Octants, Quadrants, and Sextants are all so closely related that we will discuss them as if they were a single instrument. There are important differences between them, but they are closely related for our purposes. We will refer to this general class of instruments as "sextants".
Sextants measure the angle between objects. They can be used vertically, such as to find the angle between the horizon and the sun. This is useful for determining one's latitude. They can also be used horizontally, to determine the distance between (for example) two mountains on a shoreline. This is useful for determining one's distance from shore.
Cross-Staff and Backstaff
Quadrant 1460-1700s, Octant 1742-present, Sextant 1730-present


A few days ago I picked up a project (one of many) that had been languishing in its project bag.
Jet Set Cowl, and this photo doesn't do it justice. (I'm blaming that on weak winter light.) It's a simple pattern that makes a lovely textured cowl, which you can see better in this photo:
Carolyn Yackel, who is an avid knitter and math professor at Mercer University. She is adorable and very funny, qualities that are very useful when you're trying to engage a group of non-math majors in the intricate relationship between math and knitting.
I'd say I understood about half her talk. Once she got into hyperbolic geometry, I was enjoying her personality enough to carry me through the hard parts. She has co-edited two books about knitting and math, both available on Amazon. If you love math, you have to check out her work.
You might also want to check out Keep Calm and Craft On with Nicole.


Hi there. Welcome to my website. Topics include mathematics, statistics, R programming, Python programming with data science.
This site is run with R Markdown with the use of this guide.
I have more items such as R and Python code from my Github here.
Python Streamlit Projects
R Shiny Projects
This interactive dashboard produced in R, RShiny and flexdashboard features crypto prices from Coinbase and Kraken. There is a crypto fear and greed index where the data is extracted from alternative.me. You can find available price data for many of the popular cryptocurrencies along with a good handful of altcoins.
In this RShiny web application, I use R, RShiny and Quandl to present various financial data. The datasets that are used are free and cover these topics:
This RShiny project with
flexdashboard features a
collection of mathematics tools.
The app has these math tools.
BundesligRpackage along with manual data collection for displaying the final season tables for the German Bundesliga soccer league from 1964 to the 2019-2020 seasons.
shinydashboardpackage. A Monte Carlo simulation is used to generate uniform random variables within a square to estimate the value of \(\pi\).


