const}}}$. Since $\phi$ satisfies the heat equation, we can rewrite (\[5.88\]) as $$\label{5.91}
\Delta_{xx}\phi = K_o\phi.$$ The general solution of (\[5.91\]) is obtained for $K_o \neq 0$ by putting $\phi = a^x$ and solving (\[5.91\]) for $a$. We find $$\label{5.92}
\phi = c_1(1+\sqrt{K_0}\delta_x)^{x/\delta_x} +
c_2(1-\sqrt{K_0}\delta_x)^{x/\delta_x}.$$ For $K_o = 0$ we have $$\label{5.93}
\phi = c_1 + c_2x.$$ Solutions of the discrete Burgers equation are obtained via the Cole-Hopf transformation $$\label{5.94}
u = \frac{\Delta_x\phi}{\phi}.$$ The same procedure can be followed for all other symmetries. We obtain linear second order difference equations for $\phi$ involving one variable only. However, the equations have variable coefficients and are hard to solve. They can be reexpressed as equations for $u(x, t)$, again involving only one independent variable. Thus, a reduction takes place, but it is not easy to solve the reduced equations explicitly.
For instance, Galilei invariant solutions of the discrete Burgers equation must satisfy the ordinary difference equation $$\label{5.95}
\begin{array}{c}
2tT_xu + x - K(t) + 2t\delta_xuT_xu + \delta_t\biggl(\frac 72T_xu + \frac 72
\delta_xuT_xu\\
+ xuT_xu + x\Delta_xu - \frac 32u\biggr) + \frac 32\delta_x - K(t)[\delta_xu +
\delta_t(T_x\Delta_xu\\
+ uT^2_xu -uT_xu) + T_xuT^2_xu + \delta_xuT_xuT^2_xu] = 0
\end{array}$$ where $t$ figures as a parameter.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank the organizers of the CIMPA School for giving me the opportunity to present these lectures. Special thanks go to the Tamizhmani family for making my visit to Pondicherry both pleasant and memorable.
The research reported upon in these lectures was partly supported by research grants from NSERC of Canada, FQRNT du Québec and the NATO collaborative grant n. PST.CLG.978431.
[50]{}
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| |
Q:
Proof of $L^1(\mathbb{R}) \ast f \neq L^1(\mathbb{R})$
It is known that $L^1(\mathbb{R}) \ast f$ is dense in $L^1(\mathbb{R})$ for some $f\in L^1(\mathbb{R})$.
So for such $f$ the closure of $L^1(\mathbb{R}) \ast f$ in the $L^1$ norm is $L^1(\mathbb{R})$.
But apparently
(1)$\quad\quad L^1(\mathbb{R}) \ast f \neq L^1(\mathbb{R})$ for every
$f\in L^1(\mathbb{R})$.
Is there a simple proof of (1)?
A:
I'm guessing $*$ means convolution (since this is a math forum) and not pointwise multiplication (since this is not a computer forum).
Some steps to try ... suppose $L^1 * f = L^1$
$\widehat{f} \ne 0$ a.e.
There is $g \in L^1$ so that $g * f = f$
$ \widehat{g} \widehat{f} = \widehat{f}$
$\widehat{g} = 1$ identically
$\widehat{g} \notin C_0$
$g \notin L^1$
| |
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS
FIELD OF THE INVENTION
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
This invention claims priority to German Utility Model Application 201 19 000.1, filed on Nov. 21, 2001, which is incorporated herein by reference in its entirety.
1
The present invention relates to a cylindrical covering cap for eyepiece tubes, having a conical soft eyecup, the inner aperture radius R of the covering cap being adapted to the larger outside radius RK of the cone of the eyecup.
Covering caps of this type are generally known. They usually consist of plastic and are pushed onto the eyepiece tube. Given a suitable pressure, they are clamped onto the cone of the eyecup (with the aid of the coating, which is soft like rubber) so firmly as to prevent it from falling off inadvertently. However, clamping frequently occurs that is so firm that the covering cap may be detached only by a forceful tug. In this process, the soft coating of the eyecup can also be taken off. In an attempt to detach the covering cap by rotation, rotatable eyecups for eyepieces for spectacle wearers are partially rotated out of their end click stops and detached from the eyepiece tube. It is also disadvantageous that on being gripped during detachment the covering cap is initially pressed even more firmly onto the eyecup.
An object of the present invention is to solve these and other problems by specifying a covering cap which, on the one hand, is securely seated on the eyepiece tube simply by being pushed on and, on the other hand, can be detached without damaging the eyecup.
2
1
1
The present invention provides for a cylindrical covering cap for an eyepiece tube, the eyepiece tube having a conical soft eyecup, which comprises, in a cylindrical interior of the covering cap: two mutually opposite segment regions having a segment region inside radius R that is greater than an inner aperture radius R of the covering cap; and at least two mutually opposite segment surfaces, wherein a shortest distance A from a segment surface to a cylinder axis passing through a center of the covering cap is smaller than the inner aperture radius of the covering cap, and wherein the inner aperture radius R of the covering cap is configured to adapt to a larger outside radius RK of the conical eyecup. The segment surfaces may be circumferentially spaced so that the locations of the segment surfaces are approximately symmetric about a segment region axis that passes through a center of each of the segment regions. The locations of the segment surfaces may be approximately symmetric about a perpendicular axis that is perpendicular to the segment region axis and intersects the segment region axis at the cylinder axis. The shortest distance A for each segment surface may be approximately equal for all segment surfaces.
In one aspect of the present invention, the covering cap may comprise four segment surfaces.
In another aspect of the present invention, the segment surfaces may be substantially flat or have a substantially cylindrical curvature.
In another aspect of the present invention, where the covering cap has a conical cylindrical shape, the segment surfaces may be conically inclined to a cap closure of the covering cap with respect to the cylinder axis.
2
1
2
1
1
1
In another aspect of the present invention, a difference (R−R) between the segment region inside radius R and the inner aperture radius R may be approximately equal to a wall thickness D of the covering cap. Further, a difference (R−A) between the inner aperture radius R and said shortest distance A is approximately equal to the wall thickness D of the covering cap. The wall thickness of the covering cap may be approximately 1 mm.
In another aspect of the present invention, the segment regions may be marked on an outer surface of the covering cap by a ribbed surface structure configured to be gripped.
In another aspect of the present invention, the wall thickness D of the covering cap and a material composition of the covering cap may be coordinated with one another such that the covering cap can be deformed elastically.
In another aspect of the present invention, a circumferential angle between two adjacent segment surfaces may be smaller than a circumferential angle between a segment surface and an adjacent segment region.
In another aspect of the present invention, a circumferential segment region angle, which is the largest continuous angle that includes one segment region but does not include a segment surface, may be greater than a circumferential segment surface angle, which is the largest angle including only adjacent segment surfaces and the angle between them.
The present invention also provides for a cylindrical covering cap for an eyepiece tube, the eyepiece tube having a conical soft eyecup, comprising, in a cylindrical interior of the covering cap: two mutually opposite segment regions having a segment region inside radius that is greater than an inner aperture radius of the covering cap; and at least two mutually opposite segment surfaces that are circumferentially spaced so that the locations of the segment surfaces are approximately symmetric about a segment region axis that passes through a center of each of the segment regions, wherein a shortest distance from a segment surface to a cylinder axis passing through a center of the covering cap is smaller than the inner aperture radius of the covering cap, wherein the inner aperture radius of the covering cap is configured to adapt to a larger outside radius of the conical eyecup, wherein the locations of the segment surfaces are approximately symmetric about a perpendicular axis that is perpendicular to the segment region axis and intersects the segment region axis at the cylinder axis, wherein a circumferential angle between two adjacent segment surfaces is smaller than a circumferential angle between a segment surface and an adjacent segment region, and wherein a circumferential segment region angle, which is the largest continuous angle that includes one segment region but does not include a segment surface, is greater than a circumferential segment surface angle, which is the largest angle including only adjacent segment surfaces and the angle between them.
BRIEF DESCRIPTION OF THE DRAWINGS
An exemplary embodiment of the covering cap according to the present invention is illustrated schematically in the drawings.
FIG. 1
shows a bottom view and the opening of the covering cap with flat segment surfaces;
FIG. 2
shows a side view of the covering cap with an eyecup;
FIG. 3
shows a bottom view and the opening of the covering cap with segment surfaces having a cylindrical curvature.
FIG. 4
2
shows a cross sectional view of the covering cap as shown along section Z—Z in FIG. .
FIG. 5
3
shows the angles &agr;, &bgr;, &Ggr;, and &dgr; of the embodiment as shown in FIG. .
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
1
1
1
2
3
2
1
5
2
3
1
5
1
2
3
4
4
1
4
9
4
10
9
9
5
4
5
1
1
4
FIG. 1
FIG. 2
The covering cap illustrated in is conically cylindrical in longitudinal section (as shown in ) with an inner aperture radius R and a wall thickness D. Formed in the cylindrical interior of the covering cap are two mutually opposite segment regions , having a segment region inside radius R which is larger than inner aperture radius R. (“Mutually opposite,” as used herein, means “located approximately symmetrically about the cylinder axis .” For example, two segment regions , that are center-mirrored, or mirrored about the center of the covering cap , would be mutually opposite, because they would be located approximately symmetrically about the cylinder axis , which passes through the center of the covering cap .) The part of the cylindrical interior that is situated between these segment regions , has two pairs of flat segment surfaces , each pair of which is mutually opposite. The segment surfaces are circumferentially spaced about the cylindrical interior of the covering cap so that the locations of the segment surfaces are approximately symmetric about the segment region axis (that passes through the centers of the segment regions ) and about the perpendicular axis (that is perpendicular to the segment region axis and intersects the segment region axis at the cylinder axis ). The shortest distance A from these segment surfaces to the cylinder axis may be smaller than the inner aperture radius R. The wall thickness D of the covering cap may therefore be greater in the locations of the segment surfaces than in the remaining locations.
FIG. 4
FIG. 4
FIG. 1
FIG. 4
2
8
4
2
3
4
shows a cross sectional view along section Z—Z of FIG. . is very similar to , differing primarily in that the depth of the covering cap, and its corresponding cap closure , are not shown in FIG. . is intended to clarify the shape and dimensional relationships among segment regions , and the segment surfaces .
FIG. 3
4
1
4
4
5
1
4
4
8
5
Referring now to , instead of the flat segment surfaces formed as chordal surfaces in the cylindrical interior of the covering cap , it is also possible to provide cylindrical segment surfaces ′ (having an approximately cylindrical curvature), having an inside radius A. In this embodiment inside radius A is also the shortest distance A from these segment surfaces ′ to the cylinder axis . In the case of a conical configuration of the covering cap , the segment surfaces , ′ may be conically inclined toward the cap closure with reference to the cylinder axis .
1
6
4
4
6
4
4
1
1
1
6
FIG. 2
In mounting the covering cap on a conical eyecup (as shown in ) of an eyepiece tube, the segment surfaces , ′ are situated on the soft coating of the eyecup . When pushed on further, the segment surfaces , ′ are pushed radially outward, the covering cap being deformed in cross section. The elastic stress produced in the process inside the covering cap ensures a firm seat of the covering cap on the eyecup , without damaging the latter.
1
2
3
7
2
3
1
6
2
3
1
4
4
6
FIG. 2
In order to remove the covering cap , a radial inwardly directed pressure may be exerted on the segment regions , from the outside. As shown in , owing to a longitudinal fluting or ribbed surface structure , segment regions , may be felt from outside as well as fashioned suitably for gripping in order to lift the covering cap from the eyecup . The pressure on the segment regions , likewise produces an elastic deformation of the covering cap by means of which the segment surfaces , ′ may be detached from the eyecup .
FIG. 5
FIG. 5
4
4
1
2
3
4
4
4
4
2
3
5
4
4
2
3
4
4
5
4
4
2
3
2
3
7
1
6
4
4
2
3
1
6
Referring now to , in one embodiment of the present invention, the segment surfaces , ′ may be situated closer to one another on the circumference of the cylindrical interior of the covering cap than in the direction of the segment regions , . In other words, a circumferential angle a between two adjacent segment surfaces , ′ may be smaller than a circumferential angle &dgr; between a segment surface , ′ and an adjacent segment region , . Further, a circumferential segment surface angle y may be smaller than a circumferential segment region angle &bgr;, as shown in FIG. . The circumferential segment region angle &bgr;, also known as a free angle (because it is free of a protruding segment surface , ′), is the largest continuous angle that includes one segment region , but does not include a segment surface , ′, as shown in FIG. . The circumferential segment surface angle &ggr; is the largest angle including only adjacent segment surfaces , ′ and the angles between them—i.e., as shown in , the circumferential segment surface angle &ggr; does not include any segment regions , . The relationships among angles &agr;, &bgr;, &ggr;, and &dgr; help to ensure that the necessary amount of radial, inwardly-directed pressure that must be applied to segment regions , (via ribbed surface structures ), in order to deform the covering cap sufficiently to remove it from the eyecup , is minimized. In short, the farther the segment surfaces , ′ are from the segment regions , , and the closer they are to each other, the easier it is to deform the covering cap sufficiently to remove it from the eyecup .
2
1
2
3
6
6
2
3
2
3
1
The distance (R−R) of the segment regions , from the eyecup ensures that the eyecup is not touched by the segment regions , when the segment regions , are compressed. The covering cap can easily be removed in this way.
4
4
1
1
4
4
2
3
2
1
2
3
1
The stability of the pressure surfaces (e.g., segment surfaces , ′) and the elasticity of the covering cap in all remaining parts are important for the proper operation of the covering cap . The elasticity is a function of the ratio of the thicknesses of the wall, the segment regions, and the segment surfaces. In a preferred embodiment of the present invention, the flat segment surfaces (in their middle or, otherwise, thickest points) and the cylindrical segment surfaces ′ may have a thickness approximately double the wall thickness D. Further, the segment regions , may have a thickness approximately equal to the wall thickness D. Further still, the difference (R−R), corresponding to the free space for pressing in the segment regions , , may also be approximately equal to the wall thickness D. The actual wall thickness D may depend on the material from which the covering cap is produced. A wall thickness D of approximately 1 mm has proved to be favorable for production as an injection-molded plastic part.
6
1
6
1
6
1
6
6
1
4
4
4
4
1
4
4
4
4
1
6
4
4
1
6
1
6
FIG. 2
The operation of the present invention according to a preferred embodiment will now be described. To cover an eyecup of an eyepiece tube, as shown in , a covering cap may be pushed onto the eyecup , with the opening of the covering cap facing the eyecup . As the covering cap is being pushed onto the eyecup , the outer portion of the eyecup , which may have a diameter RK that is slightly larger than the shortest distance A (shown in FIG. ), exerts a radial pressure on the segment surfaces , ′. As pressure is exerted on segment surfaces , ′, the covering cap , made of an appropriate material and having appropriate dimensions, elastically deforms, allowing the segment surfaces , ′ to be pushed radially outward. In this deformed state, the segment surfaces , ′ of the covering cap continue to exert a pressure or normal force against the eyecup , generating a friction between the segment surfaces , ′ of the covering cap and the eyecup that holds the covering cap in place on the eyecup .
1
7
2
3
1
2
3
6
4
4
6
4
4
6
1
6
6
To remove the covering cap , a radial pressure may be inwardly exerted by a human user or operator onto the ribbed surface structures (each of which corresponds to and is located on an exterior side of one of the segment regions , ). In doing so, the covering cap elastically deforms so that the segment regions , move radially inwardly (toward the eyecup ) while the segment surfaces , ′ are forced radially outwardly (away from the eyecup ), thus reducing or relieving the pressure/normal force between the segment surfaces , ′ and the eyecup . At this point, the covering cap may be lifted off the eyecup without risk of damaging the eyecup .
The foregoing description of a preferred embodiment of the invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the invention. The embodiment was chosen and described to explain the principles of the invention and as a practical application to enable one skilled in the art to utilize the invention in various embodiments and with various modifications suited to the particular use contemplated. It is intended that the scope of the invention be defined by the claims appended hereto and their equivalents. | |
Effort and Skills
In CrossFit we are constantly faced with the unknown and the unknowable. Although something that we are almost certain of is the CrossFit Open, and some of the movements and workout styles we will probably be faced with, only five short months from now.
No skill can be perfected overnight or in one training session, but the more time we commit to that skill, the easier and more proficient we will become at it. My reasoning has me believe that starting to practice our skills now and commit minutes and hours of practice in October, November, December, and January will only help us when our “test” comes out at the end of February.
A quote that I really enjoy from the most recently crowned Fittest Man on Earth, Mat Fraser, seems fitting to the point I’m trying to get across.
“When I was in school, I’d literally sit in the library for 10 hours on a Saturday. I’d read the textbook cover to cover, when I’d finish a chapter if I couldn’t tell you every single law that was in there or every single formula I’d go back and read it again. It’s not because I was addicted to reading the textbook, it’s because I was addicted to the day we got our exam, and the exam wasn’t challenging… That’s the feeling I chase.”
I’m not saying spend an 10 hours a day, but take it one chapter at a time, if you will. Spend 5-10 minutes before and or after your workout to get that extra skill workout in. It might be double unders, strict or kipping HSPU or muscle ups, toe to bar, etc. If they’ve been tested before either at Regionals or in the Open, don’t be shocked if we see them in 2017. You never know, your hard work might land you a spot on the Vic City Regional Team or a qualifying score during the Open.
We might not know exactly what the questions on the test will be, but we can prepare ourselves with the answers.
This is our second time taking on this interval workout. Each successive interval starting on a 90 second timer. You will complete 5 intervals in total. The focus is on accuracy and smoothness of movement, record your weight if you complete all the intervals in the designated time frame.
| |
This week has been so nice soaking in being a family of four. We’ve had ourselves some lazy and relaxing days filled with movies, dance parties, arts & crafts, and the occasional doctor’s appointment to attend. Today also marks one full week with sweet baby Brinlee! One week of pure joy. One week of love. One week for big brother Mason.
We are so proud of Mason and how well he is adjusting to new baby sister. He gives her hugs, checks on her and continues to be his happy self. I can tell he’s getting some cabin fever because every chance he gets he grabs Mark’s or my hand and walks us to the back door because he wants to go swimming so bad. Mark takes him out often, but we’re mostly staying low key in order for me to recover. However, after my doctor’s appointment today, it was brought to my attention that I have not been relaxing as much as I should.
I feel better each day and enjoy helping Mark take care of the kiddos…but in reality my feet are swelling, my blood pressure is up and I haven’t taken a nap in a few days. Since I’ve been feeling so well I feel like I can handle more than my body can take. So from here on out I’m going to try and glue myself to the bed or couch.
So, yesterday we dug into Mason’s sibling bag with a bunch of different goodies Mark and I picked out. I came across the sibling bag idea from Pinterest and we had so much fun choosing what to get Mason. We got him three books, a coloring book, crayons, markers with sketch book, a light up bouncy ball, a tin box, as well as a NOW Disney music CD. We initially presented Mason with his gift in the hospital when he was meeting Brinlee for the first time. I was so excited to open it this week and really get in some fun doodling time.
He LOVES the markers, but they were making a pretty decent mess so luckily we had crayons for back up. He loves everything we got him and is mesmerized when he colors. All these goodies have come in handy during these rainy days and have given us some special time with Mason, aka…Big Bro! | https://mommiemission.com/2016/09/01/big-brother-gift-bag/ |
Background of the Invention
Summary of the Invention
Brief Description of the Drawings
Detailed Description of the Preferred Embodiment
The present invention generally relates to digital speech coding at low bit rates, and more particularly, is directed to an improved method for coding the excitation information for code-excited linear predictive speech coders.
Code-excited linear prediction (CELP) is a speech coding technique which has the potential of producing high quality synthesized speech at low bit rates, i.e., 4.8 to 9.6 kilobits-per-second (kbps). This class of speech coding, also known as vector-excited linear prediction or stochastic coding, will most likely be used in numerous speech communications and speech synthesis applications. CELP may prove to be particularly applicable to digital speech encryption and digital radiotelephone communication systems wherein speech quality, data rate, size, and cost are significant issues.
In a CELP speech coder, the long term ("pitch") and short term ("formant") predictors which model the characteristics of the input speech signal are incorporated in a set of time-varying linear filters. An excitation signal for the filters is chosen from a codebook of stored innovation sequences, or code vectors. For each frame of speech, the speech coder applies each individual code vector to the filters to generate a reconstructed speech signal, and compares the original input speech signal to the reconstructed signal to create an error signal. The error signal is then weighted by passing it through a weighting filter having a response based on human auditory perception. The optimum excitation signal is determined by selecting the code vector which produces the weighted error signal with the minimum energy for the current frame.
Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
The term "code-excited" or "vector-excited" is derived from the fact that the excitation sequence for the speech coder is vector quantized, i.e., a single codeword is used to represent a sequence, or vector, of excitation samples. In this way, data rates of less than one bit per sample are possible for coding the excitation sequence. The stored excitation code vectors generally consist of independent random white Gaussian sequences. One code vector from the codebook is used to represent each block of N excitation samples. Each stored code vector is represented by a codeword, i.e., the address of the code vector memory location. It is this codeword that is subsequently sent over a communications channel to the speech synthesizer to reconstruct the speech frame at the receiver. See M.R. Schroeder and B.S. Atal, "Code-Excited Linear Prediction (CELP): High-Quality Speech at Very Low Bit Rates", , Vol. 3, pp. 937-40, March 1985, for a detailed explanation of CELP.
The difficulty of the CELP speech coding technique lies in the extremely high computational complexity of performing an exhaustive search of all the excitation code vectors in the codebook. For example, at a sampling rate of 8 kilohertz (kHz), a 5 millisecond (msec) frame of speech would consist of 40 samples. If the excitation information were coded at a rate of 0.25 bits per sample (corresponding to 2 kbps), then 10 bits of information are used to code each frame. Hence, the random codebook would then contain 2¹⁰, or 1024, random code vectors. The vector search procedure requires approximately 15 multiply-accumulate (MAC) computations (assuming a third order long-term predictor and a tenth order short-term predictor) for each of the 40 samples in each code vector. This corresponds to 600 MACs per code vector per 5 msec speech frame, or approximately 120,000,000 MACs per second (600 MACs/5 msec frame x 1024 code vectors). One can now appreciate the extraordinary computational effort required to search the entire codebook of 1024 vectors for the best fit --an unreasonable task for real-time implementation with today's digital signal processing technology.
Moreover, the memory allocation requirement to store the codebook of independent random vectors is also exorbitant. For the above example, a 640 kilobit read-only-memory (ROM) would be required to store all 1024 code vectors, each having 40 samples, each sample represented by a 16-bit word. This ROM size requirement is inconsistent with the size and cost goals of many speech coding applications. Hence, prior art code-excited linear prediction is presently not a practical approach to speech coding.
Proc. ICASSP
One alternative for reducing the computational complexity of this code vector search process is to implement the search calculations in a transform domain. Refer to I.M. Trancoso and B.S. Atal, "Efficient Procedures for Finding the Optimum Innovation in Stochastic Coders", , Vol. 4, pp. 2375-8, April 1986, as an example of such a procedure. Using this approach, discrete Fourier transforms (DFT's) or other transforms may be used to express the filter response in the transform domain such that the filter computations are reduced to a single MAC operation per sample per code vector. However, an additional 2 MACs per sample per code vector are also required to evaluate the code vector, thus resulting in a substantial number of multiply-accumulate operations, i.e., 120 per code vector per 5 msec frame, or 24,000,000 MACs per second in the above example. Still further, the transform approach requires at least twice the amount of memory, since the transform of each code vector must also be stored. In the above example, a 1.3 Megabit ROM would be required for implementing CELP using transforms.
Proc. ICASSP
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A second approach for reducing the computational complexity is to structure the excitation codebook such that the code vectors are no longer independent of each other. In this manner, the filtered version of a code vector can be computed from the filtered version of the previous code vector, again using only a single filter computation MAC per sample. This approach results in approximately the same computational requirements as transform techniques, i.e., 24,000,000 MACs per second, while significantly reducing the amount of ROM required (16 kilobits in the above example). Examples of these types of codebooks are given in the article entitled "Speech Coding Using Efficient Pseudo-Stochastic Block Codes", , Vol. 3, pp. 1354-7, April 1987, by D. Lin. Nevertheless, 24,000,000 MACs per second is presently beyond the computational capability of a single DSP. Moreover, the ROM size is based on 2 x # bits/word, where M is the number of bits in the codeword such that the codebook contains 2 code vectors. Therefore, the memory requirements still increase exponentially with the number of bits used to encode the frame of excitation information. For example, the ROM requirements increase to 64 kilobits when using 12 bit codewords.
A review of the concepts employed in vector quantization is found in "Vector Quantization in Speech Coding", Proceedings of the IEEE, Vol. 73, No. 11, November 1985, J. Makhoul et al.
A need, therefore, exists to provide an improved speech coding technique that addresses both the problems of extremely high computational complexity for exhaustive codebook searching, as well as the vast memory requirements for storing the excitation code vectors.
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According to a first aspect of the invention, there is provided a method for generating a codebook of excitation vectors for use in high speed speech analysis or synthesis, said codebook having at least 2 excitation vectors u(n), each having N elements, where 1 ≦ n ≦ N, and where 0 ≦ i ≦ 2 - 1, the method characterised by the steps of: storing a set of M basis vectors v(n) each having N elements, where 1 ≦ n ≦ N and where 1 ≦ m ≦ M; identifying a set of 2 digital codewords I, each having M bits, where 0 ≦ i ≦ 2 - 1; identifying a signal ϑ for each bit of each codeword I, such that ϑ has a first value if bit m of codeword I is of a first state, and such that ϑ has a second value if bit m of codeword I is of a second state; and calculating said codebook of 2 excitation vectors according to the equation:
where 1 ≦ n ≦ N.
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In a preferred embodiment, the method for generating a codebook further comprises the steps of: inputting at least two selector codewords; defining a plurality of interim data signals (ϑ) based upon said selector codewords; inputting a set of X basis vectors (v(n)), where X < Y; and generating said codebook vectors by performing linear transformations on said X basis vectors (v(n)), said linear transformations defined by said interim data signals (ϑ).
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In the preferred embodiment, each of said selector codewords can be represented in bits, wherein said interim data signals (ϑ) are based upon the value of each bit of each selector codeword, and said codebook vector generating step is further characterized by the steps of: multiplying said set of X basis vectors (v(n)) by said plurality of interim data signals (ϑ) to produce a plurality of interim vectors; and summing said plurality of interim vectors to produce said codebook vectors.
The set of X basis vectors is typically stored in memory in place of storing the entire codebook.
A digital memory may contain a codebook of excitation vectors, which excitation vectors have been generated by the aforedescribed method.
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In accordance with a second aspect of the invention, there is provided a codebook vector generating means for generating a codebook of excitation vectors for use in high speed speech analysis or synthesis, said codebook having at least 2 excitation vectors u(n), each having N elements, where 1 ≦ n ≦ N, and where 0 ≦ i ≦ 2 - 1, characterised by: digital memory means for storing a set of M basis vectors vm(n) each having N elements, where 1 ≦ n ≦ N and where 1 ≦ m ≦ M; means for identifying a set of 2 digital codewords I, each having M bits, where 0 ≦ i ≦ 2 - 1; means for identifying a signal ϑ for each bit of each codeword I, such that ϑ has a first value if bit m of codeword I is of a first state, and such that ϑ has a second value if bit m of codeword I is of a second state; and means for calculating said codebook of 2 excitation vectors according to the equation:
where 1 ≦ n ≦ N.
A digital radio communication device may comprise a code-excited linear predictive speech coder having codebook vector generating means for generating a codebook of excitation vectors generated as aforedescribed.
Further reductions in computational complexity may be achieved by sequencing from one codeword to the next codeword by changing only one bit of the codeword at a time in accordance with a predetermined sequencing technique, such that the calculations for the next codeword are reduced to updating parameters from the previous codeword based upon the predetermined sequencing technique.
According to the various aspects of the invention, an improved digital speech coding technique is provided that produces high quality speech at low bit rates. The invention provides an efficient excitation vector generating technique having reduced memory requirements. The reduced computation complexity allows implementation in real time utilizing present day digital signal processing technology.
The "vector sum" codebook generating approach of the present invention permits faster implementation of CELP speech coding while retaining the advantages of high quality speech at low bit rates. More specifically, the present invention provides an effective solution to the problems of computational complexity and memory requirements. For example, the vector sum approach disclosed herein requires only M + 3 MACs for each codeword evaluation. In terms of the previous example, this corresponds to only 13 MACs, as opposed to 600 MACs for standard CELP or 120 MACs using the transform approach. This improvement translates into a reduction in complexity of approximately 10 times, resulting in approximately 2,600,000 MACs per second. This reduction in computational complexity makes possible practical real-time implementation of CELP using a single DSP.
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Furthermore, only M basis vectors need to be stored in memory, as opposed to all 2 code vectors. Hence, the ROM requirements for the above example are reduced from 640 kilobits to 6.4 kilobits for the present invention. Still another advantage to the present speech coding technique is that it is more robust to channel bit errors than standard CELP. Using the vector sum excited speech coder of the present invention, a single bit error in the received codeword will result in an excitation vector similar to the desired one. Under the same conditions, standard CELP, using a random codebook, would yield an arbitrary excitation vector -- entirely unrelated to the desired one. An exemplary embodiment of the invention will now be described with reference to the accompanying drawings.
Figure 1 is a general block diagram of a code-excited linear predictive speech coder utilizing the vector sum excitation signal generation technique in accordance with the present invention;
Figure 2A/2B is a simplified flowchart diagram illustrating the general sequence of operations performed by the speech coder of Figure 1;
Figure 3 is a detailed block diagram of the codebook generator block of Figure 1, illustrating the vector sum technique of the present invention;
Figure 4 is a general block diagram of a speech synthesizer using the present invention;
Figure 5 is a partial block diagram of the speech coder of Figure 1, illustrating the improved search technique according to the preferred embodiment of the present invention;
Figure 6A/6B is a detailed flowchart diagram illustrating the sequence of operations performed by the speech coder of Figure 5, implementing the gain calculation technique of the preferred embodiment; and
Figure 7A/7B/7C is a detailed flowchart diagram illustrating the sequence of operations performed by an alternate embodiment of Figure 5, using a pre-computed gain technique.
The features of the present invention which are believed to be novel are set forth with particularity in the appended claims. The invention, together with further objects and advantages thereof, may best be understood by reference to the following description taken in conjunction with the accompanying drawings, in the several figures of which like-referenced numerals identify like elements, and in which:
Referring now to Figure 1, there is shown a general block diagram of code excited linear predictive speech coder 100 utilizing the excitation signal generation technique according to the present invention. An acoustic input signal to be analyzed is applied to speech coder 100 at microphone 102. The input signal, typically a speech signal, is then applied to filter 104. Filter 104 generally will exhibit bandpass filter characteristics. However, if the speech bandwidth is already adequate, filter 104 may comprise a direct wire connection.
The analog speech signal from filter 104 is then converted into a sequence of N pulse samples, and the amplitude of each pulse sample is then represented by a digital code in analog-to-digital (A/D) converter 108, as known in the art. The sampling rate is determined by sample clock SC, which represents an 8.0 kHz rate in the preferred embodiment. The sample clock SC is generated along with the frame clock FC via clock 112.
IEEE Trans. Commun.
The digital output of A/D 108, which may be represented as input speech vector s(n), is then applied to coefficient analyzer 110. This input speech vector s(n) is repetitively obtained in separate frames, i.e., blocks of time, the length of which is determined by the frame clock FC. In the preferred embodiment, input speech vector s(n), 1 ≦ n ≦ N, represents a 5 msec frame containing N=40 samples, wherein each sample is represented by 12 to 16 bits of a digital code. For each block of speech, a set of linear predictive coding (LPC) parameters are produced in accordance with prior art techniques by coefficient analyzer 110. The short term predictor parameters STP, long term predictor parameters LTP, weighting filter parameters WFP, and excitation gain factor γ, (along with the best excitation codeword I as described later) are applied to multiplexer 150 and sent over the channel for use by the speech synthesizer. Refer to the article entitled "Predictive Coding of Speech at Low Bit Rates," , Vol. COM-30, pp. 600-14, April 1982, by B.S. Atal, for representative methods of generating these parameters. The input speech vector s(n) is also applied to subtractor 130, the function of which will subsequently be described.
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Basis vector storage block 114 contains a set of M basis vectors v(n), wherein 1 ≦ m ≦ M, each comprised of N samples, wherein 1 ≦ n ≦ N. These basis vectors are used by codebook generator 120 to generate a set of 2 pseudo-random excitation vectors u(n), wherein 0 ≦ i ≦ 2-1. Each of the M basis vectors are comprised of a series of random white Gaussian samples, although other types of basis vectors may be used with the present invention.
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Codebook generator 120 utilizes the M basis vectors vm(n) and a set of 2 excitation codewords I, where 0 ≦ i ≦ 2-1, to generate the 2 excitation vectors u(n). In the present embodiment, each codeword I is equal to its index i, that is, I=i. If the excitation signal were coded at a rate of 0.25 bits per sample for each of the 40 samples (such that M=10), then there would be 10 basis vectors used to generate the 1024 excitation vectors. These excitation vectors are generated in accordance with the vector sum excitation technique, which will subsequently be described in accordance with Figures 2 and 3.
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For each individual excitation vector u(n), a reconstructed speech vector s′(n) is generated for comparison to the input speech vector s(n). Gain block 122 scales the excitation vector u(n) by the excitation gain factor γ, which is constant for the frame. The excitation gain factor γ may be pre-computed by coefficient analyzer 110 and used to analyze all excitation vectors as shown in Figure 1, or may be optimized jointly with the search for the best excitation codeword I and generated by codebook search controller 140. This optimized gain technique will subsequently be described in accordance with Figure 5.
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The scaled excitation signal γu(n) is then filtered by long term predictor filter 124 and short term predictor filter 126 to generate the reconstructed speech vector s′(n). Filter 124 utilizes the long term predictor parameters LTP to introduce voice periodicity, and filter 126 utilizes the short term predictor parameters STP to introduce the spectral envelope. Note that blocks 124 and 126 are actually recursive filters which contain the long term predictor and short term predictor in their respective feedback paths. Refer to the previously mentioned article for representative transfer functions of these time-varying recursive filters.
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The reconstructed speech vector s′(n) for the i-th excitation code vector is compared to the same block of the input speech vector s(n) by subtracting these two signals in subtractor 130. The difference vector e(n) represents the difference between the original and the reconstructed blocks of speech. The difference vector is perceptually weighted by weighting filter 132, utilizing the weighting filter parameters WTP generated by coefficient analyzer 110. Refer to the preceding reference for a representative weighting filter transfer function. Perceptual weighting accentuates those frequencies where the error is perceptually more important to the human ear, and attenuates other frequencies.
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Energy calculator 134 computes the energy of the weighted difference vector e′(n), and applies this error signal E to codebook search controller 140. The search controller compares the i-th error signal for the present excitation vector u(n) against previous error signals to determine the excitation vector producing the minimum error. The code of the i-th excitation vector having a minimum error is then output over the channel as the best excitation code I. In the alternative, search controller 140 may determine a particular codeword which provides an error signal having some predetermined criteria, such as meeting a predefined error threshold.
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The operation of speech coder 100 will now be described in accordance with the flowchart of Figure 2. Starting at step 200, a frame of N samples of input speech vector s(n) are obtained in step 202 and applied to subtractor 130. In the preferred embodiment, N=40 samples. In step 204, coefficient analyzer 110 computes the long term predictor parameters LTP, short term predictor parameters STP, weighting filter parameters WTP, and excitation gain factor γ. The filter states FS of long term predictor filter 124, short term predictor filter 126, and weighting filter 132, are then saved in step 206 for later use. Step 208 initializes variables i, representing the excitation codeword index, and E, representing the best error signal, as shown.
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Continuing with step 210, the filter states for the long and short term predictors and the weighting filter are restored to those filter states saved in step 206. This restoration ensures that the previous filter history is the same for comparing each excitation vector. In step 212, the index i is then tested to see whether or not all excitation vectors have been compared. If i is less than 2, then the operation continues for the next code vector. In step 214, the basis vectors v(n) are used to compute the excitation vector u(n) via the vector sum technique.
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Figure 3, illustrating a representative hardware configuration for codebook generator 120, will now be used to describe the vector sum technique. Generator block 320 corresponds to codebook generator 120 of Figure 1, while memory 314 corresponds to basis vector storage 114. Memory block 314 stores all of the M basis vectors v₁(n) through v(n), wherein 1 ≦ m ≦ M, and wherein 1 ≦ n ≦ N. All M basis vectors are applied to multipliers 361 through 364 of generator 320.
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The i-th excitation codeword is also applied to generator 320. This excitation information is then converted into a plurality of interim data signals ϑ through ϑ, wherein 1 ≦ m ≦ M, by converter 360. In the preferred embodiment, the interim data signals are based on the value of the individual bits of the selector codeword i, such that each interim data signal ϑ represents the sign corresponding to the m-th bit of the i-th excitation codeword. For example, if bit one of excitation codeword i is 0, then ϑ would be -1. Similarly, if the second bit of excitation codeword i is 1, then ϑ would be +1. It is contemplated, however, that the interim data signals may alternatively be any other transformation from i to ϑ, e.g., as determined by a ROM look-up table. Also note that the number of bits in the codeword do not have to be the same as the number of basis vectors. For example, codeword i could have 2M bits where each pair of bits defines 4 values for each ϑ, i.e., 0, 1, 2, 3, or +1, -1 , +2, -2, etc.
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The interim data signals are also applied to multipliers 361 through 364. The multipliers are used to multiply the set of basis vectors v(n) by the set of interim data signals ϑ to produce a set of interim vectors which are then summed together in summation network 365 to produce the single excitation code vector u(n). Hence, the vector sum technique is described by the equation:
where u(n) is the n-th sample of the i-th excitation code vector, and where 1 ≦ n ≦ N.
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Continuing with step 216 of Figure 2A, the excitation vector u(n) is then multiplied by the excitation gain factor γ via gain block 122. This scaled excitation vector γu(n) is then filtered in step 218 by the long term and short term predictor filters to compute the reconstructed speech vector s′(n). The difference vector e(n) is then calculated in step 220 by subtractor 130 such that:
for all N samples, i.e., 1 ≦ n ≦ N.
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In step 222, weighting filter 132 is used to perceptually weight the difference vector e(n) to obtain the weighted difference vector e′(n). Energy calculator 134 then computes the energy E of the weighted difference vector in step 224 according to the equation:
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Step 226 compares the i-th error signal to the previous best error signal E to determine the minimum error. If the present index i corresponds to the minimum error signal so far, then the best error signal E is updated to the value of the i-th error signal in step 228, and, accordingly, the best codeword I is set equal to i in step 230. The codeword index i is then incremented in step 240, and control returns to step 210 to test the next code vector.
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When all 2 code vectors have been tested, control proceeds from step 212 to step 232 to output the best codeword I. The process is not complete until the actual filter states are updated using the best codeword I. Accordingly, step 234 computes the excitation vector u(n) using the vector sum technique as was done in step 216, only this time utilizing the best codeword I. The excitation vector is then scaled by the gain factor γ in 236, and filtered to compute reconstructed speech vector s′(n) in step 238. The difference signal e(n) is then computed in step 242, and weighted in step 244 so as to update the weighting filter state. Control is then returned to step 202.
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Referring now to Figure 4, a speech synthesizer block diagram is illustrated also using the vector sum generation technique according to the present invention. Synthesizer 400 obtains the short term predictor parameters STP, long term predictor parameters LTP, excitation gain factor γ, and the codeword I received from the channel, via de-multiplexer 450. The codeword I is applied to codebook generator 420 along with the set of basis vectors v(n) from basis vector storage 414 to generate the excitation vector u(n) as described in Figure 3. The single excitation vector u(n) is then multiplied by the gain factor γ in block 422, filtered by long term predictor filter 424 and short term predictor filter 426 to obtain reconstructed speech vector s′(n). This vector, which represents a frame of reconstructed speech, is then applied to analog-to-digital (A/D) convertor 408 to produce a reconstructed analog signal, which is then low pass filtered to reduce aliasing by filter 404, and applied to an output transducer such as speaker 402. Clock 412 generates the sample clock and the frame clock for synthesizer 400.
Referring now to Figure 5, a partial block diagram of an alternate embodiment of the speech coder of Figure 1 is shown so as to illustrate the preferred embodiment of the invention. Note that there are two important differences from speech coder 100 of Figure 1. First, codebook search controller 540 computes the gain factor γ itself in conjunction with the optimal codeword selection. Accordingly, both the excitation codeword I search and the excitation gain factor γ generation will be described in the corresponding flowchart of Figure 6. Secondly, note that a further alternate embodiment would be to use predetermined gains calculated by coefficient analyzer 510. The flowchart of Figure 7 describes such an embodiment. Figure 7 may be used to describe the block diagram of Figure 5 if the additional gain block 542 and gain factor output of coefficient analyzer 510 are inserted, as shown in dotted lines.
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Before proceeding with the detailed description of the operation of speech coder 500, it may prove helpful to provide an explanation of the basic search approach taken by the present invention. In the standard CELP speech coder, the difference vector from equation {2}:
was weighted to yield e′(n), which was then used to calculate the error signal according to the equation:
which was minimized in order to determine the desired codeword I. All 2 excitation vectors had to be evaluated to try and find the best match to s(n). This was the basis of the exhaustive search strategy.
{4} p(n) = y(n) - d(n).
In the preferred embodiment, it is necessary to take into account the decaying response of the filters. This is done by initializing the filters with filter states existing at the start of the frame, and letting the filters decay with no external input. The output of the filters with no input is called the zero input response. Furthermore, the weighting filter function can be moved from its conventional location at the output of the subtractor to both input paths of the subtractor. Hence, if d(n) is the zero input response vector of the filters, and if y(n) is the weighted input speech vector, then the difference vector p(n) is:
Thus, the initial filter states are totally compensated for by subtracting off the zero input response of the filters.
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The weighted difference vector e′(n) becomes:
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However, since the gain factor γ is to be optimized at the same time as searching for the optimum codeword, the filtered excitation vector f(n) must be multiplied by each codeword's gain factor γ to replace s′(n) in equation {5}, such that it becomes:
The filtered excitation vector f(n) is the filtered version of u(n) with the gain factor γ set to one, and with the filter states initialized to zero. In other words, f(n) is the zero state response of the filters excited by code vector u(n). The zero state response is used since the filter state information was already compensated for by the zero input response vector d(n) in equation {4}.
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Using the value for e′(n) from equation {6} in equation {3} gives:
Expanding equation {7} produces:
Defining the cross-correlation between f(n) and p(n) as:
and defining the energy in the filtered code vector f(n) as:
permits simplifying equation {8} as:
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We now want to determine the optimal gain factor γ which will minimize E in equation {11}. Taking the partial derivative of E with respect to γ and setting it equal to zero permits solving for the optimal gain factor γ. This procedure yields:
which, when substituted into equation {11} gives:
It can now be seen that to minimize the error E in equation {13}, the term [C]²/G must be maximized. The technique of codebook searching which maximizes [C]²/G will be described in the flowchart of Figure 6.
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If the gain factor γ is pre-calculated by coefficient analyzer 510, then equation {7} can be rewritten as:
where y′(n) is the zero state response of the filters to excitation vector u(n) multiplied by the predetermined gain factor γ. If the second and third terms of equation {14} are re-defined as:
and:
respectively, then equation {14} can be reduced to:
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In order to minimize E in equation {17} for all codewords, the term [-2C + G] must be minimized. This is the codebook searching technique which will be described in the flowchart of Figure 7.
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Recalling that the present invention utilizes the concept of basis vectors to generate u(n), the vector sum equation:
can be used for the substitution of u as will be shown later. The essence of this substitution is that the basis vectors v(n) can be utilized once each frame to directly pre-compute all of the terms required for the search calculations. This permits the present invention to evaluate each of the 2 codewords by performing a series of multiply-accumulate operations that is linear in M. In the preferred embodiment, only M + 3 MACs are required.
Figure 5, using optimized gains, will now be described in terms of its operation, which is illustrated in the flowchart of Figure 6A and 6B. Beginning at start 600, one frame of N input speech samples s(n) is obtained in step 602 from the analog-to-digital converter, as was done in Figure 1. Next, the input speech vector s(n) is applied to coefficient analyzer 510, and is used to compute the short term predictor parameters STP, long term predictor parameters LTP, and weighting filter parameters WFP in step 604. Note that coefficient analyzer 510 does not compute a predetermined gain factor in this embodiment, as illustrated by the dotted arrow. The input speech vector s(n) is also applied to initial weighting filter 512 so as to weight the input speech frame to generate weighted input speech vector y(n) in step 606. As mentioned above, the weighting filters perform the same function as weighting filter 132 of Figure 1, except that they can be moved from the conventional location at the output of subtractor 130 to both inputs of the subtractor. Note that vector y(n) actually represents a set of N weighted speech vectors, wherein 1 ≦ n ≦ N and wherein N is the number of samples in the speech frame.
In step 608, the filter states FS are transferred from the first long term predictor filter 524 to second long term predictor filter 525, from first short term predictor filter 526 to second short term predictor filter 527, and from first weighting filter 528 to second weighting filter 529. These filter states are used in step 610 to compute the zero input response d(n) of the filters. The vector d(n) represents the decaying filter state at the beginning of each frame of speech. The zero input response vector d(n) is calculated by applying a zero input to the second filter string 525, 527, 529, each having the respective filter states of their associated filters 524, 526, 528, of the first filter string. Note that in a typical implementation, the function of the long term predictor filters, short term predictor filters, and weighting filters can be combined to reduce complexity.
{4} p(n) = y(n) - d(n).
In step 612, the difference vector p(n) is calculated in subtractor 530. Difference vector p(n) represents the difference between the weighted input speech vector y(n) and the zero input response vector d(n), previously described by equation {4}:
The difference vector p(n) is then applied to the first cross-correlator 533 to be used in the codebook searching process.
i
i
m
m
m
m
M
M
In terms of achieving the goal of maximizing [C]²/G as stated above, this term must be evaluated for each of the 2 codebook vectors -- not the M basis vectors. However, this parameter can be calculated for each codeword based on parameters associated with the M basis vectors rather than the 2 code vectors. Hence, the zero state response vector q(n) must be computed for each basis vector v(n) in step 614. Each basis vector v(n) from basis vector storage block 514 is applied directly to third long term predictor filter 544 (without passing through gain block 542 in this embodiment). Each basis vector is then filtered by filter series #3, comprising long term predictor filter 544, short term predictor filter 546, and weighting filter 548. Zero state response vector q(n), produced at the output of filter series #3, is applied to first cross-correlator 533 as well as second cross-correlator 535.
m
m
m
mj
mj
mj
In step 616, the first cross-correlator computes cross-correlation array R according to the equation:
Array R represents the cross-correlation between the m-th filtered basis vector q(n) and p(n). Similarly, the second cross-correlator computes cross-correlation matrix D in step 618 according to the equation:
where 1 ≦ m ≦ j ≦ M. Matrix D represents the cross-correlation between pairs of individual filtered basis vectors. Note that D is a symmetric matrix. Therefore, approximately half of the terms need only be evaluated as shown by the limits of the subscripts.
i
i
i
m
m
The vector sum equation from above:
can be used to derive f(n) as follows:
where f(n) is the zero state response of the filters to excitation vector u(n), and where q(n) is the zero state response of the filters to basis vector v(n). Equation {9}:
can be rewritten using equation {20} as:
Using equation {18}, this can be simplified to:
Om
i
For the first codeword, where i=0, all bits are zero. Therefore, ϑ for 1 ≦ m ≦ M equals -1 as previously discussed. The first correlation C₀, which is just C from equation {22} where i=0, then becomes:
which is computed in step 620 of the flowchart.
m
i
i
i
Using q(n) and equation {20}, the energy term G may also be rewritten from equation {10}:
into the following:
which may be expanded to be:
Substituting by using equation {19} yields:
By noting that a codeword and its complement, i.e., wherein all the codeword bits are inverted, both have the same value of [C]²/G, both code vectors can be evaluated at the same time. The codeword computations are then halved. Thus, using equation {26} evaluated for i=0, the first energy term G₀ becomes:
which is computed in step 622. Hence, up to this step, we have computed the correlation term C₀ and the energy term G₀ for codeword zero.
im
im
im
b
b
I
Continuing with step 624, the parameters ϑ are initialized to -1 for 1 ≦ m ≦ M. These ϑ parameters represent the M interim data signals which would be used to generate the current code vector as described by equation {1}. (The i subscript in ϑ was dropped in the figures for simplicity.) Next, the best correlation term C is set equal to the pre-calculated correlation C₀, and the best energy term G is set equal to the pre-calculated G₀. The codeword I, which represents the codeword for the best excitation vector u(n) for the particular input speech frame s(n), is set equal to 0. A counter variable k is initialized to zero, and is then incremented in step 626.
M
M-1
M-1
ℓ
ℓ
In Figure 6B, the counter k is tested in step 628 to see if all 2 combinations of basis vectors have been tested. Note that the maximum value of k is 2, since a codeword and its complement are evaluated at the same time as described above. If k is less than 2, then step 630 proceeds to define a function "flip" wherein the variable ℓ represents the location of the next bit to flip in codeword i. This function is performed since the present invention utilizes a Gray code to sequence through the code vectors changing only one bit at a time. Therefore, it can be assumed that each successive codeword differs from the previous codeword in only one bit position. In other words, if each successive codeword evaluated differs from the previous codeword by only one bit, which can be accomplished by using a binary Gray code approach, then only M add or subtract operations are needed to evaluate the correlation term and energy term. Step 630 also sets ϑ to -ϑ to reflect the change of bit ℓ in the codeword.
k
ℓ
ℓ
{28} C
k
= C
k-1
+ 2ϑ
ℓ
R
ℓ
Using this Gray code assumption, the new correlation term C is computed in step 632 according to the equation:
This was derived from equation {22} by substituting -ϑ for ϑ.
k
jk
Next, in step 634, the new energy term G is computed according to the equation:
which assumes that D is stored as a symmetric matrix with only values for j ≦ k being stored. Equation {29} was derived from equation {26} in the same manner.
k
k
k
k
b
b
k
b
b
k
b
b
m
m
m
Once G and C have been computed, then [C]²/G must be compared to the previous best [C]²/G. Since division is inherently slow, it is useful to reformulate the problem to avoid the division by cross multiplication. Since all terms are positive, this equation is equivalent to comparing [C]² x G to [C]² x G, as is done in step 636. If the first quantity is greater than the second quantity, then control proceeds to step 638, wherein the best correlation term C and the best energy term G are updated, respectively. Step 642 computes the excitation codeword I from the ϑ parameter by setting bit m of codeword I equal to 1 if ϑ is +1, and by setting bit m of codeword I equal to 0 if ϑ is -1, for all m bits 1 ≦ m ≦ M. Control then returns to step 626 to test the next codeword, as would be done immediately if the first quantity was not greater than the second quantity.
b
b
b
b
b
b
b
b
b
Once all the pairs of complementary codewords have been tested and the codeword which maximizes the [C]²/G quantity has been found, control proceeds to step 646, which checks to see if the correlation term C is less than zero. This is done to compensate for the fact that the codebook was searched by pairs of complementary codewords. If C is less than zero, then the gain factor γ is set equal to -[C/G] in step 650, and the codeword I is complemented in step 652. If C is not negative, then the gain factor γ is just set equal to C/G in step 648. This ensures that the gain factor γ is positive.
m
I
I
Next, the best codeword I is output in step 654, and the gain factor γ is output in step 656. Step 658 then proceeds to compute the reconstructed weighted speech vector y′(n) by using the best excitation codeword I. Codebook generator uses codeword I and the basis vectors v(n) to generate excitation vector u(n) according to equation {1}. Code vector u(n) is then scaled by the gain factor γ in gain block 522, and filtered by filter string #1 to generate y′(n). Speech coder 500 does not use the reconstructed weighted speech vector y′(n) directly as was done in Figure 1. Instead, filter string #1 is used to update the filter states FS by transferring them to filter string #2 to compute the zero input response vector d(n) for the next frame. Accordingly, control returns to step 602 to input the next speech frame s(n).
Proc. Int. Conf. Commun.
In the search approach described in Figure 6A/6B, the gain factor γ is computed at the same time as the codeword I is optimized. In this way, the optimal gain factor for each codeword can be found. In the alternative search approach illustrated in Figures 7A through 7C, the gain factor is pre-computed prior to codeword determination. Here the gain factor is typically based on the RMS value of the residual for that frame, as described in B.S. Atal and M.R. Schroeder, "Stochastic Coding of Speech Signals at Very Low Bit Rates", , Vol. ICC84, Pt. 2, pp. 1610-1613, May 1984. The drawback in this pre-computed gain factor approach is that it generally exhibits a slightly inferior signal-to-noise ratio (SNR) for the speech coder.
Referring now to the flowchart of Figure 7A, the operation of speech coder 500 using predetermined gain factors will now be described. The input speech frame vector s(n) is first obtained from the A/D in step 702, and the long term predictor parameters LTP, short term predictor parameters STP, and weighting filter parameters WTP are computed by coefficient analyzer 510 in step 704, as was done in steps 602 and 604, respectively. However, in step 705, the gain factor γ is now computed for the entire frame as described in the preceding reference. Accordingly, coefficient analyzer 510 would output the predetermined gain factor γ as shown by the dotted arrow in Figure 5, and gain block 542 must be inserted in the basis vector path as shown by the dotted lines.
m
m
b
b
b
M
2-1
b
M
Steps 706 through 712 are identical to steps 606 through 612 of Figure 6A, respectively, and should require no further explanation. Step 714 is similar to step 614, except that the zero state response vectors q(n) are computed from the basis vectors v(n) after multiplication by the gain factor γ in block 542. Steps 716 through 722 are identical to steps 616 through 622, respectively. Step 723 tests whether the correlation C₀ is less than zero in order to determine how to initialize the variables I and E. If C₀ is less than zero, then the best codeword I is set equal to the complementary codeword I=2-1, since it will provide a better error signal E than codeword I=0. The best error signal E is then set equal to 2C₀ + G₀, since C is equal to -C₀. If C₀ is not negative, then step 725 initializes I to zero and initializes E to -2C₀ + G₀, as shown.
m
k
k
k
k
k
k
k
k
k
Step 726 proceeds to initialize the interim data signals ϑ to -1, and the counter variable k to zero, as was done in step 624. The variable k is incremented in step 727, and tested in step 728, as done in step 626 and 628, respectively. Steps 730, 732, and 734 are identical to steps 630, 632, and 634, respectively. The correlation term C is then tested in step 735. If it is negative, the error signal E is set equal to 2C + G, since a negative C similarly indicates that the complementary codeword is better than the current codeword. If C is positive, step 737 sets E equal to -2C + G, as was done before.
k
b
k
b
b
k
k
m
k
m
Continuing with Figure 7C, step 738 compares the new error signal E to the previous best error signal E. If E is less than E, then E is updated to E in step 739. If not, control returns to step 727. Step 740 again tests the correlation C to see if it is less than zero. If it is not, the best codeword I is computed from ϑ as was done in step 642 of Figure 6B. If C is less than zero, I is computed from -ϑ in the same manner to obtain the complementary codeword. Control returns to step 727 after I is computed.
M
When all 2 codewords have been tested, step 728 directs control to step 754, where the codeword I is output from the search controller. Step 758 computes the reconstructed weighted speech vector y′(n) as was done in step 658. Control then returns to the beginning of the flowchart at step 702.
M
In sum, the present invention provides an improved excitation vector generation and search technique that can be used with or without predetermined gain factors. The codebook of 2 excitation vectors is generated from a set of only M basis vectors. The entire codebook can be searched using only M + 3 multiply-accumulate operations per code vector evaluation. This reduction in storage and computational complexity makes possible real-time implementation of CELP speech coding with today's digital signal processors.
While specific embodiments of the present invention have been shown and described herein, further modifications and improvements may be made without departing from the scope of the invention defined by the apended claims. For example, any type of basis vector may be used with the vector sum technique described herein. Moreover, different computations may be performed on the basis vectors to achieve the same goal of reducing the computational complexity of the codebook search procedure. | |
Problem-solving is a part of everyone’s work, whether you’re a manager or entry-level employee. A project manager may solve problems for their clients and team members, while individual contributors may solve problems for themselves or their coworkers. Hence, it is important for every employee to understand the problem-solving process and develop problem-solving skills.
In this article, we offer ways to increase your problem-solving skills and opportunities for career advancement.
What is problem-solving?
Problem-solving is the process of understanding a challenge and working toward finding an effective solution to it. Depending upon the type and complexity of the problem, it may involve the use of mathematical operations and may test your critical-thinking skills.