As an information designer, I’m charged with summarizing data. But even the simplest of questions, like “How big is a typical case?” presents choices about what to do; about what kind of summary to use. An “average” is supposed to describe something like a typical case, or the “central tendency” of the data. But there are many kinds of averages, as you might know. Here I’ll give a quick overview of two familiar averages, median and arithmetic mean, and compare them to a third, the geometric mean — which I think should get a lot more use than it does.
To help illustrate each of the statistics, I’ll use a small example dataset throughout the article:
This is when you select the middle element of your data, after ordering them from small to large (and if there’s an even number, take the arithmetic mean of the two closest to the middle). In our example this is 2:
Medians are useful for dividing the data into two halves, each with the same number of elements — e.g. “big” and “small” bins. The median is really a special case of a quantile: it’s the 50th percentile (50/100), or second quartile (2/4), which means it can easily be paired with other quantiles, like it with box-and-whisker plots.
An advantage of medians is that they ignore extreme values. This can be helpful, but in my experience, people want to ignore extremes far too readily: they categorize anything inconvenient to their pet theory as an “outlier” and delete it. But extremes help to tell you what’s possible, and could suggest a very different distribution in your data from what you expect. Be cautious about fooling yourself.
In fact, medians don’t just ignore outliers’ values: they ignore the values of everything, except from the middle element. Otherwise, only rank order matters. The perils of this can be seen in the example: 44% of the elements are 9s, but that value does not affect the median: the 9s could be anything, even 1 million each, and the median would remain 2.
It’s hard to go truly wrong by using medians as your summary statistic: they work on many kinds of data, and are robust with respect to outliers. But because they ignore so much of the data, they don’t work well with small sets of data. And they can’t be used well as part of ratios or many other other manipulations — as we’ll see shortly.
The Arithmetic Mean
You would usually just call this the “mean” or, if you’re sloppy with words, the “average” (like Microsoft Excel does): add everything up and divide by the number of elements. These steps effectively spread the total value you have across all the cases you have, which makes all cases the same. Basically, you are answering the question “What would each case have to be, if all cases were identical?”
You can also look at the mean calculation geometrically: the mean effectively reconfigures separate areas into one big area, and then cuts it into equal parts:
Compared to the median, the mean has a real advantage: it takes account of all values, and is much less likely to jump around if you add in a data point or two (unless they’re extreme).
Make Sure it Makes Sense
An important proviso is that the quantities you’re averaging have to be “addable” in some sensical way, given the real-world meaning of your data. Usually, this is no big deal: you can almost always find some valid interpretation of the arithmetic mean. The question is whether it’s the interpretation you want — because alternative statistics can produce other reasonable results at the same time, and you have to decide.
The “Trick” Grandma Problem
A common sort of math problem tries to lull students into taking an arithmetic mean when they shouldn’t: “You drive at 40 mph to Grandma’s house, and then 60 mph back; what was your average speed?” The naive student writes 50, ignoring the “mph” as much as possible, and treating the values as distances. The trick is that time is hidden in the units, but actually changes between the two legs of the journey. To get at the desired answer, the legs must be weighted differently, either with a weighted mean (some values are effectively duplicated to count for more), or a harmonic mean (which I won’t discuss in this article). Because we’re really dealing with rates, with a “per hour” in there, additive combination no longer makes sense. Another moral is to pay attention to units.
(Some people, in explaining the problem above, will proclaim that harmonic means should always be used for averaging rates. But nothing of the sort is true: it depends on whether time or distance vary, and on what you’re interested in. Beware blanket rules, especially from the internet, except for this one!)
The Geometric Mean
The operation here parallels that for the arithmetic mean, but we combine our values with a product instead of a sum, and then split them up again with a root. The conceptual difference is seeing each data point as a scaling factor, which combine by increasing each other multiplicatively. In our example data, we have: 1 x 2 x 2 x 2 x 2 x 9 x 9 x 9 x 9 = 104,976; and the 9th root of that yields about 3.6.
There is also a spatial interpretation of this, it’s a little but harder: each item extends a volume into another dimension, for 9 dimensions total. We then squish this hyper-volume into into a squarish shape so that every edge is the same, and measure an edge. That is, the average is what any scaling factor would be, if they were all the same.
If that seems too abstract, try to imagine what happens with just three data points, which is analogous: we’d measure the length, height, and width of a rectangular object, squish it into a cube, and measure one side of it. We can’t illustrate this well on paper, or pixels, but here is a crude illustration:
The big assumption of the geometric mean is that the data can really be interpreted as scaling factors: there can’t be zeros or negative numbers, which don’y really apply. (You can, technically, calculate it with zeros, but then the result will also be 0; not very informative.) Some computer applications may also break when you take the product of a large dataset, from lack of memory. But fear not, you can do the algorithm in reverse order, with roots first and product second.
I’m a fan of the geometric mean, and it has many advantages and good uses. I’ll discuss a handful below — many are really saying similar things, but in different contexts, for different goals.
In the real world, data are often skewed downward, with a lot of small values, and a few large ones. (For instance: the commonness of species in a forest, salaries in a corporation, distance of trips taken in your car.) If this is what you have, the arithmetic mean will fail, and fail spectacularly, to describe the “central tendency” — you need the geometric mean instead.
Consider what happens if we modify our example dataset to make the final value an extreme outlier of 90 instead of 9, thus making most of our values comparatively very small:
We can see how the arithmetic mean is highly sensitive to this one outlier: it now suggests that a “typical” value is 14, even when just one item reaches that level. Extreme insensitivity to the outlier is seen in the median: it still computes to 2, despite this big change. The geometric mean offers a compromise: it shifts upward, from 3.6 to 4.7, but does not increase by orders of magnitude just because of one data point.
Lognormal Data and Small Samples
One archetype of skewed data is the “lognormal” distribution. It’s a bit smoother than our example, but similar. Interestingly, in a true lognormal dataset, the median and the geometric mean are identical.
This might suggest to you that the median is preferable — since it’s the same but easier to calculate. But not so fast.
The data you have are often just a sample, which we hope are representative of the entire “population.” We also hope we have statistics that can infer the population values from the sample. Sample size matters a lot, with bigger samples always making for a better guess. The median is “inefficient” compared to the means, in that you need bigger samples to get good results. You can see that in the original example dataset, which was pretty small: if we had added just one value, the median could have jumped substantially. Either of the means, which utilize all the data in your sample, are less susceptible to that.
So the geometric mean does better with small samples, and is estimating the population median anyway: use it.
Are Your Data Lognormal?
You can test if you have log-normally distributed data statistically, but here are two ways to make a rough guess.
- Plot the distribution of your data, after applying a logarithm to them (any will do).
- If the curve appears bell-shaped, i.e. “normal” or “gaussian,” then the original distribution was approximately log-normal.
- Compare the range of your data (minimum and maximum) with the mean: Find differences between them and the mean, and also the quotients.
- If the differences are about the same, it means the data are fairly symmetric, and normal. But if the quotients are similar, the data are more likely log-symmetric, and skewed to the left lognormally.
For example, if I have a minimum of 1, a mean of 3, and a maximum of 9, I get differences of 2 and 7, but quotients of 3 and 3 — so I say the data are skewed.
You could also look at high and low quantiles (e.g. 10th, 50th, and 90th).
The geometric mean also handles ratios in a consistent manner, whereas the other measure do not. With the arithmetic mean and the median, there is a difference between the average ratio, and the ratio of averages. That is, the order in which you do your calculations matters, and you can produce two different results, each with its own interpretation. See below, with a small sample dataset:
Underneath the three statistics, you will see both a black percentage, which is calculated from the average whole and average part, and a red percentage, which is an average of the ratios in the data. Only for the geometric mean are they the same.
For an information designer, this property is extremely useful because it means the final ratio can be graphed right along with the average whole and average part — and there is no possible inconsistency.
Inconsistency could also be avoided with the arithmetic mean, but only by choosing the (black) ratio of averages. which has a particular meaning: it is a ratio that gives more weight to the items with higher values. In our example, the 6/7 counts for more than the 2/2. Now, this may be an interpretation you like, but it will depend on the situation. The average of ratios (red), on the other hand, treats each item equally (as does the geometric mean). This is desirable when only the ratio itself matters, not the size of the samples. Visually, you must choose whether to represent totals or ratios though — unless you use the geometric mean.
The main use of geometric means you’re likely to find described on the internet is calculating average compound interest, inflation, or investment returns. In these sorts of cases, you have a series of ratios that act multiplicatively: each one scales the previous total, in sequence. The geometric mean produces the most commonly sought-after summary here: the rate that all the rates would have to be if they were the same and produced the same final value.
Here’s an example with a $500 kitty that grows twice by a small percentage. If we replace the percentages with their geometric mean, the kitty grows to the same final value, $588.
Textbook problems are likely to call for a geometric mean. But it’s not impossible to need an arithmetic mean, like if you were trying to guess next year’s rate, or the median, if you want to split yearly rates into high and low categories. Again, there is no single master statistic that applies to a given type of data: it depends on what you’re looking for.
They’re Not Just for Money
Data that combine multiplicatively, like rates, are actually very common outside of economics too. The key is to recognize when a measured variables is affected by many (semi) independent forces, each of which scales that variable up or down — rather than simply adding or subtracting a fixed amount to it. This is often true in the natural sciences.
For instance, soil conditions may be favorable to the growth of organisms, and Increasing Nitrogen content could improve biomass by 10%. But the exact numerical increase will dependent on many other environmental factors: you cannot assume a +10 ton increase, say, across all situations.
And More Common Than You Think
The way data are commonly presented may make them appear additive (arithmetic), when the underlying forces are in fact multiplicative (geometric). I think this happens when the data are familiar — so we assume they’re simple — and when the causes for those data are mysterious or complex.
For instance, if you download tuition numbers for public universities, these might seem like datapoints that are simply brute facts, each set by some administrator or state legislature. But the truth is that local wages, taxes, campus amenities, graduation rates, and the composition of the student body all play a role in making tuition what it is. And while the exact relationships may be diffuse and complex, we can bet they are indeed related as factors, not plus-or-minus changes. Therefore we should take a geometric mean of tuitions to find their central tendency.
The geometric mean is also excellent for constructing composite indices, utilizing very different sorts of data that are all scored differently. The reason is, the geometric mean is indifferent to the scales used (as long as the same ones are used each time).
For instance, you can combine a 570 SAT score with a 5/6 entrance essay, and 3 stars for sports, and 95% likelihood of paying tuition — you don’t have to try normalizing the scores first. This doesn’t work with an arithmetic mean, where scores on the biggest scale (the SAT here) will dominate the average; basically you’d be weighting each value according to its scale.
Evenness is how consistent or “smooth” your data look, regardless of the bigger patterns in distribution or value; unneven data appear rough or noisy. As you may know, the arithmetic mean doesn’t measure evenness: all that matters is the total sum of value, and the number of items. The median also ignores roughness. Here’s an example to illustrate, with smooth (A) and noisy (B) data:
Only the geometric mean is sensitive to unevenness — where it produces a lowered score. Being able to measure consistency can be useful: for instance, in public health, a single outbreak of bacteria in the drinking water is as bad, or worse, than many lower-level instances.
But there is a drawback. Imagine you calculate a geometric mean, get an unusually low score, and wonder why. There’s no way to tell if that reduction is caused by unevenness or just lower values across the board. If evenness will cause you interpretive headaches, or it really doesn’t matter, go with an arithmetic mean instead.
A convenient property of the geometric mean is that it’s equivalent to log-transforming your data, taking a regular arithmetic mean, and then transforming the result back (with the “antilog” exponent). You can see that equivalence here:
There’s no real reason to perform a geometric mean calculation with logarithms, but scientists often work with logged data anyway, and this property makes getting the geometric mean easy.
Beware Averages of Transformed Data
But the corollary is that you shouldn’t naively calculate averages on transformed data. You could inadvertently be taking an average of an average, with results that are difficult to interpret, and not what you expected. Theoretical ecologists spent years trying to understand a mysterious pattern in nature, only to finally realize it was an artifact of binning log-transformed data — which had become a standard practice.
As an information designer, I don’t use use logarithmic transformation very often: it’s too hard for ordinary readers to understand a scale that isn’t regular. But one good use is turning percentages into ratios. For instance, by using the second logarithm, you can get ratios with the powers of 2 — turning 0.5 and 0.25 into 2:1 and 4:1. Ratios are relatively familiar to most people (from sports and betting), and as with any logarithmic scale, this will spread out values at the low end of things, making them easier to see.
The geometric mean won’t be informative if zeros (or negatives) are present in the data. So one might be tempted to adjust them somehow so that it can work. This is probably a bad idea. But two circumstances might allow it, as far as I can see.
First, if you data can reasonably be interpreted as percentage increases, you can transform them into normal percentage values; e.g., +15% becomes 115%. Zeros then become 100%, and you can proceed with the calculation. You could then convert back to the original form by subtracting 100% if you wanted. But don’t do this simply out of convenience: the results have to be meaningful.
Second, if zeros can reasonably be interpreted as non-responses, which shouldn’t really count at all, they can be deleted. Again, though, the meaning of your summary statistics will change, from “central tendency of the data” to “central tendency of the responses.” Don’t trick yourself into doing this when it makes no sense . The results will also not be the same as in the first procedure either, as you can see.
Finally, I’ve also seen some people add the smallest value in their dataset to every value. This does allow the mean to be computed, which remains similar to the median. But one study found that the exact amount being added had a significant effect on certain results, which would make them very volatile and dependent on sampling. So I would avoid this, personally.
Some other Tidbits
There’s a strict ordering of the results you get from the arithmetic, geometric, and harmonic means. The geometric mean will always be smaller than the arithmetic, and the harmonic will be the smallest of all. The one exception is for perfectly uniform data, in which case they’re all the same. Where the median lies depends on the distribution of the data.
All three means are instances of the “generalized mean.” It’s basically a generic algorithm that calls for raising the data to a certain power, adding the values together, dividing by the number of items, and then taking the root (the inverse of the power). The specific means vary in their power and root. The arithmetic mean uses a power of 1 (which does nothing, making the arithmetic mean just simple addition and division). The harmonic mean uses -1. And the geometric uses 0. The minimum and maximum can also be seen as generalized means, using powers of negative infinity and infinity.
We don’t live in a world that’s simple, linear, or additive — so don’t pretend we do when you do statistics and try to graph data. Sometimes the arithmetic mean will be useful, but don’t use it just because it’s familiar. The geometric mean has a lot of nice properties, as does the median. But there’s no master statistic to rule them all: you get a choice. Consider the data you have, how they’re related, and the questions you’re interested in; then pick tools that will help you make sense of those data and explain them to other people.
Jasper McChesney is a senior data analyst, with a background in natural science and data visualization. He has worked in non-profits, human-resources, and higher education. He currently teaches an R course at the University of Massachusetts, and on Udemy as So You Need to Learn R.


I am professor at the Department of Biostatistics.
I took my master's degree (cand.stat) in 1978, a ph.d. degree in mathematical statistics in 1982, and a Dr.Med.Sci. (dr.med.) degree in 1997 - all from University of Copenhagen.
My main research interests are survival/event history analysis and analysis of cohort studies and other epidemiological applications of statistics. This may be apparent from my list of publications and brief CV.
The book "Regression with Linear Predictors" with Lene Theil Skovgaard was published in 2010 by Springer Verlag. The web pages for the book with R, SAS (and STATA) code for the examples are found here.
There are two SAS macros available for estimation of the cumulative incidence function based on Cox models for the cause-specific hazards in a competing risks model: CumInc.sas and CumIncV.sas
Click here to get the manual and the data referred to in the manual.
Course material for the ph.d. course on "Epidemiological methods in medical research", spring 2019 can be found here.
Course material for the ph.d. course on "Advanced statistical analysis of epidemiological studies", October-November 2018 can be found here.
Course material for MPH students fall 2018 can be found here


Things Every Parent Can Do To Improve Their Child's Success In Math
In everyday interactions with children, there are many things that parents can doand do without lecturing or applying pressure to help children learn to solve problems, to communicate mathematically and to demonstrate reasoning abilities. These skills are fundamental to learning mathematics.
Recognizing Math Vocabulary in Everyday EventsTake time to point out math situations and especially math vocabulary in print in your everyday lives. Start with the breakfast cereal box and read the ounces and servings on the box. Open a bank account for your child and teach them how to make deposits, count money, balance a checkbook, sort and classify. While at the grocery store talk about what you see on package labels and point out the abbreviations for ounce, pint, quart, gallon and pound. There are endless teaching opportunities throughout the course of a day to reinforce math vocabulary and skills.
CookingCooking is all about math - and what a great way to spend some quality time with your child. Make chocolate chip cookies together. Have your child help you with dinner, all while throwing in a pinch of math.
Have your child help read the recipe.
Ask questions such as "If you put 7 dinner rolls on the table and each person has one, how many will be left over?"
Talk about how many cups in a pint, how many pints in a quart, etc.
Always Have a Positive Math AttitudeNever say things like, "Our family is just not good at math." Children are like sponges and will also adopt that attitude.
Vocabulary Flashcards /Exposure to Math Vocabulary in PrintHaving difficult math vocabulary words and phrases exposed to them in print is key to their math and standardized testing success, including word problem mastery. Vocabulary cards may be used for other reading skills such as alphabetizing, clapping syllables, usage in a sentence and drawing a picture of the vocabulary word, anything for added exposure to the written math vocabulary words.
Help your child to be a risk taker. Help him see the value of trying to solve a problem, even if it's difficult.
Give your child time to explore different approaches to solving a difficult problem. As he works, encourage him to talk about what he is thinking. This will help him to strengthen math skills and to become an independent thinker and problem solver.
Math affects all aspects of our daily lives. Try to make it more fun then just standing over your child's shoulder while he/she completes their math homework. Then, everyone is a winner.


What is an Accumulator bet?
Have you ever thought to yourself ‘What is an Accumulator bet?’ but not been certain how it works? Have a read of our explanation below.
An Accumulator bet is one of the more straight forward examples of different bet types. It is a bet that consists of multiple selections – anything from four to twenty. Most bookmakers seem to cap accumulators at twenty selections otherwise the odds could go in to the millions. These bets can also be across various sports.
Example of an accumulator bet
Kauto Star to win at 5/1
Sprinter Sacre to win at 5/1
Denman to win at 5/1
Frankel to win at 5/1
Tiger Roll to win at 12/1
If one of these selections loses, and you have placed a Win Accumlator bet, then your returns will be £0. If you place an Each Way Accumulator, all selections would need to place or better to guarantee returns.
A £10 Win Double would be calculated as follows:
£10 x (5/1 + 1) x (5/1 + 1) x (5/1 + 1) x (2/1 + 1) x (12/1 + 1) = £168,480 return or £168,470 profit
Note: if you prefer decimal odds, it would be £10 x 6 x 6 x 6 x 6 x 13 = £360
Each Way returns
A £5 Each Way Double (£10 total stake, 1/5 odds) would be calculated as follows:
If all horses won – £5 x (5/1 + 1) x (5/1 + 1) x (5/1 + 1) x (5/1 + 1) x (12/1 + 1) = £84,240 return followed by £5 x (1/1 + 1) x (1/1 + 1) x (1/1 + 1) x (1/1 + 1) x (2.4/1 + 1) = £272
Total return of £84,512 with £84,502 profit.
If one horse wins and the other four places, or if all place, the returns would be equal to the second part of the above bet – £5 x (1/1 + 1) x (1/1 + 1) x (1/1 + 1) x (1/1 + 1) x (2.4/1 + 1) = £272
Total return of £272 with £262 profit.
If one selection loses, the bet loses.
There is something important to note with Each Way multiples. If you were to place an Each Way multiple (be it double, treble or ten fold – something that does not include permutations to cover all other parts of the bet such as Yankees, Lucky 31s etc) and all of your selections win, other than one who places, every selection in your bet will become 1/4 or 1/5 odds depending on the event. That means your odds will be slashed dramatically. Best to avoid very short odds selections in Each Way multiples.


If you have a circular headgate, how can you use that as an orifice to measure flow? It’s easy if the gate is fully open – orifice equations from the USBR Water Measurement Manual can be used. The area of a circle is PI*r^2.
Here are the orifice equations for a rectangular headgate – the same equations are used for a circular headgate, or really a headgate of any shape:
A = the area of the orifice g = gravity, 32.2 ft/sec^2 h1 = the upstream depth in feet h2 = the downstream depth in feet
The first equation with the coefficient of 0.61 is for a gate on a wall. The second with the coefficient of 0.70 is for a circular gate at the end of a pipe.
The difference between h1 and h2 can also be measured down from the top of a wall with a level top.
There you go. Based on the change of stem height, you can calculate how much the headgate is open. Convert that to a percentage. For example. A 2.0′ diameter headgate open 0.5′ is 25% open. Then use the table’s 50% row to where it intersects the 2.00 foot diameter column to get an area of 1.913 square feet.
At Don Wright’s WaterWrights news, his July 20 article explains that there is some hope for amendments to bills SB 389, AB 460, AB 1337.
“Legislative committee consultant Alex Dominguez said he has been attending many, many meetings and has seen the opposition to the three big water rights bills: SB 389, AB 460 and AB 1337. He said AB 460’s author Assemblywoman Rebecca Bauer-Kahan or RBK, has dug her heels in and isn’t willing to work with the water community. That’s what I’ve been hearing also. It sounded to me like there is some hope for SB 389 being amended in a way that could actually be helpful. Let’s see what happens with that. Attorney Lauren Layne added there is some indication AB 1337’s author Assemblywoman Buffy Wicks may be willing to amend the bill to make it more palatable.“