When prospective employers are talking about problem-solving, they are usually trying to gauge your ability and skills to deal with difficult situations and complicated business problems. Almost all employers value problem-solving skills and seek to have employees with these traits in order to aid the decision-making process in the day-to-day functioning of the company.
Read more: Problem-Solving Skills: Definitions and Examples
Problem-solving steps
Here are the basic steps involved in problem-solving:
1. Define the problem
Analyze the situation carefully to learn more about the problem. A single situation may involve multiple problems. Identify each problem and determine the cause. Try to anticipate the behavior and response of people affected by the problem.
Then, based upon your preliminary observation, take the following steps to pinpoint the problem more accurately:
Separate facts from opinions.
Determine the process where the problem exists.
Analyze company policies and procedures.
Discuss with team members involved in order to gather more information.
Define the problem in specific terms.
Gather all the necessary information required to solve the problem.
While defining a problem, make sure you stay focused on the problem rather than trying to define it in terms of a solution at this stage. For example, “We need to rewrite the training documents” focuses on the solution rather than the problem. Instead, saying, “Training documents are inconsistent” is a better way to define a problem.
Depending upon the complexity of the problem, you may want to use tools, like flowcharts and cause-and-effect diagrams, to define the problem and its root causes.
2. Identify alternative solutions.
Brainstorm all possible ways to solve the existing problem. Invite suggestions from everyone affected by the problem and consult those who may have more experience with the type of challenge you’re experiencing. You can also use surveys and discussion groups to generate ideas.
Keep the following points in mind while exploring alternatives:
Consider every aspect that could slow down the process of solving the existing problem.
Make sure the ideas generated are consistent with relevant goals and objectives.
Check that everyone participates in the process of idea generation.
Distinguish between short- and long-term alternatives.
Write down all the proposed solutions. You should have at least five to eight of them for each problem.
3. Evaluate solutions.
Once you have a list of alternatives, it is time to evaluate them. Assess the positive and negative consequences of each alternative defined in the previous step. Analyze and compare all the alternatives in terms of the resources required for their implementation, including time, data, personnel and budget.
4. Select a solution.
After the evaluation process is over, select a solution most likely to solve the problem. Consider to what extent a solution meets the following objectives:
It solves the problem smoothly without creating another problem.
It is acceptable to everyone involved.
It is practical and easy to implement.
It fits within the company’s policies and procedures.
It is important to consider the implementation part while choosing a solution. Decide the following:
The employees responsible for executing the solution
How the employees will implement the solution
The amount of time and resources needed
5. Implement the chosen solution.
The next step involves implementing the chosen solution, which usually requires you to take the following actions:
Develop an action plan to implement the chosen solution.
Define objectives and separate them into measurable targets to monitor the implementation.
Define timelines for implementation.
Communicate the plan to everyone involved.
Develop feedback channels to use during the process.
6. Monitor progress and make adjustments.
Make sure to continuously measure progress to ensure your solution works. Gather data and feedback from others to determine if the solution meets their needs. You may need to make adjustments to the process if anything unexpected arises. If you feel the solution doesn’t work as planned, you may need to return to your alternative solutions and implement a new plan.
What are the important problem-solving skills employers look for?
Many employers seek candidates with excellent problem-solving skills. Here are some of the most important problem-solving skills:
Listening: Active listening helps you gather valuable information for problem-solving. A good problem-solver can identify everyone involved, encourage them to get involved and actively listens to different opinions to understand the problem, its root cause and workable solutions.
Analytical thinking: Analytical thinking helps you research and understand a problem and its causes. The ability to establish a cause-and-effect relationship is also essential in anticipating the long-term effects of a course of action. Those with strong analytical skills can evaluate the effectiveness of different solutions and choose the best one.
Creativity: Problem-solving requires you to create a balance between logic and creativity. You need to use your creativity to find the cause of the issue. It also requires creativity to develop innovative solutions. Creative people bring unique perspectives and give a new direction to the company.
Communication: Whether you are seeking solutions to an existing problem or want others to follow a certain course, you should be able to communicate effectively. You may need to talk with others in person, over the phone, via text or through email. You may also need to correspond with many different people, including team members, customers and managers. Effective communication across a variety of channels allows you to be a good problem solver.
Decision making: You should be able to decide what methods you should use to research the problem, which solutions you should use and how you should implement the solution. Almost every stage of problem-solving requires you to make a decision.
Teamwork: Problem-solving involves teamwork. You ask people about their perspective on the problem, involve them in developing effective solutions, seek their feedback on the chosen solution and rely on team members to implement the process. It is essential to involve and motivate all members of the team for effective problem-solving.
Related: Hard Skills vs. Soft Skills
Highlighting problem-solving skills on your resume
Showcasing your problem-solving skills on your resume can help you stand out from other candidates.
You can mention your problem-solving skills under either the skills or achievements section of your resume. Instead of simply writing that you possess problem-solving skills, try to illustrate how you have used these skills to solve specific problems in your previous positions. Consider the following examples:
“Reduced the instances of safety violations by 40% through strategic installation of railings on the production floor”
“Reduced inventory handling costs by 15% by using specialized software solutions”
“Increased customer satisfaction ratings by 25% by documenting a standard process and scripts to address general questions”
Try to tailor your resume so your problem-solving skills match the job which you are applying for. Creating a tailored resume can help you gain and maintain the attention of the recruiter or hiring manager as they review your resume.
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Linear Regression Analysis:Develop a wage question,Formulate
This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
See attached data file.
Using numerical data from the data set attached, develop one research question and formulate a hypothesis which can be tested with linear regression analysis.
Complete and describe the results of the linear regression analysis on the collected data. Be sure to include the following:
i. Use the hypothesis statement below:
The hypothesis would be that the effect of education on wages is greater than the effect of work experience (greater positive slope), or that the correlation between wages and education is greater than the correlation between wages and work experience (less variance).
ii. Perform a regression hypothesis test on the data.
iii. Interpret the results of your regression hypothesis test.
Be sure to include your raw data tables and the results of your computations, using both graphical and tabular methods of displaying data and results.
Legend for Attached Data Set:
X1: Annual wage in dollars
X2: Industry (1=manufacturing, 2=construction, 0=other)
X3: Occupation (1=management, 2=sales, 3=clerical, 4=service, 5=professional, 0=other)
X4: Years of education
X5: Southern resident (0=no, 1=yes)
X6: Nonwhite (0=no, 1=yes)
X7: Hispanic (0=no, 1=yes)
X8: Female (0=no, 1=yes)
X9: Years of work experience
X10: Married (0=no, 1=yes)
X11: Age in years
X12: Union member (0=no, 1=yes)
100 observations
https://brainmass.com/statistics/regression-analysis/linear-regression-analysis-develop-a-wage-question-formulate-278581
Solution Summary
A Complete, Neat and Step-by-step Solution is provided in the attached Excel file. | https://brainmass.com/statistics/regression-analysis/linear-regression-analysis-develop-a-wage-question-formulate-278581 |
We like to head away camping for weeks at a time, and that means doing some meal planning so we have good, healthy, tasty and some easy to prepare meals on hand.
We’ve got a number of meals that we all (or mostly all) enjoy, and we carry the ingredients for them, and cook a myriad of meals. I’ll get to what we cook often, but firstly wanted to start off with some common things that we do to make camping meals easier:
We plan meals, and then choose each morning
If we are going away for 3 weeks, Sarah will make a list of 21 dinners that we can eat, and we’ll take the ingredients for all of them, where possible. Then, each morning we will decide what we are going to have for dinner together, and we make it happen.
This allows us to choose food based around what we are doing in the day, and what we feel like. We don’t set a fixed menu in advance as things change, and you might not feel like that meal when the day arrives.
On days where we have limited travel, or we are kicking back at camp we’ll get the weber out, or do slow meals over the fire. Trying to do those meals when you get to the end of a busy day never ends well, especially when you have two young kids!
I’ve seen people plan exactly what they are going to eat each day, and it soon becomes a recipe for disaster with late arrivals to camp, you feeling like something else and the kids not being prepared for what you’ve made them.
We start off with meals that require fresh produce which will not keep well
Our go to plan for meals is to eat what will go off, or not keep too well first. For example, if Sarah brings food for nachos or tacos, we’ll generally bump them up the list to be eaten first as everyone knows your salad items don’t keep too well when travelling.
This means that meals which will last forever (potatoes, frozen meat, pasta sauce and so on) get kept until we’ve used most of our fresh produce, and then if we visit a shop and top up the cycle starts again.
Pre cooked meals are important to have in our freezer
There’s nothing better than pre cooked meals. We do a whole heap of items from pre made pasta sauce through to meat balls, soups, pre-marinated meat and so forth. Anything that allows you to whip together a meal in under 15 minutes is a life saver with two hungry, and often hangry kids at 5PM. We avoid using these unless there’s a need to, and keep them for busy days where we arrive at camp later on, or simply cannot be bothered cooking a significant meal.
It allows us some peace of mind, and we still get to have tasty, healthy food without waiting hours to make it!
We buy in bulk, and vacuum seal and freeze it
We’ve gotten into the habit of buying larger quantities of meat, and anything else that can be vacuum sealed, and we take the time to break it into meal sized quantities, and to label it, and put it away. Mince, chops, steak and whatever else it might be can then be bought much cheaper, and prepared in smaller packets for meals, ready to go.
It takes up far less space in your freezer, won’t spill anywhere, you have less rubbish to deal with when remote and makes for an easy way to have good food at hand.
You won’t find too much packet food in our pantry
We learnt pretty quickly that packet meals don’t go down to well for our family. We don’t really like it, and funny enough our kids completely hate it. Packet rice, pasta and other majorly processed meals are not something you’ll find much of in our pantry. We do keep some for emergencies and for when people really feel like that food, but we avoid it mainly.
You will find cans of spaghetti, baked beans, pasta sauce and that side of things, but rarely full meals.
We freeze a few loaves of bread
For a number of years, I would eat 10 pieces of bread a day, comfortably. I’d have two toasties for breakfast, and 3 for smoko, and then have a frozen meal for lunch. Our kids also love bread, and when you are travelling the only way to keep it is to freeze it.
We’ve found if you have it up the warmer end of the freezer it generally doesn’t freeze, and maintains a relatively fresh feel, which is great. We do toast it from time to time, and if we have bread generally this goes before having any wraps.
We buy fresh milk, then make powdered milk up as we go
For a long time, we were buying long life milk, which works just fine, but takes up a lot of space and weight, and can be a pain. Not long after getting our upright fridge, we moved to getting a bottle of fresh milk when in town, and if that runs out, we move to powdered milk that we make up with our water in the tanks. It’s a bit messier and annoying to do, but we’ve found it works really well, and avoids having to carry extensive amounts of long life milk.
As a family we can go through 5 – 7 litres a week for cereal, custard, hot drinks and so forth, and our days of carrying 30 litres of long life milk (which we did for one Kimberley trip) are long gone!
So, what meals do we often have when camping?
We’ve already covered a big chunk of this in our 50 easy camping meals post, but we are happy with simple meals that are tasty, healthy and most importantly that the kids like. Sarah isn’t a big fan of pasta, but they eat a heap of it, and we do a lot of meat and vegetables, or meat and mash. | https://www.4wdingaustralia.com/camping/meal-plan-for-camping/ |
Note: This is a required chemistry unit of study for students intending to major in chemistry.
This is one of the two core units of study for students considering majoring in chemistry, and for students of other disciplines who wish to acquire a good general background in chemistry. The unit considers fundamental questions of molecular structure, chemical reactivity, and molecular spectroscopy: What are chemical reactions and what makes them happen? How can we follow and understand them? How can we exploit them to make useful molecules? This course includes the organic and medicinal chemistry of aromatic and carbonyl compounds, organic reaction mechanisms, molecular spectroscopy, quantum chemistry, and molecular orbital theory.
The syllabus for this unit is the same as that of CHEM2401 together with special Advanced material presented in the practical program. The lectures cover fundamental consideration of molecular electronic structure and its role in molecular reactivity and spectroscopy and include applications of spectroscopy, the organic chemistry of aromatic systems, molecular orbital theory and quantum chemistry. For more details of the lecture syllabus, please read the entry for CHEM2401.
Note: The number of places in this unit of study is strictly limited and entry is by invitation only. Enrolment is conditional upon available places.
The structure and shape of organic molecules determines their physical properties, their reaction chemistry as well as their biological/medicinal activity. The determination of this structure and understanding its chemical consequences is of fundamental importance in chemistry, biochemistry, medicinal and materials chemistry. This course examines the methods and techniques used to establish the structure of organic molecules as well as the chemistry which dictates the shapes that they adopt. The first part of the course examines the use of modern spectroscopic methods (nuclear magnetic resonance spectroscopy, infrared spectroscopy and mass spectroscopy) which are used routinely to identify organic compounds. The second part of the course examines the chemical consequences of molecular shapes in more depth and looks at the inter-relationship between molecular shape and the processes by which bonds are made and broken (the reaction mechanism). An understanding of these processes allows the outcome of reactions to be predicted, which is an essential tool enabling the construction of complex molecules from simple starting materials.
The structure and shape of organic molecules determines their physical properties, their reaction chemistry as well as their biological/medicinal activity. The determination of this structure and understanding its chemical consequences is of fundamental importance in chemistry, biochemistry, medicinal and materials chemistry. This course examines the methods and techniques used to establish the structure of organic molecules as well as the chemistry which dictates the shapes that they adopt. The first part of the course examines the use of modern spectroscopic methods (nuclear magnetic resonance spectroscopy, infrared spectroscopy and mass spectroscopy) which are used routinely to identify organic compounds. The second part of the course examines the chemical consequences of molecular shapes in more depth and looks at the inter-relationship between molecular shape and the processes by which bonds are made and broken (the reaction mechanism). An understanding of these processes allows the outcome of reactions to be predicted, which is an essential tool enabling the construction of complex molecules from simple starting materials. CHEM3911 students attend the same lectures as CHEM3111 students, but attend an additional advanced seminar series comprising one lecture a week for 12 weeks.
This course concerns the inorganic chemistry of solid-state materials: compounds that possess 'infinite' bonding networks. The extended structure of solid materials gives rise to a wide range of important chemical, mechanical, electrical, magnetic and optical properties. Consequently such materials are of enormous technological significance as well as fundamental curiosity. In this course you will learn how chemistry can be used to design and synthesise novel materials with desirable properties. The course will start with familiar molecules such as C60 and examine their solid states to understand how the nature of chemical bonding changes in the solid state, leading to new properties such as electronic conduction. This will be the basis for a broader examination of how chemistry is related to structure, and how structure is related to properties such as catalytic activity, mechanical strength, magnetism, and superconductivity. The symmetry of solids will be used explain how their structures are classified, how they can transform between related structures when external conditions such as temperature, pressure and electric field are changed, and how this can be exploited in technological applications such as sensors and switches. Key techniques used to characterise solid-state materials will be covered, particularly X-ray diffraction, microscopy, and physical property measurements.
This course concerns the inorganic chemistry of solid-state materials: compounds that possess 'infinite' bonding networks. The extended structure of solid materials gives rise to a wide range of important chemical, mechanical, electrical, magnetic and optical properties. Consequently, such materials are of enormous technological significance as well as fundamental curiosity. In this course you will learn how chemistry can be used to design and synthesize novel materials with desirable properties. The course will start with familiar molecules such as C60 and examine their solid states to understand how the nature of chemical bonding changes in the solid state, leading to new properties such as electronic conduction. This will be the basis for a broader examination of how chemistry is related to structure, and how structure is related to properties such as catalytic activity, mechanical strength, magnetism, and superconductivity. The symmetry of solids will be used explain how their structures are classified, how they can transform between related structures when external conditions such as temperature, pressure and electric field are changed, and how this can be exploited in technological applications such as sensors and switches. Key techniques used to characterise solid-state materials will be covered, particularly X-ray diffraction, microscopy, and physical property measurements. CHEM3912 students attend the same lectures as CHEM3112 students, but attend an additional advanced seminar series comprising one lecture a week for 12 weeks.
At present rates of consumption, the resources of 5 planets would be needed for everyone on earth to enjoy our standard of living. Since so much of our consumption and waste involves chemical processes in some way, more efficient chemical processes are needed in a sustainable tomorrow. Catalysis is and will increasingly be at the heart of these sustainable processes. This unit examines the fundamentals of catalysis and its use to design sustainable processes. The course will initially focus on the organometallic fundamentals in order to show how they can be used to understand and design homogeneous catalytic processes from a molecular perspective, which, in turn, leads on to biocatalytic conversions where the enzyme is treated like a large ligand with a special surface, pointing towards the surface chemistry involved in supported catalysts - the next topic. Within this general discussion, the special case of the three-dimensional surface found in zeotypes will be developed and the acid/base and redox catalysis (the mainstay of the majority of industrial processes) in such confined spaces of molecular dimensions will be examined. The course will continue with examining the production of polymers as an example of a major industrial process. An introduction on polymer chemistry and polymer properties will be given, followed by the examination of the various synthetic routes and processes that yield to the production of polymers. The recent advances in polymer synthesis and the design of new materials of improved properties and function will be reviewed. The last part of this section will explore the various approaches designed to improve the sustainability of polymer synthesis, in particular for the specific case of free radical polymerization, with an emphasis on the design of novel catalysts. The course will conclude by examining a variety of case studies. All the preceding topics find their way into the discussion of the key role of catalysts in the design of sustainable chemical processes, rationalizing the choices behind catalyst design. CHEM3913 students attend the same lectures as CHEM3113 students, but attend an additional advanced seminar series comprising one lecture a week for 12 weeks.
Coordination compounds, with bonds between a central metal atom and surrounding ligands, play critical roles in biology, biochemistry and medicine, controlling the structure and function of many enzymes and their metabolism. They play similarly vital roles in many industrial processes and in the development of new materials with specifically designed properties. Building on the foundation of crystal field theory, this course offers a comprehensive treatment of the structures and properties of coordination compounds, with a qualitative molecular orbital description of metal-ligand bonds, and their spectroscopic, magnetic and dynamic effects. The exploitation of these properties in medicine and materials will be emphasized. Medical topics include descriptions of the essential and toxic elements of the Periodic Table, metal complexes as anti-bacterial, anti-inflammatory and anti-cancer drugs, and their use as tumour imaging and radiotherapeutic agents. Materials topics include metal directed self assembly into unique structures, ligand design and control of the synthesis of nanoporous materials with new electronic and magnetic properties and applications in catalysis and molecular separations.
The development of new pharmaceuticals fundamentally relies on the ability to design and synthesize new compounds. Synthesis is an enabling discipline for medicinal chemistry - without it, the development of new drugs cannot progress from design to implementation, and ultimately to a cure. This unit will tackle important factors in drug design, and will highlight the current arsenal of methods used in the discovery of new drugs, including rational drug design, high throughput screening and combinatorial chemistry. We will develop a logical approach to planning a synthesis of a particular target structure. The synthesis and chemistry of heterocycles, which comprise some 40% of all known organic compounds and are particularly common in pharmaceuticals, will be outlined. Examples will include important ring systems present in biological systems, such as pyrimidines and purines (DNA and RNA), imidazole and thiazole (amino acids and vitamins) and porphyrins (natural colouring substances and oxygen carrying component of blood). Throughout the course, the utility of synthesis in medicinal chemistry will be illustrated with case studies such as anti-influenza (Relenza), anaesthetic (benzocaine), anti-inflammatory (Vioxx), antihypertensive (pinacidil) and cholesterol-lowering (Lovastatin) drugs. CHEM3915 students attend the same lectures as CHEM3115 students, but attend an additional advanced seminar series comprising one lecture a week for 12 weeks.
Away from the covalent and ionic interactions that hold molecules and solids together is the world of fragile objects - folded polymers, membranes, surface adsorption and stable molecular aggregates - held together by weak forces such as van der Waals and the hydrophobic effect. The use of molecules rather than atoms as building blocks means that there are an enormous number of possibilities for stable aggregates with interesting chemical, physical and biological properties, many of which still wait to be explored. In this course we will examine the molecular interactions that drive self assembly and the consequences of these interactions in supramolecular assembly, lipid membrane formations and properties, microemulsions, polymer conformation and dynamics and range of fundamental surface properties including adhesion, wetting and colloidal stability.
Away from the covalent and ionic interactions that hold molecules and solids together is the world of fragile objects - folded polymers, membranes, surface adsorption and stable molecular aggregates - held together by weak forces such as van der Waals and the hydrophobic effect. The use of molecules rather than atoms as building blocks means that there are an enormous number of possibilities for stable aggregates with interesting chemical, physical and biological properties, many of which still wait to be explored. In this course we examine the molecular interactions that drive self assembly and the consequences of these interactions in supramolecular assembly, lipid membrane formations and properties, microemulsions, polymer conformation and dynamics and range of fundamental surface properties including adhesion, wetting and colloidal stability. CHEM3916 students attend the same lectures as CHEM3916 students, but attend an additional advanced seminar series comprising one lecture a week for 12 weeks.
This course will cover the fundamentals of molecular spectroscopy as a modern research tool and as a theoretical basis with which to understand everyday phenomena. This course is aimed at the student wishing a rigorous understanding of the fabric of nature -- electronic structure -- and the interaction between light and matter. The course teaches the quantum theory needed to understand spectroscopic phenomena (such as the absorption of light) at the empirical and deeper levels. A student completing this course will take with him/her an understanding of spectroscopy as both a phenomenon and a research tool. The course teaches application and theory, with descriptions of applied spectroscopic techniques. Alongside the coverage of modern spectroscopy, the course provides an accessible treatment of the science behind vision, flames, solar cells and photochemical smog.
In this unit, you will adopt a multi-disciplinary approach to solve a real-world problem in one of three research areas: i) Functional Energy Materials, ii) Self-assembled Nanomaterials and iii) Molecular Innovations in Health. You will apply your discipline expertise in chemistry to understand the challenge, design potential solutions to the problem, and then work collaboratively with students in other disciplines (science, government, business, law, marketing, engineering) to consider solutions to the problem from a broader perspective and how these could positively impact the community. This unit will allow you to understand the challenge through stories of scientific endeavour that led to the discovery of chemistry-based solutions to societal challenges, then extend that knowledge through collecting and analysing data on new technologies as you move to design innovative approaches. You will learn to work in interdisciplinary teams and communicate your findings to a broad audience. You will build key skills in problem solving, team work and written/oral communication that will equip you for future research and professional pathways in science, technology, health, business and public policy. | http://sydney.edu.au/handbooks/science/subject_areas_ae/chemistry_descriptions.shtml |
The previous page makes available both processed data files and gif images using the high resolution digital precipitation (HDP) dataset from the Miami, Florida (AMX), Key West, Florida (BYX), Melbourne, Florida (MLB), and Tampa Bay, Florida (TBW) NEXRAD sites. Data exists for 1996 (AMX only), 1997 (AMX and BYX only), and 1998/1999 (all sites).Details regarding gif images:
Images of hourly, daily, and monthly totals are available. Looping of the images is also possible if your internet browser is JAVA compatible. The gif images consist of data from a 131 x 131 array of pixels representing radar derived precipitation accumulations. The horizontal resolution is 4 km x 4 km. The color scale on the right side of the image represents the magnitude of the accumulated rainfall on a decibel scale, termed dBA. The following formula converts the units of dBA to millimeters.
where P(mm) = precipitation accumulation in mm
P(dBA) = precipitation accumulation in dBA
The double asterisks indicate that 10 is raised to the power of the quantity in square brackets. For example, a total rainfall accumulation shown on these gif images of 20 dBA would correspond to 100 millimeters or 4 inches. The ratio of hours in the plotted total to the total hours possible in a day/month, expressed as a percentage, for all radar sites is included at the top of the image.Details regarding processed data files:
The files contain a time series for accumulated radar derived rainfall over the entire radar grid for all available hours of data. The columns are as follows: The source station, the data station, latitude of data station, longitude of data station, the end time of the accumulation period (year, month, day and hour), total accumulated rainfall over the radar grid, and the number of valid pixels. | http://orca.rsmas.miami.edu/hdp/documentation.php |
Find the volume of the composite solid. Round your answer to the nearest tenth. Round your answer to the nearest tenth. A) 19,800π≈ 62,172 ft 3 C) 7,260π≈ 22,796.4 ft 3 tous les interdits de l islam pdf Volume and surface area were introduced in 5th grade. 7th grade standards do not contain volume and surface area, but if time allows, it would be a good idea to extend these concepts.
6. Determine the surface area in square inches of a rectangular prism with dimensions 4 in. by 3 in. by 2 in. 7. Determine the volume in cubic centimeters of a rectangular prism with dimensions 8 cm by 5 … networking terms and definitions pdf Volume and Surface Area Word Problems.notebook 2 December 06, 2012 Dec 97:31 AM 3)A can that has a height of 14.5 cm and a diameter of 9 cm.
Volume and surface area were introduced in 5th grade. 7th grade standards do not contain volume and surface area, but if time allows, it would be a good idea to extend these concepts. | http://apesantsandancestors.com/queensland/volume-and-surface-area-word-problems-pdf.php |
Sweet and spicy works so well, and we’ve come to love the sweet and spicy pair of watermelon and jalapeño pepper. They work perfectly together in a spicy salad or as a refreshingly tasty cocktail like you have with this watermelon jalapeño margarita recipe.
This is an excellent summer drink that’ll have you and your guests coming back for more. It’s just the right amount of spice, but if you like all things hotter, increase the amount of jalapeño and leave the seeds and membrane intact when you blend.
Ingredients
- 4 ounces watermelon juice
- 2 ounces agave tequila
- 2 ounces fresh lime juice
- 1 1/2 teaspoons sugar
- 1 jalapeño pepper sliced into circles
Instructions
- Blend watermelon slices and 1 to 2 jalapeño slices until liquified. Strain out seeds.
- Combine the tequila, watermelon juice, lime juice, and sugar over ice in a shaker. Shake vigorously for 15 seconds.
- Pour over ice in rocks glass or in a chilled martini glass.
- Garnish with 3 jalapeño slices. | https://www.pepperscale.com/watermelon-jalapeno-margarita/ |
Testimonials by outstanding senior students reflecting on their experiences at Hope College will be featured during the Hope College Senior Recognition and Women of Color Celebration on Wednesday, April 11, at 5:30 p.m. in the Haworth Inn and Conference Center.
The students scheduled to speak, and their academic majors, are seniors Diana Cortes (communication) of Holland, Michigan; Sieun Ruth Lee (chemistry) of Seoul, South Korea; Naomi Scott (nursing) of Phnom Penh, Cambodia; and Curissa Sutherland-Smith (psychology) of Chicago, Illinois. Also included in the evening celebration will be a senior recognition for all graduating students of color.
The event is hosted by the college’s Center for Diversity and Inclusion in collaboration with the Student Development Office.
Advance tickets are required by Wednesday, April 4. Tickets, which include dinner, are free for Hope students, $15 for members of the Hope faculty and staff, and $20 for the general public, and may be purchased through the college’s Center for Diversity and Inclusion. The office, located in the Bultman Student Center, is open weekdays from 8 a.m. to 5 p.m., and can be emailed at [email protected] or called at (616) 395-7867.
The Jim and Martie Bultman Student Center is located at 115 E. 12th St., at the center of the Hope campus between College and Columbia avenues along the former 12th Street. The Haworth Inn and Conference Center is located at 225 College Ave., between Ninth and 10th streets. | https://hope.edu/news/2018/campus-life/senior-recognition-and-women-of-color-celebration-is-april-11-2018.html |
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Dorothy and Kevin have always been interested in old castles. It came spontaneously since they are surrounded by abandoned castles in the place where they live. They try to find adequate literature about those castles, to find out from when they date, who lived there or if there is something left after they had been abandoned.
While they were exploring the castles, Dorothy and Kevin found out about one very interesting castle in their country. The castle is abandoned but according to the search they have made, exactly this castle hides a hidden treasure. That is something that can’t be found often, so the two friends became very excited about it. They would like to find out something more about the castle and they hope that the treasure is still there.
The mission of the two friends is to find the hidden castle treasure, so we can help them in their mission. Let’s search together with Dorothy and Kevin and try to find what has been left in the castle, many years ago. The mission can be successful, or maybe not, but it’s surely worth trying.
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Save my name, email, and website in this browser for the next time I comment. | https://games2mad.com/castle-treasure/ |
Q:
Using R, data.table, conditionally sum columns
I have a data table similar to this (except it has 150 columns and about 5 million rows):
set.seed(1)
dt <- data.table(ID=1:10, Status=c(rep("OUT",2),rep("IN",2),"ON",rep("OUT",2),rep("IN",2),"ON"),
t1=round(rnorm(10),1), t2=round(rnorm(10),1), t3=round(rnorm(10),1),
t4=round(rnorm(10),1), t5=round(rnorm(10),1), t6=round(rnorm(10),1),
t7=round(rnorm(10),1),t8=round(rnorm(10),1))
which outputs:
ID Status t1 t2 t3 t4 t5 t6 t7 t8
1: 1 OUT -0.6 1.5 0.9 1.4 -0.2 0.4 2.4 0.5
2: 2 OUT 0.2 0.4 0.8 -0.1 -0.3 -0.6 0.0 -0.7
3: 3 IN -0.8 -0.6 0.1 0.4 0.7 0.3 0.7 0.6
4: 4 IN 1.6 -2.2 -2.0 -0.1 0.6 -1.1 0.0 -0.9
5: 5 ON 0.3 1.1 0.6 -1.4 -0.7 1.4 -0.7 -1.3
6: 6 OUT -0.8 0.0 -0.1 -0.4 -0.7 2.0 0.2 0.3
7: 7 OUT 0.5 0.0 -0.2 -0.4 0.4 -0.4 -1.8 -0.4
8: 8 IN 0.7 0.9 -1.5 -0.1 0.8 -1.0 1.5 0.0
9: 9 IN 0.6 0.8 -0.5 1.1 -0.1 0.6 0.2 0.1
10: 10 ON -0.3 0.6 0.4 0.8 0.9 -0.1 2.2 -0.6
Using data.table, I would like to add a new column (using :=) called Total that would contain the following:
For each row,
if Status=OUT, sum columns t1:t4 and t8
if Status=IN, sum columns t5,t6,t8
if Status=ON, sum columns t1:t3 and t6:t8
The final output should look like this:
ID Status t1 t2 t3 t4 t5 t6 t7 t8 Total
1: 1 OUT -0.6 1.5 0.9 1.4 -0.2 0.4 2.4 0.5 3.7
2: 2 OUT 0.2 0.4 0.8 -0.1 -0.3 -0.6 0.0 -0.7 0.6
3: 3 IN -0.8 -0.6 0.1 0.4 0.7 0.3 0.7 0.6 1.6
4: 4 IN 1.6 -2.2 -2.0 -0.1 0.6 -1.1 0.0 -0.9 -1.4
5: 5 ON 0.3 1.1 0.6 -1.4 -0.7 1.4 -0.7 -1.3 1.4
6: 6 OUT -0.8 0.0 -0.1 -0.4 -0.7 2.0 0.2 0.3 -1.0
7: 7 OUT 0.5 0.0 -0.2 -0.4 0.4 -0.4 -1.8 -0.4 -0.5
8: 8 IN 0.7 0.9 -1.5 -0.1 0.8 -1.0 1.5 0.0 -0.2
9: 9 IN 0.6 0.8 -0.5 1.1 -0.1 0.6 0.2 0.1 0.6
10: 10 ON -0.3 0.6 0.4 0.8 0.9 -0.1 2.2 -0.6 2.2
I am fairly new to data.table (currently using version 1.9.6) and would like to try for a solution using efficient data.table syntax.
A:
I think doing it one by one, as suggested in comments, is perfectly fine, but you can also create a lookup table:
cond = data.table(Status = c("OUT", "IN", "ON"),
cols = Map(paste0, 't', list(c(1:4, 8), c(5,6,8), c(1:3, 6:8))))
# Status cols
#1: OUT t1,t2,t3,t4,t8
#2: IN t5,t6,t8
#3: ON t1,t2,t3,t6,t7,t8
dt[cond, Total := Reduce(`+`, .SD[, cols[[1]], with = F]), on = 'Status', by = .EACHI]
| |
Compatible Software:
Compatible Media Players:
Solutions:
Recommended Environments:
Model: QM55B
Diagonal Size: 55"
Resolution: 3840x2160 (4K UHD)
Brightness(Typ.): 500 nit
Contrast Ratio: 4000:1
Operation Hour: 24/7
Dimensions (Set): 1235.1 x 707.9 x 46.3mm
Dimensions (Package): 1402 x 858 x 163mm
Weight (Set): 18.1kg
Weight (Package): 23.6kg
VESA Mount(mm): 200 X 200
Bezel Width: 9.2mm(U/L/R), 11.2mm(B)
Warranty: 3-year manufacturer warranty
Payment & Security
Your payment information is processed securely. We do not store credit card details nor have access to your credit card information. | https://signspace.com.au/collections/55-75/products/samsung-55-qm55b-qm-series-uhd-digital-signage-display |
Rancho Santa Fe Golf Club needs the entire Rancho Santa Fe Association’s support for building membership
Since June 30 is the end of the 2012-2013 Rancho Santa Fe Association’s fiscal year and, therefore, the RSF Golf Club’s fiscal year, I am writing this letter to all of the members of the Association to help them become educated on the state of the membership of the Golf Club. As such, here is the letter from the Membership Committee that is being presented at the annual meeting of the Golf Club on Wednesday, June 26, 2013:
MEMBERSHIP COMMITTEE
2012-2013 Annual Report
Steve Dunn, Chairman
Connie Berkley, Judy Roberts, John Snyder, Larry Springer, Scott Johnson, Al Castro
This year has been both exciting and difficult at the same time. The annual budget anticipated the addition of 16 new Regular members or four more than the 12 who joined the preceding fiscal year. I am extremely proud to say that as of this writing we have 21 new members, eight of whom are in a new Junior Executive classification! We commenced the fiscal year with a total of 521 active memberships, and we now have a total of 510 active memberships, or a net
lossof 11 active memberships. So even with the exceptional efforts of Al Castro, his staff and the hard work of all of the Membership Committee we have still lost ground.
We began the fiscal year by attempting to implement some new, fresh and dynamic changes to the usual membership marketing efforts. The Membership Committee proposed a new category of Junior Executive membership along with a proposed modification to both the Former Resident and Resident Associate categories, taking these categories from 10 years down to five years to qualify. The RSF Golf Club Board of Governors unanimously approved these recommendations which were presented to the Association board in early January 2013. The Association board approved a total of 10 in the new Junior Executive category, but tabled the changes in membership term length in the other two categories until the results of an Association-wide survey could be tabulated. As of this writing this survey still has not been disseminated to the greater Association, so we wait on this modification.
The Junior Executive classification was a very successful addition and, so far, as was mentioned above, this fiscal year has garnered eight new memberships. The Former Resident classification, which still requires a minimum of 10 years of previous membership, has grown to 29 with the addition of 14 new memberships, although many of those came from regular members who have transferred out of the community. Those on Inactive status, who continue to pay the debt service over-ride, number 36 which has pretty much remained constant over the past 10 years. There has been significant growth in the Limited A and B playing privilege classifications which totals 67, up from 45 at this time last year. While it is nice to see that more of the Association residents are taking advantage of this opportunity, they have no commitment to the Golf Club’s staying power.
The long-term prospects for continued growth of our membership are challenged at best. This has been a banner year for residential sales within the Covenant; however, unlike the past where the Golf Club garnered over 30 percent of the new residents, we have been in the 8-12 percent range now for the past three years. Since we have a limited number of Association residents from which to draw members
andsince there is much more competition with adjacent golf clubs, we continue to try everything that we can to maintain a membership level of at least 500. In June of 2008, only five years ago, we had 613 members, so we have lost almost 20 percent of our members in that short period.
It is incumbent upon both the membership of the Golf Club
andthe greater Association to do everything within our power to generate new members for the Rancho Santa Fe Golf Club. We all believe that this is one of the greatest assets of the Covenant Association, but without continuing to aggressively market this esteemed venue to the “outside world” and to support the recommended membership category changes proposed by the Board of Governors, it will be difficult to keep a stable membership level which is imperative if we want to uphold our vision of having “one of the finest traditional golf clubs in the country.”
As the Review readers can see, while the membership of the Golf Club is reasonably stable at the present time, everyone needs to help in the continuing effort to market the Golf Club to all Association members, especially new property owners! Further, the Association board needs to help the Golf Club by allowing these proposed membership opportunities to be approved. The golf course has never been in better shape, the food service is beyond compare and the staff are some of the most friendly people you can find, so please take the time to come down to the Club to see what we have to offer! | https://www.ranchosantafereview.com/sdrsf-rancho-santa-fe-golf-club-needs-the-entire-rancho-2013jun24-story.html |
Q:
Would investing equally in all 30 companies which comprise the DJIA net the same performance as the DJIA?
I'm new to investing, so I'm hoping someone could clear this up for me. As I understand it, the Dow Jones Industrial Average (DJIA) comprises of 30 major American companies. Let's say, hypothetically speaking, that on January 23rd, 1976 I invested $30,000 ($1,000 each) to all of the companies listed within the DJIA. At the time, the value of the DJIA market index was 953.95. Currently, even as it is plummeting, the DJIA has a value of 15,748.01 which is an increase of approximately 1550%. Assuming that I kept track of my investment and exchanged stock in companies leaving the DJIA for companies joining the DJIA, would my initial investment now have a value of about $495,250 (that same percentage increase)? Or am I missing something?
Assuming that I calculated those numbers correctly, is this gain approximately better, equal to, or worse than an average investment for that timespan?
A:
DJIA is a price weighted index (as in the amount of each component company is weighted by its price) and the constituents change occasionally (51 times so far). With these two effects you would not get anything like the same return by equally weighting your holdings and would have to rebalance every so often. Note that your premise was most obviously flawed thinking the number of near bankruptcies there have been in that time. More details of the differing make-ups of the index are available on Wikipedia.
When you ask about the "average investment" you would have to be a lot more specific; is it limited just to US shares, to shares, to shares and fixed income securities, should I include all commodities, etc.
see also What's the justification for the DJIA being share-price weighted?
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A partial differential equation that governs potential fields (in regions where there are no sources) and is equivalent, in three dimensions, to the inverse square law of gravitational or electrical attraction. In Cartesian coordinates, the Laplace equation equates the sum of the second partial (spatial) derivatives of the field to zero. (When a source is present, this sum is equal to the strength of the source and the resulting equation is called Poisson's equation). The differential equation is named for French mathematician Pierre-Simon de Laplace (1749 to 1827), and applies to electrical, gravity and magnetic fields.
∇2u = ∂2u/∂x2 + ∂2u/∂y2 + ∂2u/∂z2 = 0,
where u(x,y,z) is a potential function. | https://glossary.oilfield.slb.com/en/Terms/l/laplace_equation.aspx |
This is a guest post by Jessica Rosevear Fox. Read our interview with her here.
I’ve been a lifelong Francophile, and two years ago, my Provence-themed wedding led to a French honeymoon. After spending a few days in Paris, my husband and I headed down to Provence, where we spent our time in lavender fields, strolling down tiny village streets, and exploring the rustic countryside.
I bought a book while I was there–I thought it was French chick lit, but it ended up being a French translation of an British novel. Titled Tout ton portrait and written by Isabel Wolff, it told the story of a portrait artist and her family relationships. The character’s mother was a former ballerina who retired after an injury. I hadn’t thought about ballet in years, and I had a moment when I thought–Oh yeah–ballet. That’s a thing.
The idea of ballet stuck with me, and when we got back to the States, I dove into the Internet for more. I’d taken ballet for a few years as a kid but was never serious about it. Now, as an adult, I became much more interested in the art form. I found beautiful ballet images on Pinterest, technique and documentary-style videos on YouTube, and helpful information in online communities like the Adult Ballerina Project and Ballerinas by Night. After engaging with ballet at a distance for a few months, I finally took the plunge and started taking adult ballet classes at a wonderful studio nearby. After a year, I advanced to pointe. I absolutely love it!
I’m also a writer. I typically do one major writing project each summer, and I knew last summer that I wanted my project to have elements of Provence and elements of ballet. They were both areas of my life in which I wanted to spend time, even if just in my imagination. That writing project became “After the Ballet,” a short story that I published through my indie imprint, Killing the Angel Press, an extension of the literary magazine I run.
“After the Ballet” was the first time I started writing from a sensory point of departure–the buzz of cicadas, the scent of the lavender in its endless purple lines, shiny pink pointe shoes. I’d been learning about the career trajectories of professional ballet dancers and found it fascinating. What really interested me was the fact that some dancers advance so quickly from the corps de ballet, to being a soloist, to being a principal dancer, while others take a very long time. Others don’t advance beyond the corps de ballet. Nothing is guaranteed. I became interested in the question of what would happen if someone decided to leave that world behind for something new after a lifetime of singular focus. I also wanted to pair that character with a sister experiencing changes of a more domestic nature. The idea of facing change against the backdrop of a lavender farm in Provence really inspired me. It was a world I just wanted to hang out in for a while. In the meantime, the theme of the story grew beyond the external details into one that might be more universal, something many different people could recognize.
A few people who read the story told me they could see it becoming a full-length novel. I think I agree, and so I’m looking forward to this summer’s writing project!
I took my first ballet class since childhood on February 16, 2015 and I haven’t stopped since. I’m quite the unlikely ballerina – I’m a bit curvy and I suffer from a rare, progressive and incurable pain disorder called Complex Regional Pain Syndrome. It causes constant, intense pain even from a gentle breeze or soft fabric. The treatments I underwent caused my left leg to feel heavier and not feel the floor.
Ballet gave me the opportunity to start taking my body back. The building of the movements throughout class, the repetition of exercises on each side allowed me to relearn how to engage the muscles on the left side of my body. The stronger I got, the less secondary pain I had started to go away. I was able to walk and get through my day with more confidence.
I want to help others who suffer from pain disorders find this strength and freedom, so I am starting an organization to do just that. Hope.Dance will be focused on creating dance classes rooted in traditional dance forms like ballet and tap for pain sufferers. Using a model similar to the English National Ballet’s Parkinson’s classes, I want to create classes that are welcoming and beneficial. A place where the pain can melt away and no one feels like a burden on the rest of the class or be afraid if there are movements they can’t do.
To get started and before I can start fundraising, I need to create the legal entity for the organization (legal fees, state fees). I’m selling shirts to help cover these costs and maybe even help rent a studio for my first class. If you don’t want a shirt, I have also set up a GoFundMe page.
The goal is to sell 100 shirts. If I hit that goal, I will raffle off a custom-made ballet skirt to one of the people who either bought a shirt or shared the campaign on social media. To enter the giveaway, send your order number or a screenshot of your social media share to [email protected] with the subject “Sweat and Pirouettes Raffle.”
Morgan has created two versions of the challenge – regular and light. She’s also created workout calendars for the next 12 weeks (although only January is available right now).
Right now, I’m probably basing most of what I’m doing off of the Light schedule, although I expect to move everything around a lot (based on class, work and other workout schedules) since my life can seem so hectic at times. I’ve also done a lot of pilates and running lately, so I don’t want those to fall to the wayside for this challenge (since there’s only a day or two of core and cardio each week).
I don’t expect my workout schedule to change too much (other than coming back from ballet after a one month hiatus), but I love that this challenge will help me be more accountable and track my workouts more. I plan on keeping a journal of it here so that I can track my progress.
Here’s what I did for Day 1 (January 4):
2 miles cardio on the treadmill
15 minutes Pilates Flow (from this DVD set — which I love!)
Plan for tomorrow is to do follow along to one of Kathryn Morgan’s barre and center videos on YouTube (check out this great YouTube playlist with video suggestions for the challenge by fellow adult ballerina Mary Fran). I’ll also probably at least throw in a quick pilates flow as well with some light stretching.
Are you taking part in Kathryn Morgan’s 12 Week Challenge? Let me know in the comments!
(Editor’s Note: These are notes from an adult ballerina’s experiences and the article was not written by a trained Physical Therapist. Please see a doctor before starting a new training regimen, and don’t push yourself beyond your limits! Read our disclaimer.) All photos by Helen Mao except #10.
Two months after surgery for Morton’s Neuroma, I recovered well enough to move around fairly normally; I could even walk one mile for exercise. However, my left foot wasn’t strong and had little flexibility. I couldn’t curl my toes without using my hands to bend them! In order to help the last part of recovery, my podiatrist sent me to physical therapy.
I didn’t know what to expect but was delighted that many of the physical therapy exercises were ones that I had done in the past for ballet and pointe. Of course I needed to keep attending physical therapy sessions to make me DO the exercises consistently. Nonetheless, I found that the following physical therapy exercises designed to rehabilitate my foot also helped prepare me for returning to ballet class.
Exercises 1-5 are done seated in a chair.
Golf Ball Rolls: Warm/loosen up your foot by rolling it forwards and backwards over a golf ball. Although the small hard golf ball helped me for physical therapy purposes, I’ve seen many dancers use a tennis ball before and after class to massage their feet.
Towel Curls: Place a towel flat on the floor. Starting on the closest end, curl your toes to pick up the towel. Lift the towel slightly off the floor and pull the towel a little toward yourself. After putting it back on the floor, place your toes a little further away on the towel and repeat until you reach the other end of the towel.
Marble Pick-up: Pour a cup or bowl of marbles on the floor but keep them in one place.Using your toes, pick up the marbles one by one and place them back in the cup or bowl. I vary the toes I use to pick up the marbles (big, middle, smaller ones) in order to strengthen all toes.
Ankle alphabet: Pretend your big toe is a pen or that you are holding a pen between your big and second toes. Keeping your ankle still, draw the alphabet A-Z (either uppercase or lowercase) with your foot.
Ankle Circles: Keeping your ankle still, slowly rotate your foot and ankle in a counter-clockwise direction and then in a clockwise direction. Repeat 10 times in each direction.
Exercises 6-9 are done while seated on the floor.
Resisted Ankle Plantar Flexion: Loop a TheraBand around your left foot and straighten your left leg. Slowly press your foot down and up (resist popping back up!) using only your ankle. Repeat 20 times.
Resisted Ankle Eversion: Straighten both legs. Loop the TheraBand around your left foot and hold the excess band with your right foot and right hand. Turn your left foot out and repeat 20 times. Switch the exercise to your right foot and repeat
Resisted Ankle Inversion: Cross your legs with the right leg underneath. Loop the Thera-Band around your right foot and hold the excess band with your left foot and right hand. Turn your right foot in and repeat 20 times. Switch the exercise to your left foot and repeat.
Calf and Achilles Tendon Stretch: Loop the Theraband around your extended leg’s foot. Position the Thera-band around the ball of the foot and gently pull on the Thera-band to stretch your calf muscles and Achilles Tendon. Keep your knee straight.
Hamstring Stretch: Can be done using Therabands or a strap, rolled towel, bungee cord, etc. Just lie on your back and wrap whatever you’re using under or around your foot. Then, trying to keep your leg straight, pull your leg up with your arms.
Thera-Band Loop Side Walk: Tie the Theraband in a loop around your legs just above the knees. Walk sideways slowly by first stepping hip-width with your right foot; then bringing your left foot in next to your right foot. Keep feet pointing straight forward. Walk about 25 yards. Repeat walking sideways the other direction.
Thera-Band Monster Walk: Use the same loop and position but this time step forward and out to the side so feet are hip distance part, alternating feet. Keep feet facing straight forward. Walk about 25 yards.
Balancing on half ball: Stand on half ball balance trainer (i.e. a Bosu Ball), first with two feet and then with one. Balance for 1-3 minutes.
Heel Lifts: Stand behind a chair (or anything stationary and releve on two feet 20 times. If desired, repeat exercises on one foot, and then the other.
Cool-down roll: Finally you’ve earned the right to sit down in a chair and cool down by rolling your foot over frozen water bottle.
Of course you can look up more detailed information on these exercises and use whichever ones help you not only in ballet but also in everyday movement. Luckily, most of these exercises can be done while watching TV!
We had a huge event at work and I had other events to go to — so much that with Halloween, HQ and I didn’t make to ballet at all. Then I was having another rough time this week (and felt sick at the beginning of it) that I didn’t think we’d get our workouts in any workouts at all — currently we’re trying to balance ballet, running, and cross-training (mostly pilates, but some cardio, too).
I hadn’t been doing much running, but after we went running when the streets were open thanks to the Pope being here, HQ and I decided that we were trying to go running a bit more. We’ve done a fair amount of running in the past, but ever since I ended up with stress fractures, we never really got into it. A great community, Run215, has also helped motivate me as well to keep going and we’re registered for the Rothman 8K later this month. After a 5 mile run last weekend, I feel more ready than we have for a race in the past.
But — the struggle has been balancing everything, especially with running and ballet feeling like they take up so much time by themselves. Trying to fit everything in has been tough, but usually my system is to schedule my workouts ahead of time, so I know what I’ve got in the upcoming week (work, events, fun) and how I can fit in workouts around that, with a balance of running, ballet, and cross-training.
But sometimes it fails — like when you’ve had a really rough day at work, daylight saving change has messed with your head, and end up quitting .66 miles into your workout. When that happens, I find the best thing to remember is your body needs rest and time to recover. Sometimes taking a break is just what you need.
So my question for you is, how do you achieve balance with all your workout goals? How do you know when it’s time to take a break? I’ll do a follow up post later this week with responses!
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Section 101(a)(15)(B)(i) of the Immigration and Nationality Act (INA).
The individual must be entering the U.S. temporarily. This means that the Business Visitor must have a foreign residence in his or her home country that he/she has no intention of abandoning. Furthermore, to be eligible for a B-1 visa, the applicant must be able to show ties to his or her home country in the form of such evidence as property, family, and a permanent job. It is up to the B-1 applicant to overcome the presumption by the U.S. Consular Official that he or she has immigrant intent.
The Business Visitor must not be coming to the U.S. to provide services or engage in business activities that are primarily for the benefit of a U.S. employer. In most cases, Business Visitors are admitted to the U.S. to conduct business for the principal benefit of their foreign employer.
Business Visitors may not be paid a salary or other remuneration from a U.S. source. However, reimbursement for per diem, travel expenses, and, in limited circumstances, an honorarium, is permitted and discussed below, in detail.
Business Visitors must depart the U.S. on or before their expiration date. There is no grace period for individuals in B-1 or WB status. Therefore, failure by the Business Visitor to depart will result in an unlawful overstay.
Who Can Come to NIH in B-1 or WB Status?
He/she must have a foreign employer and be coming to NIH on behalf of the foreign employer to consult with associates in the field.
He/she must be the recipient of a foreign grant awarded for the specific research that will be conducted at NIH (this may include students coming to conduct research as part of their dissertation).
An individual that does not meet the criteria listed in 1. above, but is coming to NIH to conduct independent research (research that clearly will not result in services or any benefit to NIH), such as a Guest Researcher.
An individual that does not meet the criteria stated above in 1. and 2., but is coming to NIH for observation only (such a scientist or a student coming to observe a technique in the lab).
An individual that does not meet the criteria stated above in 1., 2., and 3., but is coming to NIH to participate in a scientific conference or seminar (such as a scientist giving a lecture).
One of the above criterion must be met whether or not the individual is being reimbursed for travel and/or per diem expenses and/or honorarium. To assist in determining eligibility for B-1 or WB status, complete the B-1/WB Statement.
*Note Only medical students coming for elective clerkships as discussed in item 5. above, are authorized to engage in supervised patient contact. Otherwise, physicians in B-1 or WB status may only come for observation and consultation, where no element of patient contact is involved.
When Is B-1 or WB Status Inappropriate?
It is inappropriate to invite a foreign scientist to join NIH's intramural research program holding B-1 or WB status with the intention of changing immigration status (e.g., to J-1 Exchange Visitor).
This is often interpreted by the U.S. Department of Homeland Security (DHS) as misrepresentation of intent and can be cause for denial of entry into the U.S. and a bar from future admissions to the U.S. or, for those who succeed in being admitted in B-1 or WB status, a denial later by DHS of a request for a change of status.
It is inappropriate for an individual whose J-1 status at the NIH has expired to leave the U.S. and immediately return to the NIH in B-1 or WB status to finish a project, if that individual does not have an employer in the home country who will be the principal beneficiary of the research.
It is inappropriate to invite an individual to the NIH in B-1 or WB status to do work that will displace a U.S. worker.
An overview of the process can be found on the B-1 Process poster.
As previously discussed, B-1 or WB Visitors may not receive a salary or other remuneration from a U.S. source other than reimbursement for expenses incidental to the individual's temporary stay (i.e., per diem and travel) and honorarium in limited circumstances. Honorarium payments must meet strict guidelines. If honorarium is paid, the NIH Institute or Center (IC) needs to complete the following declaration at the time of the scientist’s arrival.
If the scientist is reimbursed for per diem expenses or receives honorarium during his or her stay at NIH, discretion should be used to determine an appropriate amount. It is important to remember that reimbursement for per diem or honorarium cannot be used as a mechanism to pay the Business Visitor a stipend or salary. If DOS or DHS believes that the amount being paid to the scientist is comparable to a salary, the Business Visitor will be denied a B-1 visa and/or entry into the U.S. in B-1 or WB status.
Avoid using the term "volunteer" as this can result in the scientist being given a tourist (B- 2) visa by the U.S. Consulate and/or being admitted by the DHS into the U.S. in B-2 or WT (B-2 Visa Waiver Program) status.
When being admitted into the U.S., the foreign scientist should request that the DHS Immigration Inspector indicate “B-1” on the Form I-94 (Arrival-Departure Record) and include the period of admission specified in the NIH letters of invitation. To assist the DHS Inspector, the scientist should present the letters from the NIH host, the DIS, and the home country employer (if applicable).
Although DIS instructs the foreign scientist about entering the U.S. in B-1 status, the IC sponsor and administrative Key Contact should also reinforce this, inasmuch as the visa stamp in the passport usually indicates B-1/B-2, and DHS officials may erroneously annotate “B-2” rather than “B-1” on the Form I-94. Such an error will prevent the scientist from participating in the business activities until his or her status is changed to B-1 by the DHS, a procedure that could take 2-3 months.
See “Links” at the end of this advisory for the Department of State (DOS) web site for more details about the B-1, Visitor for Business.
* NOTE: Canadian citizens do not need to apply for a B-1 visa to enter the U.S. When coming to the NIH, however, they must undergo U.S. customs and immigration inspection and obtain evidence that they were admitted in B-1 status. Therefore, when entering the U.S., Canadians must obtain either a Form I-94 OR an entry date-stamp in the passport marked “B-1” to indicate that they were admitted in B-1 status.
As a result of the Immigration Reform and Control Act (IRCA) of 1986, provisions were made for nationals of eight countries to come to the United States without obtaining a B-1 (Business) or B-2 (Tourist) visa. Since then, other countries have been added to the VWP (see list of countries currently participating below). Eligible nationals who wish to come under this program do not need to obtain a B-1/B-2 visa from a U.S. Consulate to enter the U.S.
For an extension of stay in the U.S. beyond the 90-day limit under the VWP.
Therefore, if there is any intention that the individual will remain at NIH beyond 90 days, he/she should apply for a B-1 visa at the U.S. Consulate in the home country and enter the U.S. in B-1 status.
REMINDER: WT status is NEVER appropriate for ANY foreign scientist carrying out research activities in NIH's laboratories for any period of time under any circumstances. An individual coming to NIH for an interview, who was admitted to the U.S. in WT status, CANNOT be reimbursed for travel or per diem expenses.
For additional information on the B-1, Visitor for Business, click here. | https://www.ors.od.nih.gov/pes/dis/AdministrativeStaff/Pages/B-1TemporaryVisitors.aspx |
For decades now, “mixed-income” has been a watchword throughout the affordable housing and community development worlds. Compared to the worst examples of urban design that have physically isolated low-income families, mixed-income housing seems like an intuitively healthier, more equitable way to go about designing neighborhoods.
But what specifically is the goal of mixed-income housing? To quote Pat Sharkey from his excellent essay for the Furman Center, “Making Our Assumptions About Integration Explicit,”:
“We should start by making explicit the rationale for residential integration. By being explicit about why integration might be thought of as a desirable goal, we allow all of our unstated assumptions to rise to the surface and be subjected to critique. In doing so, we can clarify what social outcomes are advanced by confronting residential segregation, and what mechanisms are most appropriate for reaching those outcomes.”
Sharkey is talking about racial integration, but the question also holds true for economic integration: goals matter, and they have implications for implementation.
For example, are we trying to improve the odds of those children who can get an affordable home as dramatically as possible, or are we trying to maximize the number of affordable homes we’re able to provide in neighborhoods with a certain baseline of amenities and opportunities? Do we believe the value in economic integration is primarily for low-income families, for all residents, or for the regions or municipalities whose neighborhoods are being integrated? Do we believe the benefit for low-income residents comes primarily from access to a better neighborhood (public safety, school quality, transit connectivity, access to jobs, green space, etc.) or from diversified social networks?
Also, what are the down sides? Are we sacrificing the total volume of affordable units? Are we overlooking the value in the social networks present in poor communities and underestimating the harm of losing those networks? Will the dislocation of displacement for some outweigh the benefits to those who can stay?
These conversations can get delicate. I regularly cringe as I hear patronizing commentary that assumes that low-income people will benefit from higher income “role models,” and even “different social norms,” as if there were something inherently virtuous about not being poor. Slipping into such formulations is shameful for equity advocates.
On the other hand, I have also heard folks who grew up in low-income communities forcefully make the case that it is harder to imagine yourself in college, for example, or leadership positions without people around you who have gone down that route and know how it works. The difference there, while important, is also subtle.
There are no universal answers to these questions, but how we approach them affects on-the-ground implementation. Reports from many mixed-income developments tell us that the reality is not yet living up to the ideals—stigma and social segregation are stubbornly hanging on. Many in low-income areas consider “mixed-income” as code for gentrification. In our “Voices from the Field” survey results you’ll see that most survey respondents saw multiple benefits to the idea of mixed-income housing, but many were also skeptical or cautious about implementation.
In this issue, our authors tackle these questions from many sides. Rick Jacobus argues that the research shows benefits to integration at a neighborhood level—amenities and access to opportunity—and says that we can get the best benefit for the most people by focusing on neighborhood, but not building-level integration. HOPE SF is trying to learn from the mistakes of past HOPE VI projects—starting with a 100 percent right to return—and it isn’t easy. David Holtzman spoke to many developers who had worked on mixed-income projects to get their insights from everything from financing to governance, while trauma-informed community building,” an approach to working with residents who have experienced various sorts of trauma that often come with being poor, isolated, and oppressed. This approach recognizes the particular effects that trauma can have that can work against community building efforts, and addresses them in ways that don’t disempower the residents along the way.
And finally—would you like to know how to stem one of the largest causes of job loss in the country? Shelterforce’s Keli Tianga explores how worker co-ops can step up as baby boomer business owners retire. | https://shelterforce.org/2016/05/03/mixing-it-up/ |
Serge Leonovich Grigoriev (1883 - 1968)
To most of the world, Serge Grigoriev is a name on a birth certificate in Tichvin, Russia, but when we go to the ballet today we are in debt to him for the classic ballets from the repertory of Serge Diaghilev's Ballets Russes. Because of his phenomenal memory, he became régisseur of the Ballets Russes on their first trip to Paris in 1909 and remained in that position for twenty years.
Grigoriev was born October 5, 1883. He studied at the St. Petersburg Imperial School, graduating in 1900, and danced with the Maryinsky Ballet until Diaghilev appointed him ballet master in 1909. Grigoriev was a friend of Mikhail Fokine, and Fokine recommended him to rehearse the ballets for the 1909 season. He was one of the few to remain until the death of Serge Diaghilev. Even when his good friend Fokine left because of the rift over Nijinsky's choreography, Grigoriev stayed with the company, although his sympathies were with Fokine.
After Diaghilev fired Vaslav Nijinsky he needed to rehire Fokine. Before he agreed to return Fokine made many demands: to dance leading roles; that all of Nijinsky's ballets be dropped from the repertoire; and that Grigoriev and his wife, the ballerina Lubov Tchernicheva, be discharged from the company. He got everything but the termination of Grigoriev. Diaghilev got Fokine, Grigoriev and Tchernicheva to reconcile.
After the death of Diaghilev in 1929 the company disbanded; Grigoriev and his wife joined Colonel de Basil's Ballet Russe in 1932, restaging the original choreography of Diaghilev's repertoire and remaining until 1948. Grigoriev mounted The Firebird in 1954 and Petrushka in 1957 for the Royal Ballet in London. He also wrote a book, The Diaghilev's Ballet. Richard Buckle, in his book Diaghilev, implied that Grigoriev minimized Diaghilev's sexual preferences, which played a large part in his hiring and firing. Diaghilev was willing to lose Pavlova, when she resented his preference for the male dancers. He also became the lover to most of his male stars.
Being able to remember a complete ballet is an art in itself. Many choreographers can't even restore their own ballets. Today we have videos and movies to make a permanent record of not only the performance, but the choreography.
"The Public never guessed how much the success of the performance was due to the work of Serge Grigoriev, director, stage-manager, ballet encyclopedia and receptacle of many artistic--and other--secrets which are now buried in the past." (Three Centuries of Ballet by Cornelius Conyn, 1953). | http://michaelminn.net/andros/biographies/grigoriev_serge/index.html |
An approach is described for an improved representation of shales in a clastic reservoir. Oil and gas reservoirs are extremely complex, and simulation of their performance is carried out using simplified models. At best, such models are crude representations of true geological complexity and they often fail to predict performance with any degree of reliability. Impermeable zones, or shales, are generally modelled as horizontal and rectangular simulation grid blocks, although natural shales dip, are irregular, and usually better modelled by a dipping ellipse, semi-ellipse or rectangle. For the estimation of an effective vertical permeability, no more than five to ten shales usually need to be included in a model. Correction factors can be applied to model the "real" shales as equivalent horizontal rectangles.
The size of shale is a critical parameter in modelling a sand-shale sequence and can be represented by a "characteristic distance" that is unique to that sequence, and that is composed of:
a dimensionless shale density
a geometric, or shape, factor
an orientation factor.
The simulation of the flow of fluids, (oil, gas or water) through a reservoir requires that a model be constructed from log, core, and outcrop analog data In such a mode, the complex, heterogenous, rock material is replaced by grid blocks with "equivalent" flow properties. One of the most important aspects of building a reservoir model is the calculation of an equivalent vertical permeability for a sequence containing impermeable layers. Models of sequences containing discontinuous impermeable layers (called shales in this paper, although they can be any impermeable unit) commonly assume that the shales:
have a regular shape;
are of the same shape as the simulation grid (usually rectangular);
are of the same size, or an integral multiple, of the simulation grid;
are horizontal and planar on the downstream flow side.
Real shales satisfy none of the above requirements, being irregular in shape, of different sizes, rarely, if ever, horizontal and unlikely to have a planar upper or lower surface. The approach described in this paper is a step towards more realistic models of reservoirs containing discontinuous shales.
The model in this paper is a development of the streamline model described by Haldorsen and Lake1 and developed further by Begg and King2, Begg, Chang and Haldorsen3, and Haldorsen and Chang4. | https://onepetro.org/PETSOCATM/proceedings-abstract/94ATMA/All-94ATMA/PETSOC-94-82/6408 |
Philanthropy has received a lot of attention in recent years since it entails helping others and serving them. Philanthropy may take several forms, the most significant of which is a monetary donation. Many individuals do so by giving education, safe food and water, or any other work that helps society. It is your values that determine your purpose and how much effect you can create with it.
The steps below can assist you on your philanthropic journey.
Understand Your Values
There are an infinite number of causes that you can aid via, so never assume that the cause you want to help is not so worth it. Consider your vision for this cause and how you envision yourself making a difference in the long run. Given the complexity of global challenges, it is easy to stray from the route; thus, never lose sight of your goal and continue making a difference via the cause you have chosen.
Establish Goals
Setting targets and sketching out everything is one of the greatest methods to keep on track. This map will assist you during the assignment and will keep you from thinking about other issues. In fact, it would be best if you simply wrote a brief statement describing the mission of your case so that you could refer to it whenever you stray from the path.
Consider Alternatives to Donations
Donations are a vital part of the cause, but there are other things you should look into so that more people may join you. Conduct thorough research to learn about other organizations fighting for the same cause so that you can be certain that your money is being sent in the right direction. Determine what other abilities are necessary to contribute to the cause and employ people who have them.
Learning From Queen Zaynab Otiti Obanor
Queen Zaynab is a well-known philanthropist who has been assisting poor children and women from remote regions. As an ambassador, she represents the United Nations Population Fund (UNFPA). It is affiliated with the United Nations and aids in the essential rights of women, men, and children to health and fairness. It also addresses difficult issues such as child marriage, female genital mutilation, and the need for family planning in controlling the world’s population.
The Arab African Economic Development Initiative was launched by Queen Zaynab (AAEDI). The fundamental goal of the non-governmental organization AAEDI is to develop and deepen economic and social links between African and Arab nations. Aside from that, she also runs The Queen Zaynab Foundation (QZF), through which she has aided many people, notably her unrivaled efforts in assisting over 10,000 families during the Covid crisis. Her #Forward initiative is spearheading the reconstruction of orphanages and training facilities in Lagos and Benin.
Because of her exceptional work, she got the Humanitarian of the Year Award during her 2017 visit to the United States during United Nations Week. Furthermore, she was named one of the top 40 trending people. Following such a great benefactor before venturing out on your own would undoubtedly teach you a lot. | https://www.atoallinks.com/2022/change-lives-by-getting-involved-in-philanthropy/ |
Kindly be advised of the following :
1) The above Company's securities will be traded and quoted "Ex - Consolidation” as from: 10 Feb 2022
2) The last date of lodgment : 11 Feb 2022
Remarks 1:-
Participating Organisations are to take note of the following Share Consolidation exercise by AIRASIA X BERHAD.
The Share Consolidation comprises the following:-
CONSOLIDATION OF EVERY 10 EXISTING ORDINARY SHARES IN AIRASIA X BERHAD ("AAX" OR THE "COMPANY") ("AAX SHARES" OR "SHARES") HELD AT 5.00 P.M. ON 11 FEBRUARY 2022 INTO 1 AAX SHARE ("CONSOLIDATED SHARE") ("SHARE CONSOLIDATION")
In relation to the Share Consolidation undertaken by AAX as a SPEEDS Corporate Exercise, Bursa Malaysia Securities Berhad would like to highlight that:
(a) on or after the Ex-date on 10 February 2022, trading of AAX shares will be based on the newly adjusted share after the Share Consolidation of AAX shares,
(b) on the basis of settlement taking place after 11 February 2022 with consolidated AAX shares, any entitled shareholder who owns AAX shares as at Ex-date may sell only up to the maximum AAX shares he expects to receive after the Share Consolidation, i.e. the reduced amount, on or after the Ex-date 10 February 2022.
With the adjustments pursuant to the Share Consolidation effected on AAX shareholders’ CDS account at the end of the Entitlement Date (“Books Closure Date”), an entitled AAX shareholder may use the following basis to estimate the maximum number of AAX shares that he may sell from the Ex-date until the Book Closing Date which is from 10 February 2022 until 11 February 2022.
Number of AAX shares that may be = Number of shares held / 10
sold from 10 February 2022 until 11 February 2022 (All fractional shares computed disregarded)
Illustration:
For example, if Mr X owns or purchases 1000 AAX shares on cum basis on 9 February 2022, his CDS account would still show 1000 AAX shares until 11 February 2022. However, as a result of the above Share Consolidation exercise, Mr X’s 1000 AAX shares in his CDS account will be adjusted to 100 AAX shares on the night of 11 February 2022 which is the Book Closing Date. Therefore, Mr X may, if he so wishes, sells only up to 100 AAX shares on or after the Ex-date i.e. from 10 February 2022 onwards.
Participating Organisations are hereby requested to caution all dealers and remisiers that, during the period from 10 February 2022 until 11 February 2022, they are only entitled to sell the maximum of 1/10 of the shares owned before the Ex-date. Participants Organisations are reminded that it is important to caution all dealers and remisiers on the above to prevent the dealers and remisiers from overselling of their client’s position.
Remarks 2:-
The Share Consolidation involves the consolidation of every 10 existing AAX Shares into 1 Consolidated Share as at 5.00 p.m. on 11 February 2022, being the entitlement date for the Share Consolidation.
The actual number of Consolidated Shares to be issued would depend on the issued share capital of the Company on the entitlement date for the Share Consolidation. Fractional entitlements arising from the Share Consolidation, in respect of the Consolidated Shares, if any, shall be disregarded and/or dealt with by the Board of Directors of the Company ("Board) in such manner and on such terms and conditions as the Board it in its absolute discretion may deem fit or expedient and in the best interests of the Company.
The Consolidated Shares will be listed and quoted on the Main Market of Bursa Malaysia Securities Berhad ("Bursa Securities") on 14 February 2022, being the next market day following the entitlement date for the Share Consolidation. The notices of allotment of the Consolidated Shares will be issued and despatched to the entitled shareholders within 4 market days after the date of listing and quotation of the Consolidated Shares on the Main Market of Bursa Securities.
The Share Consolidation will result in a reduction in the number of Shares available in the market and the trading price of the Shares will be adjusted accordingly in proportion to the basis of the Share Consolidation. The shareholders of the Company are strongly advised to trade cautiously to prevent overselling of their position in respect of the Shares held. | https://www.airasiax.com/news.rev/id/2370918 |
Despite its recent premiere on October 1, overshadowed by the furor caused by “Squid Game”, the series “Maid” has been gaining a special place in the hearts of viewers, and its plot has brought more than one tear to their eyes.
This production, one of Netflix’s most watched, is based on real-life events and revolves around the life of Alex, a 25-year-old woman who escapes with her young daughter Maddy from the abuse of her partner, with whom she lived in a trailer.
The series was created by Molly Smith Metzler and is inspired by Stephanie Land’s memoir “Maid: Hard Work, Low Pay, and a Mother’s Will to Survive”. In her work, she recounts the details of working as a nanny and dealing with the injustices faced by those who need the help of the U.S. government.
A story of struggle and overcoming
Imagine you have just turned 20, you have a perfect relationship, good grades, you live in America, and you got a scholarship to study Art and Literature at the University. However, one day you get pregnant and decide to have your daughter. Your partner doesn’t agree with your decision and your world falls apart.
So begins “Maid”, the Netflix series that tells the story of struggle and overcoming through its main character, Alex, who will do the impossible to save herself and her 3-year-old daughter from this drama.
Its author, Stephanie Land, managed to overcome a situation very similar to that of the protagonist of the series and wrote a diary narrating everything that had happened.
This production is also a critique on the failed American system and a reflection on how the class struggle is still entrenched in society. The series is honest, direct and will make you rethink many things, for example, the issue of domestic violence.
“Maid” doesn’t deal with it through physical abuse, but analyses it in a much more subtle and profound way. Alex has never been physically beaten, but she has been subjected to all kinds of psychological abuse that makes her find herself lost in a spiral of hatred and other emotions.
Without a roof over her head, Alex decides to go to a shelter for abused women. There she discovers that there are many women in the same situation and that getting any kind of financial help is an odyssey. That is why she is forced to put aside her dream of going to college and start cleaning rich people’s houses to support herself. There she learns how money does not buy everything and how people who apparently have had it much easier than her, also hide among luxury homes, expensive wines and prohibitive whims, almost equally serious dramas.
Who played the character of Alex on “Maid”?
“Maid” stars Margaret Qualley, who plays the role of Alex. We will be hearing more about her soon, as she has several interesting projects coming up. For example, the actress will be seen next year in the movie “Sanctuary”, and then she will participate in “Fred & Ginger”.
The Montana-born actress made her debut in front of the camera with a minor role in the film “Palo Alto” (2013). Subsequently, she achieved greater recognition after playing a troubled teenager in “The Leftovers”. Similarly, she garnered good reviews for her portrayal of actress and dancer Ann Reinking in the biographical miniseries “Fosse/Verdon”. Other projects include “Death Stranding” and “Once Upon a Time in Hollywood”.
The author of the novel and original story, Stephanie Land, spoke about the experience of seeing the drama of her life in the Netflix catalog.
More about the series “Maid”:
- “Maid” | Netflix official site
- “Maid” is a depiction of reality for struggling single mothers
- Will there be another series of “Maid” on Netflix? | https://mygoosebumpmoment.com/maid-an-emotional-series/ |
What is Coordinated Universal Time (UTC)?
Coordinated Universal Time (UTC) is the primary time standard by which the world regulates clocks and time. Time zones around the globe are defined as a positive or a negative offset from UTC.
Note. UTC time does not change in winter or summer. Therefore, for those places where there is daylight savings time, the UTC offset is changed instead.
What should I do if the service doesn't work?
The service time is displayed incorrectly
The time may be incorrectly set on your device. To learn how to set up the time and date on your device, please read the following instructions.
Note. If your device clock settings are inaccurate, Yandex.Time will notify you about this. | https://yandex.com/support/time/support.html?lang=en |
« PreviousContinue »
RULE.
Place the causes of the first set of quantities on one side of a vertical line, and the effects on the opposite side, and the causes and effects of the second set opposite to the corresponding quantities of the first, supplying the deficient term by a dash. Cancel the like factors on the opposite sides of the lines, and divide the product of the numbers opposite to the dash, by the product in which the dash is included. The quotient will be the term required.
f
12. How many men, in 6 months, will build a wall that 36 men will build in 8 months?
13. How many bushels of meal will serve 36 persons 12 months, if 10 persons consume 8 bushels in 2 months?
14. If 18 men build a cistern 20 feet long, 12 feet wide, and 10 feet high, in 3 weeks, by working 5 days in a week, and 9 hours a day, how many men, working 6 days in a week, and 12 hours a day, will build a cistern 32 feet long, 16 feet wide, and 15 feet high, in 12 weeks?
15. If $12.00 gain $3.00 in 5 months, how much ought $25.00 to gain in 10 months?
16. If the freight of 900lb. for 56 miles, is $2.50, how far may 2 tons be carried for $40.00?
17. If $1200 will support 24 persons 8 months, how long will $900 support 16 persons?
EXAMPLE FOR THE BOARD.
If $27 buy 4 yards of cloth that is tyd. wide, how many yards of like quality, that is yd. wide, may be bought for $13 ?
Reducing the mixed numbers to improper fractions, we transpose the denominators, writing them above the causes, then cancel and divide as before.
8
denom's.
$5 $5
19
$ $
Ans. 18-1yd.
18. If 571cwt. of sugar cost $375, what will 18/cwt.
cost?
19. If the rent of 19A. 3R. of land is £4 10s., what will be the rent of 24 A.?
20. The shadow of a stick that is 5 feet 6 inches high, measures 3ft. 4in. What is the height of tree whose shadow measures 75 feet at the same time?
21. If the expenses of a family of 8 persons are $40 in 10 weeks, how many persons can be supported 12 weeks for $100?
22. How much wheat, at $1.20 a bushel, must be given in exchange for 90 barrels of flour, at $4.75 per barrel?
23. If 10 compositors, in 4 days of 10 hours, set 663 pages of types, each page containing 45 lines of 50 letters, how many compositors will set 94 pages, each page containing 35 lines of 40 letters, in 5 days of 8 hours? 24. If 293 bushels of wheat yield 1760 bushels in 5 years, how much will 151⁄2 bushels yield in 6 years?
25. A crew of 150 men were supplied with provisions for 9 months, allowing each man 2 pounds per day. When they have been out half the time, they find it will require 6 months to finish their voyage. How much may be allowed to each man, 25 of the crew having been lost?
EXAMPLE FOR THE BOARD.
A French merchant wishes to pay in London a bill of £1500. How many francs must he pay to procure remittances through Russia, Hamburg and Spain, allowing £13=75 roubles, 5 roubles=9 mares of Hamburg; 3 marcs=1 Spanish dollar; and 9 dollars 50 francs?
Merchants often find an advantage in remitting bills circuitously, rather than directly to the place where they are due. The solution of such questions as the above, is called ARBITRATION OF EXCHANGE, and is best determined by THE CHAIN RULE, which is essentially the same as the Rule of Proportion.
We write the quantities which are equivalent to each other, as antecedent and consequent, or as cause and effect, making each effect of the same denomination with the next cause. The like factors on opposite sides are cancelled, and the products divided as in proportion, to obtain the answer.
The question may be otherwise stated in the following manner: If £13 produce 75 roubles, 5 roubles produce 9 marcs, 3 marcs produce $1.00, and $9.00 produce 50 francs, how many francs will £1500 produce? The second set, or set of demand, contains but a single cause and effect. The first, or given set, contains a number of causes and effects, but they are so connected, that all the terms may be multiplied together, as a single compound term. Thus, if £13 produce 75 roubles, and 5 roubles produce 9 marcs, £13 will produce 5 of 9 marcs, and £13×5 will produce 75×9 marcs. In the same way it may be shown that £13×5×3=$75× 9 × 1, and £13×5×3×9=75×9×1×50 francs. Then, how many francs will £1500 produce?
£13 | 1500£
rou. 5 mar. 3
9
$
fr.
75 rou.
9 mar.
1 $ 50 fr.
26. A London merchant wishing to pay 1000 milrees in Lisbon, remits as follows: to Amsterdam at 36 schillings 7 groats per £; thence to Cadiz, at 17 groats for 2 rials of plate; thence to Leghorn, at 17 pezze for 100 rials ; thence to Lisbon, at 1497 rees for 2 pezze. How many pounds did he remit?
27. If a merchant of New York remits $5000 to Havre, at 5fr. 35c. for $1.00; thence to London, at 49fr. for £2; thence to Hamburg, at 1 marc for 1s. 6d. ; and thence to St. Petersburg, at 8 roubles for 17 marcs, how many roubles can he pay with his remittance?
28. If 33 copecks are equal to 5 English pence, 11 English pence are equal to 3 piasters, 13 piasters are equal to 1 florin, and 5 florins are equal to 29 francs, how many francs are equal to 9000 copecks?
29. If a man receives $30 for building 8 rods of wall, and he can purchase 3 barrels of flour for $14, and 3cwt. of sugar for 4 barrels of flour, and 21lb. of tea for 2cwt. of sugar, how many pounds of tea could he purchase by building 17 rods of wall?
30. If $1000 gain $111 in 80 days, how much will $2500 gain in 120 days, at the same rate?
31. If 13 days' work will purchase 1 hogshead of molasses, and 2 hogsheads of molasses are worth 5 tons of hay, and 3 tons of hay are worth 4 bags of coffee, how many bags of coffee can be bought with 39 days' labour?
32. If a man, by walking 3 miles an hour, for 6 hours a day, can accomplish a journey in 12 days, in how many days would a man walk the same distance, at the rate of 2 miles an hour, for 9 hours a day?
33. If 42 bushels of corn, that weighs 514 pounds a bushel, can be bought with 23 bushels of wheat, that weighs 56 pounds a bushel, how much corn, weighing 60 pounds a bushel, would be equivalent to 100 bushels of wheat that weighs 54 pounds a bushel ?
34. If a man travels 240 miles in 8 days, when the days are 12 hours long, how many miles will he travel in 24 days, when the days are 16 hours long?
35. If the freight of 2T. 6cwt. for 28 miles, is $14.50, what will be the freight of 9T. 4cwt. for 96 miles?
36. If 4 men in 3 days of 8 hours, build 40 rods of wall, how many rods will 18 men build in 5 days of 9 hours?
37. If I pay $9.75 for carrying 23 tons 37 miles, how much ought I to pay for carrying 7T. 3cwt. 2qr., 45 miles.
38. How many men, in 24 days of 16 hours, will do three times as much work as 18 men can perform in 36 days of 12 hours?
39. If $495 will support a family of 7 persons, 11 months and 6 days, how much will maintain a family of 9 persons, for 13 months and 10 days?
40. How many persons may be supported a year, with $1014, if $140.25 will support 11 persons 8 weeks? ·
41. If $973.16 yields an interest of $91.25 in 13 months, what will be the interest of $2801 for 173 months, at the same rate?
42. If 3 tons are carried 87 miles for $151, how far may 23 tons be carried for $27.75 ?
43. If 11 men reap 16 acres of grain in 3 days, how many acres can 7 men reap in 15 days?
44. If 14 men, in 5 days, by working 8 hours a day, reap 38 acres of grain, how many men will reap 37 acres in 63 days, by working 9 hours a day?
45. If 12 men dig a trench 124yd. long, 4ft. wide, and
92
2ft. deep, in 6 days, by working 10 hours a day, how long a trench that is 7ft. wide, and 4ft. deep, will 45 men dig in 21 days, working 7 hours a day?
46. If 14 yards of cloth, that is yd. wide, cost $29. 50, what should be the price of 21 yds. of similar quality, that is lyd. wide?
47. If 300 tiles that are 9in. long and 6in. wide, will pave a court-yard, how many tiles would be required that are 6in. long and 4in. wide?
48. What is the weight of 16 iron bars, each 7ft. long, ɓin. wide, and 3 in. thick, if a bar 2ft. long, 2in. wide, and lin. thick, weighs 18 pounds?
49. How many men will build a wall 240yd. long, 6ft. high, and 3ft. thick, in 8 days of 9 hours, if 7 men can build a wall 40yd. long, 4ft. high, and 2ft. thick, in 32 days of 7 hours?
50. If 70 braces of Venice are equal to 75 braces of Leghorn, and 7 braces of Leghorn are equal to 4 yards, how many yards are there in 79.375 braces of Venice?
51. A merchant in New York orders £500 sterling, due him in London, to be sent by the following circuit: to Hamburg, at 15 marcs banco per £; thence to Copenhagen, at 100 marcs banco for 33 rix-dollars; thence to Bourdeaux, at 3 rix-dollars for 18 francs; thence to Lisbon, at 125 francs for 18 milrees; and thence to New York, at $1.25 per milree. What was the arbitrated value of a dollar by this remittance?
52. If the freight of 11 boxes of sugar, each weighing 7cwt. 3qr. 11lb., is $37.50 for 90 miles, what must be paid for the freight of 34 boxes, each weighing 818cwt., for 75 miles?
53. How much wheat, that weighs 607b. per bushel, would be required to supply a garrison of 1400 men 9 months, if 2800 bushels, weighing 581b. per bushel, supply 800 men 3 months?
54. How many hours a day must 15 men work, to dig a trench 400ft. long, 6ft. wide, and 3ft. deep, in 187 days, if 72 men can dig a trench 250ft. long, 8ft. wide, and 4ft. deep, in 314 days, by working 7 hours a day? | https://books.google.co.nz/books?id=odL-pQs64rgC&pg=PA90&focus=viewport&vq=%22the+product+of+the+means+is+equal+to+the+product+of+the+extremes.%22&dq=editions:UOM39015065320957&lr=&output=html_text |
To effectively operate and maintain a network infrastructure, network assets must be efficiently detected and managed. A network asset, as that term is used herein, refers to any resource that can be used on a network. Such a network asset may be a “hard” asset such as, for example, a personal computer or a printer, or a “soft” asset such as, for example, a software application.
In conventional systems for detecting and managing network assets, multiple independent systems are employed to acquire and store various information related to network assets. Such independent systems may include “asset detection systems” and “asset repositories.” An asset detection system, as that term is used herein, refers to a system that automatically detects and registers assets. An asset repository, as that term is used herein, refers to a system in which assets are manually registered and recorded.
An exemplary conventional system for detecting and managing network assets is shown in FIG. 1. As shown, asset detection and monitoring system 100 includes multiple independent systems 120–150 connected to network 110. Multiple independent systems 120–150 include network and system monitoring tools 120, intrusion detection tools 130, digital asset management (DAM) system 140, and grid information service (GIS) 150. Network and system monitoring tools 120 and intrusion detection tools 130 are example of asset detection systems. DAM 140 and GIS 150 are examples of asset repositories.
Network and system monitoring tools 120 are typically employed to monitor a network and/or system for the appearance of new assets. Upon detection of a new asset, network and system monitoring tools 120 automatically register the asset by acquiring registration information from the asset. For a hard asset, such registration information generally includes information such as a number of central processing units (CPU's) operating at the asset or an amount of memory available at the asset. For a soft asset, such registration information generally includes information such as licensing information and information about the device on which the soft asset can be deployed. Such information about the device on which the soft asset is deployed may be acquired by examining a digital signature that is correlated with a digital certificate issued to the corresponding soft asset. Network “sniffers” may be employed to detect such digital signatures and acquire network information to determine the actual deployed location of a soft asset. A unique identifier such as, for example, an integer or a string may also be assigned to the asset.
Intrusion detection tools 130 are typically employed to detect assets and monitor such assets for an intrusion or security breach. Like network and system monitoring tools 120, intrusion detection tools 130 monitor a network and/or system for new assets, and automatically register such new assets. Intrusion detection tools 130 also monitor detected assets for an intrusion or security breach. If an intrusion or breach is detected, detection tool 130 may generate an alert including an identification of the breached asset. A network or system administrator may then suspend current applications of the breached asset and prevent the breached asset from executing new applications.
DAM system 140 is typically employed as a centralized repository for digital files that enables digital content to be archived, searched and retrieved. Digital content may be stored in databases, which are examples of “asset repositories.” Metadata corresponding to the digital content such as photo captions, article key words, advertiser names, contact names, file names or low-resolution thumbnail images is stored in separate databases, that may be referred to as “media catalogs.” Such media catalogs refer to items in the asset repositories. Assets are manually registered in the DAM system.
GIS 150 is also typically employed as a centralized repository for assets on a grid. A grid is a collection of distributed computing infrastructure resources, such as, for example, processors, memory, storage, and services that are available over a local area network (LAN) or wide area network (WAN). GIS 150 provides information such as, for example, the availability, location, functionality and capacity of such resources, so that such resources appear to an end user or application as one large virtual computing system. GIS 150 may be used to monitor assets on the grid and to produce reports relating to selected assets.
Such independent systems 120–150 perform several identical operations. For example, both network and system monitoring tools 120 and intrusion detection tools 130 serve as asset detection systems, and both DAM 140 and GIS 150 serve as asset repositories. However, conventional asset detection and management systems do not include an interface for exchanging identical information among independent component systems 120–150. Such an interface would offer several advantages. For example, such an interface would eliminate the duplicative acquisition of identical registration information by asset detection systems 120–130. Furthermore, such an interface would enable registration information to be automatically acquired by an asset detection system 120–130 and electronically reported to an asset repository 140–150. Thus, registration information would no longer need to be manually entered into asset repositories 140–150.
In addition to electronically reporting registration information, such an interface could be used to electronically verify registration information. Specifically, asset detection systems 120–130 could compare acquired registration information with registration information stored in asset repositories 140–150, and, if inconsistencies are detected, an error message could be generated. Such verification of registration information would be particularly advantageous for asset repositories 140–150 that require registration information to be manually entered and are thus susceptible to human errors.
Furthermore, such an interface could be used to enable independent systems to query one another. For example, GIS 150 could query the DAM 140 to obtain information regarding digital licenses. Thus, there is a need in the art for such an interface between multiple independent systems 120–150.
| |
---
abstract: 'We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen [@Hughen]. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariants help to find them. It is worth noting, that the obtained conditions can be explicitly checked for a given contact sub-Riemannian surface. Also, we consider sub-Riemannian surfaces which are not contact and find their invariants along the surface where the distribution fails to be contact.'
address:
- 'Universidad de Los Andes, Bogota, Colombia'
- 'Kazan State University, Kazan, Russia'
author:
- 'José Ricardo Arteaga & Mikhail Malakhaltsev'
date: 'August 26, 2009'
title: 'Symmetries of sub-Riemannian surfaces'
---
Introduction {#introduction .unnumbered}
============
A sub-Riemannian manifold is a $k$-dimensional distribution endowed by a metric tensor on an $n$-dimensional manifold. At present sub-Riemannian geometry is intensively studied, this is motivated by applications in various fields of science (see, e.ġ. the book [@Montgomery], where many applications of sub-Riemannian geometry are presented; also, for interesting examples, we refer the reader to [@Bloch1], [@Pavlov], [@Schempp], where applications to mechanics, thermodynamics, and biology are given). At the same time, various aspects of the theory of symmetries of sub-Riemannian manifolds are widely investigated because symmetries are always of great importance for applications [@Bloch2], [@Olver]. Many papers are devoted to the theory of homogeneous (in part, symmetric) sub-Riemannian manifolds (see e.g. [@Falbel1], [@Falbel2], [@Hughen], [@Sachkov]). The main investigation tool in these papers is the Lie algebras theory as is usual when we study homogeneous spaces.
In the present paper we study symmetries of sub-Riemannian surfaces, i.ė. of sub-Riemannian manifolds with $k=2$ and $n=3$. Our main goal is to give a practical tool (or an algorithmic procedure) for investigation of symmetries of a sub-Riemannian surface. The paper is organized as follows. In the first section we give in details construction of invariants of a contact sub-Riemannian surface using the Cartan reduction procedure (here we follow [@Hughen]) and show how to calculate them. In the second section we demonstrate how to apply invariants to finding symmetries of a contact sub-Riemannian surface. Finally, in the third section we consider a sub-Riemannian surface without assumption that it is contact and find invariants along the “singular surface”, where the distribution fails to be contact.
Contact sub-Riemannian surfaces
===============================
Let $M$ be an $n$-dimensional manifold and $\Delta$ be a $k$-dimensional distribution on $M$ endowed by a metric tensor field $$\forall p \in M, \quad \langle \cdot,\cdot \rangle_p : \Delta_p \times \Delta_p \to \mathbb{R}.
\label{<eq:0_1>}$$ Then $(M,\Delta,\langle \cdot,\cdot \rangle)$ is called a *sub-Riemannian manifold* [@Montgomery].
In the present paper we consider a *sub-Riemannian surface* $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$, i.e. a two-dimensional distribution $\Delta$ on a three-dimensional manifold $M$, where $\Delta$ is endowed by a metric tensor field $\langle \cdot,\cdot \rangle$. In addition, we assume that *the distribution $\Delta$ and the manifold $M$ are oriented*. Note that we do not suppose that any metric on $M$ is given.
Throughout the paper we will denote the Lie algebra of vector fields on a manifold $N$ by $\mathfrak{X}(N)$, and the space of covector fields by $\mathfrak{X}(N)^*$. Also the space of $r$-forms on $N$ will be denoted by $\Lambda^r(N)$.
$G$-structure associated with a sub-Riemannian surface
------------------------------------------------------
### Elements of theory of $G$-structures {#subsubsec:g_structures}
Recall notions and results of the theory of $G$-structures we use in the present paper (for the details we refer the reader to [@Montgomery] and [@KN]).
#### Tautological forms, pseudoconnection form, and structure equations
Let $M$ be a smooth $n$-dimensional manifold, and $\pi : B(M) \to M$ be the coframe bundle of $M$.
On $B(M)$ the *tautological forms* $\theta^a \in \Omega^1(B(M))$ are defined as follows [@KN]. For a point $\xi \in B(M)$ ($\xi=\{\xi^a\}_{a=\overline{1,n}}$ is a coframe of $T_p M$, where $p = \pi(\xi)$), we set $$\theta^a_\xi : T_\xi (B(M)) \to \mathbb{R}, \quad \theta^a_\xi(X) = \xi^a(d\pi(X)).
\label{eq:1_11}$$
Now, on a neighborhood $U$ of a point $p \in M$, take a coframe field $\eta = \{\eta^a\}$. This gives a trivialization $\alpha : \pi^{-1}(U) \to U \times GL(n)$: to a coframe $\xi$ at $p \in U$ we assign $(p,g) \in U \times GL(n)$ such that $\xi^a = \tilde g^a_b \eta^b_p$, where $||\tilde g^a_b || = g^{-1}$.
For a coframe field $\eta$ on $U$ let us consider the pullback $1$-forms $\bar\eta^a = d\pi^*\eta^a$ on $U \times GL(n) \cong \pi^{-1}(U) \subset B(M)$. Then $$\theta^a_{(p,g)} = \tilde g^a_b \bar\eta^b_{(p,g)} = \tilde g^a_b d\pi^* \eta^b_p.
\label{eq:1_12}$$
A *$G$-structure* $P \to M$ is a principal subbundle of $\pi : B(M) \to M$ with structure group $G \subset GL(n)$. The tautological forms on $P$ are the restrictions of $\theta^a$ to $P$ and will be denoted by the same letters.
Let us denote by $\mathfrak{g}$ the Lie algebra of the Lie group $G$. A *pseudoconnection form* $\omega$ on a $G$-structure $\pi : P \to M$ is a $\mathfrak{g}$-valued 1-form on $P$ such that $\omega(\sigma(a))=a$, where $\sigma(a)$ is the fundamental vector field ([@KN], Ch. I, Sec. 5) on $P$ corresponding to $a \in \mathfrak{a}$ .
Given a pseudoconnection form $\omega$, we have *structure equations* on $P$: $$d\theta^a = \omega^a_b \wedge \theta^b + T^a_{bc} \theta^b \wedge \theta^c
\label{eq:1_13}$$ where the functions $T^a_{bc} : P \to \mathbb{R}$ uniquely determined by equations are called *torsion functions*, and the map $T : P \to \Lambda^2 \mathbb{R}^n \otimes \mathbb{R}^n$, $ \xi \to \{T^a_{bc}(\xi)\}$, is called the *torsion* of the pseudoconnection $\omega^a_b$.
#### Structure function
Let us find how the torsion changes under change of the pseudoconnection. If $\omega^a_b$, $\hat\omega^a_b$ are pseudoconnections on $P$, then $\mu^a_b = \hat{\omega}^a_b-\omega^a_b$ is a $\mathfrak{g}$-valued form on $P$ with property that $\mu(\sigma(a)) = 0$ for any $a \in \mathfrak{g}$. Then $\mu^a_b = \mu^a_{bc} \theta^c$.
$$\begin{gathered}
d\theta^{a} = \hat\omega^{a}_{b} \wedge \theta^{b} + \hat T^{a}_{bc} \theta^{b} \wedge \theta^{c} =
\left (\omega^{a}_{b} + \mu^{a}_{bc}\theta^{c}\right )\wedge \theta^{b} + \hat T^{a}_{bc} \theta^{b} \wedge \theta^{c} =
\\
\omega^{a}_{b} \wedge \theta^{b} + \left (\hat T^{a}_{bc} - \mu^{a}_{[bc]}\right ) \theta^{b} \wedge \theta^{c} =
\omega^{a}_{b}\wedge \theta^{b} + T^{a}_{bc} \theta^{b} \wedge \theta^{c}\end{gathered}$$
Hence follows that $$\hat{\omega}^a_b=\omega^a_b + \mu^a_{bc}\theta^c \Rightarrow \hat T^{a}_{bc} = T^{a}_{bc} + \mu^{a}_{[bc]}
\label{eq:1_31}$$ Let us define the Spencer operator $\delta$ from the space of tensors $T^2_1(\mathbb{R}^n)$ of type $(2,1)$ to the space $\Lambda^2(\mathbb{R}^n) \otimes \mathbb{R}^n$ as follows: $$\delta : t^a_{bc} \in T^2_1(\mathbb{R}^n) \mapsto t^a_{[bc]} = \frac{1}{2}(t^a_{bc} - t^a_{cb}).
\label{eq:1_32}$$ Note that $\mathfrak{g}\otimes(\mathbb{R}^n)^* \subset \mathfrak{gl}(n) \otimes \mathbb{R}^* \cong T^2_1(\mathbb{R}^n)$ and we will denote the restriction of $\delta$ to $\mathfrak{g}\otimes(\mathbb{R}^n)^*$ by the same letter $\delta$. Thus, can be rewritten as follows: $$\hat{\omega}^a_b=\omega^a_b + \mu^a_{bc}\theta^c \Rightarrow \hat T^{a}_{bc} = T^{a}_{bc} + \delta(\mu^{a}_{bc}).
\label{eq:1_32_1}$$ From we conclude that *if $\delta: \mathfrak{g}\otimes(\mathbb{R}^n)^* \to \Lambda^2(\mathbb{R}^n) \otimes \mathbb{R}^n$ is a monomorphism, then, pseudoconnections $\omega^a_b$, $\hat\omega^a_b$ with the same torsion $T^a_{bc}$ coincide*.
Now denote $$\mathcal{T} = \frac{\Lambda^2(\mathbb{R}^n) \otimes \mathbb{R}^n}{\delta(\mathfrak{g}\otimes(\mathbb{R}^n)^*)}.
\label{eq:1_33}$$ From it follows that one can correctly define the *structure function*: $$\mathcal{C} : P \to \mathcal{T},
\quad
\xi \mapsto [T^a_{bc}(\xi)].
\label{eq:1_34}$$
#### $G$-equivariance of structure function
The group $G$ acts on $\Lambda^2(\mathbb{R}^n) \otimes \mathbb{R}^n$ from the right as follows: $$(\bar\rho(g)T)^a_{bc} = \tilde g^a_r T^r_{pq} g^p_b g^q_c
\label{eq:1_35}$$ and one can easily prove that the subspace $\delta(\mathfrak{g}\otimes(\mathbb{R}^n)^*)$ is invariant under this action. Then we have the following $G$-action on $\mathcal{T}$: $$\forall g \in G, \quad \rho(g) :
\mathcal{T} \to \mathcal{T},
\quad
[T^a_{bc}] \mapsto [\tilde g^a_r T^r_{pq} g^p_b g^q_c]
\label{eq:1_36}$$ By cumbersome calculations, from the structure equations one can obtain that $$\mathcal{C}(\xi g) = \mathcal{C}(g^{-1}\xi) = \rho(g) \mathcal{C}(\xi), \forall \xi \in P, g \in G.
\label{eq:1_37}$$
If $\omega$ is a connection, one can prove that $T^a_{bc} (\xi g) = \tilde g^a_r T^r_{pq}(\xi) g^p_b g^q_c$, however it is wrong if $\omega$ is a pseudoconnection. In this case, we have only that $T^a_{bc} (\xi g) = \tilde g^a_r T^r_{pq}(\xi) g^p_b g^q_c + \nu^a_{bc}$, where $\nu^a_{bc} \in \delta(\mathfrak{g}\otimes(\mathbb{R}^n)^*)$.
### Cartan reduction {#subsubsec:cartan_reduction}
Let $P \to M$ be a $G$-structure. Let $\mathcal{T} = \sqcup \mathcal{T}_\alpha$ be the decomposition of $\mathcal{T}$ into orbits of the $G$-action . Assume that the structure function $c$ takes values in one orbit $\mathcal{T}_0$, only.
Fix $\tau_0 \in \mathcal{T}_0$. Then $$P_1 = \{\xi \mid \mathcal{C}(\xi) = \tau_0\}
\label{eq:1_38}$$ is the total space of a principal $G_1$-subbundle of $P$, where $$G_1 = \{ g \in G \mid \rho(g)\tau_0 = \tau_0 \}.
\label{eq:1_39}$$
They say that the $G_1$-structure $P_1 \to M$ is obtained by the *Cartan reduction* from the $G$-structure $P \to M$.
$G$-structure associated to a sub-Riemannian surface
----------------------------------------------------
Let $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ be a sub-Riemannian surface. We say that a coframe $\eta = \left(\eta^{1}, \eta^{2}, \eta^{3} \right)$ of $T_p M$, $p \in M$, is adapted to $\mathcal S$ if
- $\eta$ is positively oriented, and $(\eta^1 |_{\Delta_p}, \eta^2 |_{\Delta_p})$ is a positively oriented coframe of $\Delta_p$;
- $\eta^3 \in Ann(\Delta)_p$, or, equivalently, $\eta^3 (W) = 0$ for any $W \in \Delta_p$;
- $ \langle W , W \rangle = [\eta^{1}(W)]^{2} + [\eta^{2}(W)]^2$ for any $W \in \Delta_p$.
To a given sub-Riemannian surface $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ we associate the principal subbundle $B_0 \subset B$ consisting of adapted frames. It is clear that the structure group of $B_0$ is $$\nonumber
G_{0}=
\left \{
\left (
\begin{array}{cc}
A & B \\
0 & c
\end{array}
\right )\mid A \in SO(2), B=
\left (
\begin{array}{cc}
b_{1} \\
b_{2}
\end{array}
\right )
\in \mathbf{R}^{2},
c\in \mathbf{R}\backslash \left \{0\right \}
\right \}.$$ One can easily prove
\[prop:1\_1\] A sub-Riemannian surface $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ is equivalent to a $G_0$-structure on $M$.
### Contact sub-Riemannian surfaces
#### Contact distributions
Let $\omega$ be a $1$-form on a $(2n+1)$-dimensional manifold $M$. The form $\omega$ is said to be *contact* if $$\underbrace{\omega \wedge d\omega \wedge \ldots \wedge d\omega}_n \ne 0.
\label{eq:1_300}$$ A $1$-form is contact if and only if $\omega$ is nonvanishing (hence the Pfaff equation $\omega=0$ determines a $2n$-dimensional distribution $\Delta$), and $d\omega |_\Delta$ is nondegenerate.
Let $\Delta$ be a $2n$-dimensional distribution on a $(2n+1)$-dimensional manifold $M$. Denote by $Ann(\Delta)$ the vector subbundle in $T^* M$ of rank $1$ such that the fiber of $Ann(\Delta)$ at $p \in M$ is $$Ann(\Delta)_p = \{\omega \in T^*_p M \mid \omega(W) = 0 \ \forall W \in \Delta_p\}.
\label{eq:1_1}$$ The distribution $\Delta$ is said to be *contact* if for each $p \in M$ there exists a contact section $\omega$ of $Ann(\Delta)$ in a neighborhood of $p$. Note that this definition does not depend on the choice of $\omega$: *if $\Delta$ is contact, then any nonvanishing section of $Ann(\Delta)$ is contact*.
We say that a sub-Riemannian surface $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ is *contact* if the distribution $\Delta$ is contact.
\[Th:Theorema-1\] Any contact sub-Riemannian surface $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ uniquely determines an $SO(2)$-structure $B_{2}\to M$ and a connection on $B_2$ with the connection form $$\omega =
\left (
\begin{array}{ccc}
0 & \alpha & 0 \\
-\alpha & 0 & 0 \\
0 & 0 & 0
\end{array}
\right ).
\label{eq:1_14}$$ The $1$-form $\alpha$ in together with the tautological forms $\theta^a$ give a coframe field on $B_2$. The structure equations are written as follows: $$\left (
\begin{array}{c}
d\theta^{1}\\
d\theta^{2}\\
d\theta^{3}
\end{array}
\right )
=
\wedge
\left (
\begin{array}{c}
\theta^{1}\\
\theta^{2}\\
\theta^{3}
+
a_{1} & a_{2} & 0 \\
a_{2} & -a_{1} & 0 \\
0 & 0 & 1
\end{array}
\right )
\left (
\begin{array}{c}
\theta^{2}\wedge\theta^{3}\\
\theta^{3}\wedge\theta^{1}\\
\theta^{1}\wedge\theta^{2}
\end{array}
\right )
\label{eq:1_15}$$
This theorem was proved by K. Hughen in [@Hughen] (also a sketch of the proof is given in [@Montgomery], Ch. 7, 7.10). We present here a detailed proof of this theorem which is based on the Cartan reduction procedure as it was exposed in \[subsubsec:g\_structures\] and \[subsubsec:cartan\_reduction\].
### Proof of Theorem \[Th:Theorema-1\]
**Step 1**. We start with the $G_0$-structure $B_0 \to M$ associated with $\mathcal{S}$ (see Proposition \[prop:1\_1\]). We have $$G_{0}=\left \{
\left.
\left(
\begin{array}{ccc}
\cos\varphi & -\sin\varphi & b_{1}\\
\sin\varphi & \cos\varphi & b_{2} \\
0 & 0 & c
\end{array}
\right)
\ \right| c \ne 0
\right\}
\label{eq:1_201}$$ Then the Lie algebra of $G_0$ is $$\mathfrak{g}_{0}=\left \{
\left.
\left (
\begin{array}{ccc}
0 & \alpha & \beta_{1}\\
-\alpha & 0 & \beta_{2} \\
0 & 0 & \gamma
\end{array}
\right )
\ \right|
\alpha, \beta_1, \beta_2, \gamma \in \mathbb{R}
\right\}
\label{eq:1_202}$$ Now we will calculate $$\mathcal{T}_0 =
\frac{\Lambda^2(\mathbb{R}^3) \otimes \mathbb{R}^3}{\delta(\mathfrak{g}_0\otimes(\mathbb{R}^3)^*)},
\label{eq:1_203}$$ (see ). Let us denote the standard basis of $\mathbb{R}^3$ by $e_1$, $e_2$, $e_3$, and the dual basis by $e^1$, $e^2$, $e^3$. Then the basis of $\mathfrak{g}_0$ is $$\mathcal{E}_1 = e_1 \otimes e^2 - e_2 \otimes e^1;
\mathcal{E}_2 = e_1 \otimes e^3;
\mathcal{E}_3 = e_2 \otimes e^3;
\mathcal{E}_4 = e_3 \otimes e^3;
\label{eq:1_204}$$ and $\mathcal{E}_i \otimes e^a$, $i=\overline{1,4}$, $a=\overline{1,3}$, is the basis of $\mathfrak{g}_0 \otimes (\mathbb{R}^3)^*$. Then $\delta: \mathfrak{g}_0 \otimes(\mathbb{R}^n)^* \to \Lambda^2(\mathbb{R}^n) \otimes \mathbb{R}^n$ acts on the basis elements as follows $$\begin{cases}
\mathcal{E}_1 \otimes e^1 \mapsto - e_1 \otimes e^2 \wedge e^1,
\quad
\mathcal{E}_1 \otimes e^2 \mapsto - e_2 \otimes e^1 \wedge e^2,
\\
\mathcal{E}_1 \otimes e^3 \mapsto e_1 \otimes e^2 \wedge e^3 - e_2 \otimes e^1 \wedge e^3,
\\
\mathcal{E}_2 \otimes e^1 \mapsto e_1 \otimes e^3 \wedge e^1,
\quad
\mathcal{E}_2 \otimes e^2 \mapsto e_1 \otimes e^3 \wedge e^2,
\quad
\mathcal{E}_2 \otimes e^3 \mapsto 0,
\\
\mathcal{E}_3 \otimes e^1 \mapsto e_2 \otimes e^3 \wedge e^1,
\quad
\mathcal{E}_3 \otimes e^2 \mapsto e_2 \otimes e^3 \wedge e^2,
\quad
\mathcal{E}_3 \otimes e^3 \mapsto 0,
\\
\mathcal{E}_4 \otimes e^1 \mapsto e_3 \otimes e^3 \wedge e^1,
\quad
\mathcal{E}_4 \otimes e^2 \mapsto e_3 \otimes e^3 \wedge e^2,
\quad
\mathcal{E}_4 \otimes e^3 \mapsto 0,
\end{cases}
\label{eq:1_205}$$ From we get that $\mathcal{T}_0$ is spanned by $[e_3 \otimes e^1 \wedge e^2]$ and so is one-dimensional.
Now let us find the action of $G_0$ on $\mathcal{T}_0$. From we get that, for any $g \in G_0$, $$\rho(g) [e_3 \otimes e^1 \wedge e^2] = [ \tilde g^a_3 g^1_b g^2_c e_a \otimes e^b \wedge e^c] = [\tilde g^3_3 (g^1_1 g^2_2 - g^2_1 g^1_2) e_3 \otimes e^1 \wedge e^2 ] = c^{-1} [e_3 \otimes e^1 \wedge e^2].
\label{eq:1_206}$$ Hence the action of $G_0$ on $\mathcal{T}_0$ has two orbits: $\mathcal{O}_0 = \{0 \in \mathcal{T}_0\}$ and $\mathcal{O}_1 = \{t \in \mathcal{T}_0 \mid t \ne 0\}$.
Let us prove that, if $\mathcal{S}$ is contact, the structure function $\mathcal{C}$ takes values in $\mathcal{O}_1$. The structure equations can be written as follows : $$\left (
=
0 & \alpha &\beta \\
-\alpha & 0 & \gamma \\
0 & 0 & \delta
+
T_{23}^{1} & T_{31}^{1} &T_{12}^{1} \\
T_{23}^{2} & T_{31}^{2} &T_{12}^{2} \\
T_{23}^{3} & T_{31}^{3} &T_{12}^{3}
\label{eq:1_207}$$ Now take a section $s : U \to B_0$, that is a coframe field $\eta^a$ adapted to $\mathcal{S}$ on $U$. Then from the definition of the tautological forms we have that $ds^* \theta^a = \eta^a$, hence $ds^*(d\theta^3 \wedge \theta^3) = d\eta^3 \wedge \eta^3 \ne 0$ because $\eta^3$ is a contact form (see ). At the same time, from , we get that $d\theta^3 \wedge \theta^3 = T^3_{12} \theta^1 \wedge \theta^2 \wedge \theta^3$, hence follows that $T^3_{12} \ne 0$. Thus $\mathcal{C}(s(p)) = [T^3_{12}(s(p)) e_3 \otimes e^1 \wedge e^2] \ne 0$ for any $p \in U$. As each $\xi \in \pi^{-1}(U)$ can be written as $\xi = s(p) g$, $p = \pi(\xi)$, and the structure function $\mathcal{C}$ satisfies , we have that $\mathcal{C}(\xi) \ne 0$ for any $\xi \in B_0$, hence $\mathcal{C}$ takes values in $\mathcal{O}_1$.
Thus, we can make the Cartan reduction and pass to the $G_1$-structure $B_1 \to M$, where $$B_1 = \{\xi \mid \mathcal{C}(\xi) = [e_3 \otimes e^1 \wedge e^2]\}
\label{eq:1_208}$$ is the total space of a principal $G_1$-subbundle of $B_0$, and $$G_1 = \{ g \in G \mid \rho(g)[e_3 \otimes e^1 \wedge e^2] = [e_3 \otimes e^1 \wedge e^2] \}
0 & 0 & 1
\end{array}
\right)
\right\}
\label{eq:1_209}$$
**Step 2**. The Lie algebra of $G_1$ is $$\mathfrak{g}_{1}=\left \{
0 & 0 & 0
\end{array}
\right )
\ \right|
\alpha, \beta_1, \beta_2, \in \mathbb{R}
\right\}
\label{eq:1_210}$$ and, by the construction of $B_1$, the structure equations have the form: $$\left (
=
+
T_{23}^{3} & T_{31}^{3} & 1
\label{eq:1_210_1}$$ With notation of the previous step we have the basis of $\mathfrak{g}_1$ is $$\mathcal{E}_1 = e_1 \otimes e^2 - e_2 \otimes e^1;
\mathcal{E}_2 = e_1 \otimes e^3;
\mathcal{E}_3 = e_2 \otimes e^3;
\label{eq:1_211}$$ and $\mathcal{E}_i \otimes e^a$, $i=\overline{1,3}$, $a=\overline{1,3}$, is the basis of $\mathfrak{g}_1 \otimes (\mathbb{R}^3)^*$. Then $\delta: \mathfrak{g}_1 \otimes(\mathbb{R}^n)^* \to \Lambda^2(\mathbb{R}^n) \otimes \mathbb{R}^n$ acts on the basis elements as follows $$\begin{cases}
\\
\\
\\
\end{cases}
\label{eq:1_212}$$ From we get that $$\mathcal{T}_1 =
\frac{\Lambda^2(\mathbb{R}^3) \otimes \mathbb{R}^3}{\delta(\mathfrak{g}_1\otimes(\mathbb{R}^3)^*)},
\label{eq:1_213}$$ is spanned by $[e_3 \otimes e^2 \wedge e^3]$, $[e_3 \otimes e^3 \wedge e^1]$, $[e_3 \otimes e^1 \wedge e^2]$ and so is three-dimensional.