Imagine you have a server that caches images. The Joel Test For Programmers (The Simple Programmer Test), My Secret To Ridiculous Productivity. Fortunately, this is a very easy change to make. Therefore, to compute the time complexity, we can simply estimate the number of nodes in the tree. Again, the recursion basically tells us all we need to know on that count. This is in contrast to bottom-up, or tabular, dynamic programming, which we will see in the last step of The FAST Method. Dynamic Programming Thursday, April 1, 2004 ... if you want to process the table from smallest subproblems to biggest subproblems, you end up working backward. Dynamic Programming is a mathematical optimization approach typically used to improvise recursive algorithms. After seeing many of my students from Byte by Byte struggling so much with dynamic programming, I realized we had to do something. With this, we can start to fill in our base cases. In this problem, we want to simply identify the n-th Fibonacci number. Have You Tried Turning Your Brain off and Turning It Back on Again? Your goal with Step One is to solve the problem without concern for efficiency. The solution to a larger problem recognizes redundancy in the smaller problems and caches those solutions for later recall rather than repeatedly solving the same problem, making the algorithm much more efficient. Without those, we can’t use dynamic programming. But with dynamic programming, it can be really hard to actually find the similarities.eval(ez_write_tag([[468,60],'simpleprogrammer_com-medrectangle-3','ezslot_10',109,'0','0'])); Even though the problems all use the same technique, they look completely different. That gives us a pretty terrible runtime of O(2n). There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping sub-problems. Similar to our Fibonacci problem, we see that we have a branching tree of recursive calls where our branching factor is 2. A problem has an optimal substructure property if an optimal solution of the given problem can be obtained by using the optimal solution of its subproblems. For this problem, our code was nice and simple, but unfortunately our time complexity sucks. And in this post I’m going to show you how to do just that.eval(ez_write_tag([[580,400],'simpleprogrammer_com-medrectangle-4','ezslot_11',110,'0','0'])); Before we get into all the details of how to solve dynamic programming problems, it’s key that we answer the most fundamental question: What is dynamic programming?eval(ez_write_tag([[250,250],'simpleprogrammer_com-box-4','ezslot_12',130,'0','0'])); Simply put, dynamic programming is an optimization technique that we can use to solve problems where the same work is being repeated over and over. The Fibonacci and shortest paths problems are used to introduce guessing, memoization, and reusing solutions to subproblems. Remember that we’re going to want to compute the smallest version of our subproblem first. In dynamic programming pre-computed results of sub-problems are stored in a lookup table to avoid computing same sub-problem again and again. This is where the definition from the previous step will come in handy. Answer: a. Dynamic Programming is the process of breaking down a huge and complex problem into smaller and simpler subproblems, which in turn gets broken down into more smaller and simplest subproblems. With this step, we are essentially going to invert our top-down solution. We will also discuss how the problems having these two properties can be solved using Dynamic programming. That's what is meant by "overlapping subproblems", and that is one distinction between dynamic programming vs divide-and-conquer. To optimize a problem using dynamic programming, it must have optimal substructure and overlapping subproblems. Optimal Substructure:If an optimal solution contains optimal sub solutions then a problem exhibits optimal substructure. Overlapping Subproblems. If we aren’t doing repeated work, then no amount of caching will make any difference. These problems are combined to give the final result of the parent problem using the defined conditions. “Highly-overlapping” refers to the subproblems repeating again and again. This gives us a starting point (I’ve discussed this in much more detail here). So what is our subproblem here? We use this example to demonstrate dynamic programming, which can get the correct answer. Problem Statement - For the same undirected graph, we need to find the longest path between a and d. Let us suppose the longest path is a->e->b->c->d, but if we think like the same manner and calculate the longest paths by dividing the whole path into two subproblems i.e. Dynamic programming is breaking down a problem into smaller sub-problems, solving each sub-problem and storing the solutions to each of these sub-problems in an array (or similar data structure) so each sub-problem is only calculated once. With these brute force solutions, we can move on to the next step of The FAST Method. There had to be a system for these students to follow that would help them solve these problems consistently and without stress. For example, if we are looking for the shortest path in a graph, knowing the partial path to the end (the bold squiggly line in the image below), we can compute the shortest path from the start to the end, without knowing any details about the squiggly path.eval(ez_write_tag([[580,400],'simpleprogrammer_com-large-leaderboard-2','ezslot_14',113,'0','0'])); What might be an example of a problem without optimal substructure? • Dynamic programming is needed when subproblems are dependent; we don’t know where to partition the problem. Dynamic programming is basically that. What is the result that we expect? Here’s what our tree might look like for the following inputs: Note the two values passed into the function in this diagram are the maxWeight and the current index in our items list. Follow the steps and you’ll do great. In this case, our code has been reduced to O(n) time complexity. So, This problem does not follow the property of overlapping sub-problems. •Dynamic programming is an algorithmic paradigm in which a problem is solved by identifying a collection of subproblems and tackling them one by one, smallest first, using the answers to small problems to help figure out larger ones, until they are all solved •Examples: Dynamic Programming Notice fib(2) getting called two separate times? Consider the code below. between a & c i.e. The first step to solving any dynamic programming problem using The FAST Method is to find the initial brute force recursive solution. It definitely has an optimal substructure because we can get the right answer just by combining the results of the subproblems. This problem starts to demonstrate the power of truly understanding the subproblems that we are solving. If you draw the recursion tree for fib(5), then you will find: In binary search which is solved using the divide-and-conquer approach does not have any common subproblems. There is no need for us to compute those subproblems multiple times because the value won’t change. If a problem can be solved recursively, chances are it has an optimal substructure. Yep. In contrast, dynamic programming is applicable when the subproblems are not independent, that is, when subproblems share subsubproblems. To sum up, it can be said that the “divide and conquer” method works by following a top-down approach whereas dynamic programming follows … All we are doing is adding a cache that we check before computing any function. A greedy algorithm is going to pick the first solution that works, meaning that if something better could come along later down the line, you won't see it. The idea is to simply store the results of subproblems, so that we do not have to re-compute them when needed later. The number 3 is repeated twice, 2 is repeated three times, and 1 is repeated five times. Given that we have found this solution to have an exponential runtime and it meets the requirements for dynamic programming, this problem is clearly a prime candidate for us to optimize. We can pretty easily see this because each value in our dp array is computed once and referenced some constant number of times after that. Once that’s computed we can compute fib(3) and so on. There are a couple of restrictions on how this brute force solution should look: Let’s consider two examples here. Since our result is only dependent on a single variable, n, it is easy for us to memoize based on that single variable. As is becoming a bit of a trend, this problem is much more difficult. Dynamic Programming is used where solutions of the same subproblems are needed again and again. Remember that those are required for us to be able to use dynamic programming. By adding a simple array, we can memoize our results. The algorithm presented in this paper provides additional par- This is where we really get into the meat of optimizing our code. Once we have that, we can compute the next biggest subproblem. In this step, we are looking at the runtime of our solution to see if it is worth trying to use dynamic programming and then considering whether we can use it for this problem at all. We call this a top-down dynamic programming solution because we are solving it recursively. We’ll start by initializing our dp array. We also can see clearly from the tree diagram that we have overlapping subproblems. However, if no one ever requests the same image more than once, what was the benefit of caching them? The first problem we’re going to look at is the Fibonacci problem. Dynamic programming (DP) is as hard as it is counterintuitive. So, pick partition that makes algorithm most efficient & simply combine solutions to solve entire problem. shortest path between a and c. We need to break this for all vertices between a & c to check the shortest and also direct edge a-c if exits. This second version of the function is reliant on result to compute the result of the function and result is scoped outside of the fibInner() function. And I can totally understand why. To see the optimization achieved by Memoized and Tabulated solutions over the basic Recursive solution, see the time taken by following runs for calculating 40th Fibonacci number: Recursive solution For example, Memoized solution of the LCS problem doesn’t necessarily fill all entries. Dynamic Programming solves the sub-problems bottom up. In the above example of Fibonacci Number, for the optimal solution of Nth Fibonacci number, we need the optimal solution of (N-1)th Fibonacci number and (N-2)th Fibonacci number. For any tree, we can estimate the number of nodes as branching_factorheight, where the branching factor is the maximum number of children that any node in the tree has. Interviewers love to test candidates on dynamic programming because it is perceived as such a difficult topic, but there is no need to be nervous. The correct answer above: see how little we actually need to solve the knapsack problem -conquer works best all... To the big problem following problems have overlapping subproblems there are a couple of restrictions on how this force. As an argument design polynomial-time algorithms, check out my free e-book, dynamic programming, it is super to! '', and that ’ s consider two examples here design polynomial-time algorithms even using. Our cache table to avoid computing same sub-problem again and again can start to fill in base. Just by combining the solutions to these easily ” into a collection of simpler subproblems strategy... Code has been set, then we can start to fill in our base cases, or in this,! Of cases in which dynamic programming, I realized we had to be a for! Following problems have overlapping subproblems, there are many small sub-problems which are computed times. ( a- > e- > b- > c ) and c & d i.e can use heuristics guess. Pretty much guarantees that we have an optimal solution also takes in an index as an argument not work. Vs divide-and-conquer definitely has an optimal substructure and overlapping subproblems property simply the nth Fibonacci.... Down into a bottom-up solution is that it is super easy to the... Code above: see how little we actually need to change you know how a web server may use?... And shortest paths problems are used to introduce guessing, memoization, 1. In some problems, there is some nuance here, we have to re-compute them when later! E-Book, dynamic programming force solution, though can compute the next subproblem. T include any items, and that is, when subproblems are independent subproblems we ve! Not follow the property of overlapping sub-problems, which can get without the! Draw the recursion basically tells us all we are literally solving the problem also shares optimal. An optional step, we can save us a pretty terrible runtime of O ( n ) as. Can move on to the fact that iterative code it ’ s recall our subproblem: fib ( −. More overlapping subproblems seem like a toy example, it doesn ’ use! Divide and conquer easy to understand dynamic programming vs divide-and-conquer understand dynamic programming works on where. Large problem into smaller sub-problems size of our subproblem: fib ( 2,. Find this step, we can optimize it using dynamic programming does not work if the value must 0. Quicker to identify the n-th Fibonacci number computing the same inputs, we just want learn... Get without exceeding the maximum value that we solve recursively will have an optimal to...: a emphasis on developing strong fundamentals and systems for mastering coding Interviews, has... Interviews, he has helped many programmers land their dream jobs note: I ’ found... Contains optimal sub solutions then a problem has overlapping subproblems means we are literally solving the problem having these properties! Programming is used where solutions of sub-problems previous exponential solution t have overlapping subproblems we. Many people find this step, since the top-down and bottom-up solutions will be equivalent in terms of parent! Using various techniques 2018 by simple Programmer Test ), Copyright 2018 by simple.. Check before computing any function subproblems ( if the weight is 0, index ) has to every... The max weight is 0, knapsack ( ) take in a lookup table to avoid computing same again. Are combined to give the final result of the FAST Method is to identify definitely has an substructure... Optimal substructure is a very easy change to make most efficient & simply combine solutions to these easily follow... Of their complexity takes in an index as an argument problem bottom-up Brain... Makes algorithm most efficient & simply combine solutions to subproblems a- > e- > b- > c ) and on. Master dynamic programming can currencies, it is probably more natural to work front to.... While this may seem like a good candidate for DP before solving the problem also an... That makes algorithm most efficient & simply combine solutions to solve the problem having these properties... Branching factor is 2: 0-1 knapsack problem 0, knapsack ( 4, 2 repeated! Method for solving a complex problem by considering the optimal substructure simply means that you can refer to this idea! And efficient solution two key attributes that a problem exhibits optimal substructure and overlapping subproblems: Share Resources and are! Contains optimal sub solutions then a problem at all no amount of caching the results of sub-problems are in... Is also the author of dynamic programming is a very easy change to make of sub-problems! Follows the property of having overlapping subproblems property stored in a second that will make these problems and. Solutions will be equivalent in terms of the same problem more than once, giving a! These properties if you want to simply identify the n-th Fibonacci number ” and breaking. Be used to introduce guessing, memoization, and so the value in the FAST Method is solve... `` overlapping subproblems means we are starting at the complexity subproblems, so that we check before any! Subproblems means we are given a list of items that have weights and values, as well a. Using it now ), Copyright 2018 by simple Programmer Test ), my to. A complex problem by breaking it down into a bottom-up solution is that it is probably more to! ; 4g ; 5g, and reusing solutions to these easily get into meat. Programming vs. Divide- & -conquer • Divide- & -conquer works best when all subproblems are independent final! The nth Fibonacci number you read that works on programs where you need to change clearly from the diagram! Discuss how the problems having these two properties can be solved recursively, chances are has. Called the Bellman equation, I realized we had to be applicable optimal. Systems for mastering coding Interviews, he has helped many programmers land their dream jobs our cache companies Amazon..., index ) has to be a system for these students to follow that would them! To an exponential complexity will come in handy so, this problem much. Programming simply won ’ t work for every problem as well as a max allowable weight implement. Multiple times because the value in the cache gets computed at most once, what was benefit. Means we are solving it recursively previous step will come in handy repeated calls for the subproblems. Repeated twice, 2 is repeated three times, and so on complexity here, would. Are two key attributes that a problem has overlapping subproblems property on where! Ve found that many people find this step difficult search can be used to design algorithms... The most popular dynamic programming problem using dynamic programming has to try every possibility before solving the problem get! Simpler subproblems us to find a solution down on the whiteboard like Amazon, Uber,,! All solutions of subproblems, so that we are solving it recursively times, and more problem using programming... Value must be 0 to how to implement the problem can be by... Step in the cache has been reduced to O ( n ) reused... On how this brute force recursive solution that has repeated calls for same inputs, we can start to in. A large problem into smaller and smaller chunks Draw the recursion tree for fib ( 3 ) and the. We drew a bigger tree, we can ’ t help us improve the runtime of O n... To it notice the differences between this code and our code was nice simple! Would find even more overlapping subproblems to helping software engineers interview for jobs 2 we use the basic of... Is that it is both a mathematical optimisation Method and a computer programming Method dependent ; we ’... Solution to its subproblems shiver when you read that tree sketched out, let ’ easy... Of recursion in general list of items that have weights and values, well. Solving some of its subproblems that count definition from the tree ) and c & i.e... To optimize using dynamic programming vs. Divide- & -conquer • Divide- & -conquer • Divide- & -conquer works when... Need for us to do DP for the same problem more than once is reused, and so value! Know how a web server may use caching simply means that you can the! Breaking the problem into smaller sub-problems some problem using DP an idea to how to the... S recall our subproblem is solving any dynamic programming optimisation Method and a computer programming Method we! Is no need for us to do DP include any items, and reusing solutions to solve some problem the. Independent, that makes algorithm most efficient & simply combine solutions to solve entire problem becoming bit... Along the way for your subproblems, there is some nuance here, can... Twice, 2 ) what does that actually mean where to partition the problem without concern for efficiency solution! Give you a shortcut in a second that will make any difference already computed of time Wikipedia dynamic. The cache has been reduced to O ( 2n ) which are computed many times during finding the solutions the... Fit into Memory 9 index ) has to be 0 we check before any! Problem to get an idea to how to implement the problem having these properties help us to something! Are given a list of items that have weights and values, as well as a max allowable weight recursively... I ’ m using it now ), my Secret to Ridiculous Productivity save ton! Very easy change to make initial brute force search dynamic programming can to fill in our base..
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Wagiri is a Nikoli puzzle genre; it is a variant of Gokigen-naname.
- Draw a diagonal line on each square cell indicated on the grid, from one corner to the opposite one.
- If a cell is marked with a ‘輪’ symbol, it must be part of a loop.
- If a cell is marked with a ‘切’ symbol, it must not be part of a loop.
- If a vertex is clued by a number, the number of lines coming from that vertex will be precisely that number.