At the same time, by construction of $B_1$, the structure function $\mathcal{C}$ takes values in the affine subspace $$\mathcal{T}'_1 = \{ u [e_3 \otimes e^2 \wedge e^3] + v [e_3 \otimes e^3 \wedge e^1] + [e_3 \otimes e^1 \wedge e^2] \} \subset \mathcal{T}_1
\label{eq:1_214}$$
Let us find the action of $G_1$ on $\mathcal{T}_1$. Using , we find that $$\begin{aligned}
\rho(g) [e_3 \otimes e^2 \wedge e^3] &=& \cos\varphi [e_3 \otimes e^2 \wedge e^3] - \sin\varphi [e_3 \otimes e^3 \wedge e^1]
\\
\rho(g) [e_3 \otimes e^3 \wedge e^1] &=& \sin\varphi [e_3 \otimes e^2 \wedge e^3] + \cos\varphi [e_3 \otimes e^3 \wedge e^1]
\\
\rho(g) [e_3 \otimes e^1 \wedge e^2] &=& (-b_1 \cos\varphi - b_2 \sin\varphi) [e_3 \otimes e^2 \wedge e^3] +
\\
&& (b_1 \sin\varphi - b_2 \cos\varphi)[e_3 \otimes e^3 \wedge e^1] + [e_3 \otimes e^1 \wedge e^2]\end{aligned}$$ From this follows that $\rho(g)$ maps $\mathcal{T}'_1$ into itself, and moreover, $$\begin{gathered}
\rho(g) \big(u [e_3 \otimes e^2 \wedge e^3] + v [e_3 \otimes e^3 \wedge e^1] + [e_3 \otimes e^1 \wedge e^2]\big) =
\\
\{(u-b_1)\cos\varphi+(v-b_2)\sin\varphi\}[e_3 \otimes e^2 \wedge e^3] +
\\
\{-(u-b_1)\sin\varphi+(v-b_2)\cos\varphi\}[e_3 \otimes e^3 \wedge e^1] + [e_3 \otimes e^1 \wedge e^2]
\label{eq:1_215}\end{gathered}$$
Thus, we can make the Cartan reduction and pass to the $G_2$-structure $B_2 \to M$. We take $$\tau_1 = 0 \cdot [e_3 \otimes e^2 \wedge e^3] + 0 \cdot [e_3 \otimes e^3 \wedge e^1] + [e_3 \otimes e^1 \wedge e^2] \in \mathcal{T}_1
\label{eq:1_216}$$ and set $$B_2 = \{\xi \mid \mathcal{C}(\xi) = \tau_1\}
\label{eq:1_217}$$ is the total space of a principal $G_2$-subbundle of $B_1$, and $$G_2 = \{ g \in G_1 \mid \rho(g)\tau_1 = \tau_1 \}
=\left \{
\left(
\begin{array}{ccc}
\cos\varphi & -\sin\varphi & 0\\
\sin\varphi & \cos\varphi & 0 \\
\label{eq:1_218}$$
**Step 3**. The Lie algebra of $G_2$ is $$\mathfrak{g}_{2}=\left \{
\left.
\alpha \in \mathbb{R}
\right\}
\label{eq:1_219}$$ and the structure function $\mathcal{C}(\xi) = \tau_1$, for any $\xi \in B_2$, hence the structure equations are written as follows: $$\left (
=
+
\label{eq:1_220}$$ The basis of $\mathfrak{g}_2$ is $$\mathcal{E}_1 = e_1 \otimes e^2 - e_2 \otimes e^1;
\label{eq:1_221}$$ and $\mathcal{E}_1 \otimes e^a$, $a=\overline{1,3}$, is the basis of $\mathfrak{g}_2 \otimes (\mathbb{R}^3)^*$. Then $\delta: \mathfrak{g}_2 \otimes(\mathbb{R}^3)^* \to \Lambda^2(\mathbb{R}^3) \otimes \mathbb{R}^3$ acts on the basis elements as follows $$\begin{cases}
\mathcal{E}_1 \otimes e^1 \mapsto e_1 \otimes e^1 \wedge e^2,
\quad
\mathcal{E}_1 \otimes e^2 \mapsto - e_2 \otimes e^1 \wedge e^2,
\\
\mathcal{E}_1 \otimes e^3 \mapsto e_1 \otimes e^2 \wedge e^3 + e_2 \otimes e^3 \wedge e^1,
\end{cases}
\label{eq:1_222}$$ Hence follows immediately that $\delta$ *is a monomorphism*.
From we get that $$\mathcal{T}_2 =
\frac{\Lambda^2(\mathbb{R}^3) \otimes \mathbb{R}^3}{\delta(\mathfrak{g}_2\otimes(\mathbb{R}^3)^*)},
\label{eq:1_223}$$ is spanned by $$\begin{aligned}
&&[e_1 \otimes e^2 \wedge e^3] - [e_2 \otimes e^3 \wedge e^1], [e_1 \otimes e^3 \wedge e^1], [e_2 \otimes e^2 \wedge e^3],
\\
&&[e_3 \otimes e^2 \wedge e^3], [e_3 \otimes e^3 \wedge e^1], [e_3 \otimes e^1 \wedge e^2]
\label{eq:1_224}\end{aligned}$$ and so is six-dimensional.
However, by construction of $B_2$, the structure function $\mathcal{C}$ takes values in the affine subspace $$\begin{split}
\mathcal{T}'_2 = \{ u ([e_1 \otimes e^2 \wedge e^3] - [e_2 \otimes e^3 \wedge e^1]) + v [e_1 \otimes e^3 \wedge e^1] +
\\
w [e_2 \otimes e^2 \wedge e^3] + [e_3 \otimes e^1 \wedge e^2] \} \subset \mathcal{T}_2
\label{eq:1_225}
\end{split}$$ Hence follows that the structure equations have the following form $$\left (
=
+
u & v & 0 \\
w & - u & 0 \\
\label{eq:1_226}$$
Let us now prove that the form $\alpha$ which satisfies is unique, In $\Lambda^2(\mathbb{R}^3) \otimes \mathbb{R}^3$ consider subspace $N$ spanned by $$\begin{aligned}
&&e_1 \otimes e^2 \wedge e^3 - e_2 \otimes e^3 \wedge e^1, e_1 \otimes e^3 \wedge e^1, e_2 \otimes e^2 \wedge e^3,
\\
&&e_3 \otimes e^2 \wedge e^3, e_3 \otimes e^3 \wedge e^1, e_3 \otimes e^1 \wedge e^2.\end{aligned}$$ It is clear that we have the direct sum $$\Lambda^2(\mathbb{R}^3) \otimes \mathbb{R}^3 = N \oplus \delta(\mathfrak{g}_2\otimes(\mathbb{R}^3)^*)
\label{eq:1_227}$$ and $\{T^a_{bc}\}$ from takes values in $N$, If we have another $\hat\alpha$ and the corresponding torsion $\{\hat T^a_{bc}\}$ which satisfy , then $\{\hat T^a_{bc}\}$ also take values in $N$, so the same is true for $\{\hat T^a_{bc} - T^a_{bc}\}$. However, from it follows that $\hat T^a_{bc} - T^a_{bc} = \delta(\mu^a_{bc})$. As we have the direct sum decomposition , we obtain that $\hat T^a_{bc} - T^a_{bc} = 0$ and $\delta(\mu^a_{bc})=0$. But $\delta$ is a monomorphism (see ), hence $\mu^a_{bc} = 0$, and so $\hat\omega^a_b = \hat\omega^a_b$ (see ). Thus $\hat\alpha = \alpha$.
To finish the proof of the theorem it is sufficient to prove that in $v = w$. From we get $d\theta^3 = \theta^1 \wedge \theta^2$, then, again using , we obtain $$0 = d\theta^1 \wedge \theta^2 - \theta^1 \wedge d\theta^2 = v \theta^3 \wedge \theta^1 \wedge \theta^2 - w \theta^1 \wedge \theta^2 \wedge \theta^3 = (v-w)\theta^1 \wedge \theta^2 \wedge \theta^3.
\label{eq:1_228}$$ Now we set $u = a_1$, $v = w = a_2$ and from get the structure equations .
Thus we have proved that for any contact sub-Riemannian surface $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ there exists an $SO(2)$-structure on $M$ and a unique pseudoconnection form $\alpha$ such that the structure equations hold true. The uniqueness of the $SO(2)$-structure $B_2$ will be proved later, in Corollary \[cor:1\] of Proposition \[prop:2\].
### The functions $a_1$, $a_2$ and $1$-form $\alpha$ in terms of structure functions of a local frame
Let $\eta = \{\eta^a\}$ be a coframe field in a neighborhood $U$ of $p \in M$ which is a section of $B_2 \to M$. Let $d\eta^a = C^a_{bc} \eta^b \wedge \eta^c$ be the corresponding structure equations. Then, for $\bar\eta^a = d\pi^*\eta^a$, we have $d\bar{\eta}^a = \bar{C}^a_{bc} \bar{\eta}^b \wedge \bar{\eta}^c$, where $\bar{C}^a_{bc} = \pi^* C^a_{bc} = C^a_{bc} \circ \pi : \pi^{-1}(U) \to \mathbb{R}$.
Let $\psi : \pi^{-1}(U) \to U \times SO(2)$ be a local trivialization of $\pi : B_2 \to M$ determined by $\eta$, then $$\psi^{-1} (p,g(\varphi)) = g(\varphi)^{-1} \eta_p,$$ where $$g(\varphi)=
\right ),\qquad
\eta_p = \left( \begin{array}{c} \eta^1_p \\ \eta^2_p \\ \eta^3_p \end{array} \right).
\label{eq:1_16}$$
\[prop:2\] *a)* If a coframe field $\eta = \{\eta^a\}$ is a local section of $B_2 \to M$, then $d\eta^3 = \eta^1 \wedge \eta^2$.
*b)* The form $\alpha$ is expressed in terms of $\bar C^a_{bc}$ as follows: $$\label{eq:1_90}
\alpha = d\varphi + \bar C^{1}_{12}\bar\eta^{1}+\bar C^{2}_{12}\bar\eta^{2}-\frac{1}{2}\left (\bar C^{1}_{23}+\bar C^{2}_{31}\right )\bar\eta^{3}$$
*c)* The functions $a_{1}$ and $a_{2}$ are expressed in terms of $\bar C^a_{bc}$ as follows: $$\begin{aligned}
a_{1}&=&
\cos 2\varphi \left (\dfrac{\bar C^{1}_{23}-\bar C^{2}_{31}}{2}\right )+\sin 2\varphi\left (\bar C^{1}_{31}\right )
\\
a_{2}&=& -\sin 2\varphi \left (\dfrac{\bar C^{1}_{23}-\bar C^{2}_{31}}{2}\right )+\cos 2\varphi \left (\bar C^{1}_{31}\right )\end{aligned}$$
If a coframe field $\eta = \{\eta^a\}$ is a local section of $B_2 \to M$, then the equations are written as follows: $$\label{eq:1_17}
\begin{cases}
\theta^{1}= \cos\varphi \bar\eta^{1} + \sin\varphi\bar\eta^{2}\\
\theta^{2}= -\sin\varphi \bar\eta^{1} + \cos\varphi\bar\eta^{2}\\
\theta^{3}= \bar\eta^{3}
\end{cases}
\iff
\begin{cases}
\bar\eta^{1}= \cos\varphi \theta^{1} - \sin\varphi\theta^{2}\\
\bar\eta^{2}= \sin\varphi \theta^{1} + \cos\varphi\theta^{2}\\
\bar\eta^{3}= \theta^{3}
\end{cases}$$ Then $$\label{eq:1_18}
\begin{cases}
\theta^{2}\wedge\theta^{3}=\sin\varphi\bar\eta^{3}\wedge\bar\eta^{1}+\cos\varphi\bar\eta^{2}\wedge\bar\eta^{3}\\
\theta^{3}\wedge\theta^{1}=\cos\varphi\bar\eta^{3}\wedge\bar\eta^{1}-\sin\varphi\bar\eta^{2}\wedge\bar\eta^{3}\\
\theta^1 \wedge \theta^2 = \bar\eta^1 \wedge \bar\eta^2
\end{cases}$$ $$\label{eq:1_19}
\begin{cases}
d\bar\eta^{1}=-\sin\varphi d\varphi\wedge\theta^{1} + \cos\varphi d\theta^{1}-\cos\varphi d\varphi \wedge\theta^{2} - \sin\varphi d\theta^{2}\\
d\bar\eta^{2}=\cos\varphi d\varphi\wedge\theta^{1} + \sin\varphi d\theta^{1}-\sin\varphi d\varphi \wedge\theta^{2} + \cos\varphi d\theta^{2}\\
d\bar\eta^{3}=d\theta^{3}
\end{cases}$$ To we substitute $d\theta^a$ from and then $\theta^a$ from and use , finally we arrive at $$\label{eq:1_20}
\begin{cases}
d\bar\eta^{1}=&(\alpha-d\varphi) \wedge \bar\eta^{2}+
\left (a_{1}\cos 2\varphi - a_{2}\sin 2\varphi\right )\bar\eta^{2}\wedge\bar\eta^{3}+
\\
&\left (a_{1}\sin 2\varphi +
a_{2}\cos 2\varphi\right )\bar\eta^{3}\wedge\bar\eta^{1}
\\
d\bar\eta^{2}=&(d\varphi - \alpha) \wedge\bar\eta^{1} +
\left (a_{1}\sin 2\varphi + a_{2}\cos 2\varphi\right )\bar\eta^{2}\wedge\bar\eta^{3} +
\\
& \left (-a_{1}\cos 2\varphi + a_{2}\sin 2\varphi\right )\bar\eta^{3}\wedge\bar\eta^{1}
\\
d\bar\eta^{3}=&\bar\eta^{1}\wedge\bar\eta^{2}
\end{cases}$$ Let us set $\alpha - d\varphi = p_1 \bar\eta^1 + p_2 \bar\eta^2 + p_3 \bar\eta^3 + p_4 d\varphi$ and substitute to . We get $$\label{eq:1_21}
\begin{cases}
d\bar\eta^{1}=&
\left(a_{1}\cos 2\varphi - a_{2}\sin 2\varphi - p_3 \right)\bar\eta^{2}\wedge\bar\eta^{3}+
\left (a_{1}\sin 2\varphi + a_{2}\cos 2\varphi\right )\bar\eta^{3}\wedge\bar\eta^{1} +
\\
& p_1 \bar\eta^1 \wedge \bar\eta^2 + p_4 d\varphi \wedge \bar\eta^2
\\
d\bar\eta^{2} =& \left (a_{1}\sin 2\varphi + a_{2}\cos 2\varphi\right )\bar\eta^{2}\wedge\bar\eta^{3} +
\left (-a_{1}\cos 2\varphi + a_{2}\sin 2\varphi -
p_3 \right)\bar\eta^{3}\wedge\bar\eta^{1} +
\\
&p_2 \bar\eta^1 \wedge \eta^2 + p_4 \bar\eta^1 \wedge d\varphi
\\
d\bar\eta^{3}=&\bar\eta^{1}\wedge\bar\eta^{2}
\end{cases}$$ At the same time, $d\bar{\eta}^a = \bar{C}^a_{bc} \bar{\eta}^b \wedge \bar{\eta}^c$, hence we get that $p_4 =0$ and $\bar C^a_{bc}$ are expressed as follows: $$\begin{aligned}
&&
\bar C^1_{23} = a_{1}\cos 2\varphi - a_{2}\sin 2\varphi - p_3
\label{eq:1_22_1}
\\
&&
\bar C^1_{31} = a_{1}\sin 2\varphi + a_{2}\cos 2\varphi
\label{eq:1_22_2}
\\
&&
\bar C^1_{12} = p_1
\label{eq:1_22_3}
\\
&&
\bar C^2_{23} = a_{1}\sin 2\varphi + a_{2}\cos 2\varphi
\label{eq:1_22_4}
\\
&&
\bar C^2_{31} = -a_{1}\cos 2\varphi + a_{2}\sin 2\varphi - p_3
\label{eq:1_22_5}
\\
&&
\bar C^2_{12} = p_2
\label{eq:1_22_6}
\\
&&
\bar C^3_{23} = 0
\label{eq:1_22_7}
\\
&&
\bar C^3_{31} = 0
\label{eq:1_22_8}
\\
&&
\bar C^3_{12} = 1
\label{eq:1_22_9}
\label{eq:1_22}\end{aligned}$$
From – we get claim a).
The equations , , and the sum of equations , give us $p_1$, $p_2$, and $p_3 = -\frac{1}{2}\left (\bar C^{1}_{23}+\bar C^{2}_{31}\right)$, thus we prove claim b).
The equations , with $p_3$ substituted give us claim c),
Finally note that , imply that $\bar C^1_{31} = \bar C^2_{23}$, this also can be proved in the following way. We have $d\bar\eta^3 = \bar\eta^1 \wedge \bar\eta^2$. From this it follows that $0 = d\bar\eta^1 \wedge \bar\eta^2 - \bar\eta^1 \wedge d\bar\eta^2$, hence we obtain $\bar C^1_{31} = \bar C^2_{23}$.
\[cor:1\] The $SO(2)$-structure $B_2 \to M$, where $B_2$ is a $SO(2)$-principal subbundle of $B_0$ such that the tautological forms $\theta^a$ on $B_2$ satisfy structure equations is unique.
Let $B_2$ and $\hat B_2$ be $SO(2)$-principal subbundles of $B_0$ such that the tautological forms satisfy structure equations . Take local sections $\eta^a$ and $\hat\eta^a$ of $B_2$ and $\hat B_2$, respectively.
By Proposition \[prop:2\] a), we have $$d\eta^3 = \eta^1 \wedge \eta^2 \text{ and } d\hat{\eta}^3 = \hat{\eta}^1 \wedge \hat{\eta}^2
\label{eq:200_1}$$
Let $\Omega$ be the area form on $\Delta$ determined by the metric $\langle \cdot,\cdot \rangle$. Since $\eta^a$ and $\hat\eta^a$ are sections of $B_0$, we have $$\eta^1 \wedge \eta^2 |_\Delta = \Omega = \hat\eta^1 \wedge \hat\eta^2 |_\Delta$$ By the same reason, we have $\hat\eta^3 = e^f \eta^3$, hence $d\hat\eta^3 = e^f df \wedge \eta^3 + e^f d\eta^3$. We restrict it to $\Delta$ and from $d\hat\eta^3 |_\Delta = d\eta^3 |_\Delta$ get that $e^f = 1$. Hence $\hat\eta^3 = \eta^3$. Now $$\begin{aligned}
\hat\eta^1 &= \cos\varphi \eta^1 - \sin\varphi \eta^2 + a \eta^3
\\
\hat\eta^2 &= \sin\varphi \eta^1 + \cos\varphi \eta^2 + b \eta^3
\label{eq:3}\end{aligned}$$ But $$\hat\eta^1 \wedge \hat \eta^2 = d\hat\eta^3 = d\eta^3 = \eta^1 \wedge \eta^2,$$ so one can easily prove that $a=b=0$.
The function $$\label{Eq:formula-M}
\bar{\mathcal{M}}=\left (a_{1}\right )^{2}+\left (a_{2}\right )^{2}=\left (\dfrac{\bar C^{1}_{23}-\bar C^{2}_{31}}{2}\right )^{2}+\left (\bar C^{1}_{31}\right )^{2}$$ is a pullback of a function $\mathcal{M} : M \to \mathbb{R}$, i.e. $\bar{\mathcal{M}} = \mathcal{M} \circ \pi$, where $$\mathcal{M}=\left (\dfrac{C^{1}_{23}-C^{2}_{31}}{2}\right )^{2}+\left (C^{1}_{31}\right )^{2}.$$
From Corollary \[cor:1\] it follows that the $SO(2)$-structure $B_2$ is uniquely determined by the sub-Riemannian surface $\mathcal{S}$. As the coframe field $\left\{\theta^{1}, \theta^{2}, \theta^{3}, \alpha\right \}$ on $B_2$ is uniquely determined, we see that the functions $a_1, a_2 : B_2 \to \mathbb{R}$ as well as the form $\alpha$ are uniquely determined by $\mathcal S$. Thus we get
The function $\mathcal{M}$ is an invariant of the sub-Riemannian surface $\mathcal{S}$.
### Curvature of a contact sub-Riemannian surface
Let us write down as follows: $$\begin{aligned}
&&
d\theta^{1}=\alpha\wedge\theta^{2}+a_{1}\theta^{2}\wedge\theta^{3}+a_{2}\theta^{3}\wedge\theta^{1}
\label{eq:1_100_1}
\\
&&
d\theta^{2}=-\alpha\wedge\theta^{1}+a_{2}\theta^{2}\wedge\theta^{3}-a_{1}\theta^{3}\wedge\theta^{1}
\label{eq:1_100_2}
\\
&&
d\theta^{3}=\theta^{1}\wedge\theta^{2}
\label{eq:1_100_3}\end{aligned}$$ Take the exterior differential of , then we get $$\begin{split}
0 = d\alpha \wedge \theta^2 - \alpha \wedge d\theta^2 + da_1 \wedge \theta^2 \wedge \theta^3 + a_1 d\theta^2 \wedge \theta^3 - a_1 \theta^2 \wedge d\theta^3 + \\
da_2 \wedge \theta^3 \wedge \theta^1 + a_2 d\theta^3 \wedge \theta^1 - a_2 \theta^3 \wedge d\theta^1
\end{split}
\label{eq:1_101}$$ To we substitute – and get $$d\alpha \wedge \theta^2 + 2 a_1 \alpha \wedge \theta^3 \wedge \theta^1 - 2 a_2 \alpha \wedge \theta^2 \wedge \theta^3 + da_1 \wedge \theta^2 \wedge \theta^3 + da_2 \wedge \theta^3 \wedge \theta^1 = 0.
\label{eq:1_102}$$ In the same manner from we get $$- d\alpha \wedge \theta^1 + 2 a_1 \alpha \wedge \theta^2 \wedge \theta^3 + 2 a_2 \alpha \wedge \theta^3 \wedge \theta^1 + da_2 \wedge \theta^2 \wedge \theta^3 - da_1 \wedge \theta^3 \wedge \theta^1= 0.
\label{eq:1_103}$$ Now we consider the expansions: $$\begin{cases}
da_1 = a \alpha + A_1 \theta^1 + A_2 \theta^2 + A_3 \theta^3
\\
da_2 = b \alpha + B_1 \theta^1 + B_2 \theta^2 + B_3 \theta^3
\\
d\alpha = P_1 \alpha \wedge \theta^1 + P_2 \alpha \wedge \theta^2 + P_3 \alpha \wedge \theta^3
+ X_{23} \theta^2 \wedge \theta^3 + X_{31} \theta^3 \wedge \theta^1 + X_{12} \theta^1 \wedge \theta^2.
\end{cases}
\label{eq:1_104}$$ We will express $P_a$ and $X_{ab}$ in terms of $A_a$ and $B_b$. To do it we substitute to and get $$\begin{split}
P_1 \alpha \wedge \theta^1 \wedge \theta^2 + P_3 \alpha \wedge \theta^3 \wedge \theta^2 + X_{31} \theta^3 \wedge \theta^1 \wedge \theta^2 + 2 a_1 \alpha \wedge \theta^3 \wedge \theta^1 -
\\
2 a_2 \alpha \wedge \theta^2 \wedge \theta^3 +
a \alpha \wedge \theta^2 \wedge \theta^3 + A_1 \theta^1 \wedge \theta^2 \wedge \theta^3 +
\\
b \alpha \wedge \theta^3 \wedge \theta^1 + B_2 \theta^2 \wedge \theta^3 \wedge \theta^1 = 0.
\end{split}
\label{eq:1_105}$$ From this we get $$P_1 = 0, \quad P_3 - a + 2a_2 = 0, \quad X_{31} + A_1 + B_2 = 0, \quad 2a_1 + b = 0.
\label{eq:1_106}$$ In the same manner, substituting to , we get $$P_2 = 0, \quad -P_3 - a + 2a_2 = 0, \quad -X_{23} + B_1 - A_2 = 0, \quad 2a_1 + b = 0.
\label{eq:1_107}$$ From and we get $$\begin{split}
P_1 = P_2 = P_3 = 0, \quad a = 2a_2, \quad b = -2 a_1,
\\
X_{31} = -A_1 - B_2, \quad X_{23} = B_1 - A_2.
\label{eq:1_108}
\end{split}$$ Thus only $X_{12}$ is undetermined, and we denote it by $\bar{\mathcal{K}}$. In this way we obtain $$\begin{cases}
da_{1}=2a_{2}\alpha + A_{1}\theta^{1}+A_{2}\theta^{2}+A_{3}\theta^{3}\\
da_{2}=-2a_{1}\alpha + B_{1}\theta^{1}+B_{2}\theta^{2}+B_{3}\theta^{3}\\
d\alpha = \bar{\mathcal{K}}\theta^{1}\wedge\theta^{2}+\left (B_{1}-A_{2}\right )\theta^{2}\wedge\theta^{3}+\left (-A_{1}-B_{2}\right )\theta^{3}\wedge\theta^{1}
\end{cases}
\label{eq:1_108_1}$$ Now let us express $\bar{\mathcal{K}}$ in terms of $\bar C^c_{ab}$. To do it, we use and . Then, from we get $$\begin{split}
d\alpha = d\bar C^1_{12} \wedge \bar\eta^1 + \bar C^1_{12} \wedge d\bar \eta^1 + d\bar C^2_{12} \wedge \bar \eta^2 + \bar C^2_{12} \wedge d\bar \eta^2 -
\\
\frac{1}{2}d(\bar C^1_{23} + \bar C^2_{31})\wedge \bar \eta^3 - \frac{1}{2}(\bar C^1_{23} + \bar C^2_{31})\wedge d\bar \eta^3.
\end{split}
\label{eq:1_110}$$ Let us take the frame field $\{\bar E_1,\bar E_2,\bar E_3,\bar E_4\}$ dual to $\{\bar\eta^1,\bar\eta^2,\bar\eta^3,\alpha\}$. Note that $\bar C^c_{ab}$ depend only on the base coordinates, so $E_4 \bar C^c_{ab} = 0$. Therefore, $$\begin{split}
d \bar C^1_{23} = \bar E_1 \bar C^1_{23} \bar\eta^1 + \bar E_2 \bar C^1_{23} \bar\eta^2 + \bar E_3 \bar C^1_{23} \bar\eta^3;
\\
d \bar C^2_{23} = \bar E_1 \bar C^2_{23} \bar\eta^1 + \bar E_2 \bar C^2_{23} \bar\eta^2 + \bar E_3 \bar C^2_{23} \bar\eta^3.
\label{eq:1_111}
\end{split}$$ If we substitute to we obtain expansion $$\begin{gathered}
d\alpha = (\bar E_{1}\bar C^{2}_{12}-\bar E_{2}\bar C^{1}_{12}+\left (\bar C^{1}_{12}\right )^{2}+\left (\bar C^{2}_{12}\right )^{2}-
\\
\frac{1}{2}(\bar C^{1}_{23}+\bar C^{2}_{31})) \bar\eta^1 \wedge \bar\eta^2 + (\dots) \bar\eta^3 \wedge \bar\eta^1 + (\dots) \bar\eta^2 \wedge \bar\eta^3
\label{eq:1_112}\end{gathered}$$ where …stands for the coefficient we are not interested in now. Then, use , and from we get $$\begin{gathered}
d\alpha = (\bar E_{1}\bar C^{2}_{12}-\bar E_{2}\bar C^{1}_{12}+\left (\bar C^{1}_{12}\right )^{2}+\left (\bar C^{2}_{12}\right )^{2}-
\\
\frac{1}{2}(\bar C^{1}_{23}+\bar C^{2}_{31})) \theta^1 \wedge \theta^2 + (\dots) \theta^3 \wedge \theta^1 + (\dots) \theta^2 \wedge \theta^3
\label{eq:1_113}\end{gathered}$$ Compare and , then we finally find $$\label{Eq:formula-K1}
\bar{\mathcal{K}} = \bar E_{1}\bar C^{2}_{12}-\bar E_{2}\bar C^{1}_{12}+\left (\bar C^{1}_{12}\right )^{2}+\left (\bar C^{2}_{12}\right )^{2} -\frac{1}{2}(\bar C^{1}_{23}+\bar C^{2}_{31}).$$ It is clear that $\bar E_a$ are horizontal lifts of vector fields $E_a$ which constitute a local frame field on $U$, and $\bar E_a \bar C^b_{cd} = (E_a C^b_{cd}) \circ \pi$. As $\bar C^a_{bc} = C^a_{bc} \circ \pi$, we have $\bar{\mathcal{K}} = \mathcal{K} \circ \pi$, where $$\label{Eq:formula-K}
\mathcal{K} = E_{1} C^{2}_{12}- E_{2} C^{1}_{12}+\left ( C^{1}_{12}\right )^{2}+\left ( C^{2}_{12}\right )^{2} -\frac{1}{2}( C^{1}_{23}+ C^{2}_{31}).$$
The function $\mathcal{K}$ is called the *curvature* of $\mathcal{S}$, and it is clear that $\mathcal{K}$ is an invariant of $\mathcal{S}$.
We result our investigations of invariants of a contact sub-Riemannian surface in the following
\[Th:Theorem-2\] Let $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ be a contact sub-Riemannian surface. Then, for any $p \in M$, in a neighborhood $U$ of $p$ a coframe field $\eta = \{\eta^a\}$ exists such that $$\label{Eq:structure-functions-coframe}
\left (
\begin{array}{c}
d\eta^{1}\\
d\eta^{2}\\
d\eta^{3}
=
C^{1}_{23} & C^{1}_{31} & C^{1}_{12}\\
C^{2}_{23} & C^{2}_{31} & C^{2}_{12}\\
\eta^{2}\wedge\eta^{3}\\
\eta^{3}\wedge\eta^{1}\\
\eta^{1}\wedge\eta^{2}\\
\end{array}
\right )$$ The functions $$\begin{aligned}
&&
\mathcal{M}=\left (\dfrac{C^{1}_{23}-C^{2}_{31}}{2}\right )^{2}+\left (C^{1}_{31}\right )^{2}
\\
&&
\mathcal{K} = E_{1} C^{2}_{12}- E_{2} C^{1}_{12}+\left ( C^{1}_{12}\right )^{2}+\left ( C^{2}_{12}\right )^{2} -\frac{1}{2}( C^{1}_{23}+ C^{2}_{31})
\label{eq:invariants}\end{aligned}$$ do not depend on the choice of coframe field $\eta$ with structure equations and are correctly defined on $M$.
Symmetries of contact sub-Riemannian surfaces
=============================================
A *symmetry* of a sub-Riemannian surface $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ is a local diffeomorphism $F : M \to M$ such that, for any $p \in M$, $$\begin{aligned}
&&
F(\Delta_p) = \Delta_{F(p)},
\label{eq:1_150_1}
\\
&&
\langle dF(W), dF(W) \rangle_{F(p)} = \langle W, W \rangle_{p}, \forall W \in \Delta_p.
\label{eq:1_150_2}\end{aligned}$$
A vector field $V \in \mathfrak{X}(M)$ is called an *infinitesimal symmetry* if its flow consists of symmetries.
\[Th:Theorem-3\] Let $\mathcal S = (M,\Delta,\langle \cdot,\cdot \rangle)$ be a contact sub-Riemannian surface. Let $\eta = \{\eta^a\}$ be a coframe field in a neighborhood $U$ of $p \in M$ such that holds true $($$\eta$ exists by Theorem \[Th:Theorem-2\]$)$, and $\{E_a\}$ be the dual frame field.
a\) For any infinitesimal symmetry $V$ of $\mathcal{S}$, a unique function $f : U \to \mathbb{R}$ exists such that $$\label{Eq:form-of-function-f}
V=-E_{2}(f)E_{1} + E_{1}(f)E_{2} + fE_{3} \text{ and } E_3 f = 0.$$
b\) Let $\mathcal{M}$ and $\mathcal{K}$ be the invariants of $\mathcal{S}$ $($ see Theorem \[Th:Theorem-2\]$)$. Then, if $V$ is transversal to $\Delta$ and $E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M} \ne 0$, the function $\ln f$ satisfies the following system of partial differential equations: $$\label{Eq:system-to-find-f}
\begin{cases}
E_{1}(\ln f) = \dfrac{E_{3}\mathcal{K}E_{1}\mathcal{M}-E_{1}\mathcal{K}E_{3}\mathcal{M}}{E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M}}\\
E_{2}(\ln f) = \dfrac{E_{3}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{3}\mathcal{M}}{E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M}}\\
E_{3}(\ln f) = 0.
\end{cases}$$
Let $V$ be an infinitesimal symmetry of $\mathcal{S}$, and $\phi_t$ be the flow of $V$. As $\{E_1(p), E_2(p)\}$ is an orthonormal frame of $\Delta (p)$, we have, by definition of infinitesimal symmetry , , that $$\begin{cases}
d\phi_{t}E_{1}(p)=\cos\varphi (t) E_{1}\left (\phi_{t}(p)\right )+\sin\varphi (t) E_{2}\left (\phi_{t}(p)\right )\\
d\phi_{t}E_{2}(p)=-\sin\varphi (t) E_{1}\left (\phi_{t}(p)\right )+\cos\varphi (t) E_{2}\left (\phi_{t}(p)\right )
\end{cases}$$ Hence $$\begin{cases}
d\phi_{t}E_{1}\left (\phi_{-t}(p)\right )=\cos\varphi (t) E_{1}(p)+\sin\varphi (t) E_{2}(p)\\
d\phi_{t}E_{2}\left (\phi_{-t}(p)\right )=-\sin\varphi (t) E_{1}(p)+\cos\varphi (t) E_{2}(p)
\end{cases}$$ From this follows that $$\label{Eq:lambda-system}
\begin{cases}
[E_{1}, V] = \lambda E_{2}\\
[E_{2}, V] = -\lambda E_{1}\\
\end{cases}$$ because $$\mathcal{L}_{V}E_{1}=[V,E_{1}]=\left.\dfrac{d}{dt}\right|_{t=0} d\phi_t E_{1}\left (\phi_{-t}(p)\right )=\varphi' (0)E_{2}=-\lambda E_{2}$$ and similar for $[V,E_{2}]=\lambda E_{1}$.
On the other hand we know that the structure equations for the dual frame $E=\left (E_{1}, E_{2}, E_{3}\right )$ are $$\begin{cases}
[E_{1}, E_{2}]=c^{1}_{12}E_{1}+c^{2}_{12}E_{2}+c^{3}_{12}E_{3}\\
[E_{3}, E_{1}]=c^{1}_{31}E_{1}+c^{2}_{31}E_{2}+c^{3}_{31}E_{3}\\
[E_{2}, E_{3}]=c^{1}_{23}E_{1}+c^{2}_{23}E_{2}+c^{3}_{23}E_{3}\\
\end{cases}$$ But $c^{i}_{jk}=-C^{i}_{jk}$ from . Therefore $$\begin{cases}
[E_{1}, E_{2}]=-\left (C^{1}_{12}E_{1}+C^{2}_{12}E_{2}+E_{3}\right )\\
[E_{3}, E_{1}]=-\left (C^{1}_{31}E_{1}+C^{2}_{31}E_{2}\right )\\
[E_{2}, E_{3}]=-\left (C^{1}_{23}E_{1}+C^{2}_{23}E_{2}\right )\\
\end{cases}
\label{eq:1_140}$$ Substituting $V = V^1 E_1 + V^2 E_2 + V^3 E_3$ to the first equation in , we get $$\begin{gathered}
\lambda E_2 = [E_1, V^1 E_1 + V^2 E_2 + V^3 E_3] =
\\
E_1V^1\, E_1 + E_1 V^2\, E_2 + V^2 [E_1,E_2] + E_1V^3\,E_3 + V^3 [E_1,E_3] = \\
(E_1V^1 - V^2 C^1_{12} + V^3 C^1_{31}) E_1 + (E_1V^2 - V^2 C^2_{12} + V^3 C^2_{31}) E_2 + (E_1 V^3 - V^2)E_3.
\label{eq:1_151}\end{gathered}$$ In the same manner, substituting $V = V^1 E_1 + V^2 E_2 + V^3 E_3$ to the second equation in , we get $$\begin{gathered}
-\lambda E_1 = [E_2, V^1 E_1 + V^2 E_2 + V^3 E_3] =
\\
E_2 V^1\, E_1 + V^1 [E_2,E_1] + E_2 V^2\, E_2 + E_2 V^3\,E_3 + V^3 [E_2,E_3] =
\\
(E_2V^1 + V^1 C^1_{12} - V^3 C^1_{23}) E_1 + (E_2 V^2 + V^1 C^2_{12} - V^3 C^2_{23}) E_2 + (E_2 V^3 + V^1)E_3.
\label{eq:1_152}\end{gathered}$$ From and we obtain the following equation system: $$\begin{aligned}
&&
E_1V^1 - V^2 C^1_{12} + V^3 C^1_{31} = 0,
\label{eq:1_153_1}
\\
&&
E_1V^2 - V^2 C^2_{12} + V^3 C^2_{31} = \lambda,
\label{eq:1_153_2}
\\
&&
E_1 V^3 - V^2 = 0,
\label{eq:1_153_3}
\\
&&
E_2V^1 + V^1 C^1_{12} - V^3 C^1_{23} = -\lambda,
\label{eq:1_153_4}
\\
&&
E_2 V^2 + V^1 C^2_{12} - V^3 C^2_{23} = 0,
\label{eq:1_153_5}
\\
&&
E_2 V^3 + V^1 = 0.
\label{eq:1_153_6}\end{aligned}$$ Let us set $f = \eta^3(V) = V^3$, then and give $$V=-E_{2}(f)E_{1} + E_{1}(f)E_{2} + fE_{3}.$$ Now substitute $V^1 = -E_2 f$, $V^2 = E_1 f$, and $V^3 = f$ to and : $$\begin{aligned}
- E_1 E_2 f - C^1_{12} E_1 f + C^1_{31} f = 0,
\\
E_2 E_1 f - C^2_{12} E_2 f - C^2_{23} f = 0,\end{aligned}$$ Summing these equalities, we arrive at $$[E_2,E_1]f - C^1_{12} E_1 f - C^2_{12} E_2 f = 0$$ but, by the first equation in , this means that $E_3 f = 0$. Thus we have proved and claim a).
Let us now prove b). As $V$ is an infinitesimal symmetry of the sub-Riemannian surface $\mathcal{S}$, we have $$\label{Eq:system-VK-VM}
\begin{cases}
V\mathcal{K}=0\\
V\mathcal{M}=0
\end{cases}$$ If we substitute to , we get $$\begin{cases}
- E_{1}\mathcal{K} E_{2}f + E_{2}\mathcal{K} E_{1}f + E_{3}\mathcal{K} f = 0
\\
- E_{1}\mathcal{M} E_{2}f + E_{2}\mathcal{M} E_{1}f + E_{3}\mathcal{M} f = 0
\end{cases}$$ Since $V$ is transversal to $\Delta$, and hence $f$ does not vanish in $U$, we can divide both equations by $f$ and obtain the system of linear equations in $E_1 \ln f$ and $E_2 \ln f$: $$\begin{cases}
- E_{1}\mathcal{K}\, E_{2}\ln f + E_{2}\mathcal{K}\, E_{1} \ln f + E_{3}\mathcal{K} = 0
\\
- E_{1}\mathcal{M}\, E_{2} \ln f + E_{2}\mathcal{M}\, E_{1} \ln f + E_{3}\mathcal{M} = 0
\end{cases}
\label{eq:1_157}$$ If $E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M} \ne 0$, this system has the unique solution $$\begin{aligned}
E_{1}(\ln f) &=& \dfrac{E_{3}\mathcal{K}E_{1}\mathcal{M}-E_{1}\mathcal{K}E_{3}\mathcal{M}}{E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M}}
\label{eq:1_155_1}
\\
E_{2}(\ln f) &=& \dfrac{E_{3}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{3}\mathcal{M}}{E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M}}
\label{eq:1_155_2}\end{aligned}$$ To and we add $E_3 \ln f = 0$, which follows from , and get the system . Thus we have proved b).
If an infinitesimal symmetry $V$ of $\mathcal{S}$ lies in $\Delta$ at each point of an open set $W$, then $f$ is zero, and, by , $V$ is zero, too. So, nonvanishing $V$ should be transversal to $\Delta$ almost everywhere.
Theorem \[Th:Theorem-3\] can be used in order to prove that a sub-Riemannian surface $\mathcal{S}$ does not admit nontrivial infinitesimal symmetries. To do it, it is sufficient to prove that the integrability conditions do not hold for the system . The integrability conditions have the form: $$\begin{cases}
E_{1}(EQ2)-E_{2}(EQ1)=C^{1}_{12}EQ1+C^{2}_{12}EQ2\\
E_{3}EQ1=C^{1}_{31}EQ1+C^{2}_{31}EQ2\\
-E_{3}EQ2=C^{1}_{23}EQ1+C^{2}_{23}EQ2,
\end{cases}
\label{eq:1_156}$$ where $$\begin{aligned}
EQ1 &=& \dfrac{E_{3}\mathcal{K}E_{1}\mathcal{M}-E_{1}\mathcal{K}E_{3}\mathcal{M}}{E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M}},
\\
EQ2 &=& \dfrac{E_{3}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{3}\mathcal{M}}{E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M}}.\end{aligned}$$ However, if the integrability conditions do hold for , one can use this system in order to find $f$ and then $V$. In fact, we can take a natural frame field $\partial_a$ and write $E_{a}=B_{a}^{b}\partial_{b}$, Then the equation system can be rewritten as $\partial_{a} \ln f = g_{a}$, and the solution can be found by the well-known formula: $$\ln f(x^{a})=\int_{0}^{1}x^{b}g_{b}(tx^{a})dt.$$
The condition $\mathcal{D} = E_{1}\mathcal{K}E_{2}\mathcal{M}-E_{2}\mathcal{K}E_{1}\mathcal{M} \ne 0$, in general, does not hold. If $\mathcal{D} = 0$, the system may not have any solutions and this means that $\mathcal{S}$ does not admit any infinitesimal symmetries; or it may have infinitely many solutions, then we simply get an additional relation for $f$, which can be used in order to find infinitesimal symmetries by another method.
Examples of infinitesimal symmetries
------------------------------------
### Heisenberg distribution
Consider the Heisenberg distribution $\Delta$ given with respect to the standard coordinates in $\mathbf{R}^{3}$ by the $1$-form $$\nonumber
\eta^{3}=dz+ydx-xdy.$$ For the metric on $\Delta$ we take the metric induced from $\mathbb{R}^3$. By calculations, we get the following results:
1. The $SO(2)$-structure $B_2 \to \mathbb{R}^3$ is given by the coframe field $$\nonumber
\begin{cases}
\eta^{1}=\dfrac{\left (2+3y^{2}\right )dx}{2\sqrt{1+y^{2}}}-\dfrac{3xydy}{2\sqrt{1+y^{2}}} +\dfrac{ydz}{2\sqrt{1+y^{2}}}\\
\eta^{2}=-\dfrac{xydx}{2\sqrt{1+y^{2}}\sqrt{1+x^{2}+y^{2}}}+\dfrac{\left (2+3x^{2}+2y^{2}\right )dy}{2\sqrt{1+y^{2}}\sqrt{1+x^{2}+y^{2}}} -\dfrac{xdz}{2\sqrt{1+y^{2}}\sqrt{1+x^{2}+y^{2}}}\\
\eta^{3}=-\dfrac{y}{2}\sqrt{1+x^{2}+y^{2}} dx+\dfrac{x}{2}\sqrt{1+x^{2}+y^{2}} dy-\dfrac{1}{2}\sqrt{1+x^{2}+y^{2}} dz
\end{cases}$$
2. The orthonormal dual frame is $$\nonumber
\begin{cases}
E_{1}=
\dfrac{1}{\sqrt{1+y^{2}}}\dfrac{\partial }{\partial x}-
\dfrac{y}{\sqrt{1+y^{2}}}\dfrac{\partial }{\partial z}\\
E_{2}=
\dfrac{xy}{\sqrt{1+y^{2}}\sqrt{1+x^{2}+y^{2}}}\dfrac{\partial }{\partial x}+
\dfrac{\sqrt{1+y^{2}}}{\sqrt{1+x^{2}+y^{2}}}\dfrac{\partial }{\partial y}+
\dfrac{x}{\sqrt{1+y^{2}}\sqrt{1+x^{2}+y^{2}}}\dfrac{\partial }{\partial z}\\
E_{3}=
\dfrac{y}{\left (1+x^{2}+y^{2}\right )^{3/2}}\dfrac{\partial }{\partial x}-
\dfrac{x}{\left (1+x^{2}+y^{2}\right )^{3/2}}\dfrac{\partial }{\partial y}-
\dfrac{2+3x^{2}+3y^{2}}{\left (1+x^{2}+y^{2}\right )^{3/2}}\dfrac{\partial }{\partial z}
\end{cases}$$
3. The structure functions [^1] $C^{i}_{jk}$ are $$\nonumber
\begin{array}{lll}
C^{1}_{23}= -\dfrac{1-3y^{2}}{\left (1+y^{2}\right )\left (1+x^{2}+y^{2}\right )} & C^{2}_{23} = -\dfrac{3xy}{\left (1+y^{2}\right )\left (1+x^{2}+y^{2}\right )^{3/2}} & C^{3}_{23} =0\\
C^{1}_{31} =-\dfrac{3xy}{\left (1+y^{2}\right )\left (1+x^{2}+y^{2}\right )^{3/2}} & C^{2}_{31}=-\dfrac{1-2x^{2}+y^{2}}{\left (1+y^{2}\right )\left (1+x^{2}+y^{2}\right )} & C^{3}_{31} =0\\
C^{1}_{12}=-\dfrac{3y}{\sqrt{1+y^{2}}\sqrt{1+x^{2}+y^{2}}} & C^{2}_{12}= \dfrac{2x}{\sqrt{1+y^{2}}\left (1+x^{2}+y^{2}\right )} & C^{3}_{12}=1\\
\end{array}$$
4. The invariants $$\nonumber
\begin{cases}
\mathcal{M}=\dfrac{9}{4}\dfrac{\left (x^{2}+y^{2}\right )^{2}}{\left (1+x^{2}+y^{2}\right )^{4}}\\
\mathcal{K}=\dfrac{3\left (1+2x^{2}+4y^{2}+3x^{2}y^{2}+3y^{4}\right )}
{\left (1+y^{2}\right )\left (1+x^{2}+y^{2}\right )^{2}}
\end{cases}$$
5. The family of functions $f$ which define symmetries $$\nonumber
f=A\sqrt{1+x^{2}+y^{2}}, \; \mbox{where}\; A=const.$$
6. The connection form $$\begin{aligned}
\alpha &=&
d\varphi -
\dfrac{1}{2\left (1+y^{2}\right )\left (1+x^{2}+y^{2}\right )^{3/2}}
\{
y(4+9x^{2}+16y^{2}+12x^{2}y^{2}+12y^{4})dx\nonumber\\
&-&x(2+7x^{2}+14y^{2}+12x^{2}y^{2}+12y^{4})dy
+(-2+3x^{2}+4y^{2}+6x^{2}y^{2}+6y^{4})dz
\}\nonumber\end{aligned}$$
Cartan distribution
-------------------
Consider the Cartan distribution $\Delta$ given with respect to the standard coordinates in $\mathbf{R}^{3}$ by the $1$-form $$\nonumber
\eta^{3}=dz+ydx.$$
1. The $SO(2)$-structure $B_2 \to \mathbb{R}^3$ is given by the coframe field $$\nonumber
\begin{cases}
\eta^{1}=\dfrac{1+2y^{2}}{\sqrt{1+y^{2}}}dx+\dfrac{y}{\sqrt{1+y^{2}}}dz\\
\eta^{2}=dy\\
\eta^{3}=-\sqrt{1+y^{2}}dx-\sqrt{1+y^{2}}dz
\dfrac{1}{\sqrt{1+y^{2}}}\dfrac{\partial }{\partial x} - \dfrac{y}{\sqrt{1+y^{2}}}\dfrac{\partial }{\partial z} \\
E_{2}=\dfrac{\partial }{\partial y}
\\
E_{3}=
\dfrac{y}{\left (1+y^{2}\right )^{3/2}} - \dfrac{1+2y^{2}}{\left (1+y^{2}\right )^{3/2}}\dfrac{\partial }{\partial z}
\end{cases}$$
3. The structure functions are, $$\nonumber
\begin{array}{lll}
C^{1}_{23}= -\dfrac{1-y^{2}}{\left (1+y^{2}\right )^{2}} & C^{2}_{23} = 0 & C^{3}_{23} = 0 \\
C^{1}_{31} = 0 & C^{2}_{31}= 0 & C^{3}_{31} =0 \\
C^{1}_{12}=-\dfrac{2y}{1+y^{2}} & C^{2}_{12}= 0 & C^{3}_{12}=1\\
\end{array}$$
4. The invariants $$\nonumber
\begin{cases}
\mathcal{M}=\dfrac{1}{4}\dfrac{\left (1-2y^{2}\right )^{2}}{\left (1+y^{2}\right )^{4}}\\
\mathcal{K}=\dfrac{1+4y^{2}}{\left (1+y^{2}\right )^{2}}
\end{cases}$$
5. The family of functions $f$ which define symmetries $$\nonumber
f=f(y,z+xy)$$
6. The connection form $$\nonumber
\alpha =
d\varphi -
\dfrac{y\left (1+6y^{2}\right )}{\left (1+y^{2}\right )^{3/2}}dx +
\dfrac{y\left (1-4y^{2}\right )}{\left (1+y^{2}\right )^{3/2}}dz$$
Noncontact sub-Riemannian surfaces of stable type
=================================================
Let us consider a sub-Riemannian surface $\left (\Delta , \langle \cdot , \cdot \rangle\right )$. Now we do not assume that $\Delta$ is contact, so we admit that the set $$\Sigma = \{p \in \mathbf{R}^3 \mid (\omega \wedge d\omega)_p = 0\},$$ where $\Delta$ is the kernel of the 1-form $\omega$, is, in general, non-empty. However, we assume that *$\Sigma$ is a 2-dimensional submanifold in $\mathbf{R}^3$ and the distribution $\Delta$ is transversal to $\Sigma$*.
\[rem:2\_1\] The surface $\Sigma$ does not depend on the choice of $\omega$.
\[rem:2\_2\] Any stable germ of a Pfaffian equation on a $3$-dimensional manifold is equivalent either to the germ of the $1$-form $\omega_0 = dz + xdy$, or b) $\omega_1 = dy + x^2 dz$, at the origin [@Zhitomirskii]. For a contact distribution $\Delta$, the germ of $\Delta$ at each point is equivalent to the germ of the distribution determined by $\omega_0$. If a distribution $\Delta$ satisfies our assumption, then, for any point $p in \mathbf{R}^3 \setminus \Sigma$, the the germ of $\Delta$ at $p$ is equivalent to the germ of the distribution determined by $\omega_0$, and, for $p \in \Sigma$, the germ is equivalent to the germ of the distribution determined by $\omega_0$.
Nonholonomity function of sub-Riemannian surface
------------------------------------------------
For the sub-Riemannian surface $\left (\Delta ,\langle \cdot , \cdot \rangle\right )$ let us take a non-vanishing section $\omega$ of the bundle $Ann(\Delta)$. Then, in a neighborhood $U$ of each point $p \in M$ take an positively oriented orthonormal frame field $\{E_1, E_2\}$ of $\Delta$. Then we define the function $\lambda_U : U \to \mathbf{R}$, $\lambda_U(q) = \omega([E_1,E_2](q))$. One can easily check that $\lambda_U$ does not depend on a choice of the frame field $\{E_1, E_2\}$, therefore if $U \cap V \ne \emptyset$, $\lambda_U |_{U \cap V} = \lambda_V |_{U \cap V}$. Therefore, we have correctly defined function $\lambda_\omega$ on $M$ by setting $\lambda_\omega |_U = \lambda_U$. This function will be called the *nonholonomity function of sub-Riemannian surface*. Note that this function depends on the choice of form $\omega$ and on the metric on $\Delta$.
\[prop:2\_2\] The nonholonomity function has the following properties:
- $\lambda_{e^\varphi\omega} = e^\varphi \lambda_\omega$;
- $\lambda(p) = 0$ if and only if $p \in \Sigma$;
- $d\lambda_\omega |_p \ne 0$ for any $p \in \Sigma$.
- $d\omega |_\Delta = -\frac{1}{2}\lambda_\omega \Omega$, where $\Omega$ is the area 2-form on $\Delta$ determined by the metric.
a\) is evident from the definition of nonholonomity function.
b\) In a neighborhood $U$ of a point $p$ take an positively oriented orthonormal frame field $\{E_1, E_2\}$ of $\Delta$ and a vector field $E_3$ such that $\omega(E_3)=1$. Then, $\{E_1, E_2,E_3\}$ is a frame field on $U$. We have $d\omega(E_1,E_2) = - \frac{1}{2}\omega([E_1,E_2) = - \frac{1}{2}\lambda_\omega$, and $\omega(E_1)=\omega(E_2)=0$, from this follows $$d\omega\wedge\omega(E_1,E_2,E_3) = -\frac{1}{6}\lambda_\omega.$$ This proves b).
c\) By our assumptions, for any $p \in \Sigma$, with respect to a coordinate system, $\omega=e^\varphi \omega_1$, where $\omega_1 = dz + x^2 dy$. From a) it follows that $\lambda_\omega = e^\varphi\lambda_{\omega_1}$. Also, $dx$, $dy$, $\omega_1$ is a coframe, hence $dx \wedge dy \wedge \omega_1 (E_1,E_2,E_3) \ne 0$, then $dx \wedge dy (E_1,E_2) \ne 0$. Therefore, $\lambda_{\omega_1} = e^\psi x$, for a function $\psi$, and, hence, $\Sigma \cap U$ is given by the equation $x=0$, and $\lambda_\omega = e^{\varphi+\psi} x$. From this immediately follows the required statement.
d\) For a positively oriented orthonormal frame field $\{E_1, E_2\}$ of $\Delta$ we have $\Omega(E_1,E_2)=1$ and $d\omega(E_1,E_2)=-\frac{1}{2}\omega([E_1,E_2]) = -\frac{1}{2}\lambda_\omega$. This proves d).
Characteristic vector field
---------------------------
Let us denote by $Ann(\Delta)$ the vector subbundle in $T^* M$ of rank $1$ whose fiber at $p \in M$ consists of $1$-forms vanishing at $\Delta_p$.
\[prop:2\_1\] For each point $p \in M$ there exists a section $\omega$ of $Ann(\Delta)$ in a neighborhood $U(p)$ which admits a vector field $V$ on $U(p)$ such that $L_V \omega = 0$ and $\omega(V)=1$. For a given $\omega$ this vector field is unique.
For $p \in \Sigma$ we can take $\omega = dz + x^2 dy$, for the other $p$ we can take $\omega=dz+xdy$ with respect to an appropriate coordinate system (see Remark \[rem:2\_2\]). In both cases, the vector field $V = \frac{\partial}{\partial z}$ has the required properties.
Let us prove that, for a given $\omega$, the vector field $V$ such that $L_V \omega = 0$ and $\omega(V)=1$ is unique.
Let us take a frame field $E_1$, $E_2$, $E_3$ on $U(p)$ such that $\Delta$ is spanned by $E_1$ and $E_2$, and $E_3 = V$. Now let $W$ be a vector field with the required properties, then $W = W^1 E_1 + W^2 E_2 + W^3 E_3$. Since $\omega(V) = \omega(W) = 1$, we have $W^3=1$. As $E_3$ is an infinitesimal symmetry of $\omega$, $E_3$ is an infinitesimal symmetry of $\Delta$, too, therefore the vector fields $[E_3,E_1]$, $[E_3,E_2]$ are tangent to $\Delta$. From this follows that the vector fields $W^1 [E_1,E_2]$, $W^2 [E_1,E_2]$ are tangent to $\Delta$, but $\omega([E_1,E_2])\ne 0$ on $U(p) \setminus \Omega$, therefore $W^1 = W^2 = 0$ on $U(p) \setminus \Omega$, and $W^1$ and $W^2$ vanish on $U(p)$. Thus $W=E_3=V$ and the uniqueness has been proved.
If for $p \in M$, a nonvanishing form $\omega \in Ann(\Delta)$ on a neighborhood $U$ of $p$ for which there exists a vector field $V$ on such that $L_V \omega = 0$ and $\omega(V)=1$ will be called a *special form at* $p$, and $V$ the *characteristic vector field of* $\omega$. Note that, if $\omega$ is special at each point of an open set $U \subset M$, then on $U$ we have a unique vector field $V$ such that $L_V \omega = 0$ and $\omega(V)=1$.
Let us consider the form $\widetilde\omega =e^\varphi \omega$. In general, $\widetilde\omega$ is not special.
\[prop:2\_3\] a) If $p \in M \setminus \Omega$, then any nonvanishing form $\omega \in Ann(\Delta)$ is special.
b\) If $p \in \Sigma$ and $\omega$ is special at $p$, then $\widetilde\omega = e^\varphi \omega$ is special at $p$ if and only if on a neighborhood $U$ of $p$ we have $d\varphi|_\Delta = \lambda_w \xi$, where $\xi$ is a nonvanishing $1$-form on $\Delta$ in $U$.
First note that if $V$ is a vector field corresponding to a special form $\omega$, then $L_V \omega = d(\iota_V\omega) + \iota_V d\omega = \iota_V d\omega$ because $\iota_V\omega=1$. Therefore, $V$ is the characteristic vector field of $\omega$ if and only if $\omega(V)=1$ and $\iota_V d\omega = 0$.
Let $\omega$ be a special form on a neighborhood $U$ of $p$, and $V$ be the corresponding characteristic vector field. Let us take a form $\widetilde{\omega} = e^\varphi \omega$ and find conditions on $\varphi$ for $\widetilde{\omega}$ to be a special form.