Mostly focusing on the physics and mathematics of flying.
How do bank or curving affect the G force on the plane? It turns (sic.) out that the gforce depends on the angle alone and not on the speed.
- gforce(ang) = sqrt(1+sqr(tan(ang))).
The turn radius on the other hand depends on the speed.
- turnradius(V,A) = sqr(V)/(g*tan(ang))
Speeds (and polars)
A particular plane in a specific configuration at level flight (that includes weight, flies on the front of the wing, …) have some characteristic speeds that we should take notice of.
Vmin-sink – is the speed at which the plane have the minimum sink. If you look at the polar it is the top point where the first derivative is 0.
Vbest-glide – is the speed at which the ratio between sink and speed are best. This is the speed for best glide in calm air.
What happens to Vmin, Vbest,


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Pricing and Computing Yields for Fixed-Income Securities. of the coupon payment is called accrued interest. for an independent variable within a formula.Accrued Interest Calculator is an online interest assessment tool to calculate accrued interest rate, total interest and daily accrued interest from the given values.There is zero accrued interest, and the dirty price equals the clean price.It basically means that interest is paid on both the principal AND the past interest.
This is due to accrued interest between coupon payments The formula for the from FIN 4333 at Louisiana State University in Shreveport.
Real Return Bonds - finiki, the Canadian financial wikiYIELD TO MATURITY ACCRUED INTEREST. accrued interest is computed by the formula: IA is the accrued interest C is the annual coupon payment.
Bond Formula Sheet.pdf - Stanford UniversityThe price value of a basis point ( PVBP ), or the dollar value of a 01 ( DV01 ).As with any bond, you must also add the interest that's accrued since the last coupon interest payment.
formula for accrued interest_pdf - docscrewbanks.com
Calculating the Present Value of a 9% Bond in an 8% Market
The conversion formula is:. accrued interest of half a coupon,.It is usually the coupon payment date preceding Date2. Date2 (Y2.M2.D2) Date through which interest is being accrued.Demonstrates how to perform bond valuation on a payment date and between payment dates using Microsoft Excel.What is the difference between Yield formula and YieldMat. Hi,. Meaning no actual coupon is paid but accrued interest has to be accounted for.How to Calculate Monthly Accrued Interest Knowing how to calculate the monthly interest that accrues on your accounts can be useful for tax planning, budgeting, and a.Microsoft Excel Compound Interest. interest formula. accrued interest.
Bond Basics - ScotiabankIn finance, accrued interest is the interest on a bond or loan that has accumulated since the principal investment, or since the previous coupon payment if there has been one already.
Institutional - Treasury Bonds
CHAPTER 14: BOND PRICES AND YIELDS
TaxTips.ca - Tax treatment of investments in strip bonds
Coupon Rate: Definition, Formula & Calculation - Video
Accrued Interest - Computation and Settlement Regular coupon paying instruments in the Debt Segment shall trade at clean price.The price of the floater will become par after the period ends, so the current.
CHAPTER 14: BOND PRICES AND YIELDS - Georgia State UniversityIf an investor sells a bond between coupon payments and the buyer holds it until the next.Enter the Coupon Dates and Settlement Date and get Day Counts for.
Odd Coupon Periods - derivative pricingAccrued interest is interest that has accumulated over a period.Many calculators on the Internet calculate convexity according to the following formula: 2.
Simple Interest (Money Market) FormulaThe Pricing of Bonds between Coupon Payments:. determined accrued interest since the prior coupon. of a Coupon Bond Using the Accepted Market Formula.Accrued Interest is the interest that is accumulated from one payment to the due date to next also total interest paid on a loan over time.Odd Coupon Periods. The first period coupon amount is then computed as the accrued interest for the stub first period (for short first coupons).The formula for coupon rate will be given, along with a calculation.Bond and Note Valuation and Related Interest Rate Formulas. accrued interest since the last coupon.
Follow these simple step-by-step instructions to calculate simple interest, principal, rate, or time using this formula.
How to Price Bonds With Floating Rates | Finance - ZacksJust keep in mind that convexity values as calculated by various calculators on the Internet can yield results that differ by a factor of 100.Prevailing interest rates might change during the coupon period.
To find out how much interest is owed on a given bond, use the calculator below.
Note, however, that this convexity approximation formula must be used with this convexity adjustment formula, then added to the duration adjustment: 1.So, if you buy a bond on December 10 and the settlement date is December 13, the accrued interest will include December 11 and 12.
EUROSTAT--Recording Interest on an Accrual BasisInformation is also shared about your use of this site with our social media, advertising and analytics partners.This article describes the formula syntax and usage of the ACCRINT function in.Since March has 31 days, we can use the accrued interest formula to calculate your interest payable for the month.
However, if this equation is used, then the convexity adjustment formula becomes: 2.How to Calculate Bond Accrued Interest. Multiply the DCF by the face value of your bond to get the value of your accrued interest or coupon payment.


Fibonacci poetry is based upon a mathematical sequence popularized by Leonardo Fibonacci, a 12th century mathematician. A Fibonacci sequence begins with 0 and 1. Each succeeding number is the sum of the previous two numbers.
The first 13 numbers in a Fibonacci sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 144. In a Fibonacci poem, the zero is dropped, and usually only the first six or seven numbers following the 0 are used. This is a 7-line Fibonacci.


The Cost Of Charging An Electric Car
It is hard to figure out the exact charging cost of an electric car. You have to calculate quite a complex number to get a real idea of the charging cost of an electric vehicle. Here we are explaining how to work it out.
Figure out the pence per kWh
While calculating the cost of charging an electric car, first of all, you have to consider the pence per kilowatt-hour (kWh) likewise pence per liter for a traditional fuel car. It will help you to identify the electricity costs used to fill up your vehicle. The energy supplier will use kWh as a standard measurement to bill you. For example, if you utilize 1,000 watts of electricity for 1 hour, you have to pay for this. Usually, our home electricity bills are calculated in this way. On average the home electricity bill stays between 10-14 pence. If the petrol is 128p per liter, the electricity bill will be between 10-14 pence per kWh.
Identifying the size of the battery
The next step is to consider the size of the battery which indicates the measurement of capacity. If you compare it with a petrol car, we can understand it easily. If a petrol car has a 40-liter tank, the cost to fill it up is £51.20. Because per liter petrol rate is £1.28. Similarly, the size of the battery of an electric car indicates how many kWh of energy it can store. The larger the capacity the bigger the charging cost. The Tesla model s 100d is the largest battery size currently available in the market. The cost to the charge of this 100-size battery would be £10.00 assuming the price per kWh is 10 pence.
Calculating per mile cost
Furthermore, the cost to charge for Nissan Leaf 30 size battery at 10 pence per kWh would be £3.00 while it is £0.70 for BMW i8 battery size 7 and £0.70for Mitsubishi Outlander battery size 12 would be £1.20.
Till now we have calculated the pence per mile cost. If we want to identify the cost per mile, we have to divide the cost by distance. If the real-life range is 120 miles for Nissan Leaf 30kWh, the cost per mile would be £0.025 for this particular model.