First, $\widetilde\omega(\widetilde V) = 1$ if and only if $\widetilde V = e^{-\varphi} V + W$, where $W \in \mathfrak{X}(\Delta)$, because $\omega(V)=1$. Further, we have $$d\widetilde\omega = e^\varphi d\varphi \wedge \omega + e^\varphi d\omega =
d\varphi\wedge\widetilde\omega + e^\varphi d\omega.
\label{eq:2_1}$$ Then $$\iota_{\widetilde V} d\widetilde\omega =
\iota_{\widetilde V} d\varphi\, \omega - d\varphi\, \iota_{\widetilde V} \omega +
e^\varphi\iota_{\widetilde V} d\omega =
\widetilde V\varphi\, \omega - d\varphi\, \omega(\widetilde V) +
\iota_{V} d\omega + e^\varphi\iota_{W} d\omega;
\label{eq:2_2}$$ As $V$ is the characteristic vector field of $\omega$, we have $\iota_V d\omega = 0$, hence follows $$\iota_{\widetilde{V}} d\widetilde{\omega} = \widetilde V\varphi \widetilde\omega - d\varphi + e^\varphi \iota_W d\omega.
\label{eq:2_3}$$
Assume that $p$ does not lie in $\Sigma$, then, by Proposition \[prop:2\_2\], d), we have that $d\omega|_\Delta$ is nondegenerate form, so one can find a unique $W$ such that $d\varphi(W') = e^\varphi i_W d\omega (W')$. By , with this $W$, $i_{\widetilde{V}} d \widetilde{\omega} = 0$ on $\Delta$. Also, if we substitute $V$ to the right hand side of , then we get $\widetilde{V}\varphi\,e^\varphi - V\varphi = 0$, since $\iota_W d\omega (V) = 2 d\omega(W,V) = -\iota_V d\omega(W) = 0$. Thus, we have found $W$ such that the corresponding vector field $\widetilde{V} = e^{-\varphi} + W$ is the characteristic vector field for $\widetilde \omega$ since $\widetilde\omega(\widetilde V) = 1$ and $\iota_{\widetilde{V}} d\widetilde{\omega} =0$. Thus, the form $\widetilde\omega$ is special, and we have proved a).
For $p \in \Sigma$, $d\omega |_\Delta = \lambda_\omega \Omega$, where $\Omega$ is the area form of the metric on $\Delta$. Thus, if $\omega$ and $\widetilde\omega$ are special, by we have that $d\varphi|_\Delta = -\frac{1}{2}\lambda_\omega e^\varphi \Omega(W, \cdot)$, therefore in this case we have that $ d\varphi|_\Delta = \lambda_\omega \xi$, where $\xi(W') = -\frac{1}{2} e^\varphi \Omega(W, W')$ is a nonzero $1$-form on $\Delta$. Now, if $d\varphi|_\Delta = \lambda_\omega \xi$, then one can find $W$ such that $\xi(W') = -\frac{1}{2} e^\varphi \Omega(W, W')$ for any $W' \in \mathfrak{X}(\Delta)$. If we now set $\widetilde V = e^{-\varphi} V +W$, then $\iota_{\widetilde{V}} d\widetilde{\omega} (W') = 0$, for any $W' \in \mathfrak{X}(\Delta)$. Also, as before, we have $\iota_{\widetilde{V}} d\widetilde{\omega} (V) = 0$, hence follows $\iota_{\widetilde{V}} d\widetilde{\omega} = 0$. Thus, $\widetilde\omega$ is special and we have proved b).
It is clear that any symmetry of the nonholonomic surface maps a special form to a special form.
Adapted frame
-------------
Assume that the distribution $\Delta$ is given by a special form $\omega$. In a neighborhood $U$ of $p \in M$ take a frame field constructed in the following way. Since $\lambda_\omega$ vanishes at $\Sigma$ and $d\lambda_\omega \ne 0$ at $\Sigma$, $U$ is foliated by the level surfaces $\Sigma_c = \lambda_\omega^{-1}(c) \cap U$, $c \in (-\alpha,\beta)$, $\alpha,\beta>0$, of $\lambda_\omega$. Moreover, since $\Delta$ is transversal to $\Sigma=\Sigma_0$, then $\Delta$ is transversal to $\Sigma_c$, too. We take $E_1$ be the unit vector field of the line distribution $\Delta \cap T\Sigma_c$ on $U$, $c \in (-\alpha,\beta)$, (in fact, there are two such vector fields, they are opposite each other, we take one of them). The vector field $E_2 \in \mathfrak{X}(\Delta)$ is such that $E_1, E_2$ is positively oriented orthonormal frame field of $\Delta$. For $E_3$ we take the characteristic vector field of $\omega$.
Now, in the structure equations $[E_i,E_j] = c^k_{ij} E_k$, we have $c^3_{31} = c^3_{23}=0$ because the flow of $E_3$ maps $\Delta$ to $\Delta$, and $c^3_{12} = \lambda_\omega$ by definition of $\lambda_\omega$.
Let $\{\eta^1,\eta^2,\eta^3\}$ be the coframe dual to $\{E_a\}$. Note that $\eta^3 = \omega$.Then the structure equations are $$\begin{aligned}
&&
d\eta^1 = C^1_{23} \eta^2 \wedge \eta^3 + C^1_{31} \eta^3 \wedge \eta^1 + C^1_{12} \eta^1 \wedge \eta^2,
\\
&&
d\eta^2 = C^2_{23} \eta^2 \wedge \eta^3 + C^2_{31} \eta^3 \wedge \eta^1 + C^2_{12} \eta^1 \wedge \eta^2,
\\
&&
d\eta^3 = - \lambda_\omega \eta^1 \wedge \eta^2,
\label{eq:3_4_3}\end{aligned}$$
From the coframe construction it follows that $$d \lambda_\omega = \lambda_2 \eta^2 + \lambda_3 \eta^3,
\label{eq:3_5}$$ as $\lambda_1 = E_1 \lambda_\omega = 0$.
Applying the exterior differential to , we get that $$0 = - d\lambda_\omega \eta^1 \wedge \eta^2 + \lambda_\omega d\eta^1 \wedge \eta^2 - \lambda_\omega \eta^1 \wedge d\eta^2 = (-\lambda_3 + \lambda_\omega(C^1_{31}-C^2_{23}))\eta^1\wedge\eta^2\wedge\eta^3.
\label{eq:3_5_1}$$ hence follows that $$\lambda_3 = \lambda_\omega(C^1_{31}-C^2_{23}).
\label{eq:3_5_2}$$ Therefore $\lambda_3 = 0$ on $\Sigma$, and, as $d\lambda_\omega \ne 0$ on $\Sigma$ we have that $\lambda_2 = E_2\lambda \ne 0$ on $\Sigma$.
Change of the coframe
---------------------
It is clear that the frame $\{E_a\}$, and so the coframe $\{\eta^a\}$, is uniquely determined by the special form $\omega$. Now let us find how the coframe and the structure equations transform under a change of the special form $\omega$.
Let us take two special forms $\omega$ and $\widetilde\omega = e^\varphi \omega$, where $d\varphi |_\Delta = \lambda_\omega \xi$ (see Proposition \[prop:2\_3\], b)). Let $\{\eta^a\}$ and $\{\tilde\eta^a\}$ be the corresponding coframe fields. Then from construction we have: $$\begin{aligned}
&&
\eta^1 = \cos\alpha\, \tilde\eta^1 + \sin\alpha\, \tilde\eta^2 + a\, \tilde\eta^3
\notag
\\
&&
\eta^2 = -\sin\alpha\, \tilde\eta^1 + \cos\alpha\, \tilde\eta^2 + b\, \tilde\eta^3
\\
&&
\eta^3 = e^{-\varphi} \, \tilde\eta^3.
\notag
\label{eq:3_6}\end{aligned}$$
From Proposition \[prop:2\_3\], b) it follows that $d\varphi = \lambda_\omega \xi_1 \eta^1 + \lambda_\omega \xi_2 \eta^2 + \varphi_3 \eta^3$, Then $$\begin{split}
d\, \tilde\eta^3 = d (e^\varphi \eta^3) =
e^\varphi d\varphi \wedge \eta^3 + e^\varphi d\eta^3 = e^\varphi \lambda_\omega (\xi_1 \eta^1 + \xi_2 \eta^2) \wedge \eta^3 + e^\varphi (-\lambda_\omega \eta^1 \wedge \eta^2)
\\
=\lambda_\omega [(\xi_1 \eta^1 + \xi_2 \eta^2) \wedge \tilde\eta^3 - e^\varphi \eta^1 \wedge \eta^2].
\label{eq:3_7}
\end{split}$$ From it follows that $$\xi_1 \eta^1 + \xi_2 \eta^2 = (\xi_1 \cos\alpha - \xi_2 \sin\alpha)\tilde\eta^1 + (\xi_1 \sin\alpha + \xi_2 \cos\alpha)\tilde\eta^2 + (a\xi_1 + b\xi_2) \tilde\eta^3.
\label{eq:3_8}$$ then $$(\xi_1 \eta^1 + \xi_2 \eta^2)\wedge \tilde\eta^3 = (\xi_1 \cos\alpha - \xi_2 \sin\alpha)\tilde \eta^1\wedge \tilde\eta^3 + (\xi_1 \sin\alpha + \xi_2 \cos\alpha)\tilde \eta^2\wedge \tilde\eta^3.
\label{eq:3_9}$$ Also $$\eta^1 \wedge \eta^2 = \tilde\eta^1 \wedge \tilde\eta^2 + (b\cos\alpha + a\sin\alpha )\tilde\eta^1 \wedge \tilde\eta^3 + (b\sin\alpha - a\cos\alpha )\tilde\eta^2 \wedge \tilde\eta^3.
\label{eq:3_10}$$ The equation written for the coframe $\{\tilde\eta^a\}$ gives $d\tilde\eta^3 = -\lambda_{\tilde{\omega}} \tilde\eta^1 \wedge \tilde\eta^2$, and from Proposition \[prop:2\_2\] we have $\lambda_{\tilde{\omega}} = e^\varphi \lambda_\omega$. Hence follows $d\tilde\eta^3 = - e^\varphi\lambda_{\omega} \tilde\eta^1 \wedge \tilde\eta^2$. Thus , , and together give the equation system $$\left\{
\begin{array}{l}
\lambda_\omega[\xi_1 \cos\alpha - \xi_2\sin\alpha - e^\varphi(b\cos\alpha + a\sin\alpha)]=0
\\
\lambda_\omega[\xi_1 \sin\alpha - \xi_2\cos\alpha - e^\varphi(b\sin\alpha - a\cos\alpha)]=0
\end{array}
\right.
\label{eq:3_11}$$ As $\lambda_\omega \ne 0$ almost everythere in $U$, we have that the expressions in brackets in vanish. Therefore, we arrive at the system $$\left\{
\begin{array}{l}
(\xi_1 - e^\varphi b) \cos\alpha - (\xi_2+e^\varphi a)\sin\alpha = 0
\\
(\xi_1 - e^\varphi b) \sin\alpha - (\xi_2+e^\varphi a)\cos\alpha = 0
\end{array}
\right.
\label{eq:3_12}$$ Thus we have found $$a = - e^{-\varphi} \xi_2, \qquad b = e^{-\varphi} \xi_1.
\label{eq:3_13}$$
Now it remains to find the function $\alpha$ in . To this end we use . We have $$\begin{split}
d \lambda_{\widetilde\omega} = d (e^\varphi \lambda_\omega) = e^\varphi \lambda_\omega d\varphi + e^\varphi d\lambda_\omega =
e^\varphi\lambda_\omega(\lambda_\omega \xi_1 \eta^1+\lambda_\omega \xi_2 \eta^2+\varphi_3\eta^3) + e^\varphi(\lambda_2 \eta^2 + \lambda_3 \eta^3) =
\\
e^\varphi [ \lambda_\omega^2 \xi_1 \eta^1 + (\lambda_\omega^2 \xi_2 + \lambda_2) \eta^2 + (\lambda_\omega \varphi_3 + \lambda_3)\eta^3].
\end{split}
\label{eq:3_14}$$ To we substitute and get that $$\begin{split}
d\lambda_{\widetilde\omega} = e^\varphi [\lambda_\omega^2 \xi_1 \cos\alpha - (\lambda_\omega^2\xi_2 + \lambda_2) \sin\alpha] \tilde\eta^1 +
e^\varphi[\lambda_\omega^2 \xi_1 \sin\alpha + (\lambda_\omega^2\xi_2 + \lambda_2) \cos\alpha] \tilde\eta^2 +
\\
(\lambda_\omega \varphi_3 + \lambda_2 \xi_1 + \lambda_3)\tilde\eta^3.
\end{split}
\label{eq:3_15}$$ From this follows that $$d\lambda_{\widetilde\omega} = \widetilde\lambda_1 \tilde\eta^1 + \widetilde\lambda_2 \tilde\eta^2 + \widetilde\lambda_3 \tilde\eta^3,
\label{eq:3_16}$$ where $$\begin{aligned}
&&
\widetilde\lambda_1 = e^\varphi [\lambda_\omega^2 \xi_1 \cos\alpha - (\lambda_\omega^2\xi_2 + \lambda_2) \sin\alpha],
\label{eq:3_17_1}
\\
&&
\widetilde\lambda_2 = e^\varphi[\lambda_\omega^2 \xi_1 \sin\alpha + (\lambda_\omega^2\xi_2 + \lambda_2) \cos\alpha]
\label{eq:3_17_2}
\\
&&
\widetilde\lambda_3 = \lambda_\omega \varphi_3 + \lambda_2 \xi_1 + \lambda_3
\label{eq:3_17_3}\end{aligned}$$ From it follows that $\lambda_3 = \lambda_\omega f$, in the same way, $\widetilde\lambda_3 = \lambda_{\widetilde\omega} \tilde f$, where $f$, $\tilde f$ are functions. Then $\widetilde\lambda_3 = e^\varphi \lambda_\omega \tilde f$. As $\lambda_2 \ne 0$ at $\Sigma$ (see reasoning below ), from we have that $$\xi_1 = \lambda_\omega \mu.
\label{eq:3_18}$$ Also, by , $\widetilde\lambda_1 = 0$, hence and give us the expression for $\alpha$: $$\tan\alpha = \frac{\lambda_\omega^3 \mu}{\lambda_\omega^2\xi_2+\lambda_2}.
\label{eq:3_19}$$ Thus we have proved
Let $\left\{ \eta^a\right\}$ be the adapted frame determined by a special form $\omega$, and $\left\{ \tilde\eta^a\right\}$ be the adapted frame determined by a special form $\widetilde\omega = e^\varphi\omega$. Then $$\begin{aligned}
&&
\eta^1 = \cos\alpha\, \tilde\eta^1 + \sin\alpha\, \tilde\eta^2 - e^{-\varphi}\xi_2 \, \tilde\eta^3
\label{eq:3_20_1}
\\
&&
\eta^2 = -\sin\alpha\, \tilde\eta^1 + \cos\alpha\, \tilde\eta^2 + e^{-\varphi}\xi_1\, \tilde\eta^3
\label{eq:3_20_2}
\\
&&
\eta^3 = e^{-\varphi} \, \tilde\eta^3,
\label{eq:3_20_3}\end{aligned}$$ Here functions $\xi_1$, $\xi_2$, and $\alpha$ are determined by $\varphi$ in the following way: $$\begin{aligned}
&&
E_1\varphi = \varphi_1 = \lambda_\omega \xi_1 = \lambda_\omega^2\mu
\label{eq:3_21_1}
\\
&&
E_2\varphi = \varphi_2 = \lambda_\omega \xi_2
\label{eq:3_21_2}
\\
&&
\tan\alpha = \frac{\lambda_\omega^3 \mu}{\lambda_\omega^2\xi_2+\lambda_2},
\label{eq:3_21_3}\end{aligned}$$ where $\left\{ E_a \right\}$ is the adapted frame dual to $\left\{ \eta^a \right\}$, $\mu$ is a function, and $\lambda_2 = E_2\lambda_\omega$. \[prop:2\_5\]
Invarians of sub-Riemannian surface along the singular surface
--------------------------------------------------------------
Let $\left\{ \eta^a\right\}$ be the adapted frame determined by a special form $\omega$, and $\left\{ \tilde\eta^a\right\}$ be the adapted frame determined by a special form $\widetilde\omega = e^\varphi\omega$. Then we have the structure equations: $d\eta^a = C^a_{bc} \eta^b \wedge \eta^c$ and $d\tilde\eta^a = \tilde C^a_{bc} \tilde\eta^b \wedge \tilde\eta^c$. Let us denote the restrictions of the structure functions $C^a_{bc}$ and $\tilde C^a_{bc}$ to the surface $\Sigma$ by $Q^a_{bc}$ and $\tilde Q^a_{bc}$, respectively. Let us find relation between $Q^a_{bc}$ and $\tilde Q^a_{bc}$,
The surface $\Sigma$ is given by the equation $\lambda_\omega = 0$. Using Proposition \[prop:2\_5\], we get that at the points of $\Sigma$ the equalities – are written as follows: $$\begin{aligned}
&&
\eta^1 = \tilde\eta^1 - e^{-\varphi}\xi_2 \, \tilde\eta^3
\label{eq:3_22_1}
\\
&&
\eta^2 = \tilde\eta^2
\label{eq:3_22_2}
\\
&&
\eta^3 = e^{-\varphi} \, \tilde\eta^3,
\label{eq:3_22_3}\end{aligned}$$ Now, let us take the exterior derivative of $$\begin{split}
d\eta^1 = -\sin\alpha d\alpha\wedge\tilde\eta^1 + \cos\alpha d\tilde\eta^1 + \cos\alpha d\alpha\wedge\tilde\eta^2 + \sin\alpha d\tilde\eta^2 +
\\
e^{-\varphi} \xi_2 d\varphi\wedge\tilde\eta^3 - e^{-\varphi} d\xi_2\wedge\tilde\eta^3 - e^{-\varphi} \xi_2 d\tilde\eta^3.
\end{split}
\label{eq:3_23_1}$$ and take the result at a point of $\Sigma$, then we have, by , that $\cos\alpha=1$, $\sin\alpha=0$, $d\alpha = 0$. Also, by and , $$d\varphi \wedge \tilde\eta^3 = (\varphi_1 \eta^1 + \varphi_2 \eta^2 + \varphi_3 \eta^3)\wedge\tilde\eta^3 = \lambda_\omega \xi_1 \eta_1 \wedge \tilde\eta^3 + \lambda_\omega \xi_2 \eta_2 \wedge \tilde\eta^3,
\label{eq:3_24}$$ hence, at points of $\Sigma$, $d\varphi \wedge \tilde\eta^3 = 0$. In addition, from it follows that at $\Sigma$, $d\tilde\eta^3=0$. Therefore, on $\Sigma$ we have $$d\eta^1 = d\tilde\eta^1 - e^{-\varphi} d\xi_2 \wedge \tilde\eta^3 = d\tilde\eta^1 - d\xi_2 \wedge \eta^3.
\label{eq:3_25}$$ From – it follows that on $\Sigma$ we have $$\begin{aligned}
&&
\tilde\eta^1 = \eta^1 + \xi_2 \,\eta^3
\label{eq:3_26_1}
\\
&&
\tilde\eta^2 = \eta^2
\label{eq:3_26_2}
\\
&&
\tilde\eta^3 = e^{\varphi} \,\eta^3,
\label{eq:3_26_3}\end{aligned}$$ Then, at points in $\Sigma$ we have $$\begin{gathered}
d\tilde\eta^1 = \tilde Q^1_{23} \tilde\eta^2 \wedge \tilde\eta^3 + \tilde Q^1_{31} \tilde\eta^3 \wedge \tilde\eta^1 + \tilde Q^1_{12} \tilde\eta^1 \wedge \tilde\eta^2 =
\\
(e^\varphi \tilde Q^1_{23} - \xi_2 \tilde Q^1_{12}) \eta^2 \wedge \eta^3 +
e^\varphi \tilde Q^1_{31} \eta^3 \wedge \eta^1 +
\tilde Q^1_{12} \eta^1 \wedge \eta^2.
\label{eq:3_27}\end{gathered}$$ Set $d\xi_2 = \xi_{21}\eta^1 + \xi_{22}\eta^2 + \xi_{23}\eta^3$, and substitute it together with to . Then we get $$d\eta^1 =
(e^\varphi \tilde Q^1_{23} - \xi_2 \tilde Q^1_{12} - \xi_{22}) \eta^2 \wedge \eta^3 +
(e^\varphi \tilde Q^1_{31} + \xi_{21}) \eta^3 \wedge \eta^1 +
\tilde Q^1_{12} \eta^1 \wedge \eta^2.
\label{eq:3_28}$$ Thus, $$Q^1_{23} = e^\varphi \tilde Q^1_{23} - \xi_2 \tilde Q^1_{12} - \xi_{22};
\quad
Q^1_{31} = e^\varphi \tilde Q^1_{31} + \xi_{21};
\quad
Q^1_{12} = \tilde Q^1_{12}.
\label{eq:3_29}$$ In the same manner we prove that $d\eta^2 = d\tilde\eta^2 + d\xi_1\wedge\eta^3$, By , we have $$d\xi_1 = \mu d\lambda_\omega + \lambda_\omega d\mu = \mu(\lambda_2 \eta^2 + \lambda_3\eta^3) + \lambda_\omega d\mu
\label{eq:3_30}$$ By , we get that $\lambda_3 = 0$ on $\Sigma$, so at points of $\Sigma$ we have $$d\xi_1 = \mu\lambda_2 \eta^2.
\label{eq:3_31}$$ Then we get $$Q^2_{23} = e^\varphi \tilde Q^2_{23} - \xi_2 \tilde Q^2_{12} + \mu\lambda_2;
\quad
Q^2_{31} = e^\varphi \tilde Q^2_{31};
\quad
Q^2_{12} = \tilde Q^2_{12}.$$ Thus we have proved
Let $d\eta^a = C^a_{bc} \eta^b \wedge \eta^c$ be the structure equations of the adapted frame of the sub-Riemmanian surface in a neighborhood of a point in $\Sigma$. The functions $Q^1_{12} = C^1_{12}|_\Sigma$ and $Q^2_{12} = C^2_{12}|_\Sigma$ do not depend on the choice of adapted frame, and so these functions are invariants of the surface.
If $V$ is an infinitesimal symmetry of the sub-Riemannian surface, then $V Q^1_{12}= 0$, and $V Q^2_{12} = 0$.
Any symmetry $f$ of the sub-Riemannian surface maps $\Sigma$ onto itself, and sends a special form to a special form and the corresponding adapted frame to the corresponding adapted frame. Therefore, $f^* Q^1_{12} = Q^1_{12}$ and $f^* Q^2_{12} = Q^2_{12}$. From this follows that an infinitesimal symmetry $V$ is tangent to $\Sigma$ and $V Q^1_{12} = V Q^2_{12} = 0$.
[99]{}
Yu. Aminov, The geometry of vector fields, Gordon and Breach Publishers, Amsterdam, 2000. P. Bieliavsky, E. Falbel, C. Gorodski, The classification of simple-connected contact sub-Riemannian symmetric spaces, Pacific J. of Math. Vol. 188, 1999 A.M. Bloch, J.E. Marsden, D. Zenkov, Nonholonomic Dynamics, Notices of AMS, Vol 52,3, 2005. A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden, R. Murray, Nonholonomic Mechanical Systems with Symmetry, Arch. Rational Mech. Anal. 169, 1996. K. Ehlers, Geometric Equivalence of Nonholonomic three-manifolds, Proceedinds of the fourth International Conference on Dynamical System and Differential Equations, May 24 – 27, (2002), Wilmington, NC, USA pp. 246–255. K. Ehlers, J. Koiller, P. M. Rios, Nonholonomic Systems: Cartan’s equivalence and Hamiltonization, Vienna, Preprint ESI 1389 (2003) E. Falbel, C. Gorodski, Sub-Riemannian Homegeneous Spaces in Dimensions 3 and 4, Geometriae Dedicadta 62, 1996. E. Falbel, C. Gorodski, On contact sub-Riemannian symmetries spaces, Annales scientifiques de l’É.N.S. 4e série, tome 28, nº5, 1995, p.571-589 M. Fels, P.J. Olver, Moving Coframes: I. A Practical Algorithm, Acta Applicandae Mathematicae 51, Kluwer Academic Publishers. Printed in the Netherlands, 1998, p.161–213. R.B. Gardner, The Method of Equivalence and Its Applications, SIAM, Philadelphia, 1989. P.Ya. Grozman, D.A. Leites, Nonholonomic Riemann and Weyl tensor for flag manifolds, Theoretical and Mathematical Physics, 153, 2007. K. Hughen, The Geometry of Subriemannian Three-Manifolds, PhD. Thesis, Duke Univ. 1995. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Interscience, London, 1963. R. Montgomery, A tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, V. 91, AMS, Providence, 2002. P.J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. V.P. Pavlov, V.M. Sergeev, Thermodynamics from the differential geometry standpoint, Theoretical and Mathematical Physics, 157, 2008. K.V. Pereira, Vector fields, invariant varieties and linear system, Annales de L’Institute Fourier Tome 51, nº5, 2001. W. Schempp, Sub-Riemannian Geometry and Clinical Magnetic Resonance Tomography, Math. Meth. Appl. Sci., 22, 1999. B.N. Shapukov, Connections on dlfferentiable bundles, Itog. Nauki Tekh. Probl. Geom., 15, 61-93, 1983. R.S. Strichartz, Sub-Riemannian Geometry, J. Differential Geometry, 24, 1986. Y. L. Sachkov. Symmetries of flat rank two distributions and sub-riemannian structures, Trans. AMS, V. 356(2), 2003, 457–494.
A.M. Vershik, V. Gerhskovich, Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems, Dynamical Systems VII. Ed. V.I. Arnol’d and S.P. Novikov, vol. 16 of the Encyclopedia of Mathematical Sciences series, Springer-Verlag, NY, 1994 M. Zhitomirskii, Typical singularities of Differential 1-forms and Pfaffian equations, Translation of Mathematical Monographs, Vol. 113, AMS, 1992
[^1]: $d\eta^{i}=C^{i}_{jk}\eta^{j}\wedge\eta^{k}$, $[E_{j}, E_{k}]=c^{i}_{jk}E_{i}$, $C^{i}_{jk}=-c^{i}_{jk}$
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CONDLEY: Daphne Oz’s fudgey brownies are Snicket approved
While off work because of the pandemic, I was flipping through the television and stopped on the Dr. Oz Show.
His daughter, Daphne, was in the kitchen with her children making something, and I wanted to see what was happening. About half way through the segment, Brad came in and I told him they were making “good for you” brownies.
He finished watching the baking portion of the show with me, and said, “We should try those.”
Brad had been in the mood for something sweet, which rarely happens, so I decided to give the brownies a try.
Just a few days before we had been looking through our pantry and found a can of black beans. Neither of us really knew why we had them; I suspect we bought them thinking Brad would use them in his chili for something different. This was the perfect time to use those beans and try something different along with satisfying Brad’s not-so-sweet tooth.
I had not written anything down while Daphne was in the kitchen with her kids making these brownies, so I headed to the computer to find the recipe. I just searched “Daphne Oz brownies,” and lots of websites popped up. I looked at three or four of the recipes, and they were all alike except for one which called for more coffee than the others.
I opted for the recipe I found on doctoroz.com, which listed less coffee.
Daphne Oz’s Fudgey Brownies
INGREDIENTS
— 1 cup sweet potato
— 2 tablespoons butter, melted
— 2 eggs
— 1/2 cup brown sugar
— 1 teaspoon vanilla
— 1/2 cup black beans, smashed
— 1/2 cup semi-sweet chocolate chips
— 1/3 cup hot coffee
— 3/4 cup unsweetened cocoa powder
— 1/2 teaspoon baking soda
— 1 teaspoon kosher salt
INSTRUCTIONS
Preheat oven to 350 degrees.
Microwave sweet potato until tender, about seven minutes, and set aside to cool. Peel off skin and mash.
Whisk the sweet potato, eggs, melted butter, brown sugar, black beans and vanilla together in a large bowl.
Place the coffee and the chocolate into a heat-proof glass measuring cup and microwave until melted.
In a separate bowl, combine the dry ingredients. Stir the melted chocolate into the sweet potato mixture. Then fold in the dry ingredients.
Pour into a greased pan or muffin tin and bake for eight to 10 minutes. Let cool and serve.
The first thing I did was microwave the sweet potato. I let it cool while Brad made some coffee.
I put the chocolate chips into a glass two-cup measuring cup and poured the coffee in. This started the melting process, but I ended up putting the coffee/chocolate mixture in the microwave and heating it a bit to completely melt the chocolate.
By this time, the sweet potato was slightly cool, and I removed the peel and mashed it.
Next, it was time to tackle the black beans. I wasn’t sure if I needed to drain the beans, so I headed back to the computer and looked up Daphne’s recipe again. One of the websites said to drain and rinse the beans. Brad took care of that for me.
To puree the beans he placed them in the blender. They were so dry that he ended up adding a tiny bit of water and blending them a little more just so there wouldn’t be chunks of beans in the batter.
By this time, the sweet potato was cool enough to add the eggs, melted butter, brown sugar, black beans and vanilla. After a little mixing, I poured in the coffee/chocolate mixture, stirring just to combine.
I folded in the dry ingredients and was ready to scoop the mixture into a pan.
One of the recipes I had reviewed said this recipe would make 18 mini muffins. Since I think mini muffins are cute, I sprayed one of my 24-cup mini muffin pans with cooking spray. I sprayed all 24 because it looked like there was plenty of batter to make that many.
After filling all 24 cups, there was still batter left over. I decided we’d try larger muffins the second go round, and filled a regular size six-cup muffin pan with the remaining batter.
When the mini brownies came out of the oven, we let them cool slightly before Brad popped one in his mouth.
“They are moist and fudgey, but they don’t satisfy my sweet tooth,” he said.
By now you know Brad isn’t crazy about sweets, so to me, this meant they weren’t sweet at all. Before trying a second one he sprinkled it with powdered sugar, and he said that did the trick.
Since we had so many “good for you” brownies, and I don’t care for chocolate or things that are good for you (aka healthy), I texted our neighbor, Rachel, and asked if she’d like to have some.
She said, “Of course,” and I headed to her house. I handed the plate of brownies to her through the crack in the door and told her to let me know what she thought of them.
A couple of hours later, I received a photo from Rachel with the empty plate sitting on the counter and the message “You see this?” It’s a running joke with us that her husband, Chris, will sometimes eat all of something without sharing with her or their daughter, Maggie. They also have a 10-month-old Labradoodle named Snicket, and I replied, “Bad Snicket;” as a joke, thinking she and maybe Maggie had eaten all the brownies before Chris got home from work.
After several texts back and forth I realized Snicket really had eaten all of the brownies. You see, Rachel and Chris were in another part of the house when Snicket came in the room, laid down and started licking his lips and smacking his mouth.
Rachel wondered what in the world he was doing. She thought maybe he’d eaten another sock that had been laying around the house. After her talk with Chris, she headed to the kitchen for a brownie bite and discovered the empty plate and the crumbs on the counter. She immediately realized why Snicket had been smacking his mouth.
When I figured out Rachel didn’t get to try the brownies, I told her I still had some, and she headed to our house. I slipped them through her car window and she headed home, vowing to eat one on the way.
A little later Rachel texted me again and said she liked the brownies a lot; that they actually tasted healthy and she liked that they weren’t too sweet. She also mentioned that they were very rich and she could eat them a couple of times a month. The next day, Rachel informed me that she’d hidden the remaining brownies but Snicket somehow found them and ate the rest of them. I guess she doesn’t have a very good hiding place, or Snicket just really likes brownies.
At first, we were worried he would get really sick from the chocolate, but he must have an iron clad stomach because he’d eaten 10 brownies, three pencils, a tube of ointment, a half cup of coconut oil and a My Little Pony head, all within 48 hours. I guess he’s still a puppy, though a big one, and is testing the waters with his family.
We still had brownies, and Brad took a few to Sandra, his physical therapist, the next day. Sandra also really liked the brownies.
That evening I asked Brad what he thought about the brownies and if he’d want them again. He had a funny look on his face and said they were better the second day, but he’d rather have a chocolate chip pie.
My mom tried the brownies and agreed with Brad, they weren’t very sweet.
With that being said, I’ll have to say this is a failed it recipe at our house. But Sandra, Rachel and Snicket all say it’s a nailed it recipe.
I guess you’ll just have to try this one your self and you be the judge.
Just watch out for a Snicket in your house.
Sarah Condley is an amateur baker and chef who is compiling a cookbook of her favorite recipes. | https://www.winchestersun.com/2020/05/01/daphne-ozs-fudgey-brownies-are-snicket-approved/ |
Q:
Merge similar lists of lists in Haskell
I need to combine multiple lists with common elements to one. And i should do it inside list.
Eg:
INPUT: [[1,2,3], [5,6], [8,3,11], [4,9,1]]
MY output: [[1,2,3,8,3,11,1,2,3,4,9,1],[]]
Needed OUTPUT: [[1,2,3,8,11,9], [5,6]]
Another example:
INPUT: [[4],[0],[7,10],[6],[6]]
MY output: [[6,6],[]]
OUTPUT: [[4],[0],[7,10],[6]]
My code:
mergeAllLists :: [[Integer]] -> [[Integer]]
mergeAllLists (x:[]) = [x]
mergeAllLists (x:[]:y) = [x]
mergeAllLists (x:y:[]:_) = mergeOneToAll_ x [y]
mergeAllLists (x:xs) = mergeAllLists (mergeOneToAll x xs)
mergeOneToAll :: [Integer] -> [[Integer]] -> [[Integer]]
mergeOneToAll _ [] = [[]]
mergeOneToAll list (y:list_of_list) = (mergeLists list y) : ( mergeOneToAll list list_of_list )
A:
mergeAllLists :: [[Integer]] -> [[Integer]]
mergeAllLists = foldl mergeOneToAll []
mergeOneToAll :: [[Integer]] -> [Integer] -> [[Integer]]
mergeOneToAll [] xs = [xs]
mergeOneToAll (as:acc) xs =
if null $ intersect xs as then
as : mergeOneToAll acc xs
else
(as ++ xs) : acc
It's not very efficient though (not tail recursive). Let me know if it needs to be more efficient and I will try and improve on it.
| |
In order to motivate the students and have them begin to explore the real world problems of area and perimeter, I will read the book "Spaghetti and Meatballs for All" by Marilyn Burns. It is a funny story about a couple, Mr. and Mrs. Comfort, hosting a dinner party. At the party, the guests begin to move the tables around and that ruins the number of seats available.
The story is a bit long, so I suggest you take two days to read it, with activities as you go. On the first day, read the first half and have the students look for patterns and really try to figure out what the main problem is, which is that the perimeter, or places the guests can sit, changes as the combined areas remain the same.
Remember, that reading the book as an engaging activity does not "teach" the mathematical concepts of area and perimeter. However, in reading this book, you will now have a common referent as you explore area and perimeter.
As you read, allow the students to enjoy the story and the illustrations.
Boys and Girls, Mr. and Mrs. Comfort have a real problem in the story "Spaghetti and Meatballs for All". They are having a dinner party for 32 people and when everyone comes, the company rearranges their tables!
Have you ever seen your parents do that in a restaurant or at your home? Sometimes they push tables together or pull them apart so different people can sit together? That's what happens in the story. As you enjoy, watch for the problems that moving the tables causes.
I display my book on the document camera so all students can take in the illustrations, which helps them connect to the math in this story. I think it is important to show the class the illustrations, as the story is read, so students pick up on that chairs are eliminated every time the same tables are rearranged.
I found it helpful to pause at every page spread and have the students just observe the illustrations. They get a kick out of the antics of the cat! Use questions to guide them to see the visual clues that the perimeter (or seating space) is getting smaller. Have them look for stacked chairs, people standing, Mrs. Comfort acting upset. All of this interaction will help when they work on their partner activity.
Boys and girls, can you turn and talk to your partner about what is happening to the seating? Mrs. Comfort looks uncomfortable. Why do you think she feels that way?
Why are there extra chairs? At the beginning of the story there weren't enough, now there are too many?
Listen to student ideas to pull out the understanding that everywhere the tables touch, two chairs must be removed. Mathematically, they will begin to understand that the number of tables (or area) remains the same, but the perimeter changes. In this case, it decreases as the tables are pushed together. In focusing on this, you will really be promoting Mathematical Practice 4: Look for and express regularity in repeated reasoning. As you read the next section, you will notice the students really grabbling with this.
I stop reading halfway through the book, at the point when I know the students understand that every time a new set of guests arrive, the tables are moved around again. Now is the perfect time to engage them in the problem solving of this book!
To do this activity, I've created sets of 8 square inch blocks for the students to use. Their use can first be modeled on the projector to re-create the original setting of the book, which is 8 separate tables. Ask some mathematical questions:
After I've established that students are recognizing the connection between the arrangement and the number of places people can sit, I send the students off with their partners to continue to figure out more seating arrangements. Give them these specific rules:
I challenged my students to look for an arrangement that would seat the most people, one that's different than the original arrangement. You may choose to do this, or have them explore arrangements and the seating that would go with it, for discussion later at the close.
This group is working on how to rearrange one of their models to get more seating, and then trying to figure out a way of counting the seats.
This partnership works on figuring out how to count the perimeter and realizes there is a way to increase it.
For closing today, I have the students share their favorite configuration and strategy. As the students share their area and perimeter findings, my other students are expected to use our math talking moves to respond.
I had my students explain whether they agreed or disagreed with the findings of their peers. In doing this, both the presenter and the questioner must once again think through their strategies and often they find their own mistakes. This is much more powerful than me telling them they made an error, which they may or may not grasp. This way they internalize their strategies. It is also a wonderful way to develop students' mathematical practices. In today's lesson I'm focusing on MP3: Construct viable arguments and critique the reasoning of others and MP7: Attend to precision, which means students advance their mathematical communication skills as they try to use the precise language to speak with each other.
Mathematicians, your thinking today was wonderful. I am impressed with the patterns and solutions you found for our problem! Many of you began to develop strategies that you can use every time you are working to find area or perimeter. Tomorrow we will again work with these concepts and share out more of our thinking. | https://betterlesson.com/lesson/538299/area-and-perimeter-in-real-life-day-1?from=breadcrumb_lesson |
Battery capacity is rated in amp-hours. This is a measure of how much amperage can be drawn from a fully charged battery over time until it is discharged (for a 12 volt battery this is when it reaches 10.5v). Then the amps are multiplied by the amount of time taken to get the Ah rating.
As an example of this, if you had a device that pulled 25 amps and you used it for 30 minutes then the amp-hours used up would be 25(amps) x 0.5(hours) = 12.5 amp-hours.
The standard and most widely accepted rating period for deep cycle batteries is 20 hours. This means that the battery was discharged down to 10.5 volts over 20 hours while measuring the total amp hours it supplies.
Sometimes however the time period can differ, in some circumstances knowing the 6 hour amp rating may be more useful and in others the 100 amp hour rating may be used. Due to something called the Peukert effect a battery gives higher amp hours when it is discharged over a longer time period. For instance the rated Capacity for the Trojan T105-RE is 225 at 20 hours and 250 at 100 hours.
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1. Introduction {#sec1-sensors-17-00649}
===============
Location-based services (LBSs) have been popular for many years. Although global navigation satellite systems (GNSSs) can provide good localization services outdoors, there is still no dominant indoor positioning technique \[[@B1-sensors-17-00649]\]. Therefore, an alternative technology is required that can provide accurate and robust indoor localization and tracking. Moreover, the spatial structures of indoor spaces are usually more complex than the outdoor environment, and thus, distinctive information is needed to better describe locations for the LBS-based applications.
With the wide availability of smartphones, a large amount of research has been conducted in recent years targeting indoor localization. Most of the existing indoor localization technologies require additional infrastructure, such as ultra-wideband \[[@B2-sensors-17-00649]\], laser scanning systems (LSSs), radiofrequency identification (RFID) \[[@B3-sensors-17-00649]\] and Wi-Fi access points \[[@B4-sensors-17-00649]\]. However, these approaches often require extensive labor and time. To solve this problem, pedestrian dead reckoning (PDR) has recently been proposed as one of the most promising technologies for indoor localization \[[@B5-sensors-17-00649]\]. Differing from the above approaches, PDR uses the built-in smartphone inertial sensors (accelerometer, gyroscope and magnetometer) to estimate the position. However, PDR suffers from error accumulation when the travel time is long. To achieve improved localization results, a number of studies have been conducted under particular circumstances, but the applicability and accuracy are still limited.
In addition to the direct application in indoor localization, the built-in smartphone sensors can also be used to understand the user's movements \[[@B6-sensors-17-00649]\], as well as to identify the indoor environment. Sensing the implied location information about the user moving in the corresponding environment provides a new opportunity for indoor mobile localization. To exploit this underlying information, some studies have been presented based on human activity recognition (HAR) \[[@B7-sensors-17-00649],[@B8-sensors-17-00649],[@B9-sensors-17-00649]\], which uses these sensors to identify user activity and then infers information about the context of the user's location. Therefore, it is worth exploring how to use this information to assist with indoor localization.
Recently, semantic information in the indoor environment has received increased attention. In many cases, semantic information is as valuable as the location. For example, from a human cognition perspective, in comparing the position coordinates, it is more valuable to know if a location is a room, a corridor or stairs \[[@B10-sensors-17-00649]\]. Furthermore, it is also more convenient for a user to obtain semantic information (e.g., "turn left", "turn right", "go upstairs", "go downstairs" and "go into a room") than information about a route. However, the extraction and description of the necessary semantic information remains an open challenge.
In this paper, a method that combines PDR, HAR and landmarks is developed to accurately determine indoor localization. The proposed method requires no additional devices or expensive labor, and the user trajectory can be corrected and displayed. In addition, to solve the initial position determination problem, a hidden Markov model (HMM) that considers the characteristics of the indoor environment is used to match the continuous trajectory. Furthermore, to describe the user's indoor activities and trajectories, an indoor semantic landmark model is also constructed by detecting the user's activities.
[Figure 1](#sensors-17-00649-f001){ref-type="fig"} shows an overview of the proposed approach.
The remainder of the paper is organized as follows. The related works are briefly reviewed in [Section 2](#sec2-sensors-17-00649){ref-type="sec"}. The primary methods are then introduced in [Section 3](#sec3-sensors-17-00649){ref-type="sec"}. [Section 4](#sec4-sensors-17-00649){ref-type="sec"} presents the experimental process, and [Section 5](#sec5-sensors-17-00649){ref-type="sec"} discusses and analyzes the experimental results. Finally, the conclusions and recommendations for future work are presented in [Section 6](#sec6-sensors-17-00649){ref-type="sec"}.
2. Related Works {#sec2-sensors-17-00649}
================
Most of the existing indoor localization technologies require additional infrastructure or expensive labor and time. How to achieve reliable and accurate localization in indoor environments at a low cost is still a challenging task \[[@B11-sensors-17-00649]\]. Compared to other methods of indoor localization, using the built-in smartphone sensors provides a more convenient and less expensive indoor localization method, which has the advantage of providing continuous localization across the whole indoor space \[[@B12-sensors-17-00649]\]. Smartphone-based pedestrian dead reckoning (SmartPDR) \[[@B13-sensors-17-00649]\] has been the subject of increased attention, and it is now considered a promising technology for low-cost and continuous indoor navigation \[[@B1-sensors-17-00649]\]. However, a number of parameters, such as step length and walking direction, can easily affect PDR's localization accuracy \[[@B14-sensors-17-00649]\]. Furthermore, PDR suffers from error accumulation over time because of the low-cost sensors, and thus, it is necessary to combine it with other methods, such as Wi-Fi, indoor map assistance or landmark matching.
With Wi-Fi routers widely deployed in most buildings \[[@B15-sensors-17-00649]\], many studies have combined Wi-Fi and PDR to complement the other's drawbacks. An early attempt can be found in \[[@B16-sensors-17-00649]\], in which a PDR-based particle filter was used to smooth Wi-Fi-based positioning results, and a Kalman filter based on Wi-Fi was used to correct the PDR errors. Furthermore, a barometer was used to identify upstairs and downstairs. Another study \[[@B17-sensors-17-00649]\] used a Bayes filter to combine PDR and Wi-Fi fingerprinting, and PDR was used to update the motion model. The Wi-Fi fingerprinting method was used to correct the model. In order to improve the efficiency of Wi-Fi-based indoor localization, some new techniques have been recently adopted, such as the Light-Fidelity-assisted approach \[[@B18-sensors-17-00649]\] and received signal strength estimation based on support vector regression \[[@B19-sensors-17-00649]\].
Aside from Wi-Fi, indoor maps can also be used to reduce the accumulated error of PDR. A number of studies have used map information for location correction. In \[[@B20-sensors-17-00649]\], the user's location, stride length and direction were used as the state values of the particle filter. To achieve localization with less computational resources, a conditional random field (CRF)-based method was proposed in \[[@B21-sensors-17-00649]\]. Maps as constraints were used in this work, and the Viterbi algorithm was used to generate a backtracked path. In \[[@B22-sensors-17-00649]\], maps were also considered as constraints, and the impossible paths were eliminated when the user walks for a sufficient length. The data from the trajectory were then used to construct a Wi-Fi training set. A wider integration can be found in \[[@B23-sensors-17-00649],[@B24-sensors-17-00649]\], where PDR, Wi-Fi and map information were combined to achieve pedestrian tracking in indoor environments. Particle filter-based approaches were used to match the maps in these approaches.
Like indoor maps, landmarks, which can be detected by the unique patterns of smartphone sensor data, can be used to correct the PDR trajectory \[[@B10-sensors-17-00649]\]. Indoor landmarks detected by HAR provide a new opportunity for indoor localization \[[@B15-sensors-17-00649]\]. Activities, such as going upstairs (or downstairs), turning or opening doors, can be treated as landmarks. In \[[@B25-sensors-17-00649]\], an accelerometer was used to recognize standing, walking, stairs, elevators and escalators, achieving accurate recognition. In \[[@B15-sensors-17-00649]\], Wi-Fi, PDR and landmarks were combined to provide a highly accurate localization system. In this work, a Kalman filter was used to combine the Wi-Fi- and PDR-based localization techniques with landmarks. Without relying on Wi-Fi infrastructure, the built-in accelerometer and magnetometer in a smartphone were used to record pedestrians' walking patterns in \[[@B26-sensors-17-00649]\], which were then matched to an indoor map.
Semantic information has also received attention from researchers. An integrated navigation system that considers both geometrics and semantics was presented in \[[@B27-sensors-17-00649]\]. This work proposed a semantic model that can be used to describe indoor navigation paths. Similarly, a semantic description model derived from a spatial ontology was used to describe the basic elements of navigation paths in \[[@B28-sensors-17-00649]\]. A human-centered semantic indoor navigation system was also proposed in \[[@B29-sensors-17-00649]\]. In order to provide services involving human factors, the system used ontology-based knowledge representation and reasoning technologies.
Although many works have been done in this field, achieving an accurate and semantic-rich trajectory in mobile environments is still a challenging task \[[@B10-sensors-17-00649],[@B11-sensors-17-00649]\]. To improve the accuracy of trajectory locations, the user's motion information and the indoor map were also exploited to reduce the localization errors, such as in \[[@B30-sensors-17-00649]\]. In their work, a sequence of navigation-related action were extracted from sensor data, and an HMM was used to match the user's trajectory with the indoor map. Inspired by this idea, the semantic acquisition and utilization were further improved in the localization process in this work. The rich location-based semantic information was extracted based on the user's activity recognition with HAR, and a semantic model was described and constructed for indoor navigation. The semantic model can be used to not only describe the user's location, but also to improve the user's localization efficiency. The simultaneous localization and semantic acquisition can be considered as a significant contribution of the proposed method.
3. Methods {#sec3-sensors-17-00649}
==========
3.1. Location Estimation and Activity Recognition {#sec3dot1-sensors-17-00649}
-------------------------------------------------
In this section, PDR is first introduced to estimate the user's location, and then, landmarks are applied to correct the location. Next, with the help of multiple phone sensors, HAR is used to identify the user's activity. Finally, an HMM is proposed in order to estimate the user's initial position.
### 3.1.1. Landmark-Based PDR {#sec3dot1dot1-sensors-17-00649}
The smartphone-based PDR system uses the inbuilt inertial and orientation sensors to track the user's trajectory \[[@B31-sensors-17-00649]\]. The main processes include step detection, step length estimation, direction estimation \[[@B20-sensors-17-00649]\] and trajectory correction.
\(a\) Step detection:
Peak step detection \[[@B32-sensors-17-00649]\], or the zero-crossing step detection algorithm \[[@B33-sensors-17-00649]\], is the most frequently-used method to detect the user's steps. To improve the robustness of the detection result, as in \[[@B34-sensors-17-00649],[@B35-sensors-17-00649]\], the synthetic acceleration magnitude of a three-axis accelerometer is used. This is calculated as follows: $${a\left( t \right) = \sqrt{a_{x}^{2}\left( t \right) + a_{y}^{2}\left( t \right) + a_{z}^{2}\left( t \right)} - g},$$ where a(t) is the synthetic acceleration reading at time t, and the constant component g represents the Earth's gravity.
A low-pass filter is applied to smooth the data and to remove the spurious peaks, as shown in [Figure 2](#sensors-17-00649-f002){ref-type="fig"}. The step detection process is conducted according to the following conditions \[[@B10-sensors-17-00649],[@B35-sensors-17-00649]\]: $a\left( t \right)$ is the local maximum and is larger than a given threshold $\mathsf{\delta}_{thr}$.The time between two consecutive detected peaks is greater than the minimum step period $t_{\min}$.According to human walking posture, the start of a step is the zero-crossing point before the peak.
[Figure 2](#sensors-17-00649-f002){ref-type="fig"}b shows the detected peaks marked with red circles, and the blue circles represent the start and end points of the steps.
\(b\) Step length estimation:
The length of a step depends on the physical features of the pedestrian (height, weight, age, health status, etc.) and the current state (walking speed and step frequency) \[[@B35-sensors-17-00649]\]. Although step length varies from step to step, even in the same person, step length can be estimated by its corresponding acceleration. A nonlinear model \[[@B34-sensors-17-00649]\] is used to effectively estimate step length. $$l_{k} = \mu\sqrt[4]{a_{max}\left( k \right) - a_{min}\left( k \right)},\ 0 < k \leq {num}\left( {steps} \right)$$ where $a_{max}\left( k \right)$ and $a_{min}\left( k \right)$ are the maximum and minimum values of the synthetic acceleration during step *k*. The coefficient $\mu$ is the stride length parameter, and it can be corrected by the landmarks.
\(c\) Direction estimation:
Direction estimation is a challenging problem for PDR using a smartphone. The gyroscope and magnetometer in the smartphone are normally used to estimate the pedestrian's walking direction \[[@B10-sensors-17-00649]\]. The gyroscope and magnetometer obtain the steps' direction during walking. An external environment can easily affect the magnetometer, which may lead to short-term heading estimation errors. Magnetic fields do not affect gyroscopes; however, gyroscopes do accumulate drift error over time \[[@B25-sensors-17-00649]\]. In order to resolve each sensor's drawbacks, both sensors are combined to enhance the direction estimation \[[@B13-sensors-17-00649],[@B34-sensors-17-00649]\]. $${\theta_{k} = \omega^{mag}\theta_{k}^{mag} + \omega^{gyro}\theta_{k}^{gyro},~0 < k \leq {num}\left( {steps} \right)},$$ where $\omega^{mag}$ and $\omega^{gyro}$ are the weighting parameters on the magnetometer's estimated direction and the gyroscope's estimated angle, respectively. The weight value changes according to the magnitude and correlation of the gyroscope and magnetometer.
\(d\) Trajectory correction:
The raw pedestrian trajectory obtained through the above methods may encounter some bias because of the accumulated error of the PDR. To solve this problem, landmarks are used to recalibrate the errors. Depending on the location and angle of the landmarks, the step length and the angle in the raw trajectory are scaled to form a new corrected trajectory. In the process of a user's indoor walking, two situations occur when passing a landmark. One is when a user goes straight through a landmark, as shown in [Figure 3](#sensors-17-00649-f003){ref-type="fig"}a. The other is when a user turns (see [Figure 3](#sensors-17-00649-f003){ref-type="fig"}b). As shown in [Figure 3](#sensors-17-00649-f003){ref-type="fig"}, the blue lines are the raw PDR trajectory, the green lines are the corrected trajectory and the red dots indicate the landmarks.
### 3.1.2. Multiple Sensor-Assisted HAR {#sec3dot1dot2-sensors-17-00649}
As in PDR, the synthetic three-axis accelerometer data are used as the base data in HAR. In addition, the smartphone's magnetometer and barometer provide information about direction and height, respectively, which helps to improve the classification accuracy.
\(a\) Segmentation:
Three different windowing techniques have been used to divide the sensor data into smaller data segments: sliding windows, event-defined windows and activity-defined windows \[[@B36-sensors-17-00649],[@B37-sensors-17-00649]\]. Since some specific events, such as the start and the end of a step length, are critical for pedestrian location estimation, the event-defined window approach is applied in our work. To use this approach, each step's start and end points are detected, and then, the samples between them are regarded as a window. If no steps are detected over a period of time, the sliding window approach is used, in which two-second-long time windows with 50% overlap are selected \[[@B26-sensors-17-00649],[@B38-sensors-17-00649]\].
\(b\) Feature extraction:
Two main types of data features are extracted from each time window. Time-domain features include the mean, max, min, standard deviation, variance and signal-magnitude area (SMA). Frequency-domain features include energy, entropy and time between peaks \[[@B8-sensors-17-00649]\]. Two time-domain features are selected---the mean and standard deviation---because they are computationally inexpensive and sufficient to classify the activities.
\(c\) Classification:
A supervised learning method is adopted to infer user activities from the sensory data \[[@B7-sensors-17-00649]\]. A number of different classification algorithms can be applied in HAR, such as decision tree (DT), *k*-nearest neighbor (KNN), support vector machine (SVM) and naive Bayes (NB) \[[@B8-sensors-17-00649],[@B9-sensors-17-00649],[@B38-sensors-17-00649]\]. Due to the simplification and high accuracy of KNN, the KNN algorithm (see Algorithm 1) is selected to classify four activities: standing, going up (or down) stairs, walking and opening a door.
Algorithm 1.
KNN.
Input:
Samples that need to be categorized:
X
j
; the known sample pairs: (
X
i
,
y
i
)
Output:
Prediction classification:
y
j
1:
for
every sample in the dataset to be predicted
do
2:
calculate the distance between (
,
) and the current sample
X
j
3:
sort the distances in increasing order
4:
select the k samples with the smallest distances to
X
j
5:
find the majority class of the k samples
6:
return the majority class as the prediction classification
y
j
7:
end For
In order to improve the classification accuracy of indoor activities, a barometer is used to determine upstairs or downstairs and to locate the user's floor. A magnetometer is used to assist in identifying the door-opening activity. Different ways of opening the door correspond to different magnetometer reactions. However, their similar performance patterns can be extracted by detecting the peak value change of the magnetometer in the sliding window. As shown in [Figure 4](#sensors-17-00649-f004){ref-type="fig"}, when the user opens a door, the magnetometer readings change significantly within a short time and then quickly return to the previous readings. Thus, the door-opening activities can be effectively identified.
### 3.1.3. The Hidden Markov Model {#sec3dot1dot3-sensors-17-00649}
When the user's initial location is unknown, HMM is used to match the motion sequence with indoor landmarks. PDR and HAR also provide useful information for matching and location estimation. As a widely-applied statistical model, HMM has a unique advantage in processing natural language, and it can capture the hidden states in a sequence of motion observations \[[@B30-sensors-17-00649],[@B36-sensors-17-00649]\]. There are five basic elements in HMM: two sets of states (N, M) and three probability matrices (A, B, π).
Because of the unique indoor environment, HMM is presented as follows: (1)N represents the hidden states in the model, which can be transferred between each other. The hidden states in HMM are landmark nodes in the indoor environment, such as a door, stairs or a turning point.(2)M indicates the observations of each hidden state, which are the user's direction selection (east, south, west and north) and the activity result from HAR.(3)A and B state the transition probability and the emission probability, respectively. The pedestrian moves indoors from one node to another, and when the direction of the current state is determined, the reachable nodes are reduced. In order to reduce the algorithm's complexity, A and B are combined to give a transition probability set $C$. \[$C_{e}$, $C_{s}$, $C_{w}$, $C_{n}$\] represent the transition probabilities of different directions.(4)π is the distribution in the initial state. The magnetometer and barometer provide direction and altitude information when the user starts recording, which helps to reduce the number of candidate nodes in the initial environment. If the starting point is unknown, the same initial probability is given.
The Viterbi algorithm uses a recursive approach to find the most probable sequence of hidden states. It calculates the most probable path to a middle state, which achieves the maximum probability in the local trajectory. Choosing the state's maximum local probability can determine the best global trajectory. However, in the indoor environment, using a partial maximum probability to obtain the global path is not appropriate, because the probability between hidden states could be zero, and a local best trajectory could become a dead trajectory in the next moment. In this study, the distance information from PDR and the activities information from HAR are combined with the Viterbi algorithm to compute the most likely trajectory. The improved Viterbi algorithm (Algorithm 2) is proposed as follows:
Algorithm 2.
Improved Viterbi algorithm.
Input:
The proposed HMM tuples
\<
N
=
{
n
i
\|
i
=
1
,
2
,
...
,
N
N
}
,
M
=
{
m
\|
=
,
,
...
,
M
}
,
C
,
π
\>
; HAR classification results
H
=
{
h
\|
=
,
,
...
,
H
}
; PDR distance information
D
=
{
d
\|
=
,
,
...
,
D
}
; Initial direction of magnetometer
O
; Initial pressure of barometer
F
;
d
σ
is the distance threshold.
Output:
Prediction trajectory.
1:
O
s
t
a
r
t
←
O
,
F
←
F
/\* Determine the initial orientation and floor
2:
for
i
from 1 to
N
M
do
3:
for
each path pass through
n
i
−
1
to
n
i
do
4:
if
((Distance(
d
(
n
i
−
1
),
d
(
n
i
)) -
d
i
)\<
d
σ
) and (
P
(
n
i
)
\>0)
then
/\* Determine whether the distance between two landmark nodes coincides with the distance information estimated by PDR
5:
Path
(
N
s
,
P
(
N
s
)
)
←
Obtain the subset data
6:
end if
7:
end for
8:
end for
9:
for
path
j
in
(
,
(
)
)
do
10:
H
(
path
j
)
=
{
\|
=
,
,
...
,
}
←
Obtain the landmark data set
11:
if
H
(
path
j
)
match with HAR data
H
then
12:
Path
(
N
f
,
P
(
N
f
)
)
←
Add this trajectory to the final trajectory data set
13:
end for
14
return Max(
(
,
(
)
)
)/\* Return the trajectory of the maximum probability
With the determination of the user's trajectory, the initial position can be obtained through the first landmark point and the PDR information.
3.2. Semantic Landmark Model {#sec3dot2-sensors-17-00649}
----------------------------
### 3.2.1. Trajectory Information Collection {#sec3dot2dot1-sensors-17-00649}
**Definition 1.** *Trajectory information: A trajectory is defined as a six-tuple* $\Gamma:\left\langle {I,\ T,\ D,\ A,\ U,L} \right\rangle$, *where I is the ID of the trajectory, and* $T$ *and* $D$ *are the timestamp and position information, respectively, of each step.* $U$ *is the direction change list;*$A$ *is the activities information list; and L is the landmark list. [Figure 5](#sensors-17-00649-f005){ref-type="fig"} shows the trajectory information collection process*.
For example, if a user went from Entrance (ET) to Room 108, I is assigned ET--R108.
From PDR, the timestamp and xyz coordinate value for each step can be obtained and can be represented as: $$T = \left\lbrack {t_{1},t_{2}\ldots\ t_{n}} \right\rbrack,$$ $$D = \left\lbrack {\left( {x_{1},y_{1},z_{1}} \right),\ \left( {x_{2},y_{2},z_{2}} \right),\ldots,\quad\left( {x_{n},y_{n},z_{n}} \right)} \right\rbrack,$$ where n denotes the number of steps detected.
Using the HAR method, the user's activity information can be collected. Hence, $A$ can be given as follows: $$A = \ \left\{ {{Standing},\ {Walking},\ {Going}\ {up}\ {stairs},\ {Opening}\ a\ {door}} \right\}$$
The direction change list $U$ can be obtained by the gyroscope. It should be noted that we detected only large directional changes (\>15°), and thus, walking along a smaller arc was not detected. Because most of the turns could be completed in less than five steps, a five-step turn detection method (see Algorithm 3) is proposed to determine the direction change activity.
Algorithm 3.
Five-step turn detection algorithm.
Input:
Angle value sequence
θ
= \[
θ
1
,
θ
2
,
...
θ
n
\]
Output:
Direction change list
U
.
1:
θ
max
←
Findpeaks(
θ
)/\* Find the local maximum sequence
2:
for
θ
i
in
θ
max
do
3:
if
(
θ
i
\>
15
or
θ
i
\<
−
15
) and (
i
\>2)
then
4:
θ
sum
=
sum
\[
θ
i
−
2
:
θ
i
\+
2
\]
5:
if
(
θ
sum
\>
30
a
n
d
θ
sum
≤
60
)
then
6:
U
.
add
(
"
Go
left
"
)
7:
else if
(
θ
sum
\>
60
θ
≤
120
)
then
8:
U
.
add
(
"
Turn
left
"
)
9:
(
θ
\>
−
θ
≤
−
30
)
then
10:
U
.
add
(
"
Go
right
"
)
11:
else if
(
θ
sum
\>
−
120
θ
≤
−
)
12:
U
.
add
(
"
Turn
right
"
)
13:
else if
(
θ
sum
\>
120
o
r
θ
sum
≤
−
120
)
then
14
U
.
add
(
"
Turn
around
"
)
15:
end for
16:
return
U
For example, $U$ can be described as follows: $$U = \left\{ {Go}\ {left},\ {Turn}\ {left},{Turn}\ {right} \right\}$$
Since the landmarks are used as the key points in a trajectory, a landmark list can be used to denote a trajectory. According to the time series, a landmark list L is detected in a trajectory. Three types of landmarks (stairs, turns and doors) are added to list L according to the following rules: If the going-up-(or -down)-stairs activity is detected, the nearest stairs landmark is added to L.If direction change activity (see Algorithm 3) is detected, the nearest turn landmark is added to L.If a door-opening activity is detected, the nearest door landmark is added to L.
### 3.2.2. Semantics Extraction {#sec3dot2dot2-sensors-17-00649}
**Definition 2.** *Semantic landmark: A semantic landmark* $S\left\lbrack l \right\rbrack$ *consists of five parts: Id, attribute, adjacent segments, direction information and semantic description. Id is the landmark identifier. Attribute is one of the three types of landmarks: stairs, turn, or door. Adjacent segments contain the distance and semantic information between the current landmark and the next landmarks. Direction information and semantic description indicate the direction information and the semantic information when the user passes the landmark, respectively, as shown in [Figure 6](#sensors-17-00649-f006){ref-type="fig"}*.
A sequence of landmarks can denote a trajectory. Therefore, adding semantic information to the landmarks and their adjacent segments can describe the trajectory. The semantic description of trajectories is expressed as follows: $${S\left\lbrack t \right\rbrack = \left\{ {S\left\lbrack {seg_{s - 1}} \right\rbrack,~S\left\lbrack l_{1} \right\rbrack,S\left\lbrack {seg_{1 - 2}} \right\rbrack,S\left\lbrack l_{2} \right\rbrack,~\ldots~S\left\lbrack l_{n} \right\rbrack,~S\left\lbrack {seg_{n - e}} \right\rbrack} \right\}},$$ where $S\left\lbrack t \right\rbrack$ indicates the semantics of trajectory $t$, and $n$ is the number of landmarks for this trajectory. $S\left\lbrack l_{n} \right\rbrack$ represents the semantics of landmark $l_{n}$. $S\left\lbrack {seg_{s - 1}} \right\rbrack$ denotes the semantics of the region from the start point to the first landmark point. Similarly, $S\left\lbrack {seg_{n - e}} \right\rbrack$ denotes the semantics of the region from the last landmark point to the end.
A semantic landmark or an adjacent segment can store multiple semantics and provides semantics based on the detected activity. According to the trajectory information $\mathsf{\Gamma}:\left\langle {I,\ T,\ D,\ A,\ U,L} \right\rangle$, the semantic information can be obtained as shown in [Table 1](#sensors-17-00649-t001){ref-type="table"}.
Detected ($U$) and Find ($L\left( {turn} \right)$) indicate that turn activity is detected and that a nearby turn landmark is found. If the user's activity information is detected, but there are no corresponding landmarks nearby, this activity's semantics is added to the corresponding adjacent segment, as shown in [Table 2](#sensors-17-00649-t002){ref-type="table"}.
According to the semantics acquisition rules shown in [Table 1](#sensors-17-00649-t001){ref-type="table"} and [Table 2](#sensors-17-00649-t002){ref-type="table"}, the semantic landmarks are constructed as shown in [Figure 7](#sensors-17-00649-f007){ref-type="fig"}.
As the above process shows, both semantic information and distance information are added to the semantic model. The order information (e.g., "Turn left at the 2nd turning point") is added according to the following rule: If the current landmark's adjacent segments contain multiple turn or door landmarks and they have the same semantics, sort them by distance and then provide them with the order information ${Order}_{turn}$ or ${Order}_{door}$.
Set S denotes the semantics obtained, such as {S1: \['Go left'\], S2: \['Go up the steps'\], S3: \['Turn left'\]}. If the order information is obtained simultaneously, it can be expressed as {S_order: \['at ${Order}_{turn}$ turn point'\] or \['at ${Order}_{door}$ door'\]}.
4. Experiment {#sec4-sensors-17-00649}
=============
The experiment was performed at the State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing (LIESMARS) at Wuhan University, China. An Android mobile phone and the indoor floor plans of LIESMARS were used in the experiment. It should be noted that, in this study, only the hand-held situation was considered. The experimental process is shown in [Figure 8](#sensors-17-00649-f008){ref-type="fig"}.
In the following, the pre-knowledge is first provided. Multiple user trajectories are then presented, including a trajectory on a single floor, a trajectory on multiple floors and a trajectory without knowing the starting point. Finally, the semantics acquisition process and results are described.
4.1. Pre-Knowledge {#sec4dot1-sensors-17-00649}
------------------
Door points, stair points and turn points were used as landmarks in the experiment, as shown in [Figure 9](#sensors-17-00649-f009){ref-type="fig"}. In indoor spaces, pedestrians tend to walk along a central line, and they tend to go in a straight line between places. The intersection of the corridor area's centerline was therefore selected as a landmark. Most turn points near door points were not assumed to be landmarks, because the door points could replace them as landmarks. The above principles were used to generate the landmarks. The proposed approach does not focus on landmark extraction, but on trajectory generation and the steps in the semantics acquisition.
The smartphone's barometer and magnetometer were used to determine the initial orientation and locate the user's floor, respectively. However, they need to be analyzed first. We collected barometer data at eight different locations on each layer, as shown in [Table 3](#sensors-17-00649-t003){ref-type="table"}. As the change in barometer readings in the same layer was not significant, we used the average of the collected barometer readings as a benchmark to determine the user's floor. If the current barometer reading is within ±0.1 hpa of B(f1) or B(f2), the corresponding floor is determined.
Compared to the barometer, the magnetometer is more unstable, so 80 north-facing magnetometer readings were collected at various locations within the building. The distribution of the difference between the collected magnetometer data and True North is shown in [Figure 10](#sensors-17-00649-f010){ref-type="fig"}a. Most of the magnetic differences are between −5° and 15° and occasionally more than 20°. Based on the above data, a threshold of 30° was chosen to determine the direction semantics of the initial position. Direction information like 'north' (330 \< θ \< 360, 0 ≤ θ \< 30), 'east' (60 ≤ θ \<120), 'south' (150 ≤ θ \<210) and 'west' (240 ≤ θ \< 300) can be obtained ([Figure 10](#sensors-17-00649-f010){ref-type="fig"}b) when the magnetometer reading $\mathsf{\theta}$ of the user's initial position is acquired.
In addition, the user's step length parameters need to be determined over a short distance (0.45 in our experiment). When the user goes upstairs, the horizontal and vertical distances of each step are given as fixed values (0.3 m and 0.15 m in the experiment).
4.2. Trajectory Generation and Correction {#sec4dot2-sensors-17-00649}
-----------------------------------------
To present a user's trajectory, HAR is performed to identify landmarks, and then, these landmarks are used to correct the PDR trajectory.
Because the HAR training set requires a variety of activities, 25 trajectories from the entrance to each room on the second floor were chosen. If a room has multiple doors, each door corresponds to a trajectory. In order to obtain information from standing activity samples, the user needs to stand for a while at the beginning point and the end point of each trajectory. A training sample is shown in [Figure 11](#sensors-17-00649-f011){ref-type="fig"}.
Landmarks that the user passes can be determined using the activity classification results provided by HAR, and then, the user's trajectory can be corrected. [Figure 12](#sensors-17-00649-f012){ref-type="fig"} shows the trajectory from the entrance to Room 108. The blue points indicate the raw trajectory, and the red points indicate the corrected trajectory. The corrected trajectory is extremely close to the ground truth trajectory. In addition, only four landmarks were used: stairs landmarks s0 and s1, turn landmark u0 and door landmark r8.
For trajectories on multiple floors, height information is added to each step point. [Figure 13](#sensors-17-00649-f013){ref-type="fig"} shows the trajectory from the entrance to Room 201. The blue points and red points indicate the raw trajectory and the corrected trajectory, respectively.
When the user's initial location is unknown, the direction observation sequence is obtained from the direction sensors. Information about position and activities is obtained from PDR and HAR, respectively. When the user trajectory is determined, the starting position can be inferred from PDR.
The user went from point $S$ to point $E$. The trajectory when using only PDR is shown in [Figure 14](#sensors-17-00649-f014){ref-type="fig"}a, and the direction observation sequence is obtained.
M
=
{
south
,
east
,
north
,
west
,
north
,
east
,
north
,
west
}
From PDR, the distance between the landmarks of two adjacent observation sequences is obtained, which is denoted by $D$. For example, $D_{s}$ represents the distance from the starting point to the first landmark, and $D_{1 - 2}$ denotes the distance from the first landmark to the second landmark. $D_{e}$ represents the distance from the end point to the last landmark. $$D = \left\{ {D_{s},~D_{1 - 2},~D_{2 - 3},\ldots,~D_{e}} \right\}$$
Information about activities is obtained from HAR. $$A = \left\{ {{Standing},\ {Opening}\ a\ {door},\ {Walking},\ {Opening}\ a\ {door},\ {Walking},\ {Standing}} \right\}$$
The matching process used in the algorithm is shown in [Table 4](#sensors-17-00649-t004){ref-type="table"}. To simplify the proposed model, a flag was abstracted to express similar landmarks. For example, $DN$ stands for the adjacent doors at the north side of the corridor: ${dn} =$\[$d0$,$d2$,$d4$,$d6$,$d8$,$d10$\], ${ds} =$\[$d1$,$d3$,$d5$,$d7$,$d9$\], $dw$ = \[$d15$,$d17$,$d19$,$d21$\], ${de} =$ \[$d14$,$d16$,$d18$,$d20$\] (see [Figure 9](#sensors-17-00649-f009){ref-type="fig"}). The virtual landmark E indicates the connecting points between the doors and the corridor. For example, $E\left( {d3} \right)$ indicates the connecting point between door d3 and the corridor.
A trajectory is represented by a list, and the elements in the list represent the points that have been passed. The HAR results are denoted by $A$. $A_{s}$ = (s, w, o) indicates the sequence of activities from the start point to the first landmark, which is "Standing-Walking-Opening a door".
It should be noted that we used the real landmark coordinates to correct the PDR results when landmarks were detected. As the trajectory ended, d~e~ was given to estimate the final position, and the matching trajectory was obtained.
4.3. Semantics Extraction {#sec4dot3-sensors-17-00649}
-------------------------
According to the proposed semantics extraction method described in [Section 3.2](#sec3dot2-sensors-17-00649){ref-type="sec"}, the semantics for landmarks in trajectory ET--R108 (in [Figure 12](#sensors-17-00649-f012){ref-type="fig"}) were obtained as shown in [Table 5](#sensors-17-00649-t005){ref-type="table"}. The start and end points were considered as virtual landmarks, which have no specific attributes or fixed locations.
After sufficient trajectories were acquired, the landmarks' full semantics could be obtained. Taking the turn landmark (u0) as an example, the complete semantics are as shown in [Table 6](#sensors-17-00649-t006){ref-type="table"}. In addition, the order of the landmarks could be obtained using the method described in [Section 3.2.2](#sec3dot2dot2-sensors-17-00649){ref-type="sec"}.
According to the above tables and our semantics model, the three trajectories presented in [Section 4.2](#sec4dot2-sensors-17-00649){ref-type="sec"} can be described as follows:
ET--R108: {\['Go left'\], \['Go up the steps'\], \['Turn right'\], \['Go straight',' Turn right (at 5th door)'\], \['Go into the door'\]}
ET--R201: {\['Go left'\], \['Go up the steps'\], \['Go straight', 'Go upstairs'\], \['Turn left\], \['Turn left (at 1st door)'\], \['Go into the door'\]}
R204--R213: {\['Go out the door'\], \['Turn left'\], \['Go straight'\], \['Turn left'\], \['Turn left'\], \['Turn right'\], \['Go straight'\], \['Turn right'\], \['Turn left'\], \['Turn left'\], \['Go straight'\], \['Go into the door'\]}
The construction of complete semantics for all of the indoor landmarks requires a large amount of trajectory data. However, the proposed approach only considers the complete semantics of key landmarks and the partial semantics of non-key landmarks, because they can describe most of the user's activities.
5. Discussion {#sec5-sensors-17-00649}
=============
Firstly, the performance of the HAR classification is evaluated. The location errors in a trajectory are then described. Finally, the accuracy of the landmark matching is analyzed.
5.1. Error Analysis {#sec5dot1-sensors-17-00649}
-------------------
### 5.1.1. HAR Classification Error {#sec5dot1dot1-sensors-17-00649}
In order to evaluate the performance of the HAR classification, 10-fold cross-validation \[[@B39-sensors-17-00649]\] was used. In this method, the dataset is divided into 10 parts: nine parts are used for training, and one part is used for testing each iteration. The classification accuracy of the common classifiers was compared with the proposed classifier, and two different window segmentation approaches were compared. Since the error rate of our step detection is quite low, at 0.19% (total steps: 3092; error detection steps: six), it is more convenient to use the event-defined window approach to sense the user's activity. As shown in [Table 7](#sensors-17-00649-t007){ref-type="table"}, the results show that the event-defined window approach performs better than the sliding window approach, which applies two-second-long time windows with a 50% overlap.
Many different performance metrics could have been be used to evaluate the HAR classification \[[@B8-sensors-17-00649]\]. A confusion matrix was adopted, which is a method commonly used to identify error types (false positives and negatives) \[[@B40-sensors-17-00649]\]. Several different performance metrics---accuracy (the standard metric to express classification performance), precision, recall and F-measure---could be calculated based on the matrix \[[@B9-sensors-17-00649]\]. [Table 8](#sensors-17-00649-t008){ref-type="table"} shows the confusion matrix used to evaluate the results of the KNN classification activities.
As shown in [Table 8](#sensors-17-00649-t008){ref-type="table"}, the proposed method achieves an extremely high accuracy (\>99%) in detecting stairs and walking activities. Some errors occur in identifying door-opening and standing activities. However, by detecting the magnetometer change, it is easy to distinguish between these two activities, thereby reducing the amount of errors.
### 5.1.2. Localization Error {#sec5dot1dot2-sensors-17-00649}
The localization error of trajectory ET--R108 is shown in [Figure 15](#sensors-17-00649-f015){ref-type="fig"}a; the blue line indicates the original PDR trajectory, and the orange line indicates the location errors after only the landmarks were corrected. The results show that the PDR errors increase with distance, and a high average localization accuracy (0.59 m) is achieved when we use the landmarks to correct the cumulative errors. [Figure 15](#sensors-17-00649-f015){ref-type="fig"}b shows the cumulative error distribution of the 25 test trajectories. We can see that the proposed approach is more stable than using only PDR, and the average error is reduced from 1.79 m to 0.52 m.
### 5.1.3. Landmark Matching Errors {#sec5dot1dot3-sensors-17-00649}
The shortest distance method was used to match the landmarks. As shown in [Figure 16](#sensors-17-00649-f016){ref-type="fig"}, the result matches the partial trajectories.
Although the trajectories were corrected at the turn landmark (the red point), the PDR-estimated user location still introduced errors, particularly when the user was far from the previous landmark. In the experiment, an error occurred because the distances of d18 and d20 were extremely close and far from the turning point landmark (t6). We can also see that a similar error occurred in turn landmark $t5$, which is matched to landmark $t6$ (see [Table 9](#sensors-17-00649-t009){ref-type="table"}).
5.2. Comprehensive Comparison {#sec5dot2-sensors-17-00649}
-----------------------------
Some similar indoor localization schemes, which require no additional devices or expensive labor, are compared in terms of requirement, sensors, user participation, accuracy, expression and extensibility in [Table 10](#sensors-17-00649-t010){ref-type="table"}. Each technique has its own advantages. Zee \[[@B22-sensors-17-00649]\] tracked a pedestrian's trajectory without user participation, and a Wi-Fi training set was simultaneously collected, which can be used in Wi-Fi fingerprinting-based localization techniques. UnLoc \[[@B25-sensors-17-00649]\] only needs a door location as the basic input information and simultaneously computes the user's location and detects various landmarks. Compared to the above localization schemes, the proposed approach needs more basic information; however, the information allows us to obtain a better localization accuracy. Moreover, a semantic landmark model was constructed during the localization process, which can be used not only to describe the user's trajectory, but also to improve the localization efficiency. The overall scores of the three approaches are shown in [Figure 17](#sensors-17-00649-f017){ref-type="fig"}.
5.3. Computational Complexity {#sec5dot3-sensors-17-00649}
-----------------------------
In order to verify the validity of the semantic model for localization and better analyze the computational complexity of semantics-assisted localization method, a further experiment was conducted ([Figure 18](#sensors-17-00649-f018){ref-type="fig"}a). In this experiment, the user started from any place on the second floor, and we wanted to determine the user's trajectory as soon as possible. The overall error and time complexity in the trajectory matching process are used to evaluate the proposed method.
As shown in [Figure 18](#sensors-17-00649-f018){ref-type="fig"}b, although only a few semantics are provided, the trajectory error drops rapidly (the average error drops from 9.25 m to 0.48 m). When the first semantic information is obtained from the trajectory, there are five trajectories satisfying the condition. In order to match the semantic information, each landmark point needs to be traversed once, so the time complexity is O(N). The next search only needs to traverse the semantics of the trajectory segments that satisfy the previous condition. These trajectory segments are '$l_{t8}l_{t7}$', '$l_{t8}l_{t6}$', '$l_{t7}l_{t6}$', '$l_{t4}l_{t2}$', '$l_{t1}l_{t0}$', '$l_{t1}l_{s2}$', '$l_{t0}l_{s2}$', and the time complexity is O(7). The trajectory is determined after matching the third semantic information, and the trajectory error is similar to the previous trajectory localization experiment, where the initial location was known. The above description is provided in [Table 11](#sensors-17-00649-t011){ref-type="table"}. Compared to the traditional trajectory matching method, which yields a time complexity of O(NT) or O(N2T), the proposed semantic matching method is more efficient. Since it does not need to traverse all of the states every time, the time complexity is much less than O(NT).
6. Conclusions {#sec6-sensors-17-00649}
==============
In this paper, PDR, HAR and landmarks have been combined to achieve indoor mobile localization. The landmark information was extracted from indoor maps, and then, HAR was used to detect the landmarks. These landmarks were then used to correct the PDR trajectories and to achieve a high level of accuracy. Without knowing the initial position, HMM was performed to match the motion sequence to the indoor landmarks. Because semantic information was also assigned to the landmarks, the semantic description of a trajectory was obtained, which has the potential to provide more applications and better services. Moreover, the experiment was implemented in an indoor environment, to fully evaluate the proposed approach. The results not only show a high localization accuracy, but also confirm the value of semantic information.
More extensive research can be studied in the future. For example, phone sensing could be used to recognize more activities, particularly complex activities, and more semantic information could be extracted. In addition, complex experimental conditions, such as various trajectories, a location-independent mobile phone and all kinds of users, could be included in future studies. The semantic model used in this study does not contain all possible semantics, and rich semantic information could be obtained by using the data from crowdsourced trajectories. Furthermore, real-time localization (including semantic information) is also the priority of our future work.
This work is supported by The National Key Research and Development Program of China \[grant number 2016YFB0502203\], Mapping geographic information industry research projects of public interest Industry \[grant number 201512009\] and the LIESMARS special Research Funding.
Sheng Guo conceived of and designed the study, performed the experiments and wrote the paper. Yan Zhou helped to improve some of the experiments. Hanjiang Xiong and Xianwei Zheng supervised the work and revised the paper.
The authors declare no conflict of interest.
{#sensors-17-00649-f001}
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{#sensors-17-00649-f010}
{#sensors-17-00649-f011}
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sensors-17-00649-t001_Table 1
######
Semantics acquisition rules for landmarks.
Id Conditions (C) Semantics (S)
------ ------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------------
SL-1 $A$ = 'Walking', Detected ($U$) and Find ($L\left( {turn} \right)$) 'Go left', 'Go right', 'Turn left', 'Turn right', 'Turn around'
SL-2 $A$ = 'Opening a door', Find ($L\left( {door} \right)$) and $l_{pre} = \varnothing$ and $l_{next} = \varnothing$ 'Opening a door'
SL-3 $A$ = 'Opening a door', Find ($L\left( {door} \right)$) and $l_{pre}! = \varnothing\ {or}\ l_{current}$ 'Go into the door'
SL-4 $A$ = 'Opening a door', Find ($L\left( {door} \right)$) and $l_{next}! = \varnothing\ {or}\ l_{current}$ 'Go out of the door'
SL indicates the identity of the rule. $A$ = {'Standing', 'Walking', 'Going up (or down) stairs', 'Opening a door'}, $U$ = {'Go left', 'Go right', 'Turn left', 'Turn right', 'Turn around'}. *L* is the landmark list: $L\left( {turn} \right)$, $L\left( {stairs} \right)~{and}$ $L\left( {door} \right)$ are the turn, the stairs and the door landmarks. $l_{current}$, $l_{pre}$ and $l_{next}$ represent the current landmark, the previous landmark and the next landmark.
sensors-17-00649-t002_Table 2
######
Semantics acquisition rules for landmark segments.
Id Conditions (C) Semantics (S)
------ -------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------
SK-1 $A$ = 'Standing', Detected ($U$) 'Turn left', 'Turn right', 'Turn around'
SK-2 $A$ = 'Walking', Detected ($U$) and Unfound ($L\left( {turn} \right)$) 'Go left', 'Go right', 'Turn left', 'Turn right', 'Turn around'
SK-3 $A$ = 'Walking', Undetected ($U$), $D_{walking}$ \> $D_{thresold}$ 'Go straight'
SK-4 $A$ = 'Going up(or down) stairs', Find ($L\left( {stairs} \right)$) and $Z_{current}$ \< $Z_{next}$, $T_{stairs}$ \< 5 s 'Go up the steps'
SK-5 $A$ = ' Going up(or down) stairs, Find ($L\left( {stairs} \right)$) and $Z_{current}$ \> $Z_{next}$, $T_{stairs}$ \< 5 s 'Go down the steps'
SK-6 $A$ = ' Going up(or down) stairs, Find ($L\left( {stairs} \right)$) and $Z_{current}$ \< $Z_{next}$, $T_{stairs}$ \> 5 s 'Go upstairs'
SK-7 $A$ = ' Going up(or down) stairs, Find ($L\left( {stairs} \right)$) and $Z_{current}$ \> $Z_{next}$, $T_{stairs}$ \> 5 s 'Go downstairs'
SK indicates the identity of the rule. $A$ = {'Standing', 'Walking', 'Going up (or down) stairs', 'Opening a door'}, $U$ = {'Go left', 'Go right', 'Turn left', 'Turn right', 'Turn around'}. *L* is the landmark list: $L\left( {turn} \right)$, $L\left( {stairs} \right)~{and}$ $L\left( {door} \right)$ are the turn, the stairs, and the door landmarks. The duration of 'Standing', 'Walking' and 'Going up (or down) stairs' activities are represented by $T_{standing}$, $T_{walking}$ and $T_{stairs}$, respectively. From *D*, we can obtain the distance and height information$;\ D_{walking}$ indicates the walk distance; $Z_{current}$ represents the *z* value of the current landmark; and $Z_{next}$ represents the *z* value of the next landmark.
sensors-17-00649-t003_Table 3
######
Barometer readings.
Floor 1 2 3 4 5 6 7 8 Average (hpa)
------- --------- --------- --------- --------- --------- --------- --------- --------- -----------------
f1 1020.91 1020.92 1020.92 1020.9 1020.88 1020.86 1020.85 1020.87 B(f1) = 1020.89
f2 1020.32 1020.34 1020.34 1020.33 1020.29 1020.3 1020.33 1020.32 B(f2) = 1020.32
sensors-17-00649-t004_Table 4
######
Trajectory matching results. E = east, S = south, W = west, N = north. s = standing, w = walking, u = going up stairs, d = going down stairs, o = opening a door.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Observation Sequence Trajectories Distance and Activities Information Trajectories after
-------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------- ---------------------------------------------------------------------------------
{'S', 'E', 'N'} $\left\lbrack {d4,\ E\left( {d4} \right),\ t0} \right\rbrack$, $\left\lbrack {t5,t3,\ t2} \right\rbrack$ {$D_{s}$ = 1.86, $D_{1 - 2}$ = 1.69, $D_{2 - 3}$ = 10.9}\ $\left\lbrack {d4,\ E\left( {d4} \right),\ t0} \right\rbrack$
{$A_{s}$ = (s, w, o), $A_{1 - 2}$ = (w), $A_{2 - 3}$ = (w)}
{'S', 'E', N','W'} $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1} \right\rbrack$ $D_{3 - 4}$ = 6.73, $A_{3 - 4}$ = (w) $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1} \right\rbrack$
{'S', 'E', N','W', 'N'} $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3} \right\rbrack$ $D_{4 - 5}$ = 2.1, $A_{4 - 5}$ = (w) $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3} \right\rbrack$
{'S', 'E', N','W', 'N', 'E'} $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,\ t5} \right\rbrack$ $D_{5 - 6}$ = 13.73, $A_{5 - 6}$ = (w) $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,t5,\ t6} \right\rbrack$
{'S', 'E', N','W', 'N', 'E', 'N'} $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,\ t5,\ t6} \right\rbrack$ $D_{6 - 7}$ = 4.1, $A_{6 - 7}$ = (w) $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,t5,\ t6} \right\rbrack$
{'S', 'E', N','W', 'N', 'E', N','W'} $\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,\ t5,\ t6,\ t8} \right\rbrack$,$\lbrack d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,\ t5,\ t6,$$E\left( {dw} \right)\rbrack$ $D_{7 - 8}$ = 4.1, $A_{7 - 8}$ = (w) $\lbrack d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,t5$,$\ t6,\ t8\rbrack$
$\left\lbrack {d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,\ t5,\ t6,\ t8} \right\rbrack$ $D_{8 - 9}$ = 10.94, $D_{e}$ = 2.8 $A_{8 - 9}$ = (w, o, w) $\lbrack d4,\ E\left( {d4} \right),\ t0,\ t1,\ t3,t5$,$\ t6,\ t8,\ d13\rbrack$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
sensors-17-00649-t005_Table 5
######
Landmark semantics.
Landmark Name Expression
----------------------- ----------------------------------------------------------------------------------------------------------------- ------------
Entrance Id ET
Attribute Virtual landmarks
Adjacent segments {'$l_{ET}l_{s0}$':{'semantics': 'Go left', 'distance': 9.12, 'direction': 'West-South'}}
Direction information 'West'
Semantic description $\varnothing$
Stairs s0 Id s0
Attribute Stairs
Adjacent segments {'$l_{s0}l_{s1}$':{'semantics': 'Climb the steps', 'distance': 0.99, 'direction': 'South'}}
Direction information 'South'
Semantic description $\varnothing$
Stairs s1 Id s1
Attribute Stairs
Adjacent segments { '$l_{s1}l_{u0}$':{'semantics': $\varnothing$, 'distance': 5.05, 'direction': 'South'}}
Direction information 'South'
Semantic description $\varnothing$
Turn u0 Id u0
Attribute Turn
Adjacent segments {'$l_{u0}l_{r8}$':{'semantics': \['Go straight', 'Turn right'\], 'distance': 19.13, 'direction': 'West-North'}}
Direction information 'South - West'
Semantic description 'Turn right'
Door r8 Id r8
Attribute Door
Adjacent segments {'$l_{r8}l_{E}$':{'semantics': $\varnothing$, 'distance': 2.03, 'direction': 'North}}
Direction information 'North'
Semantic description 'Go into the door'
sensors-17-00649-t006_Table 6
######
Complete semantics of turn u0.
Name Expression
----------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Id u0
Attribute Turn
Adjacent segments {'$l_{u0}l_{r1}$': {'semantics': 'Turn left (at 1st door)', 'distance': 5.19, 'direction': 'West-South'}, '$l_{u0}l_{r2}$': {'semantics': 'Turn right (at 1st door)', 'distance': 5.21, 'direction': 'West-North'}, '$l_{u0}l_{r3}$': {'semantics': 'Turn left (at 2nd door)', 'distance': 6.87, 'direction': 'West-South'}, '$l_{u0}l_{r0}$': {'semantics': 'Turn right (at 2nd door)', 'distance': 7.19, 'direction': 'West-North'}, '$l_{u0}l_{r4}$': {'semantics': 'Turn right (at 3rd door)', 'distance': 12.13, 'direction': 'West-North'}, '$l_{u0}l_{r5}$': {'semantics': 'Turn left (at 3rd door)', 'distance': 12.27, 'direction': 'West-South'}, '$l_{u0}l_{r7}$': {'semantics': 'Turn left (at 4th door)', 'distance': 14.11, 'direction': 'West-South'}, '$l_{u0}l_{r6}$': {'semantics': 'Turn right (at 4th door)', 'distance': 15.96, 'direction': 'West-North'}, '$l_{u0}l_{r9}$': {'semantics': 'Turn left (at 5th door)', 'distance': 17.75, 'direction': 'West-South'}, '$l_{u0}l_{r8}$': {'semantics': 'Turn right (at 5th door)', 'distance': 19.13, 'direction': 'West-North'}, '$l_{u0}l_{r10}$': {'semantics': 'Go straight', 'distance': 20.28, 'direction': 'West'}, '$l_{u0}l_{s1}$': {'semantics': 'Turn left', 'distance': 5.09, 'direction': 'North}, '$l_{u0}l_{s2}$': {'semantics': 'Turn right', 'distance': 2.84, 'direction': 'South'}}
Direction information 'South-West', 'North-West', 'East-South', 'East-North'
Landmark semantic 'Turn right', 'Turn left'
sensors-17-00649-t007_Table 7
######
Classification accuracy.
Classifier Accuracy Accuracy
------------ ---------- ----------
DT 98.62% 98.69%
SVM 96.55% 97.73%
KNN 98.83% 98.95%
sensors-17-00649-t008_Table 8
######
Confusion matrix.
Actual Class Predicted Class Accuracy (%)
--------------------------- ----------------- -------------- ----- ---- --------
Standing 284 0 0 10 96.60%
Walking 0 1981 3 0 99.85%
Going up (or down) stairs 0 4 787 0 99.49%
Opening a door 3 0 0 56 94.92%
sensors-17-00649-t009_Table 9
######
Landmark matching errors.
Landmark Total Wrong Match Error Rate
---------- ------- ------------- ------------
Doors 24 1 4.17%
Stairs 96 0 0
Turns 63 1 1.59%
sensors-17-00649-t010_Table 10
######
Comparison with other localization systems.
Name Zee UnLoc The Proposed Approach
-------------------- ---------------------------- ---------------------------- ----------------------------------
Requirement Floorplan A door location Floorplan, landmarks
Sensors Acc., Gyro., Mag., (Wi-Fi) Acc., Gyro., Mag., (Wi-Fi) Acc., Gyro., Mag., Baro.
User participation None Some Some
Accuracy 1--2 m 1--2 m \<1 m
Expression Trajectory Trajectory Trajectory, semantic description
Extensibility Wi-Fi RSS distribution Landmark distribution Semantic landmark model
sensors-17-00649-t011_Table 11
######
Semantic matching results.
Trajectory Trajectory Segment Semantic Time Complexity Numbers ^1^
--------------------------- ----------------------------- ----------------------------- ----------------- -------------
Trajectory information Segment 1 (red points) 'Turn right' ('East-South') O(N) 5
Segment 2 (blue points) 'Go straight' ('South') O(7) 2
Segment 3 (purple points) 'Turn right' ('South-West') O(3) 1
^1^ The number of trajectories after semantic matching.
| |
BACKGROUND
SUMMARY OF THE INVENTION
0001 Inkjet printers generally operate by ejecting ink onto media, such as paper. One type of inkjet printer utilizes stationary staggered inkjet pens, which are also more generally referred to as fluid ejector assemblies. The inkjet pens are immobile, and are arranged in a staggered fashion over one axis referred to as the inkjet pen axis. Media is moved past the assemblies along another axis, referred to as the media axis, which is perpendicular to the inkjet pen axis. As the media moves past the inkjet pens, the pens accordingly eject ink onto the media. This type of inkjet printer is customarily, but not necessarily, used in industrial settings that require fast printing performance.
0002 The inkjet pens can be or become misaligned in two ways. Along the inkjet pen axis, the inkjet pens may not be aligned correctly, leading to gaps between output from adjacent pens, or leading to overlapping output from adjacent pens. Along the media axis, too, the inkjet pens may not be aligned correctly. Because the pens are staggered, such misalignment may result from the fluid ejection delays of the inkjet pens not being properly set with respect to one another. An inkjet pen may thus begin outputting ink too soon or too late, resulting in misalignment along the media axis.
0003 A method of one embodiment of the invention reduces misalignment of a pair of staggered fluid ejector assemblies positioned along a first axis perpendicular to a second axis along which media moves past the assemblies. The method reduces misalignment of the pair of staggered fluid ejector assemblies along the first axis. Fluid bands are output by different series of nozzles of each assembly. The method then selects as a series of active nozzles of each assembly one of the different series of nozzles outputting one of the fluid bands that is substantially aligned with one of the fluid bands output by the other assembly.
BRIEF DESCRIPTION OF THE DRAWINGS
0004 The drawings referenced herein form a part of the specification. Features shown in the drawings are meant as illustrative of only some embodiments of the invention, and not of all embodiments of the invention, unless otherwise explicitly indicated, and implications to the contrary are otherwise not to be made.
0005FIG. 1 is a diagram of the side view of an inkjet printer, according to an embodiment of the invention.
0006FIG. 2 is a diagram of the top view of the inkjet pens of an inkjet printer under which media moves past, according to an embodiment of the invention.
0007FIG. 3 is a diagram of the top view of a pair of inkjet pens of an inkjet printer and their corresponding nozzles, according to an embodiment of the invention.
0008FIGS. 4A and 4B are diagrams illustrating an example of one type of misalignment of a pair of inkjet pens along the inkjet pen axis, and the correction of such misalignment, according to an embodiment of the invention.
0009FIGS. 5A and 5B are diagrams illustrating an example of another type of misalignment of a pair of inkjet pens along the inkjet pen axis, and the correction of such misalignment, according to an embodiment of the invention.
0010FIGS. 6A and 6B are diagrams illustrating the alignment of a pair of inkjet pens along the media axis, according to an embodiment of the invention.
0011FIGS. 7A and 7B are diagrams illustrating examples of different types of misalignment of a pair of inkjet pens along the media axis, according to differing embodiments of the invention.
0012FIG. 8 is a flowchart of a method for correcting misalignment between a pair of inkjet pens along the inkjet pen axis, according to an embodiment of the invention.
0013FIG. 9 is a flowchart of a method for correcting misalignment among a number of inkjet pens along the inkjet pen axis, according to an embodiment of the invention.
0014FIG. 10 is a flowchart of a method for correcting misalignment between a pair of inkjet pens along the media axis, according to an embodiment of the invention.
0015FIG. 11 is a diagram showing lines printed by a first inkjet pen at a first period, and lines printed by aligned or misaligned second inkjet pens at a second period greater than the first period, according to an embodiment of the invention.
0016FIG. 12 is a flowchart of a method for correcting misalignment among a number of inkjet pens along the media axis, according to an embodiment of the invention.
0017FIG. 13 is a flowchart of a method according to an embodiment of the invention.
DETAILED DESCRIPTION OF THE DRAWINGS
0018 In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration how specific embodiments of the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice them. Other embodiments may be utilized, and logical, mechanical, and other changes may be made without departing from the spirit or scope of the present invention. For example, whereas an embodiment of the invention is partially described in relation to an inkjet printer dispensing ink, it is more broadly applicable to other kinds of fluid ejection systems. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the invention is defined only by the appended claims.
0019 Overview
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0020FIG. 1 shows the side view of a printer according to an embodiment of the invention. Media , such as paper, is supplied by a media supply component from a media supply roll . The media is moved over a chassis of the printer , and then is taken up by a media take-up component to a media take-up roll . While the media moves over the chassis , stationary inkjet pens eject ink onto the media . An ink supply provides ink to the inkjet pens . A heater may optionally be included as part of the printer to dry the ink being ejected from the inkjet pens after the ink is dispensed onto the medium . More generally, the ink is fluid, and the pens are fluid ejector assemblies.
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0021 The chassis includes a controller that controls movement of the media from the media supply component to the media take-up component , and controls ejection of ink from the inkjet pens . The controller includes a component that at least partially aligns the inkjet pens . Alternatively, the component may be separate from the controller . The controller and the component may each be a combination of software and/or hardware. The component may provide for automatic alignment of the inkjet pens , without user intervention, and/or manual alignment of the inkjet pens , with user intervention. The component may be considered the means for performing its respective functionality.
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0022 For automatic alignment of the inkjet pens , a sensor is optionally included as part of the printer to detect the ink output by the inkjet pens on the media . More specifically, the sensor detects the position of the ink output by the inkjet pens on the media , to determine whether the inkjet pens are aligned with one another. By interacting with the sensor , the component realigns the inkjet pens when they are misaligned.
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0023 For manual alignment of the inkjet pens , a user input/output (I/O) is optionally included as part of the printer . The user I/O includes a display mechanism to display information to the user, and a user input mechanism to receive information from the user. The user examines the output by the inkjet pens on the media , and if the user determines that the inkjet pens are misaligned, interacts with the component via the user I/O to realign the inkjet pens .
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0024FIG. 2 shows the top view of the inkjet pens over the media in detail, according to an embodiment of the invention. The inkjet pens includes the inkjet pens A, B, . . . N. The inkjet pens are positioned in a stationary and/or staggered formation over the media that moves past and under the pens from right to left, as indicated by the arrow . The inkjet pens as shown in FIG. 2 constitute one set of inkjet pens staggered from right to left. Alternatively, additional set(s) of stationary staggered inkjet pens may be included. In addition, two axes and are identified in FIG. 2. The media axis is the axis along which the media travels, in the direction identified by the arrow . The inkjet pen axis is the axis along which the inkjet pens are positioned in a staggered fashion.
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0025FIG. 3 shows the top view of the pair of inkjet pens A and B in detail, according to an embodiment of the invention. The inkjet pen A includes a number of nozzles. The nozzles are divided into a series of active nozzles , and inactive nozzles A and B above and below, respectively, the series of active nozzles . Ink is actually dispensed from the series of active nozzles . The inactive nozzles A and B do not normally dispense ink. They are present for aligning the inkjet pen A relative to the inkjet pen B along the pen axis , as will be described.
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0026 Similarly, the inkjet pen B includes a series of active nozzles , and inactive nozzles A and B above and below, respectively, the series of active nozzles . In one embodiment, there can be active nozzles within each of the series and , and there are a total of twelve inactive nozzles between the inactive nozzles A and B, and between the inactive nozzles A and B. In other embodiments, there can be more or less than active nozzles and more or less than a total of twelve inactive nozzles. Furthermore, preferably the last active nozzle of the series of the inkjet pen A is aligned with the first active nozzle of the series of the inkjet pen B, as indicated by the dotted line .
0027 Alignment and Misalignment of Inkjet Pens Along the Pen and Media Axes
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0028FIGS. 4A and 4B show an example of one type of misalignment of the inkjet pens A and B along the pen axis , and the correction of this misalignment, according to an embodiment of the invention. The inkjet pens and B of FIGS. 4A and 4B are staggered, and may also be stationary. In FIG. 4A, the series of active nozzles of the inkjet pen A prints the ink band , whereas the series of active nozzles of the inkjet pen B prints the ink band . However, the inkjet pens A and B are misaligned along the pen axis , resulting in a gap between the ink bands and printed by the series of active nozzles and . Particularly shown in FIG. 4A is that there is an inactive nozzle immediately adjacent to the active nozzle of the inkjet pen B, and that the last active nozzle of the series of active nozzles is the nozzle .
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0029 In FIG. 4B, the inkjet pens A and B are now aligned along the pen axis . Thus, the ink band printed by the series of active nozzles of the inkjet pen A aligns with the ink band printed by the series of active nozzles of the inkjet pen B, without any intervening gaps, such as the gap of FIG. 4A. The alignment along the pen axis is accomplished by shifting the series of active nozzles down by one nozzle. As a result, the series of active nozzles includes the nozzle in FIG. 4B, which was previously inactive in FIG. 4A. Furthermore, the nozzle is inactive in FIG. 4B, whereas it was part of the series of active nozzles in FIG. 4A.
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0030FIGS. 5A and 5B show an example of another type of misalignment of the inkjet pens A and B along the pen axis , and the correction of this misalignment, according to an embodiment of the invention. The inkjet pens A and B of FIGS. 5A and 5B are staggered, and may also be stationary. In FIG. 5A, the series of active nozzles of the inkjet pen A prints the ink band , whereas the series of active nozzles of the inkjet pen B prints the ink band . However, the inkjet pens A and B are misaligned along the pen axis , resulting in an area of overlap between the ink bands and printed by the series of active nozzles and . Particularly shown in FIG. 5A is that there is an inactive nozzle immediately adjacent to the active nozzle of the inkjet pen B, and that the first active nozzle of the series of active nozzles is the nozzle .
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0031 In FIG. 5B, the inkjet pens A and B are now aligned along the pen axis . Thus, the ink band printed by the series of active nozzles of the inkjet pen A aligns with the ink band printed by the series of active nozzles of the inkjet pen B, without any areas of overlap, such as the area of overlap of FIG. 5A. The alignment along the pen axis is accomplished by shifting the series of active nozzles up by one nozzle. As a result, the series of active nozzles includes the nozzle in FIG. 5B, which was previously inactive in FIG. 5A. Furthermore, the nozzle is inactive in FIG. 5B, whereas it was part of the series of active nozzles in FIG. 5A.
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0032 The inkjet pen misalignment along the inkjet pen axis in FIGS. 4A and 5A that is corrected in FIGS. 4B and 5B, respectively, is a one pixel-in-height misalignment, where the height of the output by a nozzle of an inkjet pen corresponds to one pixel. As can be appreciated by those of ordinary skill within the art, inkjet pens can become misaligned by more than one pixel in height as well. In such instances, the series of active nozzles of one of the pens can be adjusted by the number of nozzles corresponding to the number of pixels in height of the misalignment.
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0033FIGS. 6A and 6B show alignment of the inkjet pens A and B along the media axis , according to an embodiment of the invention. The inkjet pens A and B are shown in FIGS. 6A and 6B as staggered. However, these pens are at least stationary, and may also be staggered as shown in FIGS. 6A and 6B. In FIG. 6A, the media is moving from right to left, as indicated by the arrow . The inkjet pen A has printed a one pixel-in-width ink line . The media continues to move from right to left, such that in FIG. 6B, when the ink line printed by the inkjet pen A is aligned with the inkjet pen B, the inkjet pen B prints a one pixel-in-width ink line . For illustrative clarity, the dotted line separates the ink line from the ink line . The inkjet pens A and B are aligned along the media axis , resulting in the ink lines and they output themselves being aligned.
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0034 The inkjet pens A and B are aligned relative to one another by proper calibration of their respective fluid ejection delays. In particular, once the inkjet pen A has output the line in FIG. 6A, the inkjet pen B delays a length of time, commensurate with the speed of the media as it moves from right to left, before it outputs the line . If the relative fluid ejection delay between the two inkjet pens A and B are not aligned with one another, then the inkjet pen B will not output the line directly in line with the line output by the inkjet pen A.
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0035FIGS. 7A and 7B show examples of the different types of misalignment of the inkjet pens A and B along the media axis , according to different embodiments of the invention. The inkjet pens A and B are shown in FIGS. 7A and 7B as staggered. However, these pens are at least stationary, and may also be staggered as shown in FIGS. 7A and 7B. In FIG. 7A, the fluid ejection delay of the inkjet pen B is too great. After the inkjet pen A has printed the ink line , the inkjet pen B waits too long before printing the ink line , resulting in a gap . The ink line , in other words, is printed too late. To correct this misalignment, the fluid ejection delay of the inkjet pen B is decreased commensurate with the speed at which the media travels the width of the gap .
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0036 Conversely, in FIG. 7B, the fluid ejection delay of the inkjet pen B is too small. After the inkjet pen A has printed the ink line , the inkjet pen B does not wait long enough before printing the ink line , resulting in a gap . The ink line , in other words, is printed too soon. To correct this misalignment, the fluid ejection delay of the inkjet pen B is increased commensurate with the speed at which the media travels the width of the gap .
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0037 The inkjet pen misalignment along the media axis in FIGS. 7A and 7B is a one pixel-in-width misalignment, where the width of the output by a nozzle of an inkjet pen corresponds to one pixel. As can be appreciated by those of ordinary skill within the art, inkjet pens can become misaligned by more than one pixel in width as well. In such instances, the fluid ejection delays of the pens can be adjusted commensurate with the speed at which the media travels the number of pixels in width of the misalignment.
0038 Correcting Misalignment of Inkjet Pens Along the Inkjet Pen Axis
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0039FIG. 8 shows a method for correcting the misalignment between a pair of inkjet pens along the inkjet pen axis, according to an embodiment of the invention. Misalignment between pens along the inkjet pen axis is generally defined herein as misalignment of the output of the pens along this axis, as can be appreciated by those of ordinary skill within the art. Of the number of inkjet nozzles within each of a first inkjet pen nand a second inkjet pen nof the pair of inkjet pens, a contiguous I of them are used as the series of active nozzles. The method shifts the series of active nozzles of the second pen of the pair so that the second pen is aligned with the first pen. The method effectively performs the misalignment correction described in conjunction with and displayed in FIGS. 4A and 4B, and FIGS. 5A and 5B, and reference can be made thereto for an illustrative explanation as to the correction performed by the method . Furthermore, like other methods of embodiments of the invention, the method can be implemented as a computer program storable on a computer-readable medium.
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0040 A value k is first selected so that the center range of nozzles k . . kI within either of the inkjet pen represents the current series of active nozzles (). Next, the value m is set equal to k (). A gray ink band is printed with the nozzles k . . . kI of the inkjet pen no, and with the nozzles m . . . mI of the inkjet pen n(). The gray band is more generally an ink band printed with less than maximum intensity by the nozzles of the inkjet pen. The two bands printed by the two inkjet pens allow for detection of gaps and overlap between the bands, indicative of misalignment between the two pens. For instance, a gap between the bands is displayed as a lack of ink, whereas an overlap between the bands is displayed as a greater intensity of ink than that at which either band is individually printed.
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0041 The bands are examined for alignment (). For automatic alignment correction of the two inkjet pens, a sensor may determine whether a gap or an area of overlap is present between the two bands printed by the two inkjet pens. For manual alignment correction, the user determines whether a gap or an area of overlap exists between the two bands. If the no gap and no area of overlap are present, then the two inkjet pens are aligned with one another, and the method is finished (). In other embodiments of the invention, the gap is at least substantially reduced, but may not be totally eliminated.
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0042 Otherwise, if there is overlap between the bands (), then the value m is incremented (). Increasing m by one effectively shifts the active series of nozzles of the second inkjet pen nup, away from the active series of nozzles of the first inkjet pen n. That is, the series of active nozzles of the second inkjet pen is adjusted so that ink output thereby is farther away from the ink output of the first inkjet pen. This shifting of the series of active nozzles of the second pen is more specifically accomplished by adding a nozzle to the series, and removing another nozzle from the series. The nozzle added to the series of nozzles of the second pen is the inactive nozzle adjacent to the end of this series farthest away from the series of active nozzles of the first pen. The nozzle removed from the series of nozzles of the second pen is the nozzle of this series closest to the series of active nozzles of the first pen.
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0043 Next, verification is performed as to whether the series of active nozzles of the second inkjet pen nwas not shifted past the last nozzle of this pen (). That is, verification is performed to ensure that mI is not greater than the last nozzle of the second inkjet pen n. If not, then the method repeats , et seq., as has been described, to determine whether the adjustment performed results in alignment of the inkjet pens., However, if the verification fails, then the method shifts the starting nozzle of the series of active nozzles of each of the pens nand ndown by one nozzle (), such that both series of active nozzles are shifted down, so that the series of active nozzles of the second pen nis no longer shifted past its last nozzle. That is, the value k is decremented, as is the value m. The method then repeats , et seq., as has been described.
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0044 However, if the type of misalignment between the bands output by the inkjet pens does not result in overlap (), then the value m is instead decremented (), because the type of misalignment instead results in a gap between the bands. Decreasing m by one effectively shifts the active series of nozzles of the second inkjet pen ndown, towards the active series of nozzles of the first inkjet pen n. That is, the series of active nozzles of the second inkjet pen is adjusted so that ink output thereby is closer to the ink output of the first inkjet pen. This shifting of the series of active nozzles of the second pen is more specifically accomplished by adding a nozzle to the series, and removing another nozzle from the series. The nozzle added to the series of nozzles of the second pen is the inactive nozzle adjacent to the end of this series to the series of active nozzles of the first pen. The nozzle removed from the series of nozzles of the second pen is the nozzle of this series farthest from the series of active nozzles of the first pen.
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0045 Next, verification is performed as to whether the series of active nozzles of the second inkjet pen nwas not shifted past, or before, the first nozzle of this pen (). That is, verification is performed to ensure that m is not less than the first nozzle of the second inkjet pen n. If not, then the method repeats , et seq., as has been described, to determine whether the adjustment performed results in alignment of the ink pens. However, if the verification fails, then the method shifts the starting nozzle of the series of active nozzles of the pens no and nup by one nozzle (), such that both series of active nozzles are shifted up, so that the series of active nozzles of the second pen nis no longer shifted before its first nozzle. That is, the value k is incremented, as is the value m. The method then repeats , et seq., as has been described.
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0046 Other embodiments to the method can also be utilized. For instance, whereas the method describes repeatedly selecting active nozzles, printing ink bands, and determining whether the bands are in alignment, until the bands are in alignment, in another embodiment a number of ink bands can be printed by each pen, using different nozzles of each pen. Determining which of the ink bands of the first inkjet pen matches, or is aligned with, which of the ink bands of the second inkjet pen thus determines which of the nozzles of each pen should be used as the active series of nozzles so that the pens are aligned along the inkjet pen axis.
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0047 The method can be extended to correct the misalignment along the inkjet pen axis between each successive rolling pair of inkjet pens of a number of inkjet pens. FIG. 9 shows such a method for correcting misalignment among a number of inkjet pens along the inkjet pen axis, according to an embodiment of the invention. For each successive rolling pair of inkjet pens, the method shifts the series of active nozzles of the second pen of the pair so that the second pen is aligned with the first pen of the pair.
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0048 A value k is first selected so that the center range of nozzles k . . . kI within an inkjet pen represents the current series of active nozzles (). Next, an inkjet pen counter i is reset to zero (), and the value m is set equal to k (). A current rolling pair of the inkjet pens is defined as the pens nand n, where the first pen of the rolling pair is nand the second pen is n. A gray ink band is printed with the nozzles k . . . kI of the inkjet pen nand with the nozzles m . . . mI of the inkjet pen n(). The bands are manually or automatically examined for alignment (). If no gap and no area of overlap between the bands exists, then the current rolling pair of pens are aligned with one another, and the current rolling pair of pens is advanced by one pen within the inkjet pens (). That is, the counter i is incremented by one.
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0049 If the counter i is equal to the last inkjet pen (), then the method is finished (). Otherwise, the value k is set to the value m (). The value m is the starting nozzle within the range of nozzles for the second pen of the rolling pair of pens, whereas the value k is the starting nozzle within the range of nozzles for the first pen of the rolling pair of pens. Because the rolling pair of pens has been advanced by one pen, the first pen of the current rolling pair is the second pen of the previous rolling pair. Therefore, the starting nozzle m that was determined for the second pen of the previous rolling pen is now to be the starting nozzle k for the first pen of the current rolling pair. The value m is then set so that the center nozzles m . . . mI represents the active series of pens for the second pen of the current rolling pair (), and the method repeats at , et seq., as has been described, to align the newly current rolling pair of inkjet pens.