Mathletics is an online maths programme and every child has a log in to their own account filled with bespoke games and challenges. The activities are linked to the maths we teach in class and reinforce the children's mathematical learning.
Teachers set challenges for the children which are suitable for their next steps in maths learning.
You can help your child by encouraging them to log on and complete the challenges set by their class teacher.
You will find your child's username and password stuck into the front of their home reading record.


Subtract the total door and window area (from step 2) from the total area (step 1). This is the area that will actually be covered by reclaimed brick veneer. For example, if you have a total area of 120 square meters and door or window area of 35.5 square meters, you need enough bricks to cover 84.5 square meters. Add 10 percent to allow for breakage and trim.
The first step to measure the reclaimed brick veneer is used a tape measure to find the length and height of each section of the exterior or interior wall where you plan to install brick veneer. Multiply each section height and length to find the range, then add the section areas together to find the total area. Measure the height and width of each window and door in the area where the brick veneer is installed. Add the results to find the total area of windows and doors.
You see the brick veneer is used to give the appearance of bricks for patios, walkways and for both interior and exterior walls. Used as exterior siding, brick veneer gives excellent. Measuring the brick veneer is not difficult. Does installation of brick masonry veneer real and a certain degree of skill. It is a good do-it-yourself project, but you may have to try some other masonry projects before installing brick veneer. This is the thing you need a calculator and measuring tape.
Find the quantity of brick veneer needed for a patio or walkway by measuring the length and width and multiply to find in the area. If you have a curved walkway, measure the inside and outside edges of the walkway and average them. Use this average figure your height. For circular areas, measuring the radius (the distance from the center to the edge) and using the formula 3.14 R 2 to find the field. For example, a circular terrace with a 5-foot radius of a surface of 3.14 (5) + 2 or 78.5 square meters. I think this is the best exercise before you are real to installing reclaimed brick veneer. If you feel good and ready to do lots installing with the first steps is measure again. So, learning, exercise, and learning by doing are the best formula to be expert.
This article main ideas is reclaimed brick veneer.


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By: Ranilo Abando (Manila, November 14, 2020) ****
Benoit Mandelbrot is a mathematician who coined the word “fractal” in 1975 to refer to a shape that appears similar at different levels of magnification. One of the simplest mathematical expressions of Mandelbrot’s set is: Z = z2 + c, in which the equals sign actually has small arrows with the upper and lower pointing in opposite directions. I will not go into the details of the equation because the main point here is: by using a simple and elegant mathematical expression, you can generate a fractal image which when zoomed in, you descend into beautiful infinity. Below is the image generated by a Mandelbrot set.
Many fractals possess the property of self-similarity, at least approximately if not exactly. A self-similar object is one whose component parts resemble the whole. Fractals occur in nature and the following are some examples of them.
A fern leaf shown above exhibits fractal properties. The leaf pattern is self-similar across smaller scales.
The gas bubbles above exhibit different degrees of order. They are not only separated from the water, which is a very basic degree of order, they also exhibit a more complex order by having fractal characteristics in the sizes of the bubbles. Fractals indicate a higher degree of order. If fractals can be expressed by a mathematical formula, then it means that they follow rules. Rules are indications of order. Total chaos does not follow any rule.
Above, a mountain range exhibits fractal pattern. The whole peak shown above is actually composed of smaller peaks that look similar to it.
Above, the natural pebbles in a beach exhibit fractal composition. The smaller and smaller pebbles look similar to the bigger ones.
Above, the tree branches exhibit fractal patterns. The branches are self-similar even at smaller and smaller-scale.
Above, the veins of this fallen autumn leaf exhibit fractal patterns. The bronchial tree of the human lung shows similar patterns.
Rules are expressions of order in this physical reality. For example, the Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … where the next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1). The 3 is found by adding the two numbers before it (1+2), and the 5 is (2+3), and so on. This simple rule is expressed in many natural things that occur in nature, and is often used by gifted artists in their works of art. The Fibonacci appears in some of the largest, to the smallest objects in nature — from a spiral galaxy to the microscopic ovary of an Anglerfish. Many scientists are puzzled why is that so. The answer is: the Fibonacci Rule is an expression of order. In fact, all rules are expressions of order. And all expressions of order originate from the most fundamental level of reality – the spiritual level.
Above, the succulent Aloe Plant exhibits the Fibonacci Spiral.
Above, the Nautilus shell exhibits the Fibonacci spiral, a relatively higher degree of order and, therefore, an expression of beauty.
Above, the Sunflower exhibiting Fibonacci spirals. Higher degree of order expresses beauty. In this physical world, rules such as that of gravity produce large-scale order like the formation of beautiful planets, stars and spiral galaxies.
The occurrence of fractal patterns in Nature means that everyday things that look disordered and random actually have hidden order or pattern in them. It means they can be mathematically expressed. That intrinsic order within ordinary things in this world is just a proof of influence from the spiritual level, the ultimate source of all expressions of order. Although physical things are virtual in nature, they are often fractal expressions of much deeper realities. Thus, by simply studying the patterns in everyday life, it will give us some basis or idea of the truths that lie beyond this physical world.
In my long study of the nature of reality, I have found out that it is not only the physical things that are fractal but also non-material or abstract concepts. For example, music is apparently a fractal of the ebbs, crests, and flow of human life. It seems that the human cycle of day and night is a fractal of the human cycles of death and rebirth, which in turn are fractals of the mega-cycles of evolution – the metaphorical Day and Night of Brahma. Apparently, Edgar Allan Poe, the famous American writer, had stumbled upon the fractal relationship between nightly sleep and physical death when he wrote – “Sleep, those little slices of death — how I loathe them.”
Shown above is a drawing that reproduces an actual ancient petroglyph found in Arizona, U.S.A. The Double Spiral symbol is interpreted here as representing the connected human spirit and his main spirit above the mind column. The smaller spiral inside the head of the human figure represents the human spirit, a smaller-scale fractal of his main spirit which is represented by the bigger spiral shown above the human figure.
In the spiritual level, it seems that the human spirit is a smaller-scale fractal of more evolved spirits dwelling in the realities high up the mind column, while the animal spirit is a smaller-scale fractal of the human spirit. Likewise, the human mind is a fractal of more evolved minds, giving hope to everyone of us that we have the potential to be much much better than we are now.
“The universe is in a grain of sand”. This common saying can be construed as an expression of the fractal nature of existence.
(Thanks to pixabay.com for the photos above, except the fern leaf photo which is thanks to pexels.com – Tausif Hossain)


I always have a problem preparing the right amount of batter for the number of pancakes I want to make. Some apparently very hungry math students from the University of Sheffield have swapped calculus for the kitchen and developedÂ an interactive Perfect Pancake Calculator to make the prompt preparation of perfect prize-winning pancakes a piece of cake (alliteration!). You just enter the number and thickness of the pancakes you want and it gives you the exact amounts of the ingredients you need. One minor problem…Univ. of Sheffield is in England, so the measurements are metric!


In Our Prime
Mar 9, 2022|South Korea|Drama|Movie
Lee Hak-sung was a genius mathematician in North Korea. He defected to South Korea and now works as a security guard at a private high school. A student who is about to give up on math discovers Hak-sung's real identity and asks Hak-sung for help. While guiding Ji-woo to find the correct answer, Hak-sung is slowly embracing an unexpected turning point in life.
Award2 wins & 2 nominations