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0050 If the current rolling pair of inkjet pens are misaligned (), however, and if the misalignment results in the two bands output by the pens overlapping (), then the value m is incremented (), shifting the active series of nozzles of the second inkjet pen nup, away from the active series of nozzles of the first inkjet pen n. Verification is performed as to whether the series of active nozzles of the second inkjet pen nwas not shifted past the last nozzle of this pen (). That is, verification is performed to ensure that mI is not greater than the last nozzle of the second inkjet pen n. If not, then the method repeats , et seq., as has been described, to determine whether the adjustment performed results in alignment of the current rolling pair of pens.
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0051 However, if the verification fails, then the method shifts the starting nozzles of the series of active nozzles of each of the pens nand ndown by one nozzle (), such that both series of active nozzles are shifted down, so that the series of active nozzles of the second pen nis no longer shifted past its last nozzle. That is, the value k is decremented, as is the value m. Furthermore, because shifting the series of active nozzles of each of the pens nand nof the current rolling pair affects the series of active nozzles of any inkjet pens n. . . nthat have already been adjusted, the series of active nozzles of these pens are also shifted down one nozzle (). The method then repeats , et seq., as has been described.
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0052 If the type of misalignment between the bands output by the current rolling pair of inkjet pens does not result in overlap (), then the value m is instead decremented (), because the type of misalignment instead results in a gap between the bands. Decreasing m by one effectively shifts the active series of nozzles of the second inkjet pen ndown, towards the active series of nozzles of the first inkjet pen n. Verification is performed as to whether the series of active nozzles of the second inkjet pen nwas not shifted past, or before, the first nozzle of this pen (). That is, verification is performed to ensure that m is not less than the first nozzle of the second inkjet pen n. If not, then the method repeats , et seq., as has been described, to determine whether the adjustment performed results in alignment of the ink pens.
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0053 However, if the verification fails, then the method shifts the starting nozzle of the series of active nozzles of the pens nand nup by one nozzle (), such that both series of active nozzles are shifted up, so that the series of active nozzles of the second pen nis no longer shifted before its first nozzle. That is, the value k is decremented, as is the value m. Furthermore, because shifting the series of active nozzles of each of the pens nof the current rolling pair affects the series of active nozzles of any inkjet pens n. . . nthat have already been adjusted, the series of active nozzles of these pens are also shifted up by one nozzle (). The method then repeats , et seq., as has been described.
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0054 As with the method , other embodiments to the method can also be utilized. For instance, whereas the method describes repeatedly selecting active nozzles, printing ink bands, and determining whether the bands are in alignment, until the bands are in alignment, in another embodiment a number of ink bands can be printed by each pen, using different nozzles of each pen. Determining which two of the ink bands of each adjacent pair of pens thus determines which of the nozzles of these pens should be used as the active series of nozzles so that they are aligned along the inkjet pen axis.
0055 Correcting Misalignment of Inkjet Pens Along the Media Axis
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0056FIG. 10 shows a method for correcting the misalignment between a pair of inkjet pens along the media axis, according to an embodiment of the invention. Misalignment between pens along the media axis is generally defined herein as misalignment of the output of the pens along this axis, as can be appreciated by those of ordinary skill within the art. Furthermore, whereas the method is described in relation to inkjet pens that are stationary and staggered, it is generally applicable to pens that are stationary, regardless of whether they are staggered. The method adjusts the fluid ejection delay of a second inkjet pen nso that it outputs a line along the media axis that is aligned with a line output along the media axis by a first inkjet pen n. The method accomplishes this by having the first inkjet pen no print a number of lines along the media axis at a period p, and the second inkjet pen nprint a number of lines along the media axis at a period pgreater than p. The method adjusts the fluid ejection delay of the second inkjet pen nbased on which of the lines printed by the second inkjet pen nis aligned with which of the lines printed by the first inkjet pen n.
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0057 First, the method sets psuch that it and/or the time delay to which the it corresponds is preferably, but not necessarily, greater than the maximum absolute timing error between the inkjet pens nand n(). pmore precisely specifies the interval in pixels at which one-pixel wide lines will be printed by the first inkjet pen n. Therefore, pis greater than the distance corresponding to the maximum absolute timing error between the pens. That is, pis greater than the distance the media moves, in pixels, within a length of time equal to the maximum absolute timing error between the pens. pis correspondingly the interval in pixels at which one-pixel wide lines will be printed by the second inkjet pen n. pis set equal to pplus one (). A number of lines p*Pare printed by each of the inkjet pens nand n(), with the first inkjet pen no printing its lines at intervals of ppixels, and the second inkjet pen nprinting its lines at intervals of Ppixels.
0
1
0
0
1100
1102
1102
1102
1102
1100
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
0058FIG. 11 shows a rudimentary example of the lines printed by the first inkjet pen n, and three rudimentary examples of the lines printed by the second inkjet pen n, according to an embodiment of the invention. The lines printed by both inkjet pens have a nominal alignment line , with respect to which alignment of the pens is analyzed. The first inkjet pen nprints the lines at a period pof three, such that at every third pixel-wide spacing, indicated by dotted lines in FIG. 11, there is one of the lines . Seven such lines are shown in FIG. 11: the zeroth line A at the alignment line , the first lines B and B printed to either side of the zeroth line A, the second lines C and C printed to either side of zeroth line A, and the third lines D and D printed to either side of the zeroth line A. The lines B, C, and D are left lines because they are to the left of the zeroth line A, and the lines B, C, and D are right lines because they are to the right of the zeroth line A.
1
0
1
1
0
1
0
1
1104
1104
1104
1104
1100
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1104
1102
1104
0059 In the case where the second inkjet pen nis aligned with the first inkjet pen nalong the media axis, the pen nprints the lines , at a period pof four, such that at every fourth pixel-wide spacing, there is one of the lines . Five such lines are shown in FIG. 11: the zeroth line A at the alignment line , the first lines B and B printed to either side of the zeroth line A, and the second lines C and C printed to either side of the zeroth line A. The first lines B and B are referred to as the first lines, or the lines having the count number one, because they are the first lines to either side of the zeroth line A. The second lines C and C are likewise named. Furthermore, the lines B and C are left lines because they are to the left of the zeroth line A, and the lines B and C are right lines because they are to the right of the zeroth line A. Because the pens nand nare aligned, the first line printed by the pen n, the zeroth line A, is aligned with the first line printed by the pen n, the zeroth line A.
1
1
1
1106
1106
1106
1100
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
1106
0060 In the case where the second inkjet pen nis misaligned with the first inkjet pen no along the media axis, such that it prints its first line after (with respect to position) the first inkjet pen no prints its first line, the pen nprints the lines , at a period p. Five such lines are shown in FIG. 11: the zeroth line A which should be at the alignment line , the first lines B and B printed to either side of the zeroth line A, and the second lines C and C printed to either side of the zeroth line A. The first lines B and B are referred to as the first lines, or the lines having the number one, because they are the first lines to either side of the zeroth line A. The second lines C and C are likewise named. Furthermore, the lines B and C are left lines, because they are to the left of the zeroth line A, whereas the lines B and C are right lines, because they are to the right of the zeroth line A.
1106
1102
1106
1102
1
0
1
1
1
1
1
0
0061 The zeroth line A printed by the second inkjet pen nis printed one pixel width after the zeroth line A printed by the first inkjet pen n. The first line B is aligned with the first line B. To align the second inkjet pen nwith the first inkjet pen no, the fluid ejection delay of the pen nis decreased by a length of time corresponding to one pixel width, so that the inkjet pen nprints its first line sooner. That is, the delay of the pen nis decreased by the length of time it takes for the media to move one pixel width. This delay is equal to the line number countoneof the line to the right of the zeroth line printed by the second inkjet pen nthat is aligned with one of the lines to the right of the zeroth line printed by the first inkjet pen n.
1
1
1
1108
1108
1108
1100
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
1108
0062 In the case where the second inkjet nis misaligned with the first inkjet pen no along the media axis, such that it prints its first line before (with respect to position) the first inkjet pen no prints its first line, the pen nprints the lines , at a period p. Four such lines are shown in FIG. 11: the zeroth line. A which should be at the alignment line , the first lines B and B printed to either side of the zeroth line A, and the second lines C and C printed to either side of the zeroth line A. As before, the first lines B and B are referred to as the first lines, or the lines having the number one, because they are the first lines to either side of the zeroth line A. The second lines C and C are likewise named. Furthermore, the lines B and C are left lines, because they are to the left of the zeroth line A, and the lines B and C are right lines, because they are to the right of the zeroth line A.
1108
1102
1108
1102
1
0
0
0
1
1
1
1
0
0063 The zeroth line A printed by the second inkjet pen nis printed one pixel width before the zeroth line A printed by the first inkjet pen n. The first line B is aligned with the first line B. To align the second inkjet pen nwith the first inkjet pen n, the fluid ejection delay of the pen nis increased by a length of time corresponding to one pixel width, so that the inkjet pen nprints its first line later. That is, the delay of the pen nis increased by the length of time it takes for the media to move one pixel width. This delay is equal to the line number countoneof the line to the left of the zeroth line printed by the second inkjet pen nthat is aligned with one of the lines to the left of the zeroth line printed by the first inkjet pen n.
1
0x
1x
10
00
10
1000
1008
1000
1010
0064 Referring back to FIG. 10, the lines printed by the first inkjet pen no and the second inkjet pen nare referred to as tand trespectively. The method automatically or manually examines whether the first lines printed by the inkjet pens, too and t, are aligned with one another (). For automatic alignment correction of the two inkjet pens, a sensor may determine whether these two lines are in alignment. For manual alignment correction, the user determines whether these two lines are in alignment. If the two lines tand tare in alignment with one another, then the method is finished ().
0
00
1
10
1
1
0(−k)
0
1
1(−k)
1
0
0
1
1012
1014
1016
0065 Otherwise, if the zeroth line printed by the first inkjet pen n, t, was printed before the zeroth line printed by the second inkjet pen n, t(), then this means that the fluid ejection delay of the second inkjet pen nis too slowthat is, the delay is too long (). The fluid ejection delay of the pen nis decreased by the time corresponding to the number of pixels k (), where the line tis a line printed by the first inkjet pen nthat is aligned with, or matches, a line printed by the second inkjet pen n, t. That is, the first kth line printed to the right of the zeroth line by the pen nthat matches the kth line printed to the right of the zeroth line by the pen nis determined, such that the fluid ejection delay of the pen nis decreased by the number of pixels k, where the periods of the lines printed by the inkjet pens differ by one pixel. Thus, the fluid ejection delay of the pen nis decreased by the time that it takes for the media to move the number of pixels k.
1
1
0
1
0066 More generally, if the periods of the lines printed by the inkjet pens differ by a number of pixels y>1, the fluid ejection delay of the pen nis decreased by a number of pixels between ((k1)*y) and k*y. For instance, where the periods of the lines printed by the inkjet pens differ by two pixels, and the first line printed to the right of the zeroth line by the pen nmatches the first line printed to the right of the zeroth line by the pen n, the fluid ejection delay of the pen nis decreased by a number of pixels between zero or two. This is because the resolution of the fluid ejection delay mismatch between the two pens that can be detected, as it corresponds to a number of pixels, is no greater than the difference in pixels of the periods of the lines printed by the inkjet pens.
1106
1102
1102
1106
1102
1102
1106
1106
0
1
0
0(−1)
0
1(−1
1
1
0067 For example, in FIG. 11, the lines and the line represent the scenario in which the zeroth line is printed by the first inkjet pen n, the line A, before the zeroth line is printed by the second inkjet pen n, the line A. The line B, the first line printed by the first inkjet pen nto the right of the zeroth line A, is referred to as t, and matches the first line printed by the second inkjet pen nto the right of the zeroth line A, which is the line B and which is referred to as t). Thus, k1, and the fluid ejection delay of the pen nis decreased by the time corresponding to one pixel. That is, the fluid ejection delay of the pen nis decreased by the time it takes for the media to move one pixel.
0
00
1
10
1
1
0k
0
1
1k
1
0
1
1
1012
1018
1020
0068 Referring back to FIG. 10, if the zeroth line printed by the first inkjet pen n, t, was printed after the zeroth line printed by the second inkjet pen n, t(), then this means that the fluid ejection delay of the second inkjet pen nis too fastthat is, the delay is too short (). The fluid ejection delay of the pen nis increased by the time corresponding to the number of pixels k (), where the line tis a line printed by the first inkjet pen nthat is aligned with, or matches, a line printed by the second inkjet pen n, t. That is, the first kth line printed to the left of the zeroth line by the pen nthat matches the kth line printed to the left of the zeroth line by the pen nis determined, such that the fluid ejection delay of the pen nis increased by the number of pixels k, where the periods of the lines printed by the inkjet pens differ by one pixel. Thus, the fluid ejection delay of the pen nis increased by the time that it takes for the media to move the number of pixels k.
1
1
0
1
0069 More generally, if the periods of the lines printed by the inkjet pens differ by a number of pixels y>1, the fluid ejection delay of the pen nis increased by a number of pixels between ((k1)*y) and k*y. For instance, where the periods of the lines printed by the inkjet pens differ by two pixels, and the first line printed to the left of the zeroth line by the pen nmatches the first line printed to the left of the zeroth line by the pen n, the fluid ejection delay of the pen nis increased by a number of pixels between zero or two. This is because the resolution of the fluid ejection delay mismatch between the two pens that can be detected, as it corresponds to a number of pixels, is no greater than the difference in pixels of the periods of the lines printed by the inkjet pens.
1108
1102
1102
1108
1102
1102
1108
1108
0
1
0
01
1
11
1
1
0070 For example, in FIG. 11, the lines and the line represent the scenario in which the zeroth line is printed by the first inkjet pen n, the line A, after the zeroth line is printed by the second inkjet pen n, the line A. The line B, the first line printed by the first inkjet pen nto the left of the zeroth line A, is referred to as t, and matches the first line printed by the second inkjet pen nto the left of the zeroth line A, which is the line B and which is referred to as t. Thus, k1, and the fluid ejection delay of the pen nis increased by the time corresponding to one pixel. That is, the fluid ejection delay of the pen nis increased by the time it takes for the media to move one pixel.
1000
1200
1200
1200
0071 The method of FIG. 10 can be extended to correct misalignment along the media axis between each successive rolling pair of inkjet pens of a number of inkjet pens. FIG. 12 shows such a method for correcting misalignment among a number of inkjet pens along the media axis, according to an embodiment of the invention. Whereas the method is described in relation to inkjet pens that are stationary and staggered, it is generally applicable to pens that are stationary, regardless of whether they are staggered. For each successive rolling pair of inkjet pens, the method adjusts the fluid ejection delay of the second inkjet pen of the pair so that it outputs a line along the media axis that is aligned with a line output along the media axis by the first inkjet pen of the pair.
1200
1202
1204
1206
0
0
0
0
k
k−1
k
k
k+1
k
k
kx
k
(k+1)
0072 First, the method sets psuch that it is greater than the maximum absolute timing error between any two adjacent inkjet pens (). As before, pmore precisely specifies the interval in pixels at which one-pixel wide lines will be printed by the first inkjet pen n. Therefore, pis greater than the distance corresponding to the maximum absolute timing error between any two adjacent inkjet pens. Next, each Pis set to (P1), where k1 . . . m1, and where there are m total pens numbered 0 . . . m1 (). Furthermore, for k0 . . . m1, each inkjet pen nprints P*Plines at intervals of Ppixels (). The lines printed by the inkjet pen nare referred to as t, where x ranges from 0 . . . (P*P)1.
k
k
0073 In another embodiment of the invention, each inkjet pen that has two adjacent pensan adjacent pen over the current pen and an adjacent pen below the current penprints a bottom set of lines and a top set of lines, at different intervals. The bottom set of lines is used to align the current pen with the adjacent pen below the current pen, and the top set of lines is used to align the current pen with the adjacent pen above the current pen. In this embodiment, the intervals Pdo not have to be increased for each pen nas has been indicated. Rather, it is sufficient for the intervals to alternate between sets of lines of the pens. For example, the bottom most pen may print lines at intervals y, and the top most pen may print lines at intervals y1. Intervening pens then print two sets of lines, the bottom set at intervals y1, and the top set at intervals y.
1208
1200
1210
1200
1212
1214
1200
1216
1200
1210
k
k+1
k0
(k+1)0
k0
(k+1)0
0074 k is subsequently used as a counter, and set to zero (). The method then automatically or manually examines whether the first lines printed by the rolling pair of inkjet pens nand n, tand t, are aligned with one another (). If the two lines tand tmatch, then the method increments k to proceed with the next rolling pair of inkjet pens (). However, if k has been incremented to the last pen m1 (), then there are no more rolling pairs of inkjet pens, and the method is finished (). Otherwise, the method repeats at , et seq., as has been described, to determine whether the new rolling pair of inkjet pens is aligned with one another along the media axis.
k
k0
k+1
(k+1)0
k+1
k+1
k(−r)
k
k+1
(k+1)(−r)
I
1218
1220
1222
1212
1224
0075 However, if the zeroth line printed by the first inkjet pen nof the current rolling pair, t, was printed before the zeroth line printed by the second inkjet pen nof the current rolling pair, t(), then this means that the fluid ejection delay of the second inkjet pen nis too slowthat is, the delay is too long (). Therefore, the fluid ejection delay of the pen n, and the fluid ejection delays of all the inkjet pens subsequent to this pen, are decreased by the time corresponding to the number of pixels r (), where the line tis the first line printed by the first inkjet pen nto the right of the zeroth line that is aligned with, or matches, a line printed by the second inkjet pen n, t, to the right of the zeroth line. That is, the fluid ejection delay of each pen n, where Ik1 . . . m1, is decreased by the time it takes for the media to move the number of pixels r. The method then proceeds to (), as has been described.
k
k0
k+1
(k+1)0
k+1
k+1
kr
k
k+1
(k+1)r
i
1218
1226
1228
1212
1224
0076 Conversely, if the zeroth line printed by the first inkjet pen nof the current rolling pair, t, was printed after the zeroth line printed by the second inkjet pen nof the current rolling pair, t(), then this means that the fluid ejection delay of the second inkjet pen nis too fastthat is, the delay is too short (). Therefore, the fluid ejection delay of the pen n, and the fluid ejection delays of all the inkjet pens subsequent to this pen, are increased by the time corresponding to the number of pixels r (), where the line tis a line printed by the first inkjet pen nto the left of the zeroth line that is aligned with, or matches, a line printed by the second inkjet pen n, t, to the left of the zeroth line. That is, the fluid ejection delay of each pen n, where Ik1 . . . m1, is decreased by the time it takes for the media to move the number of pixels r. The method then proceeds to (), as has been described.
0077 Conclusion
1300
1300
1302
1300
1304
0078FIG. 13 shows a method that summarizes the stationary staggered inkjet pen alignment over the inkjet pen axis and the media axis that has been described, according to an embodiment of the invention. The method first aligns a pair of stationary staggered inkjet pens over the inkjet pen axis (). The method then aligns the pair of stationary staggered inkjet pens over the media axis ().
1306
1308
1312
1314
1316
0079 To align the pair of pens along the inkjet pen axis, ink bands are printed by both pens (). The series of nozzles that output aligned ink bands are selected as the active series of nozzles for the inkjet pens, such that the pens are aligned (). To align the pair of pens along the media axis, the first pen of the pair outputs lines along the media axis at a first period (). The second pen of the pair outputs lines along the media at a second period greater than the first period (). The fluid ejection delay of either or both of the inkjet pens is then adjusted, based on which of the lines output by the second pen is aligned with, or matches, which of the lines output by the first pen ().
0080 It is noted that, although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that any arrangement is calculated to achieve the same purpose may be substituted for the specific embodiments shown. This application is intended to cover any adaptations or variations of the present invention. For example, whereas an embodiment of the invention is partially described in relation to an inkjet printer dispensing ink, it is more broadly applicable to other kinds of fluid ejection systems. Therefore, it is manifestly intended that this invention be limited only by the claims and equivalents thereof. | |
Just as numbers 1 to 999 can be turned into a picture, so too can all the numbers from 00 to 099.
We are including keywords for these numbers because it’s surprising how often they come in useful.
For instance:
“Don’t forget to phone me on 734 008!”
Our favourite keyword for 734 is camera, and for 008 it’s his sofa.
Let’s pretend the person who wants you to phone has lost his camera (734), but after a long search finds it under his sofa (008).
What about this number:
000402030400
Can you imagine trying to remember this one without having made image associations for number 000 to 099?
Taking our favourite keywords for each set of three digits gives us seesaws, war zone, Siamese and roses. Now make up a memorable sentence using these words, e.g:
The seesaws were like a war zone as the Siamese cats fought over who would have the bunch of roses.
Vital numbers like the PINs for credit cards and debit cards, and National Insurance Numbers, may also contain zeros. | https://mammothmemory.net/memory/remembering-numbers/remembering-numbers/remembering-numbers.html |
The house always wins.
Auntie Mabel, we're home!
O, ooh, I'm so glad you're finally here, dear. Auntie Mabel purred as she turned in her garden chair. I've just been dying for the two of us to talk. I had my heart set on going to that lecture hall last night.
Why?
Well, dear, it's about time you came to see just how interesting this particular part of Boston is, Mabel answered, as she rose from her chair. And just last night, as I was leaving the lecture hall, a man asked me for directions to this very house. So, I just thought it would be the most wonderful way to celebrate your arrival for dinner tonight. I knew you'd love being home to meet your new aunt.
I can't believe you asked him directions, T. Rex told Auntie Mabel. "That's just too cool." T. rex stood at the foot of his aunt's garden chair. Are you ready for me to give you some of these? That's the closest thing I have to a proper greeting. Not to be stingy with my presents.
Auntie Mabel blushed and rubbed her hands together. Well, I certainly hope you'll be happy with what I've got. I'd have brought more, honey, but my bags were just too heavy.
T. rex leaned over Auntie Mabel's garden chair and licked her fingers, one by one. He looked into her eyes and smiled and purred as he started to lick her hands again. His eyes closed in pleasure as he licked two times around the outside of both of her hands, and then he purred once before he stepped back. If he wasn't such a great house cat, he'd be a great dog.
T. Rex's aunt smiled and winked before she took T. Rex's paw. Oh, how sweet that you'd find my present a pleasure, Auntie Mabel purred as she looked up at T. rex. It was almost as if Mabel could read his thoughts before T. Rex spoke. T. rex could only stare in wonder as Mabel purred, and how nice to meet you. I'm Auntie Mabel. | https://original.newsbreak.com/@jahid1975-1591385/2474127294345-zen-pet-journal-of-a-regular-ol-house-cat-2022 |
What is 50 kg in US pounds?
So you want to convert 50 kilograms into pounds? If you’re in a rush and just need the answer, the calculator below is all you need. The answer is 110.23122100919 pounds.
Is 50kg underweight?
Is 50 kg for 5’2 underweight? – Quora. Based on the BMI chart, you have a BMI of 21 so you are all good. You would be considered underweight if you were around 100. So your in good shape, just never only focus on your BMI number/the scale too much.
Is 50kg a good weight?
Hello, As per the details of your height your weight should be in the range of 50-60 kgs to be in Ideal BMI range that is 18.5-22.9. BMI 18.5-22.9 for Asians is an indicator of health, which depends on your height & weight.
Is 110 pounds fat for a 13 year old?
According to the CDC, most 13-year-old girls weigh between 76 and 148 pounds (lb). The 50th percentile for weight in this group is around 101 lb. This means that about 50% of girls this age weigh less than 101 lb. If a 13-year-old girl weighs above the 95th percentile, the doctor may diagnose obesity.
Why is 14 pounds a stone?
In the 14th century England’s exportation of raw wool to Florence necessitated a fixed standard. In 1389 a royal statute fixed the stone of wool at 14 pounds and the sack of wool at 26 stones. The stone is still commonly used in Britain to designate the weights of people and large animals.
Is 55 kg heavy for a girl?
Percentage of women classified as underweight (75–90 kg), and obese (>90 kg) by time period, 1988–2006. The obesity epidemic requires the development of prevention policy targeting individuals most likely to benefit.
Is 121 pounds overweight for a 12 year old?
Overweight children fall between the 85th and 95th percentile, and obese children have a BMI equal to or greater than the 95th percentile. A healthy weight for a 12-year-old girl, therefore, can generally fall anywhere between 65 and 120 pounds.
Is 1 kg more than 1 pound?
A kilogram (kg) is stated to be 2.2 times heavier than a pound (represented as lbs). Thus, one kilo of mass is equal to 2.26lbs.
Is 50kg heavy for a 12 year old?
The CDC also reports that a 12-year-old girl’s weight is usually between 68 and 135 pounds, and the 50th percentile weight for girls is 92 pounds. If your child is in the 50th percentile for weight, it means that out of 100 children their age, 50 may weigh more than they do and the other 50 may weigh less.
What weight is heavy for a girl?
The National Heart, Lung, and Blood Institute indicates that a healthy weight for a woman who is 5 feet, 4 inches tall ranges from 110–140 pounds with a BMI of 19–24. A woman whose BMI score is above 25 is considered overweight and 30 and above is considered obese.
What’s the average weight for a 12 year old?
The averages for 12-year-olds are 89 pounds, for males, and 92 pounds, for females. However, beyond biological sex, many other factors influence someone’s weight at this age, including their height, body composition, the onset of puberty, environmental factors, and underlying health issues.
What is underweight for a 14 year old?
Teens can have a high BMI if they have a large frame or a lot of muscle, not excess fat. Underweight: BMI is below the 5th percentile age, gender, and height. Healthy weight: BMI is equal to or greater than the 5th percentile and less than the 85th percentile for age, gender, and height.
Is 55kg healthy for a 14 year old?
For example, it places you at about the the 77th percentile for girls aged 14. That means around 77% of girls your age are skinnier than you. Another way to look at it, is that at 59 kg you would be overweight, so you might want to be careful not to get to much heavier until you grow more.
Is 55kg a healthy weight for a 17 year old?
To answer the question, as long as the 17yr old is healthy and she doesn’t have any medical problems at that weight, she’s perfectly fine.
Is 55kg skinny?
Yes, that’s very skinny. If you’re struggling to put on weight, contact a doctor or start using meal-replacement drinks in between your actual meals.
Is 180 pounds overweight for a 14 year old?
the average weight for a 14 year old is about 140–150, so youre about 20–30lbs above, but it isnt enough to cause health issues.
How much should a 67 year old woman weigh?
Average weights of U.S. women across the adult lifespan are: Ages 20-39: 167.6 pounds. Ages 40-59: 176.4 pounds. Ages 60 and up: 166.5 pounds. | https://frequentlyasked.net/what-is-50-kg-in-us-pounds/ |
Sneak Peek at the First Easter Dress!
I designed this dress for my oldest daughter from various elements of other dresses I have seen and liked. I am still thinking of a name for it, too. I will give the full tutorial later, but in the mean time, here are some of her modeling shots.
It has a cross-front bodice, just because I have never made one and wanted to try it. I also fell in love with the idea of a wide band overlaid with crochet lace trim.
What little girls wouldn't love a pretty laced up back?
The challenge was getting the lacing to work with a full circle skirt. She specifically asked for one that twirled, which meant that I had to make a dash to the store for more fabric last night.
Oh, but it twirls! I don't know how people get such cute pictures of their little girls twirling in circle skirts. Then again, my girls are half falling over when they try to twirl so maybe it isn't all the photography. I am going to have to make them practice twirling.
Hmmm... So if you can think of a good name for this one, I would love to hear your ideas! UPDATE: Combining a few of the suggestions, I came up with the Twirly Lace Dress and posted the tutorial HERE. Thanks for all of your help!! | https://pacountrycrafts.com/blog/sneak-peek-at-the-first-easter-dress |
Ethan Fineshriber is a 6x World Champion in Martial Arts. His weapon of choice is the bo staff. He has appeared on the hit shows Lip Sync Battle Shorties on Nickelodeon, SuperKids in Munich, Germany, Fantastic Baby in China, Wonderama TV, and currently is one of the stars of the Ninja Kidz TV Power Rangers series on Youtube in which he plays Tommy Oliver the Green Power Ranger. Ethan is also a vlogger and has had a couple of vlogs go viral on what it is like to live life as someone who is on the Autism Spectrum. Ethan was diagnosed at 3 years old and is on the high functioning end of the spectrum. He has followers on Instagram, Youtube, Facebook, and Twitter totally over 110K. He loves martial arts, tricking, Roller Coasters, Origami, Rubik's Cubes, and magic.
Demographic data for @ethanfineshriberofficial's audience is available for free to influence.co users.
"He's learned a lot and loves to give back." | https://influence.co/ethanfineshriberofficial |
A: I like the way you're thinking. As energy costs have risen over the years, it can make sense to get rid of older appliances and buy new energy-efficient ones that will save you energy on your monthly utility bill. The true cost of an appliance is how much it costs to buy plus its lifetime maintenance costs and utility bills.
There are a couple of ways to answer your question.
First, you can estimate how much energy an appliance uses with a simple formula: the daily kWh (kilowatt-hour) usage can be calculated by multiplying the product's wattage times the number of hours it is used each day. Multiply this result by the number of days you use the product each year, convert the result to kilowatt-hours, and then multiply this by the kWh rate you pay your utility company (this rate is available on your monthly bill).
For example, say you've got a 200-watt fan that you use about four hours a night for four months a year (120 days): 200 watts times 4 hours a day times 120 days equals 9,600 watts. Divide this by 1,000 to convert the watts to kWh, then multiply the result by the rate your company charges (let's say it's 10 cents per kWh). The answer is $9.60 per year for the electricity to power the fan. You can use this formula to estimate energy usage for any product in your home (the only exception is when you calculate energy use of refrigerators. The U.S. Dept. of Energy recommends that you divide the total time you use the refrigerator by 3 because even though the appliance is plugged in all the time, it actually cycles on and off to maintain the required temperature).
Most appliances have their wattage stamped on the back or bottom of the unit, so it's usually easy to find the information you need. Keep in mind this is just an estimate, since many appliances are used at different power levels so the actual energy use is hard to determine precisely.
If you really want to get the exact figure, though, there is a second answer to your question. Put words like "power meter" or "power monitor" into your computer's search engine and you'll find a number of fairly low-cost products on the market that are specifically designed to let you measure the actual power use of products in your home. Just plug the television or toaster or electric fan or any other electricity-using appliance into the meter and plug that unit into the wall. You'll get an easy-to-read measurement of current energy use, and you can then compare this number with the energy use of a new appliance you're looking at in a store. Many new products have labels telling what their estimated annual energy use is, so a comparison will quickly let you know if you'll save enough money by buying a new one to make the purchase a smart one to do.
Knowing how much energy you use lets you change your lifestyle to save energy at various times. Once you find out how much energy that big- screen TV actually uses, you'll possibly think twice before leaving it on when no one is watching it. That electric space heater you keep near your desk may use so much energy when it's in use that wearing a sweater might keep you just as comfortable in cold weather without running up the electric bill.
The United Kingdom recently announced that they plan to put "smart" energy meters in every home and business in the country by 2020 that will help homeowners and renters easily track how much electricity and gas they are using. This information can help the consumer pay more attention to ways they are now wasting energy. If you don't believe that this really works, go outside and take a look at your home's electric meter. If that dial is spinning quickly, it's letting you know that you're using a lot of electricity right now. Turn off some unneeded lights and appliances, and another look at the meter will show that it's slowed significantly.
Knowing how much energy you're using is an important first step in reducing energy waste. | https://www.thehour.com/entertainment/article/Home-energy-Q-A-8282888.php |
1 Mr. Ernest Aubee, Chairman of the Regional Ecological Organic Agriculture (EOA) Steering Committee of the ECOWAS Commission, said on Thursday that organic agriculture generates many resources in the world.
2 Aubee said this in Abuja during a two-day national stakeholder meeting on communicating the achievements of EOA in Nigeria to government and national stakeholders.
3 “The ECOWAS Commission promotes organic agriculture in West Africa.
4 “We as a region want to benefit from the high resource turnover so that our farmers can also benefit,” the president said.
5 He said organic farming is the best way to adopt as it promotes good health for the people of West Africa in terms of protecting the environment and improving livelihoods.
6 Aubee said most people were used to conventional farming, as such it would take a long time to convince them about organic farming which he said was one of the main challenges faced in practice. .
seven He said organic farming is a specialized farming method that requires certain standards, regulations and laws.
8 “A number of our countries in the ECOWAS region are working hard to have these laws, regulations and standards.
9 “The ECOWAS Commission supports them in developing the standards,” Aubee said.
ten He said that for organic farming to thrive in the region, the focus should be on smallholder farmers.
11 “ECOWAS has projects in all 15 member states to support organic farming.
12 “We also support various research activities so that the technology developed meets the requirements of small-scale farmers,” Aubee said.
13 The Executive Secretary of the Agricultural Research Council of Nigeria (ARCN), Prof. Garba Sharabotu, while welcoming the participants, said the council was at the forefront of promoting organic agriculture in Nigeria. .
14 Sharabutu was represented by ARCN Plant Resources Department Director, Dr. Oluwafemi Salako.
15 He said the stakeholder meeting should report on developments in organic agriculture in Nigeria with a view to seeing how it can be improved and scaled up to ensure food security.
16 “We want to see how this meeting will improve the lives of every Nigerian,” Sharabotu said.
17 The National Coordinator of the AEO Initiative in Nigeria, Dr. Olugbenga Adeoluwa commended all stakeholders at the meeting for their efforts to improve organic agriculture in the country.
18 Adeoluwa said the Ministry of Agriculture is working effortlessly to put in place the draft organic agriculture policy, assuring that it will be approved by the executive arm of government very soon.
19 “We hope that when this policy is out, Nigerians can do well in the organic farming business,” he said.
20 Mr. Salimonu Oladipo, a farmer from Oyo State, applauded the practice of organic farming describing it as the best way to farm.
21 “Organic farming is very profitable and sustainable, at least I’ve been doing it for 10 years,” Oladipo said.
22 He urged the federal government to urgently roll out a policy to support organic farming practices, as soon as possible. | https://thefutureisorganic.net/organic-agriculture-generates-a-lot-of-resources-on-a-global-scale-chairman-of-the-committee-ecowas/ |
IMAGE: Interface for upper-limb motor rehabilitation. During the BMI therapy, the patient with upper-limb paralysis would be asked to imagine/attempt to move his/her paralyzed arm, and those intentions would be translated...
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Credit: The Shirley Ryan Ability Lab, Chicago, Ill., USA
Amsterdam, NL, Aug. 13, 2018 - Stroke remains a leading cause of adult disability, and the global burden of stroke continues to grow with devastating consequences for patients, families, and caregivers. In this special issue of NeuroRehabilitation leading international experts on stroke rehabilitation provide theoretical and practical insights into the steps necessary to push beyond merely compensatory training and onto a level of recovery that is satisfactory for patients.
"Stroke rehabilitation is at a crossroads," explains Guest Editor Richard Harvey, MD, Professor, Physical Medicine and Rehabilitation and Physical Therapy and Human Movement Sciences, Northwestern University Feinberg School of Medicine, and Clinical Chair, Brain Innovation Center, Wesley and Suzanne Dixon Stroke Chair, The Shirley Ryan Ability Lab, Chicago, IL, USA. "This issue of NeuroRehabilitation explores novel concepts and approaches to the rehabilitation of stroke that will help point the direction for the next wave of neurorehabilitation research."
A promising area of research is the use of biomarkers to predict motor recovery and outcomes after stroke. Cathy M. Stinear, PhD, of the Department of Medicine, University of Auckland, Auckland, New Zealand, and colleagues, consider how algorithms to predict motor recovery and outcomes after stroke might be implemented in clinical practice.
"Since 2011 there have been eight large randomized controlled trials of motor rehabilitation that recruited all participants within 30 days of stroke. However, none were able to detect a benefit of the tested intervention," notes Dr. Stinear. "Using biomarkers to select and stratify patients in rehabilitation trials could increase the sensitivity of trials to intervention effects, which might be particularly important for detecting these effects against the background of recovery experienced by most patients during the initial days and weeks after stroke."
Biomarkers of the functional and structural integrity of the corticomotor system can predict recovery from motor impairment and motor function outcomes in individual patients. There are two broad categories of motor system biomarkers that have received the most research attention to date: transcranial magnetic stimulation (TMS) and magnetic resonance imaging (MRI). Dr. Stinear and colleagues describe the accumulating evidence for the use of these motor system biomarkers during the initial days and weeks after stroke and then discusses the potential challenges and benefits of implementing these biomarkers in clinical practice using the PREP2 algorithm as an example.
The PREP2 algorithm combines clinical assessment with biomarkers in an algorithm to predict upper limb functional outcomes for individual patients. It is the first algorithm to be tested in clinical practice. It is the standard of care in two Auckland hospitals and is being rolled out in several other hospitals in New Zealand, North America, Singapore, and Europe. Other biomarker-based algorithms are likely to follow. The researchers describe several potential facilitators and barriers to implementing biomarkers in clinical practice, including characteristics of the algorithm, the clinical setting, and the clinicians themselves.
The researchers conclude that active, theoretically underpinned implementation strategies are needed to ensure that biomarkers are successfully used in clinical practice for predicting motor outcomes after stroke and should be considered in parallel with biomarker development.
"Implementing biomarkers in stroke rehabilitation practice has been shown to help patients leave hospital sooner, with no negative effects on their outcomes or wellbeing," says Dr. Stinear. "Knowing what to expect for their recovery can also help patients and families adjust more readily to life after stroke. However, principled strategies for implementing biomarkers in clinical practice are needed to produce effective and sustainable improvements in clinical practice."
Although therapies have improved in recent years, traditional rehabilitation still fails in patients with severe paralysis. Ander Ramos-Murguialday, PhD, MSc, of the Institute of Medical Psychology and Behavioral Neurobiology, University of Tübingen, Tübingen, Germany, and TECNALIA Research and Innovation, San Sebastián, Spain, and colleagues, review brain-machine interfaces (BMIs) that have emerged as a promising tool to guide motor rehabilitation interventions and promote recovery because they can be applied to patients with no residual movement. A BMI is a system that records, decodes, and ultimately translates brain signals into an effector action or behavior, without necessarily involving the motor system.
The researchers reviewed a total of 13 studies. Following a pilot study in 2008, the first double-blinded controlled clinical trial using a BMI for completely paralyzed stroke patients was published in 2013. The experimental group showed significant motor learning. Subsequent studies confirmed these positive results. The duration of the interventions ranged from one to eight weeks and each session lasted between 30 and 90 minutes. Twelve out of the studies targeted the upper limb, while only one focused on the lower limb. All the studies reported improvements of motor function in the experimental group after the use of the BMI. Six studies demonstrated higher improvements in the intervention group than in the control group. Only two of the studies reported no improvements at all in the control group.
Although significant, Dr. Ramos-Murguialday and colleagues conclude that functional motor recovery achieved with novel BMI technology remains modest. "Motor rehabilitation based on BMIs is still in a preliminary stage, and further improvements are required to boost its efficacy," comments Dr. Ramos-Murguialday. "Invasive and hybrid approaches are promising and might set the stage for the next generation of stroke rehabilitation therapies."
"As we stand at this crossroad, the direction we need to take is becoming clearer," concludes Dr. Harvey. "Be open to new approaches to care beyond task-oriented training. Utilize new technology to extend therapeutic approaches beyond the mat, treadmill, and hi-lo table. Critique new research based on whether it suggests just another form of compensatory training versus expansion of functional capacity. Consider the incorporation of biomarkers into clinical research and bedside care. We know where we need to go. I hope we can successfully negotiate the pathways that push beyond merely 70% recovery."
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Disclaimer: AAAS and EurekAlert! are not responsible for the accuracy of news releases posted to EurekAlert! by contributing institutions or for the use of any information through the EurekAlert system. | |
Concord Art presents Clay Has Its Say: Narrative Ceramics, curated by David Duddy. This special exhibition features approximately 50 works by a group of talented and inventive contemporary artists who bring their vision and skill to the creation of stories and worlds for viewers to interpret and reflect upon. It will be on view through December 13, 2020.
Historically, depictions of stories on ceramic objects were most often recognizable narratives for the cultures that produced them—myths, battles, religious rituals, or scenes from everyday life. More recent artistic movements have offered alternatives to that practice by leaving narratives open-ended and subject to the viewer’s own perspective.
Says curator David Duddy, “These ceramic works invite us to imagine stories from the clues offered—from the ambiguous details presented by these artists and the objects they create. I hope that each artist’s works will speak to viewers in new and stimulating ways, sparking dialogs and stories yet to be made and told.”
The ceramics in Clay Has Its Say span a variety of materials, techniques, and sizes, but all of them tell a story—or invite the viewer to craft a story themselves. Some works depict a narrative through the imagery or patterns painted or etched onto the surface: plates featuring figurative scenes, vases with characters circling the perimeter, or entire surfaces covered with words. Others hint at a story through the sculpture itself: fossil-like bones of fantastic animals, pairs of altered hybrid creatures, or bizarre shapes of shells and unknown forms.
Artists featured in the exhibition include Marilyn Andrews (MA), Bruce Barry (MA), Ashley Benton (GA), Ariel Bowman (TX), Tim Christensen (ME), Angela Cunningham (MA), Holly Curcio (MA), Rebecca Doughty (MA), Lulu Fichter (NH), Kathleen O’Hara (MA), Claudia Olds Goldie (MA), Andrea Olmstead (MA), Frank and Francine Ozereko (MA), and Jeanée Redmond (MA).
Concord Center for the Visual Arts was founded a century ago by Elizabeth Wentworth Roberts, an American Impressionist and philanthropist whose mission—to promote and advance the visual arts and artists, and to sustain our cultural community—still stands today. With more than 850 members, Concord Art provides a place for contemporary art exhibitions, art education and relevant programming for everyone. | https://www.ceramicsnow.org/2020/11/13/clay-has-its-say-narrative-ceramics-is-on-view-at-concord-art-concord-ma/ |
A business plan sets out in writing the business, its objectives, strategies, the market it operates in and financial forecasts. For many businesses, they are only prepared when requested, for example by a lending institution or investors. However, the business plan should also be used as a tool to focus the mind of senior management on the goals of the organisation and as a roadmap to success.
By having a formal business plan in place progress can be measured, pitfalls identified and targets set.
Considerations
A business plan should be flexible, reviewed at regular intervals and updated as and when required.
A good plan will address more than just the financial forecasts. Matters to be incorporated in the plan could include:
The nature of the business
The mission statement & ethos
The current business structure
Key personnel
The target market – size, sustainability, strategy and access
The offering - products, services, customers and marketing
Major suppliers and trusted advisors
Operational requirements
Key Performance Indicators – Quantitative and Qualitative
Risks and how they are minimised or addressed
Financial projections
Financing requirements to achieve the stated goals
How we can help
HLB Sheehan Quinn have decades of experience in assisting clients with their business plans. We work with our clients to ensure that the plan reflects their strategies and goals, addressing the key considerations whilst being flexible enough for them to review and update as necessary.
Whether the business plan is to be used to support lending from a financial institution, to attract investment, to be a tool of management or all three, we can help to ensure that it has the right focus and ticks all the boxes.
If you want help to formalise your business plan and have a flexible, structured, ambitious but achievable document to focus the efforts of your organisation then contact our team today and we’d be delighted to assist. | https://www.hlbsheehanquinn.ie/services/business-plans/ |
A pinched nerve (nerve entrapment) in or near the elbow can cause elbow pain, numbness, tingling, or weakness of the arm, wrist, or hand. The nerve that most commonly gets pinched in or near the elbow is the ulnar nerve. It is located in the elbow area, on the little finger side when the palm is facing up. Less often, the median or the posterior interosseous nerve, a branch of the radial nerve next to the elbow area on the thumb side when the palm is facing up, may get pinched.
Examples of nerve entrapment syndromes that affect the elbow include:
Treatment for these nerve entrapment syndromes includes rest, stretching, taking anti-inflammatory medicines, and occasionally surgery.
ByHealthwise Staff
Primary Medical Reviewer William H. Blahd, Jr., MD, FACEP - Emergency Medicine
Adam Husney, MD - Family Medicine
Kathleen Romito, MD - Family Medicine
Specialist Medical Reviewer H. Michael O'Connor, MD, MMEd, FRCPC - Emergency Medicine
Current as ofNovember 20, 2017
Next Section:Related Information
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Current as of: November 20, 2017
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© 1995-2018 Healthwise, Incorporated. Healthwise, Healthwise for every health decision, and the Healthwise logo are trademarks of Healthwise, Incorporated. | https://www.healthlinkbc.ca/health-topics/aa136625 |
Willoughby Town Centre is situated in the vibrant and growing Township of Langley. Willoughby has a variety of existing restaurants, grocery stores, and retail. The area was targeted by the municipality as fundamental to local development, and the Township of Langley’s urban planners were firm in their intention to make it a lively, walkable area.
After realizing the potential of other developments in Willoughby Town Centre, Qualico and Chow & Li wanted to create a mixed-use project to serve as a programmatic pillar of Langley’s suburban centre, contributing to the neighbourhood’s development.
This mixed-use development fronts Willoughby Town Centre Drive, the main access road and ‘gateway’ to the centre of town. The project needed to provide a diversity of functions to address the corresponding local building codes.
The high-profile location also provided the opportunity to set a shining example of how a fundamental piece of design can escalate future developments in the area.
Given the pivotal gateway location, our team was required to be creative and nimble in their architectural response, while empowering the building codes themselves to make significant contributions to the design of the project.
Our team built a strong relationship with our development partners, Qualico and Chow & Li. We also maintained constant communication with the client to make sure their needs were being realized. In turn, this built trust in our collaboration.
This development promotes the Township of Langley’s urban planning goals by enhancing pedestrian walkability and local vibrancy and provides a prominent gateway to the community. The project features 91 residential units, 19 commercial retail units, a streetscape-fronting restaurant, and a two-storey office space.
Our Commercial & Mixed-Use team designed the building front to feature many material changes, volumes that push-outward and pull-inward, and cantilevered spaces. As a result, the building front varies in depth rather than being a single surface and presents a colourful face with an abundance of material choices.
In further support of urban vibrancy, our team placed the residential and office parking below grade. This move reduced vehicle presence and enhanced the neighbourhood’s walkability. Additionally, the team worked attentively to retain all of the street’s mature trees, existing sidewalks, and an existing water feature.
As one contractor told our project team, he was proud to be a part of a project that was a colourful, active, and vibrant participant in an important area of Langley’s community.
Ultimately, Willoughby Town Centre has strengthened and expanded the diversity of the community by adding a modern and contemporary piece of urban design, one that can be used as a model for future developments in the area. Since its completion, Willoughby Town Centre has sold the majority of its residential units, identified nearly all of its commercial tenants, and is poised to provide gateway vibrancy to Langley’s town centre for years to come.
We have over 35 years of experience designing mixed-use projects across Canada and internationally. Get in touch to discuss how we can help you with your mixed-use project. | https://kasian.com/project/willoughby-town-centre/ |
These payoffs are graphed in Figure 6.10, as a function of the expected stock price.
Unlike the prior two cases, the option to abandon takes on the characteristics of a put option.
Assume that Disney is considering taking a 25-year project which requires an initial investment of $ 250 million in a real estate partnership to develop time-share properties with a South Florida real estate developer, and where the present value of expected cash flows is $ 254 million. While the net present value of $ 4 million is small, assume that Disney has the option to abandon this project anytime by selling its share back to the developer in the next 5 years for $ 150 million. A simulation of the cash flows on this time-share investment yields a standard deviation in the present value of the cash flows from being in the partnership of 20%.
Call Value = 254 exp(004)(5) (0.9194) -150 (exp(-004)(5) (0.8300) = $ 89.27 million Put Value= $ 89.27 - 254 exp(004)(5) +150 (exp(-004)(5) = $ 4.13 million The value of this abandonment option has to be added on to the net present value of the project of $ 4 million, yielding a total net present value with the abandonment option of $ 8.13 million. | https://www.ajjacobson.us/finance-analysis/the-option-to-abandon-a-project.html |
Abdulmejid I was the Ottoman Empire's 31st Sultan, succeeding Mahmud II on 2 July 1839. The development of nationalist groups within the empire's borders was significant during his rule. Abdulmejid intended to promote Ottomanism among separatist subject countries and quell increasing nationalism movements within the kingdom. Still, his attempts failed despite new laws and reforms to better integrate non-Muslims and non-Turks into Ottoman culture. He attempted to form alliances with Western European great countries like the United Kingdom and France, who fought with the Ottoman Empire in the Crimean War against Russia. The Ottoman Empire was officially admitted to the European family of states at the Congress of Paris on 30 30 March 1856. Abdulmejid's most tremendous success was the declaration and implementation of his father's Tanzimat reforms, which essentially began the Ottoman Empire's modernisation in 1839. The March of Abdulmejid, one of the Ottoman Empire's Imperial anthems, was named after him to recognise his achievements.
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Abdulmejid I
31st Sultan of the Ottoman Empire
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Sovereignty
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2 July 1839 – 25 June 1861
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Ancestor
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Mahmud II
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Inheritor
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Abdulaziz
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Grand Viziers
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Born
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25 April 1823
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Died
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25 June 1861 (aged 38)
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Committal
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Yavuz Selim Mosque, Fatih, Istanbul
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Consorts
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Dynasty
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Ottoman
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Father
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Mahmud II
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Mother
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Bezmiâlem Sultan
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Religion
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Sunni Islam
Abdulmejid was born in Istanbul on 25 25 April 1823, in either the Beşiktaş Sahil Palace or the Topkapı Palace. Valide Sultan Bezmiâlem, originally called Suzi (1807 to 1853), his father's first wife in 1839, was possibly a Circassian or Georgian slave. Abdulmejid was the first Sultan to obtain a European education and speak fluent French. He, like his successor Abdülaziz, was a fan of literature and classical music. Like his father, Mahmud II, he was a reformer and benefited from the backing of progressive viziers such as Mustafa Reşit Pasha, Mehmet Emin Ali Paşa, and Fuad Pasha. Abdulmejid was also the first Sultan to listen to the public's concerns without any middlemen on special reception days, which were generally held every Friday. In addition, Abdulmejid took a tour of the empire's regions to examine how the Tanzimat reforms were being implemented firsthand. In 1844, he visited İzmit, Mudanya, Bursa, Gallipoli, Çanakkale, Lemnos, Lesbos, and Chios, and in 1846, he visited the Balkan regions.
The Ottoman Empire's affairs were in a precarious situation when Abdulmejid ascended to the throne on 2 2 July 1839, at the age of sixteen, when he was young and inexperienced. The word reached Istanbul that the empire's army had just been beaten at Nizip by the army of the rebel Egyptian viceroy, Muhammad Ali after his father died at the commencement of the Egyptian–Ottoman War. At the same time, the empire's navy was on its route to Alexandria, where its commander, Ahmed Fevzi Pasha, turned it over to Muhammad Ali under the guise that the young Sultan's counsellors had sided with Russia. Muhammad Ali was forced to come to terms with the Ottoman Empire after European countries intervened during the Oriental Crisis of 1840. The Ottoman Empire was rescued from future invasions while its lands in Syria, Lebanon, and Palestine were recovered. At the London Convention, the terms were finalised (1840). The Sultan and vükela lavished lavish hospitality to Egyptian governor Mehmed Ali Pasha, who arrived in Istanbul on the monarch's official invitation on 19 19 July 1846. So much so that in 1845, the previous vizier constructed the Galata bridge to allow him to drive between Beşiktaş Palace and Babıali. Abdulmejid promptly carried out the changes to which Mahmud II had dedicated himself, as per his father's specific orders. The Hatt-ı Șerif of Gülhane, also known as Tanzimat Fermanı, was declared in November 1839, consolidating and implementing these changes. The decree was reinforced by similar legislation known as the Hatt-ı Hümayun, published in February 1856 at the end of the Crimean War. These enactments guaranteed that all classes of the Sultan's subjects' lives and property would be safeguarded, that taxes would be imposed equitably, and justice would be administered impartially, and that everyone would enjoy complete religious liberty and equal civil rights. The program was greeted with fierce hostility from the Muslim ruling classes and the ulema, or religious authorities, and was only partially implemented, particularly in the empire's outlying regions. As a result, more than one plot against the Sultan's life was devised. Abdulmejid advocated for the following policies:
Another significant development was the outlawing of the turban in favour of the fez for the first time during Abdulmejid's rule. The Court embraced European trends as well. (The same Republican National Assembly overthrew the sultanate and declared the Turkish Republic in 1923 also banned the fez in 1925.) According to Cyrus Hamlin's memoirs, Sultan Abdulmejid awarded Samuel Morse an Order of Glory for his contributions to the telegraph, which he got after personally testing Morse's new invention. After the failed Hungarian insurrection in 1849, Kossuth and others took sanctuary in Turkey, where Austria and Russia pressured the Sultan to hand them up, but he resisted. He would also not let the assassins who plotted against his life to be executed. According to the 1911 Encyclopædia Britannica, "He had the appearance of a kind and honourable man, although one who was weak and easily led. However, his excessive spending, especially at the end of his life, must be weighed against this." He established the Ottoman lira in 1844 and the Order of the Medjidie in 1851. During the Crimean War, the Ottoman Empire received the first of its foreign loans on 25 25 August, 1854. Those following this significant foreign loan in 1855, 1858, and 1860, all of which ended in default, alienating European sympathies from the Ottoman Empire and partly leading to Abdulmejid's brother Abdülaziz's dethronement and death.
On the one hand, financial difficulties, and on the other, resentment generated by the extensive privileges granted to non-Muslim citizens, threw the kingdom into disarray once more. In 1857, there were incidents in Jeddah, and in 1858, there were incidents in Montenegro. The central European countries have taken advantage of the situation to act in their interests. Amid this crisis, the Ottoman leaders panicked and adopted a strategy that granted them every demand. The fact that Abdulmejid was powerless to avert this circumstance further added to the Tanzimat Edict's unpopularity. To prevent the European powers from acting as a guardian, the opponents planned to depose Abdulmejid and install Abdulaziz on the throne. This attempted uprising, known in history as the Kuleli Foundation, was put down before it began on 14 14 September 1859, thanks to a notification. Meanwhile, the treasury's financial situation deteriorated, and foreign debts, which had been obtained under duress to fund war expenditures, became a burden. All of the obligations owed to Beyoğlu customers totalled more than eighty million gold liras. Foreign merchants and bankers took some of the debt securities and hostages. His achievements in foreign affairs were not as remarkable as his home achievements. His reign began with his armies being defeated by the Viceroy of Egypt and the subsequent signing of the London Convention (1840), which spared his empire from further disgrace. The Ottomans were victorious in the Crimean War and gained signatories to the Treaty of Paris (1856). In 1861, the Concert of Europe obliged him to give up Lebanon after building his foundation in the Balkans failed in Bosnia and Montenegro. At the rituals outside, he stressed his adherence to the ceremonial traditions set by his predecessors, yet he made significant changes in the palace's life. For example, he entirely abandoned the Topkapı Palace, a centre of worship for the Ottoman family for four centuries. Even middle-class families were influenced by the traditions of the British, French, and Italian troops, commanders, and diplomats that arrived in Istanbul during the Crimean War (1853-1856). He oversaw the renovations to the Hagia Sophia mosque and the construction of the Dolmabahçe Palace between 1847 and 1849. In Istanbul, he also established the first French Theatre. During Abdulmecid's rule, many reconstruction projects were undertaken. Some of the borrowed funds were used to construct palaces and mansions. The notable architectural works of the era are Dolmabahçe Palace (1853), Beykoz Pavilion (1855), Küçüksu Pavilion (1857), Küçük Mecidiye Mosque (1849), and Teșvikiye Mosque (1854). The new Galata Bridge was placed into operation on the same date as Bezmiâlem Sultan's Gureba Hospital (1845-1846) during this period. A large number of fountains, mosques, lodges and other social facilities were either renovated or reconstructed.
Abdulmejid died of TB (like his father) on 25 25 June 1861, in Istanbul, at 38 and was buried at Yavuz Selim Mosque. His younger half-brother Sultan Abdülaziz, son of Pertevniyal Sultan, succeeded him. Abdulmejid had several concubines and just one legitimate wife, Perestu Kadın when he died.