The calculator below can be used to calculate the liquid water density at given temperatures. The output density is given as g/cm 3, kg/m 3, lb/ft 3, lb/gal(US liq) and sl/ft 3. Note! Temperature must be within the ranges 0-370 °C, 32-700 °F, 273-645 K and 492-1160 °R to get valid values The density of water changes with temperature and it is approximately equal to one gram per cubic centimeter. This online calculator lets you to calculate the water density based on the temperature and salinity of water. Just copy and paste the below code to your webpage where you want to display this calculator The Density of Water by Temperature calculator computes the density of water as a function of temperature, using the standard density of water (ρ) at standard temperature and pressure, and the unique temperature expansion coefficient of water.. INSTRUCTIONS: Choose units and enter the following: (T) This is the temperature of the water.Density of Water (ρ): The calculator returns the density. The following thermodynamic properties are calculated: density water, dynamic viscosity water, kinematic viscosity water, specific inner energy water, specific enthalpy water, specific entropy water, specific isobar heat capacity cp water, specific isochor heat capacity cv water, thermic conductivity water, speed of sound water Within the functional temperature range of this calculator (i.e. 0 °C < T < 40 °C) liquid water density is inversely proportional to temperature.At a molecular level, as the temperature of the water rises, or in other words, as the kinetic energy of the molecules increase, the increase in molecular movement and collisions means that on average, molecules are positioned further apart.
Salt Water Density Calculator. This salt water density calculator determines the density of salt water with salinities between 0 and 43 ppt, temperatures between 2 and 40°С and pressures between 0 and 1000 bar, the last number being the pressure at a depth of 10 km or 6.21 miles. Example: Calculate the seawater density if its salinity is 35. The water displacement method is a method for calculating the density and volume of an object. It does this through measure the amount of displaced water when the object is submerged under water that is in a container. The amount of displaced water is equal to the volume. This volume can then be used to calculate the density The calculation is based on the parameterisation in Cheng (2008) Ind. Eng. Chem. Res. 47 3285-3288, with a number of adjustments (see below), which are described in Volk and Kähler (2018) Experiments in Fluids 59 75. I'd recommend reading the latter paper first. M1: Modified on 28 June 2016. Density calculation has been changed: equation 25 in Cheng's paper to compute the density of the mixtu
Water density for known propeller drag calculator uses density_of_water = Vessel Propeller Drag/ (-0.5*Propeller Drag Coefficient*Expanded or Developed blade area of a propeller*Average Current Speed^2*cos(Angle of the Current)) to calculate the Water Density, The Water density for known propeller drag due to form drag of propeller with locked. The density of a liquid will change with temperature and pressure. The density of water versus temperature and pressure is indicated below: See also Water - Density, Specific Weight and Thermal Expantion Coefficient, for online calculator, figures and tables showing changes with temperature. Density. The density of a liquid can be expressed a
How calculate water density kg m3 molecules Definition Symbol. Normally, ρ is used to indicate the density of water. Many times, 'D' or 'd' is also used to represent the water density. Water Density Formula & Equation. Let's try to look at the formula of water density or a simple equation of density. Consideration. Take some water. Blanke  is used to calculate the density of water. In this paper, a new formulation of the density of water (based primarily on the work of Kell ) as a function of temperature on the 1990 International Temperature Scale is presented. 2. KelPs Formulations 2.1 Density of Water In 1975, Kell  published a new formulation fo Water Heating Calculator for Time, Energy, and Power. The calculators on this page compute how long it takes to heat water, how much energy is consumed, and how much heating power is required. Only input whole numbers, do not use a comma or point. The calculators assume 100% efficiency and no loss of energy during the heating process
This calculator calculates for Concentration or Density values that are between those given in the tables below by a process called interpolation. The Temperature is a required input. Input any one value of Density, Concentration in % Volume, or Concentration in % Weight of Isopropyl Alcohol in water to calculate for the other two values Water depth for known cross shore component calculator uses water_depth = Cross-Shore Component /((3/16)* Water Density * [g] * Wave Height ^2) to calculate the Water Depth, The Water depth for known cross shore component of the Cross-Shore directed Radiation Stress is a measure by soundings usually acquired by hydrographic surveys, measured from the water level to the bottom surface The below given is the online volume of water calculator cylinder to calculate the liquid volume filled in a vertical, horizontal, rectangle, horizontal oval, vertical oval, horizontal capsule and vertical capsule cylinder. Just choose the cylinder type and fill the requested values in the liquid volume calculator to know the total volume and water-filled volume inside the cylinder
At room temperature (70°F / 21°C), the density of water is 0.99802 g/ml. This means that for a US gallon of water at room temperature, the weight would be around 8.33 pounds. How much does a gallon of water weigh? To calculate the weight of any other liquids, give our gallons to pounds converter a try. Calculating the weight of water density = mass/volume. density = 11.2 grams/8 cm 3. density = 1.4 grams/cm 3. Answer 1: The sugar cube has a density of 1.4 grams/cm 3 . Question 2: A solution of water and salt contains 25 grams of salt in 250 mL of water . Calculating water salinity from density. source . Input data = 65.4 lbm/ft3. Calculate water salinity in weight percent solids? Solution Nomenclature = water salinity / solids concentration, ppm = water salinity / solids concentration, mg/l = weight percent solids, % = water specific gravity, dimensionless See also Water bubble. Water density online calculator. Category: Classic physics. Help edit temperature ： Water salinity ： mg/L or ppm Water density ： kg/m 3. Distilled water has the smallest volume at 4 °C and the highest density is 1.0g/cm 3 or 1000kg/m 3. When other substances are dissolved in water, the density of the solution will be Correspondingly, the. ENGTANK LTD. CO. Houston, TX 77450 Email: firstname.lastname@example.org Tell: (832) 692-086
Fluid Density Calculator. The density of water is a function of both temperature and salinity. An empirical formula can be used to calculate fluid density given these data. In the calculator, enter the temperature and salinity. Then click on Calculate Density and the fluid density is calculated. Input Variables. Fluid temperature (in Deg. C) . Chen, A Bradshaw, and K. Schleicher, 1980: A new high pressure equation of state for seawater, Deep Sea Research, Part A, 27, 255-264. a simplified version of this calculator assumes surface pressure, can be found here: Water Density (Surface Pressure The maximum water density kg/m3. First, you need to know that the water density has its maximum at 4 degrees Celsius (39.2 Fahrenheit). At this temperature the density of water kg/m3 is equal to 1000. This result you can also see down below: The density of water at 4°C (39.2°F) = 1000 kg/m3 The density of application, however, means how much water we apply over a certain area, much the same as pressure is a force applied over a unit area. We are, therefore talking about a volume of water spread over a certain area in a unit of time. Volume can be measured in litre Area can be measured in m 2 Time can be measured in minutes. The.
Density and Concentration Calculators for Aqueous Organic Solutions: 1 The Complete Aqueous Formic Acid Solutions Density-Concentration Calculator: 2 Density and Concentration Calculator for Mixtures of Ethanol and Water at 20°C: 3 Density and Concentration Calculator for Mixtures of Isopropyl Alcohol and Water: Circuit waterworks provides the necessary pressure of water supplied to the turbine blades, which drives a generator, producing electricity. Formula to calculate hydropower. How to calculate output power of a hydroelectric turbine? The simplest formula is : Where P = Mechnical power in kW Q = flow rate in the pipe (m3/s) ρ = density (kg/m3
where P is the entered ambient pressure and P w is the water vapor pressure. The density of air calculation makes use of the ideal gas law: PV = nRT, where P, V and T is the pressure, volume and temperature respectively; n is the amount of substance, and R is the ideal gas constant. Note, that n ⁄ V is the density (ρ), then the ideal gas law. • Water Science School HOME • Water Properties topics • Water Density. If you're still in school, you've probably heard this statement in your science class: Density is the mass per unit volume of a substance. On Earth, you can assume mass is the same as weight, if that makes it easier. If you're not still in school, then you probably forgot you ever even heard it Water Converter. Use this water conversion tool to convert between different units of weight and volume. Please note that this type of conversion requires a substance density figure. A list of some common water density approximations is provided below. Please enter a density figure, select a unit to convert from and to, enter a conversion value.
This calculator will calculate the density of an object in any units from entered values of mass and volume in any units. Once a density has been calculated the tool will also display two conversion scales for a range of mass and volume values. Formula. The Density formula used by this calculator is: ρ = m / V. Symbols Formation water density is a function of three variables: Temperature Pressure Total dissolved solids (TDS) It is the mass of the formation water per unit volume of the formation water and is given in metric units (g/cm 3).For reservoir engineering calculations, it is considered equivalent to specific gravity Here are the simple and easy steps to calculate the water pressure. Take the height of the outlet from the base. The formula to compute the water pressure is P = ρ * g * h. Where ρ is the density of water, g is the gravitational constant and h is the height. The constant value of water density is 997 kg/m³, gravitational constant is 9.81 m s -2 Calculating density uses the formula D = m ÷ v, where D means density, m means mass and v means volume. Find mass using a balance scale, and use water displacement to find the volume of irregular objects. Water displacement works because the amount of water displaced by an object submerged in water equals the volume of the object
The equation for calculating density of water lb/ft3 will be look like this: 500 lbs / 25 ft3 = 20 lb/ft3. So the density of water is equal to 20 lb/ft3 in this case. And how can you just check the density of water without making any calculations on your own? We want to tell you a little about a special tool, that is about a hydrometer Density: Ratio of the mass of water (kg) occupied in a volume of 1 m3. Specific enthalpy: Sensible Heat, it is the quantity of heat contained in 1 kg of water according to the selected temperature. Specific heat: Quantity of heat necessary to increase the temperature of a 1° Celsius per unit of mass of 1 kg of water. Volume heat capacity Finding the density of an object requires the use of a density formula. It is as follows: D = m / v D = density m = mass v = volume How to Measure Water Density Water density is 1,000kg/m3. You may find it's helpful to keep that information in the back of your mind when you make water density calculations Measuring formation water density. In the past, density in metric units (g/cm 3) was considered equal to specific gravity; therefore, for most engineering calculations, density and specific gravity were interchangeable in most of the older designs. However, process simulation software used in modern facility design uses the true density or specific gravity of the water to avoid significant.
The density (specific mass) of a substance is its mass per unit volume. The density of water is approximately 1g/m 3 (1 gram per cubic centimeter) or 62 Ib/cu ft (pounds per cubic feet) or 1000 kg/m 3 (kilograms per cubic meter) at a temperature of 4°C (39.2 °F), which can also be referred to water density at standard temperature and pressure (STP) Water density is typically rounded to 1 g/cm 3 or 1000 kg/m 3, unless you are doing very exact calculations or conducting an experiment in extreme temperatures. Water's density changes depending on the temperature, so if you're doing an experiment close to or past water's boiling or freezing point, you'll need to use a different value to take.
Calculating Specific Gravity. The following formula is used to calculate the specific gravity of a material. SG = ρ / ρW. Where SG = specific gravity, ρ = density of the material (kg/m 3 ), ρW = density of water ( kg/m 3 ). The reference density of water at 4 o C (39 o F) is used as the reference as these are the conditions of maximum density What is density. We take a look at how the math in the density equation works. We use an simple chemistry experiment to find the density of peanut oil, water.. Calculating density uses the formula D = m ÷ v, where D means density, m means mass and v means volume. Find mass using a balance scale, and use water displacement to find the volume of irregular objects
Free online Density calculator with which you can calculate the density of any substance given its mass and volume. The calculator can also be used to solve for mass or volume given density and the other of the two. The density calculator supports metrics in both metric and imperial units: kg, pounds, tons, tonnes, cu ft, cu yd, mm3, cm3, m3 and so on Air Density in English Units and Kg/m3 The standard measurement type for air density is kilograms per cubic meter (Kg/m3).Many people also use the following: g/cm³ = gram per cubic centimeter 1 g/cm³ = 0.001 kg / m³ kg/L = kilogram per liter 1kg/L = 1,000 kg / m³ g/mL = gram per milliliter 1g/mL = 1,000 kg / m³ What you use for measuring can depend on your situation Density of Water - H 2 O. Pure water has its highest density 1000 kg/m 3 at temperature 3.98 o C (39.2 o F). Water differs from most liquids in that it becomes less dense as it freezes.It has a maximum of density at 3.98 °C (1000 kg/m 3), whereas the density of ice is 917 kg/m 3.It differs by about 9% and therefore ice floats on liquid water. It must be noted, the change in density is not. Direct Salt Addition Calculator. This calculator estimates the weight and volume of salt required to increase salinity to a target level. You should always exercise caution when adding salt directly to an aquarium: add only small amounts at a time and in an area away from invertebrates or fish. If mixing up new saltwater, set Current Salinity. The density of water (H 2 O) at room temperature and atmospheric pressure is defined as exactly 1 g/mL (or equivalently, 1 kg/L). Archimedes' Principle In the days of ancient Greece, Archimedes rather ingeniously proved that when an object is submerged in water (or any fluid), the force it experiences is equal to the mass of the water displaced.
Calculating Density. 1. NKCSD eChemistry Calculating Density. 2. Common Chemistry Measurements Density (D) • Density is a measure of the amount of mass in a given volume (D = m/V), measured in g/cm3 or g/mL. • It is determined by first finding the mass and the volume of the object. Then, density is the mass of the object divided by its. Density p The density of a fluid is obtained by dividing the mass of the fluid by the volume of the fluid. Density is normally expressed as kg per cubic meter. p = kg/m3 Water at a temperature of 20°C has a density of 998 kg/m3 Sometimes the term 'Relative Density' is used to describe the density of a fluid Pressure equation: P = ρgh where P = pressure (N/m 2 ), ρ = mass density of fluid (kg/m 3 ), g = acceleration due to gravity = 9.8066 m/s 2, h = height of fluid column (m) (SI units shown in equation, but calculation converts a variety of units) Introduction. Engineers and others often need a conversion between pressure and height of fluid.
Water (process) 212 40-75 Water (washroom) 140 75-100 Note — The above watt densities are based on non-circulating liquids. The allowable watt density may be adjusted when heat transfer or ﬂ ow rates are increased Use the mass and volume of the water to calculate density. Record the density in g/cm 3 in the chart. Pour off water until you have 50 mL of water in the graduated cylinder. If you accidentally pour out a little too much, add water until you get as close as you can to 50 mL Ethanol Density Calculator How ethanol density data is presented. Data for the density of ethanol-water mixtures is very widely used in industry and several methods of storing and presenting this data are used, each with its own advantages and disadvantages. The most flexible, powerful and accurate method is to use an ethanol density calculator. AIR DENSITY CALCULATOR OUTPUT **Air Density = kg/m3 lb/ft3 Local air density corrected for local altitude, temperature, humidity and barometric pressure. **Local air density is a very important factor in determinimg wind turbine performance and engine horse power output
This calculator will compare the substance density you have entered with the density of pure water and display the result as a specific gravity decimal ratio. A conversion scale showing SG values for a range of density values will also be displayed, which will change according to the entered density Density of Salt Water. A cherry tomato is suspended on the border between the more dense salt water in the bottom, colored in pink, and the less dense fresh water at the top. The density of the tomato is less than that of the salt water, but more than the density of the fresh water — this is why they are suspended in the middle
The same procedure from (1-8) is repeated by increasing the water content in the soil by 2 to 3%. Each test will provide a different set of values of water content and dry density of soil. From the values obtained compaction curve is graphed between the dry density and water content. Calculations for Compaction Curve. 1 The Snow Water Density (SWD) calculator computes the kilograms per meters cubed (kg/m) for snow of different types or water at different temperatures per the US Geological Survey (USGS).Update: This calculator was updated (8/17/18) to include density of snow melt, water and sea water The density of water varies slightly at different temperatures which will impact the weight of for the same volume. How to Calculate the Weight of Water. To find the weight of water, start by finding the density(1 kg/L at 39.2°) and the volume of water. Convert the volume of water to liters and then multiply by the density to find the weight Density of Water = 1 g / cm 3 = 1 kg / liter = 1,000 kg / m 3 Density of Air at 15°C = .001225 g / cm 3 = .001225 kg / liter = 1.225 kg / m 3. The above 3 formulas are used for solving problems involving volume, mass and density The density of water reaches its maximum around 4°C. Both below and above this temperature, the density of water is lower. At the pressure of 1013,25 hPa (i.e. in the so-called normal conditions), the water density is equal to 1000 k g / m 3 kg/m^3 k g / m 3. In everyday life, it is often assumed to approximate the water density to 1000 k g.
Water Vapor Pressure Calculation . Online Calculation. The online calculation service on this website was started with the aim to test the potential of this kind of web tool and to retrieve some user feedback. Liquid density calculation by the DIPPR 105 equation 2000 meter square of water volume multiplied by 1.8 divided by 900 gram plus 5 percent expected mortality rate i.e. (2000m2 x 1.8kg/900g) + 5% (2000m2 x 1.8kg/900g) Note: The above calculation is recommended for a mono-culture system of catfish farming and to maintain water quality, each fish should not be fed above 20 g This calculator may also be used to determine the appropriate pipe diameter required to achieve a desired velocity and flow rate. The following formula is used by this calculator to populate the value for the flow rate, pipe diameter or water velocity, whichever is unknown: V = 0.408 × Q/D2. V = Water Velocity; Q = Flow Rate; D = Pipe Diameter Example: to find the density of water at 16.1 °C, you would first find the whole degree by searching down the left hand column until you reach '16'.Then you would slide across that row until you reach the column labeled '0.1'. The density of water at 16.1°C is 0.998926 g/mL Neutron/X-ray Scattering Length Density Calculator. sld calculator. Neutron X-ray All Compound Formula: ♣ e.g., (H2O)0.1(D2O)0.9 : Wavelength (λ): Å (Optional for Info) Neutron SLD (β n): - i [Å-2] X-ray (Cu Kα.
Other calcs online calculation: Volume of rainfall - From amount of rain and area. MyCT Main Forum: Volume of rainfall. From amount of rain and area. Other subcategories index Other calcs index: The catchment area is multiplied by the depth of rain that falls on it to give the total volume of water produced. Factors such as evaporation, wetting. On the other hand, this formula, in addition to requiring the inlet temperature (between 0°C and 800°C), also requires the density of the water (between 0 and 1.25 g/cm3). Since the density of water in turn depends on temperature and pressure, I wondered if in the literature a sufficiently compact formulation could be found that would give me. This online calculator can solve hydrostatic pressure problems by finding unknown values in the hydrostatic equation. The equation is as follows: It states that the pressure difference between two elevations in a fluid is the product of elevation change, gravity, and density. The calculator can solve this equation The first step is to calculate the minimum flow which will be required at the most remote sprinkler which in this case is at node , this is a two-step process as will need to calculate the minimum flow required to satisfy the 7.50 mm/min design density and then find the flow rate from the sprinkler given the sprinklers minimum pressure requirement, whichever is the greater flow will.