In Jason Goodwin's 2008 novel The Bellini Card, a fictitious version of Abdulmejid I appears. | http://www.alonereaders.com/article/details/134/abdulmejid-i-31st-sultan-of-the-ottoman-empire |
This qualification does not replace any other qualification and is not replaced by any other qualification.
|PURPOSE AND RATIONALE OF THE QUALIFICATION|
|Why do we need a master craftsmanship qualification?
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South Africa has a critical shortage of skilled practitioners in most technical occupations. A strong cadre of master artisans and craftspeople would have a significant impact on the ability of South African industry to build on the improved financial environment and create sustainable economic growth.
Many of those who took on this role originally came from overseas in the 1960's and 1970's. This generation of technically proficient people has by now either moved on to higher positions, retired, been retrenched, or is approaching retirement age. The reduction in the number of apprentices, from approximately 25 000 per annum in 1985 to approximately 5000 in 2002, has substantially reduced the pool of skilled people. Those apprentices have also been further reduced by emigration as the result of economic conditions locally and active recruiting by overseas countries.
The decline in the number of people taking up practical and technical occupations has meant that many such functions are performed by superficially trained workers and those gravitating to the work through redeployment and retrenchment. This has resulted in a significant reduction in the quality of workmanship and levels of service. Large organisations report that up to 70% of the work being done during annual maintenance shutdowns has to be redone (so called re-work). Some component manufacturers, for instance, have found it easier to order their tooling from Portugal:
The master craftsmanship series of qualifications could be used to improve those very aspects (quality, quick delivery and cost-effectiveness) to create sustainable economic activity. The qualifications would also give past and current artisans and craftspeople a way of having their skills recognized and targeted to the needs of the economy. For industry, these skills would fill the gap between engineering design and shop floor operations; and between new systems and technological concepts, and practical implementation.
The National Training Board investigation into the apprenticeship system in 1986 revealed that the category of persons most likely to succeed in a new business start-up were artisans and craftspeople. The decline in the number of artisans and craftspeople emerging from the training system has had a significant impact on the number of people who could successfully start up new businesses to provide general or specialised practical services to the industry or the public. This in turn has had a negative impact on economic growth and ultimately on employment opportunities. A further benefit of the master craftsmanship projects would be to assist new business start-ups to have a greater chance of success.
Experienced artisans and craftspeople also played a role in developing the next generation of people in the occupation. The apprenticeship system in its strongest form was built on the transfer of knowledge and expertise from the artisans and craftspeople to the apprentices. A further function of master craftsmanship is to transfer skills, knowledge and values. This role will support the quality assurance of apprenticeship and learnership systems, ensuring the development of people with high quality and relevant skills, knowledge and values.
This and related qualifications will act as a framework for providers, assessors and learners to plan, implement and measure the outcomes of suitable learning programmes, or the recognition of prior learning, in this new discipline.
The specific purpose of this qualification, the second in the series, represents the skills, knowledge and understanding required by competent practitioners to:
This qualification is conceptualised as a generic qualification that can be used for a wide range of trades and technical and service occupations. However, current SAQA regulations do not permit the registration of generic qualifications. This qualification will, therefore, initially be focused only on electrical trades and occupations. This qualification can be obtained in the context of a variety of electrical, maintenance, installation and manufacturing processes.
This qualification together with the National Certificate and the National First Degree in Master Craftsmanship are conceptualised as an integrated set of building blocks. The credits for the National Certificate qualification are required to fulfil all the requirements for this National Diploma. The credits for this National Diploma will, in turn, be required to fulfil the requirements of the National First Degree in Master Craftsmanship.
Rationale for the qualification:
The concept of master craftsmanship represents a career path for people involved in practical and technical occupations. While the development of the Master Craftsmanship qualifications will initially use the traditional trades as a basis, the career path is equally appropriate for a range of other occupations, both for traditional occupations as well as for new occupations emerging as the result of changing technology.
In South Africa there was previously no formal career path for artisans and craftspeople once they had acquired the initial trade qualification. Either they:
The proposed series of master craftsmanship qualifications combines aspects of these career options into a fully-fledged qualification pathway, allowing master craftspeople to perform a variety of roles within industry or in the economy.
The primary roles of master artisans or craftspeople are:
|LEARNING ASSUMED TO BE IN PLACE AND RECOGNITION OF PRIOR LEARNING|
|The credits and the related unit standards assume that the learner is either formally qualified in an NQF Level 5 Certificate in Master Craftsmanship or has extensive experience in the installation, repair, maintenance or manufacture of electrical equipment, components and control systems and has some experience with instrumentation. If a learner does not have such experience or qualifications, the learning time will be increased.
|
Recognition of prior learning:
This qualification may be obtained through the process of RPL. The learner should be thoroughly briefed prior to the assessment and support should be provided to assist the learner in the process of developing a portfolio. While this is primarily a work-based qualification, evidence from other areas of endeavour may be introduced if pertinent to any of the exit level outcomes
|RECOGNISE PREVIOUS LEARNING?|
|Y|
|QUALIFICATION RULES|
|N/A|
|EXIT LEVEL OUTCOMES|
|The exit level outcomes for this qualification reflect a combination of specific outcomes and critical cross-field education and training outcomes. The way in which the critical outcomes have been advanced through the learning required for this qualification is embedded in the way in which the unit standards have been constructed. Critical outcomes form the basis of acquiring the skills and knowledge and values. The application of these in a specific context results in the achievement of specific outcomes. The integration of specific outcomes from a variety of unit standards results in the ability to achieve the exit level outcomes.
|
1. Provide products and services which meet or exceed customer expectations
2. Develop and achieve key performance indicators for the section or the contractors
3. Resolve disputes, conflicts and grievances in the workplace
4. Maintain and improve systems, procedures and processes to enhance the quality and safety of work processes and practices
5. Facilitate and assess learning in the workplace
|ASSOCIATED ASSESSMENT CRITERIA|
|1.
|
2.
3.
4.
5.
Integrated Assessment:
The integrated assessment must be based on a summative assessment guide. The guide must spell out how the assessor will assess different aspects of the performance and will include:
The learner may choose in which language he/she wants to be assessed. This should be established as part of a process of preparing the learner for assessment and familiarising the learner with the approach being taken.
While this is primarily a workplace-based qualification, evidence from other areas of endeavour may be presented if pertinent to any of the exit level outcomes.
The assessment process should cover the explicit tasks required for the qualification as well as the understanding of the concepts and principles that underpin the activities. The assessment process should also establish how the learning process has advanced the critical outcomes.
Assessors should also evaluate evidence that the learner has been performing consistently over a period of time.
|INTERNATIONAL COMPARABILITY|
|The best-known master qualifications are those in German-speaking countries. The master qualifications are a requirement within these countries for:
|
The German system is however different and there is no qualification framework like the NQF. The master qualification is a single qualification and can only be acquired based on the following:
The master qualifications in other countries such as the United Kingdom and New Zealand focus primarily on advanced technical skills and knowledge.
The development of these qualifications was largely based on the contextualisation of the German qualifications in South Africa. German-qualified master artisans who operate in both small and large companies in South Africa assisted in the process to ensure the qualifications would have the same value as those in German-speaking countries.
|ARTICULATION OPTIONS|
|This qualification has been designed and structured so that qualifying learners can move from one context to another. Employers or institutions should be able to evaluate the outcomes of this qualification against the needs of their context and structure top-up learning appropriately. Equally, holders of other qualifications may be evaluated against this qualification for the purpose of RPL.
|
Overview of the proposed qualifications pathway and articulation possibilities:
Level----Other Specialisations
7---Engineer-Quality assurance or Education, Training and Development, Technical sales and marketing, General management
6--First Degree Master Craftsmanship-Engineering technologist or equivalent
5--Nat Diploma Master Craftsmanship-Engineering technician or equivalent
5--Nat Certificate Master Craftsmanship
4-NQF technical or supervisory qualification-NQF 4 trade
3-NQF 3 trade
Note: the actual articulation will be determined by the institutional and professional entry requirements. The articulation to engineering qualifications is being explored with the Engineering SGB but has not yet been finalised.
|MODERATION OPTIONS|
|Moderators for the qualification should be qualified and accredited with an appropriate ETQA.
|
To assure the quality of the assessment process, the moderation should cover the following:
Moderators should be qualified assessors in their own right.
|CRITERIA FOR THE REGISTRATION OF ASSESSORS|
|The following criteria should be applied by the relevant ETQA:
|
1. Appropriate qualification in the field of electrical engineering, maintenance or manufacture with a minimum of 2 years' experience in a small business environment. The subject matter expertise of the assessor can be established by recognition of prior learning.
2. Appropriate experience and understanding of assessment theory, processes and practices.
3. Good interpersonal skills and ability to balance the conflicting requirements of:
4. Registration as an assessor with a relevant ETQA.
5. Any other criteria required by a relevant ETQA.
Since this is a new field, it may be some time before there are sufficient qualified assessors. The relevant ETQAs should allow interim arrangements to be made. It is envisaged that holders of this and related qualifications will eventually form a professional association. The members of this association will then support the quality assurance and assessment processes. Assessors would then be required to be registered members of this association.
|
|
REREGISTRATION HISTORY
|As per the SAQA Board decision/s at that time, this qualification was Reregistered in 2012; 2015.|
|NOTES|
|N/A|
|UNIT STANDARDS:|
|ID||UNIT STANDARD TITLE||PRE-2009 NQF LEVEL||NQF LEVEL||CREDITS|
|Core||13942||Demonstrate a basic understanding of the role of a business strategy in managing a small business or a business unit||Level 4||NQF Level 04||5|
|Core||116783||Analyse trends and implement continuous improvements||Level 5||Level TBA: Pre-2009 was L5||10|
|Core||15237||Build teams to meet set goals and objectives||Level 5||Level TBA: Pre-2009 was L5||3|
|Core||116779||Develop and implement specifications to achieve the desired product or service||Level 5||Level TBA: Pre-2009 was L5||10|
|Core||116781||Develop and implement sustainable processes and procedures||Level 5||Level TBA: Pre-2009 was L5||10|
|Core||10043||Develop, implement and manage a project/activity plan||Level 5||Level TBA: Pre-2009 was L5||5|
|Core||15224||Empower team members through recognising strengths, encouraging participation in decision making and delegating tasks||Level 5||Level TBA: Pre-2009 was L5||4|
|Core||14214||Evaluate and improve the project team`s performance||Level 5||Level TBA: Pre-2009 was L5||8|
|Core||116785||Manage requirements related to quality and other standards||Level 5||Level TBA: Pre-2009 was L5||10|
|Core||116787||Plan, monitor and control the financial resources for a small company or business unit||Level 5||Level TBA: Pre-2009 was L5||10|
|Fundamental||114600||Apply innovative thinking to the development of a small business||Level 4||NQF Level 04||4|
|Fundamental||15234||Apply efficient time management to the work of a department/division/section||Level 5||Level TBA: Pre-2009 was L5||4|
|Fundamental||15231||Create and use a range of resources to effectively manage teams, sections, departments or divisions||Level 5||Level TBA: Pre-2009 was L5||4|
|Fundamental||15238||Devise and apply strategies to establish and maintain relationships||Level 5||Level TBA: Pre-2009 was L5||3|
|Fundamental||15215||Identify and interpret Best Practice guidelines, and plan for and implement Best Practice within the team, department or division||Level 5||Level TBA: Pre-2009 was L5||4|
|Elective||114884||Co-ordinate the improvement of productivity within a functional unit||Level 4||NQF Level 04||8|
|Elective||115753||Conduct outcomes-based assessment||Level 5||Level TBA: Pre-2009 was L5||15|
|Elective||14803||Facilitate Technical/Practical skills learning to adult learners||Level 5||Level TBA: Pre-2009 was L5||20|
|Elective||15229||Implement codes of conduct in the team, department or division||Level 5||Level TBA: Pre-2009 was L5||3|
|Elective||114716||Manage installation and maintenance contractors||Level 5||Level TBA: Pre-2009 was L5||16|
|LEARNING PROGRAMMES RECORDED AGAINST THIS QUALIFICATION:|
|NONE|
|PROVIDERS CURRENTLY ACCREDITED TO OFFER THIS QUALIFICATION:|
|This information shows the current accreditations (i.e. those not past their accreditation end dates), and is the most complete record available to SAQA as of today. Some Primary or Delegated Quality Assurance Functionaries have a lag in their recording systems for provider accreditation, in turn leading to a lag in notifying SAQA of all the providers that they have accredited to offer qualifications and unit standards, as well as any extensions to accreditation end dates. The relevant Primary or Delegated Quality Assurance Functionary should be notified if a record appears to be missing from here. | http://allqs.saqa.org.za/showQualification.php?id=49059 |
1. Where does Pip stay when he reaches his village?
2. Does Pip go to see Joe, Biddy, and his sister while he is in town?
3. Who rides on the coach with Pip?
4. What does Pip overhear the convicts discussing?
5. When Pip arrives in his village, who does he find has taken all the credit for his good fortune?
6. Who admits Pip into Miss Havisham’s gate and is now working for her?
7. How has Estella changed since the last time Pip saw her?
8. What does Miss Havisham tell Pip to do to Estella?
9. How does Pip recognize Estella when he first arrives?
10. Who does Pip envision restoring Satis House to its former glory?
Answers
1. Pip decides to stay at the Blue Boar. | https://www.enotes.com/topics/great-expectations/quiz/part-2-chapters-28-29-questions-answers |
The Centre for Geohazard Observations (CGO) was tasked to install several stations all over the region to continuously record the sea level. This is a relatively new method of recording sea level rise through the use of the Global Navigation Satellite System (GNSS), instead of traditional tide gauges that we have normally engaged. By analysing the GNSS data that was recorded over a long period of time, researchers will be able to determine the sea level in the area where direct-and-reflected GNSS signals are recorded. To achieve optimum usability of the data, the stations are installed near the coastline, where the reflected GNSS signal from the water can be received by the antenna.
In doing so, EOS has collaborated with local research institutions in every geographic location where the GNSS stations are installed. Our first station was installed in Taiwan in 2019, which was achieved with the collaboration of the Institute of Earth Science – Academia Sinica, Taiwan.
Currently, we are focusing on the installation of these stations in Singapore. We aim to install two stations at this stage with the possibility to increase the number of stations in the future. The data from these stations can be used for multiple earth science and climate research projects, as well as land survey control points for government agencies. Hence, with these multiple capabilities, GNSS technology has offered significant and relatively cheap solutions to environmental data acquisition.
In Singapore, we collaborated with local government agencies, and for the installation of GNSS stations, we are collaborating with Singapore Land Authority (SLA). Both EOS and SLA will benefit from the data collected from these stations. While EOS researchers will use the data to study sea-level change, climate, among other research areas, the SLA will be able to incorporate the data from these stations into the existing GPS network and will densify the GPS network.
As the goal is to get the coastal GNSS stations around the South China Sea, we are working closely with local collaborators in Indonesia, Vietnam, and other Southeast Asian countries to install these stations in their land. | https://www.earthobservatory.sg/facilities/field-installations/gnss-stations |
NOT FOR PUBLICATION FILED
UNITED STATES COURT OF APPEALS MAR 23 2020
MOLLY C. DWYER, CLERK
U.S. COURT OF APPEALS
FOR THE NINTH CIRCUIT
PHILIP BOBBITT, individually and on No. 18-17250
behalf of all others similarly situated,
D.C. No. 4:09-cv-00629-FRZ
Plaintiff-Appellant,
LANCE LABER, MEMORANDUM*
Intervenor-Plaintiff-
Appellant,
and
JOHN J. SAMPSON; et al.,
Plaintiffs,
v.
MILBERG LLP; et al.,
Defendants-Appellees.
Appeal from the United States District Court
for the District of Arizona
Frank R. Zapata, District Judge, Presiding
Argued and Submitted March 4, 2020
Phoenix, Arizona
Before: CLIFTON, OWENS, and BENNETT, Circuit Judges.
*
This disposition is not appropriate for publication and is not precedent
except as provided by Ninth Circuit Rule 36-3.
Philip Bobbitt appeals the district court’s order denying his Federal Rule of
Civil Procedure (“Rule”) 60(b)(6) motion based on a change in the law governing
the appealability of class certification denials. We have jurisdiction under 28
U.S.C. § 1291 to review the Rule 60(b) denial. See United States v. Sierra Pac.
Indus., Inc., 862 F.3d 1157, 1166 (9th Cir. 2017). We reverse the district court’s
denial and remand with directions to grant the Rule 60(b)(6) motion and for further
proceedings.
We review “the denial of a motion for relief from judgment under Rule
60(b) for an abuse of discretion.” Henson v. Fidelity Nat’l Fin., Inc., 943 F.3d 434,
443 (9th Cir. 2019). While this case was pending on appeal, we decided Henson.
In Henson, we reiterated that in deciding whether to grant a Rule 60(b) motion
based on a change in the law, a court must “intensively balance numerous factors.”
Id. at 444 (quoting Phelps v. Alameida, 569 F.3d 1120, 1133 (9th Cir. 2009)). We
also clarified the factors a court should consider in deciding whether to grant Rule
60(b) relief under analogous circumstances. Id. at 446–55. Here, the district court
abused its discretion because it failed to conduct the required intensive balancing
based on the facts. We note, however, that the district court did not have the
benefit of Henson when it denied the motion. Because the facts relevant to the
merits of the Rule 60(b) motion are in the record, we exercise our discretion and
decide the merits of the motion. See Phelps, 569 F.3d at 1134–35.
2
The Henson factors weigh in favor of granting Rule 60(b) relief. There was
a change in the law because before Microsoft Corp. v. Baker, 137 S. Ct. 1702
(2017), Ninth Circuit case law established that the court could review interlocutory
orders after a plaintiff voluntarily dismissed his claims with prejudice. See
Omstead v. Dell, Inc., 594 F.3d 1081, 1085 (9th Cir. 2010); see also Henson, 943
F.3d at 447 (citing Omstead to support its statement that “Plaintiffs reasonably
relied on well-established Ninth Circuit law”). Though Bobbitt knew that there
was a circuit split on the issue, he reasonably relied on Ninth Circuit precedent,
and there is no indication that he should have known that the law would change.
We find that the circumstances relevant to the change-in-the-law factor are much
like those in Henson, and therefore like in Henson, we find that this factor is
neutral or slightly favors granting Rule 60(b) relief.
We next consider Bobbitt’s diligence in seeking to avoid or mitigate the risk
of an unfavorable change in the law. Other than petitioning this court to review the
class certification denial under Rule 23(f) before voluntarily dismissing his claims,
Bobbitt did nothing else to mitigate the risk that the case would be over if Lance
Laber’s appeal were dismissed for lack of jurisdiction. Thus, this factor weighs
against granting relief.
As for Milberg LLP’s (“Milberg”) reliance interest in the finality of the case,
the record does not show, nor does Milberg show, that it changed its legal position
3
in reliance on the district court’s 2013 judgment. Indeed, after the district court’s
2013 judgment, we held in the now-vacated opinion Bobbitt v. Milberg LLP, 801
F.3d 1066 (9th Cir. 2015) (“Milberg I”), that the district court had erroneously
denied class certification, and we remanded “for further proceedings.” 801 F.3d at
1072. And because Milberg sought certiorari challenging Milberg I, the case
remained pending until we issued the mandate dismissing Laber’s appeal. On the
same day we issued the mandate, Bobbitt asked the district court to reinstate his
claims. Thus, Milberg could not have reasonably believed that the case was over
after the 2013 judgment. In Henson, we found that this factor weighed heavily in
favor of granting relief because the defendant showed no reliance interest in the
finality of the judgment. 943 F.3d at 451. The same is true here, and we therefore
find that this factor heavily favors granting relief.
The delay factor “examines the delay between the finality of the judgment
and the motion for Rule 60(b)(6) relief.” Id. at 451–52 (internal quotation marks
omitted) (quoting Phelps, 569 F.3d at 1138). In Henson, we clarified that the delay
is measured from the date when the appeal from the dismissal became final. Id. at
452. Laber’s appeal was finally decided when this court issued the mandate on
October 11, 2018. That same day, Bobbitt sought relief by moving to reinstate his
individual claims in the district court. The district court denied the motion four
days later, and Bobbitt moved for Rule 60(b) relief 24 days after that denial.
4
Because the delay was relatively short, we find that this factor favors granting Rule
60(b) relief.
We next examine “the closeness of the relationship between the decision
resulting in the original judgment and the subsequent decision that represents a
change in the law.” Id. (quoting Jones v. Ryan, 733 F.3d 825, 840 (9th Cir. 2013)).
In Henson, we found that this factor favored relief because there was a close
connection as “the voluntary dismissal was explicitly predicated on the law that
Microsoft changed.” Id. Although Bobbitt stated in his motion for voluntary
dismissal that he wanted to dismiss his claims because it was “not economically
feasible” for him to litigate his individual claims, he also explained that “[i]f the
Court grants a dismissal, a member of the putative class is prepared to seek
intervention for the limited purpose of appealing the class-certification denial.”
The connection here may not be as close as the connection in Henson because
Bobbitt’s decision to voluntarily dismiss his claims was not solely predicated on
the law that Microsoft changed. However, Bobbitt’s decision was predicated, in
part, on the law that Microsoft changed, and we thus find that this factor slightly
favors Rule 60(b) relief.
Of the two additional factors identified in Henson, we find that one applies
here—“the importance of heeding the intent of the rulings of federal appellate
5
courts.”1 Id. at 453. In analyzing this factor, the Henson court noted that treating
the voluntary dismissal as a final, irrevocable judgment would put plaintiffs in a
“catch twenty-two because, under Microsoft, the dismissal was not a final
judgment from which [plaintiffs] could appeal the denial of class certification, but
in the district court, the voluntary dismissal was treated as having finally ended the
case.” Id. at 454. The court reasoned that granting Rule 60(b) relief would avoid
creating this “contradiction,” and thus this was another consideration that weighed
in favor of granting relief. Id. The same is true here, and thus we find this
reasoning weighs in favor of granting relief.
Finally, we believe it is also appropriate to consider the fact that Bobbitt
voluntarily dismissed his claims regardless of the outcome of Laber’s appeal. This
fact weighs against granting relief because Bobbitt had no expectation that he
would be able to revive his individual claims. Though this consideration and the
diligence factor weigh against granting relief, all other factors weigh in favor of
granting relief (except for the change-in-the-law factor which is neutral or slightly
weighs in favor of granting relief). Thus, on balance, we hold that the relevant
1
The second additional factor in Henson dealt with the parties’ stipulation.
943 F.3d at 454. We do not consider this factor because Bobbitt and Milberg did
not have a stipulation.
6
considerations favor granting Bobbitt’s Rule 60(b) motion.2 The district court is
directed to grant Bobbitt Rule 60(b) relief.
The parties shall bear their own costs on appeal.
REVERSED and REMANDED.
2
Bobbitt’s request to proceed on his individual claims because his voluntary
dismissal was ineffective under Microsoft, 137 S. Ct. 1702, is moot given our
decision to direct the district court to grant his Rule 60(b) motion seeking relief
from his voluntary dismissal. The district court’s 2013 and 2018 judgments on
Bobbitt’s individual claims cannot stand after Bobbitt is granted relief because
those judgments were based on Bobbitt’s voluntary dismissal. For that reason, we
do not address Bobbitt’s argument that we have jurisdiction to review the class
certification denial based on the district court’s 2018 judgment. On remand,
Bobbitt may seek reconsideration of the class certification denial, and the district
court is not precluded from revisiting the issue by the law of the case or any other
similar doctrine. Laber can also decide whether he wants to pursue his pending
motion to intervene, and if he chooses to, the district court is similarly not barred
from considering any such motion or the merits. We express no view on the merits
of these issues.
7
| |
Cursor values into excell, a forum discussion on Cleverscope Mixed Signal USB Oscilloscopes. Join us for more discussions on Cursor values into excell on our Interesting Questions forum.
Is there an easy way to get the cursor values into excell?
Since I am using the exponential average, the samples are not independent of each other.
The exponential average can have a new value every time an FFT is finished. However, it will be highly correlated to the previous average since the decay rate of samples is small. Is there an easy way to know what the "decay" rate between successive displayed values is?
Is there an easy way to get the cursor values into excell? I am using the cursors on the maths graph. I need to take several values to average the noise power. I would like successive values of the "dv" (cursor 1 - cursor 2) to appear in a spread sheet column. Since I am using the exponential average, the samples are not independent of each other. The exponential average can have a new value every time an FFT is finished. However, it will be highly correlated to the previous average since the decay rate of samples is small. Is there an easy way to know what the "decay" rate between successive displayed values is?
Yes you can use the Signal Information to log values as seen by the markers. If you place the markers on the part of the signal you want to save, you can select Marker Y to show the amplitude under the Marker, and Marker X to show the X axis value. Marker 1 maps to Chan A and Marker 2 maps to Chan B. Note that the markers can be on any channels. Make sure you have set the Information Source to Maths.
New Average = [New Value + (N-1) Old Value]/N, allowing you to calculate the difference.
Hello, Yes you can use the Signal Information to log values as seen by the markers. If you place the markers on the part of the signal you want to save, you can select Marker Y to show the amplitude under the Marker, and Marker X to show the X axis value. Marker 1 maps to Chan A and Marker 2 maps to Chan B. Note that the markers can be on any channels. Make sure you have set the Information Source to Maths. I hope this helps. The decay rate is set by the Exponential Averaging count. This acts as an RC delay delay. Let N = Number of Averages, then we calculate New Average = [New Value + (N-1) Old Value]/N, allowing you to calculate the difference.
You are correct, the updated exponential average is dependent on both the past accumulation and the new value. Can you please explain what the problem is here?
You need to Register or Log In before posting on these forums. | https://cleverscope.com/topics/463/cursor-values-into-excell/ |
Q:
Is the linear combination of vectors in a vector space subject to the rules of addition/multiplication of that vector space?
I'll explain with this example that I'm working on:
Let vector space $V = \{ x \in R| x > 0\} $ with addition defined by $ x + y = xy$ and scalar multiplication as $ a * x = x^a $. Find a basis.
To do this, I should find that some arbitrary vector in V can be written as a linear combination of vectors in V that are linearly independent, and span V.
So let $u \in V$, so that $u > 0$, and
$u = \alpha_1x_1+\alpha_2x_2+...+\alpha_nx_n$
Then
$u = {x_1}^{\alpha_1}+{x_2}^{\alpha_2}+...+{x_n}^{\alpha_n}$
$= {x_1}^{\alpha_1}{x_2}^{\alpha_2}...{x_n}^{\alpha_n}$
Then I can take $x_1 = 2$, and see that $2^\alpha$ for $\alpha \in R$ will be in the span of V. So {2} is a basis of V, dimension 1. This feels wrong to me, which is why I ask: is the linear combination of vectors in a vector space subject to the rules of addition/multiplication of that vector space, as I have applied them here?
A:
The map $\log : \mathbb R^+ \to \mathbb R$ is a vector space isomorphism since you have
$$\log(x^{\lambda}y^{\mu}) = \lambda \log(x)+\mu\log(y)$$
and the map is bijective. So your vector space is one dimensional and every $x\in\mathbb R^+\setminus\{1\}$ is a possible basis. In particular, $x=2$ is a basis.
| |
The ever-changing role of a startup CTO
Tech startups are only as successful as their tech. With fast-paced growth, a CTO must build upon and constantly iterate with the changing times to succeed.
Thinkstock
Companies in this day are only as successful as their technology. Those that build and change with the times are usually the ones that succeed. And as technology and businesses change, so must the role of the CTO.
As the primary driver of all executive decisions regarding the tech that powers, the CTO’s role is wide; they’re responsible for building and execute against the vision of a company, implementing strategies and ensuring resources are available to achieve business goals.
Making matters more difficult, getting ahead of the competition and continuously innovating becomes even harder as an engineering team grows from zero to fifty-more employees.
In addition to being a member of the executive team and formulating and driving the company and business strategy forward, the role of the CTO typically falls into four buckets of responsibility.
1. Execute a product roadmap
This means strictly executing on the roadmap as promised, on time and keeping key stakeholders informed along the way.
2. Scale engineering team and the technology stack
Scale engineering such that the team is able to execute on the product vision today and for one to two years down the line (while not getting too far ahead of growth and overhiring). This is especially important in a high-growth company. This includes both hiring as well as the development of existing employees to build a strong engineering culture. Scaling technology implies being able to keep up with the needs of the business and its customers, for example, as the traffic to a service or a website grows, the technology should be able to keep up with it
3. Improve execution speed
This involves ensuring the right processes and tools are in place on teams so that they are able to execute with speed and also includes making sure companies are leveraging technology in every aspect of the business to have a competitive advantage.
4. Optimize technology spend
As a customer base grows, so will the costs. The CTO should be to keep a close eye on dollar spend and look for strategic areas where spending can be reduced. An example includes renegotiating cloud provider contracts or changing the way a company handles file storage.
While these four buckets will always be on a CTO’s mind, the priority given to each bucket changes based on the company’s growth.
For a team of 10
Sandeep Chouksey
When you’re working with a team of 10 people or fewer, a CTO’s main focus should be on executing on the product roadmap. In the early stages of a company, moving fast in the market is super important. Execution is also a No. 1 priority because during this stage it is important that the CTO becomes intimately familiar with the tech stack, the way the team works and the products so as to be able to start thinking about how things might need to evolve in the future.
After execution, the CTO’s No. 2 area of focus should be scaling engineering and tech stack, with an emphasis on scaling engineering. A company’s goal should always be to move fast, and if the engineering team is the bottleneck, a CTO must solve that problem quickly.
For a team of 20
Sandeep Chouksey
At an organization the size of 20 people, you can expect the presence of at least two to three managers on staff. Because of this, a CTO should be giving a lot more headspace to growing the engineering and technology stack, while relying on the engineering managers to drive the day-to-day execution. As such, executing on the product roadmap would drop secondary to scaling engineering and the technology stack with it.
Scaling the tech stack is not just about making sure you can support the business growth as it relates to website traffic or customers, it’s also about things like ensuring you can onboard engineers at a fast pace or that your build systems scale with the number of developers at the company.
If the next growth step is to scale the engineering organization to 40 people, making key investments here is going to be crucial. During this time, it’s important to ensure the right engineering culture is in place. If not, there is a risk of attrition during rapid hiring and growth stages. Before joining Care/of, at my time with Animoto, I found that implementing engineering team values (in addition to the company values already in place) was a good way to establish a good engineering culture. These values represented the engineering organization and helped establish a strong foundation for a good engineering culture that all of us aligned with. I’m also doing this with Care/of now that we have hit the 20-person engineering team mark.
For a team of 40
Sandeep Chouksey
At an organization of 40 people, it’s safe to assume that some components of the tech stack are about four to five years old. The company is clearly doing well, but as engineers, we want to make sure that we are beginning to address some of the cracks in the tech foundation.
With this in mind, the CTO should rank scaling engineering & tech stack at No. 1, improving execution speed at No. 2, optimizing spend at No. 3 and product roadmap execution at No. 4. This is no way means execution is less important, but at this stage, there are typically senior managers and directors of engineering that drive the day-to-day execution.
With all of the roles and responsibilities listed, what doesn’t change for a CTO as startups grow is their need to be nimble, quick and focused on ensuring there are plenty of opportunities and support available to innovate across an organization. While immediate obligations will change, we must always be able to innovate and stay ahead of trends.
This article is published as part of the IDG Contributor Network. Want to Join? | |
I had the pleasure of listening to David Rubenstein, FACHE, Major General, U.S. Army (Retired) at the AZ HIMSS Conference, April 11, 2019. General Rubenstein’s closing keynote, titled, “Leading Oneself in Order to Lead Others” resonated with the audience of healthcare professionals. Here are a few of the key points he shared.
- Healthcare is incredibly complex. Leaders must navigate the patient experience, politics, suppliers, technology, financial constraints, community health, payment models, payor demands. They need to be savvy about the marketplace, healthcare delivery models, innovation, knowledge management, patient safety and organizational designs.
- To lead others, we have to lead ourselves, that means creating a personal mission and vision statement.
- Get in touch with your core personal values.
- He assigned us all homework, you may find enriching and meaningful. Find a quiet spot and take about 5 minutes to reflect on “What is my mission and how does it support the team?” This should answer the question, what is your purpose? Why are you doing what you’ve chosen to do?
- As an executive coach and consultant, this is the work I’ve partnered with countless leaders to do. It increases their clarity and commitment when they can articulate what they do and who they do it for, both personally and organizationally. My mission is “to help healthcare leaders be their best, to positively impact their teams and their patients.”
- Next, ask yourself, “What is my vision and how does it support the team?” Your vision is what you aspire to be. My vision is “developing leaders for healthcare’s changing landscape.”
- Finally, ask yourself “What are my personal values and how do they support the team? Notice the clarity and focus you are feeling as you put your values to paper. If you’re not feeling excited, let’s talk, you may need to find ways to increase living your values at work.
- Once you’ve generated your mission, vision and values, share them with your team. Challenge them to identify how their MVV align to the work of your organization.
Rubenstein was the first Medical Service Corps officer in the history of the Army to be selected as a Major General. He ended his 35-year Army career as both Commanding General of the Army Medical Department Center and School and Chief of the Army Medical Service Corps. Prior to that, he was the Army’s Deputy Surgeon General and the Commanding General of Europe Regional Medical Command/Command Surgeon for United States Army, Europe, and 7th Army. In addition, he served as President of the American College of Healthcare Executives. He is still actively speaking and serving as Clinical Associate Professor of Health Administration at Texas State University. | https://www.risingstarsllc.com/uncategorized/lessons-david-rubenstein-fache-major-general-u-s-army-retired/ |
Will The 2018 Budget Affect Property Investors?
The Commonwealth and NSW Government’s 2018 Budgets did not target property investors directly, but some of the measures the government announced may have an impact on them.
The big news in real estate about the recent federal and NSW Budget announcements were that neither of them really addressed real estate, or property investors. That was a little surprising given how prominent issues such as housing affordability and population growth have become.
However, we believe some state and federal budget measures will affect investors, even if it’s indirectly. So here’s everything you need to know about the recent budgets if you’re a property investor or landlord in Sydney’s East.
NSW State Government
Housing affordability was front and centre of last year’s state Budget and it remains firmly on the NSW state government’s written agenda. However, the NSW Budget for 2018-19 didn’t give it much attention at all, except for the expected news that NSW stamp duty revenues have fallen, due to a cooling market.
In fact, it was no news at all for investors, renters or buyers more generally.
While investors could interpret this as 'no news is good news', first home owners hoping for more assistance to get on the property ladder could be disappointed that this Budget doesn’t build on initiatives from last year, such stamp duty concessions and restrictions for investors and foreign buyers. In short, it’s business as usual in NSW.
Federal Government
Just like the NSW State budget, the Federal Government made little explicit mention of housing or real estate in its recent Budget. But some measures could affect investors indirectly, over the longer term in four key ways:
1. Will Tax Cuts Make Negative Gearing Less Attractive?
The government has no plans to change negative gearing. But the big news from the federal budget included proposed changes to personal income tax, which - if the Coalition continues to win elections - will lead to a complete overhaul the tax system by 2024.
Should this eventuate, the proposed lower rates of income tax could potentially make negative gearing - and property overall - less attractive to investors, as there will be less incentive to reduce personal income tax. But a lot could happen before 2024, so we’ll have to wait and see how it plays out.
2. Flow On Effects From Infrastructure Investment For Investors
One thing which will affect property investors indirectly is the federal government’s infrastructure spending. All up, the government is set to spend $75 billion on upgrading infrastructure on key projects across the country. It used the Budget to announce another $25 billion in new infrastructure spending.
Infrastructure investment is typically great thing for property investors, as buyers and renters look to take advantage of better services and amenities, such as new rail or public transport upgrades.
Projects announced in the Budget that are most likely to affect Sydney’s East include the Port Botany line duplication and the first stage of the North South Rail Link. Outside of this Budget, Eastern Suburbs investors already set to gain from new transport infrastructure that is almost complete - the light rail and Westconnex.
3. Reverse Mortgages For Pensioners Over 65 Years Old
Another Budget measure allows Australians over 65 who draw a pension to access the equity in their home through a reverse mortgage. This, the government rationalises, could help asset rich but cash poor retirees better meet their living expenses. Under the scheme, pensioners could take a loan against the value of their home for up to $11,799 a year for singles or $17,787 for couples, while still receiving their pension.
This measure could have the flow on effect of deterring some downsizers from making a property move and thereby reducing overall stock at a time when it is often already in low supply. The result could be that it will become even harder for first home buyers to get onto the property ladder, particularly in Sydney’s already competitive Eastern Suburbs. For investors, this could help keep prices high and place rental properties in greater demand.
4. Land Banking Could Become Less Appealing
The federal Budget also revealed that council rates and maintenance rates will no longer be able to be deducted where the property is vacant land. This shouldn’t affect developers, as it won’t apply where there is an operating business on the land.
With vacant land a very scarce commodity in Sydney’s East, it’s unlikely to affect many investors in this area.
Talk To Us For More Info
These are some of the ways the state and federal government’s Budget 2018 could impact property investors. If you’re interested in finding out more or investing in Sydney’s East contact our team of specialists today. | https://taylors.com.au/articles/will-the-2018-budget-affect-property-investors |
The interactive map contains information you may find helpful to plan your Alabama nearshore fishing trips. The interactive map displays all of Alabama’s reefs between 0 and 9 nm offshore, provides a table that details the year each reef was initially constructed and the type(s) of materials used to construct each reef. Click here or on the image to use the interactive nearshore reef map.
Nearshore 0-9 nm Reef Coordinates (Current as of Feb. 28, 2019) – To properly download files, right click the link and select “Save”
- 0-9 Mile Reef Coordinates.csv
- 0-9 Mile Reef Coordinates.xls
- 0-9 Mile Reef Coordinates.kmz
- 0-9 Mile Reef Coordinates.gpx
The Alabama Marine Resources Division (AMRD) creates hard bottom habitat within 9 miles of Alabama’s coastline to increase the ecological production potential of water bottoms. Hard bottom substrate is very limited along the water bottoms offshore of Alabama and the addition of purposely designed artificial reefs along this predominantly featureless landscape of sand and muddy substrates has proven to be extremely effective at increasing the biomass of reef fish populations including red snapper, gray triggerfish, and gray snapper. Furthermore, constructing high quality reefs within nearshore waters increases the connectivity between inshore and offshore habitats. For example, red drum, sheepshead, gray snapper, and southern/gulf flounder utilize inshore and nearshore habitats. Red drum and gray snapper migrate from Alabama’s coastal rivers, bayous, and bays as juveniles and early adults to the nearshore sand bars, gas platforms, and artificial reefs as spawning adults. On the other hand, sheepshead and flounder migrate to and from inshore waters and nearshore hard bottom habitats each year during spawning migrations. Reef construction projects within this region also increases habitat utilization opportunities of juvenile triggerfish and red snapper before migrating further offshore to inhabit reefs in deeper water with more vertical complexity. Therefore, increasing the availability of a limited resource (eg. hard bottom habitat) will increase foraging opportunities, shelter, and spawning potential of numerous fish.
Not only do these nearshore reefs have an enormous habitat value, they also provide fishing and S.C.U.B.A. diving opportunities for outdoor enthusiasts. Families, including those with young children and the elderly, and anglers with relatively small vessels can easily access many of the nearshore reefs during a short fishing trip. Similarly, reefs in close proximity to shore provide angling opportunities during inclement weather. Other anglers can use the nearshore reefs as areas to catch bait while on their way offshore, or even an opportunity to patrol for cobia during the Spring.
Historically, the majority of nearshore reef construction projects have been focused offshore of Baldwin County rather than Mobile County due to the areas off Baldwin County having fewer oil/gas pipelines, less dynamic sediments, and less multipurpose water bottoms (i.e. dredge spoil areas and shrimping grounds) compared to areas off Mobile County. A number of gas platforms and a handful of artificial and natural reefs are easily accessible for Mobile County anglers within 9nm from shore. DI Bridge 07 and 08, Buffalo Barges I and II, Lipscomb Tug, and Southwest Rock are all within 9 miles from shore. The reefs within 9 nm from Baldwin County’s shoreline are highlighted by the Allen Liberty Ship, the 240+ structures that comprise the ‘Trolling Corridor’, and the 138 reef sites within the R.V. Minton East and West Nearshore Reef Zones.
The ADCNR/MRD recognizes the need for additional reef construction projects within 9 miles of Alabama’s coast and continues to work toward that goal. Not only would additional reefs in this region provide ecosystem level benefits they would also increase fishing opportunities and provide additional economic benefits to Alabama’s coastal communities. Therefore, the next major reef project (pending authorization by United States Army Corps of Engineers) will be creation of approximately 30 mi2 of new reef zones between 6 and 9 nm offshore of Mobile and Baldwin Counties and deployment of over 100 additional artificial reefs. | https://www.outdooralabama.com/artificial-reefs/nearshore-reef-zone |
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BSc Tourism
2019-06-05T08:16:52+00:00
Bas Amelung
Assistant Professor
I am an environmental economist. My research focus is on the projected impacts of climate change on tourism. I develop quantitative projections for tourism based on climate model output.
Bas Arts
Professor
My research focus is on new modes of governance in environmental politics, mainly with regard to forests, biodiversity, and climate change.
Marcel Bastiaansen
Lecturer & Researcher
I am an expert in quantitative data analysis. With a background in cognitive neuroscience, my research focuses on consumer behavior. Current projects include neuromarketing, big data, and experience marketing.
Raoul Beunen
Assistant Professor
My research explores the potentials and limitations of environmental policy and planning in the perspective of adaptive governance and sustainability.
Harald Buijtendijk
Lecturer & International Tourism Consultant
I specialise in tourism destination development. My current research focuses on value chain governance, multi-stakeholder collaboration, and business innovation.
Erdinc Cakmak
Senior Lecturer
I am a marketing researcher, interested in cross cultural marketing topics. I currently focus on place branding and image formation of tourism destinations.
Miranda Cornelisse
Lecturer & Researcher
I am specialised in product and destination development in complex situations. In my research I focus on the usage of images and visual storytelling in the creation of memorable tourist experiences.
Pieternel Cremers
Lecturer & Study Advisor
I combine my two passions and fields of expertise -tourism & working with people with intellectual disabilities – to contribute to the debate on improving the lives of people with disabilities in our society. I am also the programme’s study adviser.
Ynte Van Dam
Assistant Professor
I am a marketing researcher with special interest in consumer behaviour and sustainable marketing. I’m always looking for job opportunities for my graduates.
Domenico Dentoni
Assistant Professor
My research is inter-disciplinary across the fields of economics, sociology, psychology and political sciences, often conducted as a team effort and very often out of my office.
Art Dewulf
Associate Professor
I the study of complex problems of natural resource governance, through combining insights from organization science and policy science, with a focus on interactive processes of framing, conflict and collaboration.
Karolina Doughty
Lecturer & Researcher
I am a human geographer with a special interest in health/wellbeing and place-making. My current research applies this interest to urban spaces, coastal & water environments, and in relation to mobility & sensory experiences.
Clemens Driessen
Lecturer & Researcher
I am a philosopher/cultural geographer with an interest in science, technology, ethics and the arts. Currently I am working on human-animal relations, technological design and nature conservation.
Rene Van Der Duim
Special Professorship Tourism and Sustainable Development
I am a sociologist with special interest in actor-network theory. My research focuses on tourism, conservation and development in sub-Saharan Africa. | https://www.bsctourism.com/our-experts/ |
In addition to our own longstanding privacy policies, BCRHoF is also is required to ensure that the personal information that we collect, use and disclose is processed in accordance with the applicable law, including, without limitation, the Canadian Personal Information Protection and Electronic Documents Act and similar provincial legislation.
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Affiliated Businesses: We work closely with our affiliated businesses. We employ other companies and individuals to perform functions on our behalf. Examples include fulfilling orders, delivering packages, sending postal mail and e-mail, removing repetitive information from customer lists, analyzing data, providing marketing assistance, providing search results and links (including paid listings and links), processing credit card payments, and providing customer service. They have access to personal information needed to perform their functions, but may not use it for other purposes.
With Your Consent: Other than as set out above, our customers will receive notice when information about them might go to third parties, and they will have an opportunity to choose not to share the information.
Questions, Comments and Complaints
Customers concerned about their privacy protection should contact us by sending a written request. Customers are required to provide enough detail to allow us to identify the information they want; for example, include dates, account numbers or the names or positions of people dealt with at BCRHoF. We will provide the information requested within a reasonable time and at minimal or no cost. Send a self-addressed, stamped envelope to:
Acquisition or Changes in Ownership of BCRHoF
In the event that BCRHoF (or a substantial portion of its assets) is acquired, customer information would be considered part of those assets, and may be part of those assets that are transferred. | http://bcrhof.ca/privacy.html |
GRAND RAPIDS, MI — Single family home values in Grand Rapids increased 10.5 percent this year, a jump that officials say reflects a strong real estate market in Michigan’s second largest city.
The median assessed property value for 2020 is $73,900, up from $66,900 last year, according to data provided by the city of Grand Rapids. A property’s assessed value is roughly half its market value.
“You have a low unemployment rate, you have a booming economy, you have a very vital city,” said Grand Rapids City Assessor Paula Grivins-Jastifer. “People want to live here.”
Increasing property values can be seen throughout Grand Rapids, rising in each of the city’s zip codes. It’s a trend that reflects increased demand for housing in the city but also raises concern about affordability.
The zip code with the biggest jump: 49507.
(This map shows the median assessed single family home value in 2020 for Grand Rapids zip codes, and compares it to last year’s assessed value. Zip codes do not align with municipal boundaries. Some zip codes included above stretch beyond the city’s boundaries. The map only includes the homes in those zip codes that are within the city of Grand Rapids. Zip codes with less than 30 single family homes were not included. If you’re reading this story on the MLive app, click here to see the map.)
The median property value in the 49507 zip code increased to $54,900, up 15.3 percent from last year. The zip code covers a significant chunk of the city’s Southeast Side and encompasses several low-income neighborhoods that have not shared in the new investment playing out elsewhere in the city.
But it’s also home to the Alger Heights neighborhood, a popular residential area that in December was ranked the second most competitive neighborhood in the country for home buyers by the real estate firm Redfin.
Pam Patterson, a realtor and member of the Alger Heights Neighborhood Association board, said she’s not surprised to hear property values are rising quickly in the 49507 zip code. She said there’s a limited inventory of homes on the market, and competition for those homes can be fierce.
“The houses that are in move in ready condition are selling like hot cakes, and there’s typically bidding wars on them,” Patterson said. “It’s a hot area.”
But the increase in property values in Alger Heights also raises concerns that the neighborhood is becoming unaffordable for some residents, she said.
“It’s really extremely difficult to find anything under $150,000,” Patterson said. Later, she added: “You’re probably going to have to have two incomes or at least one very good income to be able to live in this neighborhood now.”
While property values are increasing by double digits, residents won’t necessarily see those increases wind up in their property tax bill.
Under state law, the annual growth of a property’s taxable value is limited to the inflation rate or 5 percent, whichever is less. The inflation rate for 2020 is 1.9 percent.
Grivins-Jastifer said 2020 marks the sixth consecutive year that property values have increased in Grand Rapids.
And they’re rising elsewhere in Kent County, too.
Values increased in each of the county’s 21 townships and nine cities, according to data provided by Kent County Equalization Director Matt Woolford.
The increases ranged from a low of 5 percent in Grattan Township, north of Lowell, to 10.9 percent in Nelson Township, near Cedar Springs.
Woolford said the latest property value numbers marked the eighth consecutive increase in property values countywide in the wake of the great recession.
He attributed the increases to a robust economy and population gain. Kent County’s population was estimated at 653,786 in 2018, up 8.5 percent from 2010.
Mike Childress, owner of Grand Rapids-based Childress and Associates Reality, said he’s not surprised to hear that property values are increasing. He said a low-inventory of homes on the market has resulted in strong demand from potential buyers. | |
With the FashionBrain project, we are working to improve the fashion industry value chain through novel online shopping experiences, the detection of influencers and the prediction of fashion trends.
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Information exploration and discovery in big digital archives
We are working to create a more joined up experience for the exploration of the Government Web Archive.
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Personalised Access to Cultural Heritage Spaces
We worked on developing methods to assist users with searching, exploring and using content from digital library collections.
Our research themes
- The study of human computer and information interaction to understand user cognition and behaviour with respect to the interactivity involved in information access, use and re-use.
- The development of novel solutions to information access problems, ranging from the development of specific algorithms to the design of entire prototype systems, with a particular focus on web-scale systems.
- The study and design of methods and techniques for evaluating information access systems for a variety of applications and search scenarios.
Read more about our research themes
Current projects and research areas
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Find out about the impact of some of our research and how we have engaged with the community here. | https://www.sheffield.ac.uk/is/research/groups/ir |
At first, no one was quite sure what to make of it. A gunman was shooting, and often killing, seemingly random victims in the vicinity of our nation's capital.
As we tried to make sense of the situation, we looked for patterns and connections that would give us some kind of an idea about what was going on. The cable news channels packed prime time lineups with profilers, criminologists, retired investigators and other so-called experts. Each seemed full of suggestions about the shooter's, or shooters', background and motives, and as time passed, the picture of a very complex, mastermind criminal evolved.
It was possible, the experts said, that police would find connections to terrorist organizations. The shooter, they said, carefully thought out each scenario, developing elaborate plans of escape and likely was enjoying the cat-and-mouse game he was playing with law enforcement officials. Then the clues starting coming in.
The task force investigating the shooting spree gave the media bits and pieces about a letter found in the woods after one of the last shootings. There were cryptic messages going back and forth between officials and the shooter over the airwaves. Over the weekend, the pundits credited the shooter with even greater intelligence for his elusiveness and scheming.
On Wednesday, much of that changed, though. To many people's chagrin, news helicopters hovered over a home in Tacoma, Wash. Police were using metal detectors to search for bullet fragments and shell casings. They removed a tree trunk that we were told might have been used for target practice by the suspect. While not confirmed by officials at the time, we were told that the suspected sniper had likely once lived in the house. Many people were infuriated by the coverage, fearing that the suspects would be tipped off and able to elude capture again. As they sat glued to the coverage, they were fully prepared to blame the media for interfering with the investigations and disrupting any progress the police had made.
What most people don't realize, though, is that in situations like this one, law enforcement and the media often work hand in hand. While most of us have never been involved in covering a situation like the one that unfolded in Maryland and Virginia over the past couple of weeks, experience with local officials in a town as small as Brewton can give you an idea of how that relationship works.
In the history of journalism, the investigative reporter is a relatively new concept. While over time reporters have made great impacts on society through undercover research and the resulting articles, it was not until the turmoil of the 1960s and 1970s and events like Watergate, that the general public really got a taste of it. But even still, the media rely heavily on official spokespeople as they gather information.
That was the case with the sniper situation, even on that final day. It's almost a certainty that investigators knew exactly who they were going to arrest and likely had a pretty good idea of where they were, even as news crews were filming them gathering evidence. More than doing anything to endanger the search, the media was doing its job, using information that was provided by officials leading the effort.
As more information is being made available, it appears that the two suspects in custody are less brilliant criminal minds than misguided thugs. Some of the clues they left behind, as well as previous crimes, indicate that the pair was guided as much by greed as by the "thrill" of their actions.
As is the case with many serial killers, we may never know exactly what drove the two men to do what they did. Their crimes are horrific and they deserved the harshest punishment available to prosecutors. And while it was the tireless work of law enforcement officials across this country that made their arrest possible, the media did play a supporting role in this story. In the end, it was the partnership between the two that ended the terror that gripped residents of two states. | https://www.brewtonstandard.com/2002/10/30/media-police-often-work-together/ |
The goal of the BatsLIVE: A Distance Learning Adventure is to raise the awareness, understanding, and appreciation of bats and the unique karst and cave ecosystems that many bats rely on.
Raise appreciation of bats for their ecological role, and their unique and diverse life histories.
Increase understanding of bats and their importance to the environment.
Confront myths and common misunderstandings about bats, leading to increased appreciation and understanding.
Increase understanding of karst and cave ecosystems.
Raise understanding of White-nose Syndrome (WNS) and of the actions being taken to combat WNS.
As appropriate for various audiences, take action for bat conservation, whether it is building bat boxes, becoming community bat “champions”, or becoming more actively involved in bat conservation efforts.
Understand the role of citizens, public land management agencies, and non-governmental organizations in protecting and conserving habitat. | https://batslive.pwnet.org/webcast/goals.php |
- Manage the end to end fulfillment of online orders and related processes on a daily basis.
- Ensure customer satisfaction by timely deliveries and meeting their service expectations.
- Ensure smooth processes in purchasing, receiving, checking, listing, storing, selling, picking, packing and delivering. Avoid no-stock, wrong-product or late-delivery situations.
- Generate and follow up on quotations, purchase orders, invoices and other documents.
- Handle customers’ communications via phone, email, social media and other means.
- Ensure stock inventory accuracy, address any shortages or discrepancies.
- Perform annual stock take and regular stock counts.
- Hire and guide interns, part-timers and other manpower for daily operations and events.
- Negotiate with vendors regarding pricing and other terms.
- Continuously review all parts of the supply chain and processes, and improve on them.
- Generate weekly and monthly performance reports, to analyze sales and other elements.
- Assist to develop specific targets, budgets and forecasts, and monitor performances.
- Perform any other relevant tasks to support the operations of the company.
Job Requirements
- Diploma holder with at least 1 year of working experience. Prior experience with enterprise e-commerce solution such as SAP is ideal but not required.
- Preferably with experience as a junior executive in clerical/admin support or equivalent.
- Technically savvy, with proficiency in MS Excel.
- Highly organized, meticulous with sharp attention to details.
- Possess effective interpersonal and communications skills, in both verbal and written forms.
- Possess good etiquette when communicating over the telephone and written messages.
- Possess positive work attitude, good time management and able to multi-task.
- Demonstrate integrity, dependability, responsibility, team player ethnic, and empathy.
- Able to work in a fast changing environment and adapt to needs of the market.
- Able to work and handle office operations independently without supervision.
- Work 5 ½ days week with alternate Saturdays off, must be able to work on rest day and public holidays when required. | https://www.findsgjobs.com/job/94954/operations-assistant/ |
This workshop is live online via Zoom. A recording will also be available to registrants to view at their convenience.
How do you have healthy interactions and build authentic relationships with those who are experiencing the world very differently than you? Often it can be challenging to understand different perspectives and how to best provide care and friendship to individuals in different cultures.
In this workshop, Rev. Carla Christopher Wilson will provide an overview on some primary cultural values of multiple Asian, African, South American, and Native Cultures. We will work to appreciate cultural differences without appropriating or exploiting them. Attendees will be encouraged to identify areas of privilege in their lives and given ways to de-center their own cultural experiences. Learn tangible ways to open doors for dialogue without tokenizing the other. Explore spiritual practices that create space and encourage open heart connections.
This workshop is crucial to anyone looking to increase their cultural competency to build authentic relationships. It is a good baseline for increasing your own understanding to be more effective in personal and professional relationships. In volunteer work, you will encounter people of many different cultures. PRC volunteers are especially encouraged to attend, and can register for free by contacting PRC.
Rev. Carla Christopher Wilson, MDiv, is Associate Pastor at Lutheran Church of the Good Shepherd. She is also the Assistant to the Bishop in Charge of Justice Ministries of Lower Susquehanna Synod in the Evangelical Lutheran Church in America. She has been a community organizer, cultural consultant, and advocate. Carla was the former Poet Laureate of York and performs occasionally with a funk fusion band. | https://parishresourcecenter.org/events/cultural-humility/ |
Author:
Access:OpenAccess
Citation:N.Sams, K.I. Robertson, 'Role of full-time union officer', Economic and Social Research Institute, Economic and Social Review, Vol. 8, No. 1, 1976, 1976, pp23-42
Download Item:
v8n11976_2.pdf (PDF) 1.064Mb
Abstract:
Many commentators on industrial relations postulate a more ambitious and elaborate role for the trade unions. Leading figures within the movement itself hold similar views. Such developments might be expected to place additional burdens on the professional administrators o f unions, the full-time officers. Yet the full-time officer has been a relatively neglected figure in the study of industrial relations. The main purpose of the present study has been to investigate the role of the full-time officer, and the extent to which his capacities seem suited to that role both now and as it may develop. It will be argued that the nature o f the job and the personal characteristics of the officer have increasingly diverged. The consequences for the organisation and administration of unions, and for public policy relating to industrial relations, are explored.
Author: Robertson, N.Sams, K.I.
Publisher:Economic & Social Studies
Type of material:Journal article
Collections:
Series/Report no:Economic and Social Review
Vol. 8, No. 1, 1976
Availability:Full text available
Keywords:Trade Union, Industrial Relations
ISSN:0012-9984
Licences: | http://www.tara.tcd.ie/handle/2262/69095 |
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