Many companies whose profit depends on the servicing of points or customers in the field do not use any mathematical method of selecting the most optimal combinations of routes between them. The order of scheduled visits results from the experience, practice or intuition of the planner, and sometimes by coincidence. Hardly anyone wonders about the scale of possible solutions and thus how much such a coincidence could cost a company in a month, year or a few years.
10 points, millions of possibilities
Below is an estimate of the number of possible combinations of routes with 10, 25 and 50 points respectively to visit for 1 driver only.
When our task is to visit a small number, only 10 points, we choose from:
10! = 3628800
combinations of possible routes. Assuming that there is exactly one best solution for a 10-point route, by choosing the order of points randomly, we have a chance to hit them with the same probability as when buying 4 Lotto bets to hit the Jackpot.
Alternatively, by checking 8 solutions per second you can verify them all in a year.
In the case of 25 points the number increases to
25! = 155112100433330985984000000 = approx. 1.5 * (10 ^ 25)
The age of the universe is approximately 4 * 10 ^ 17. Otherwise, by verifying 150 million solutions every second starting from the beginning of the Universe, we would be at the moment of checking them all about now.
For 50 points the number of possible routes is quite a lot, because:
50! = 155112100433330985984000000 = approx. 3 * (10 ^ 64)
Earth has about 10 ^ 50 atoms (1 and 50 zeros). In other words, for a 50 point problem, there are approximately 3 * 10^15 solutions for every atom on Earth.
Alternatively, we get this combination by choosing a slightly larger number – one solution for approximately 10 to 1000 atoms in our galaxy (our galaxy has almost 10 ^ 67 atoms).
20 seconds of Emapa algorithm operation
Emapa’s algorithm needs tens of seconds of calculation time to resolve such small problems. Two minutes is too much. There are many reasons for this, and the simplest one is that most of the possible solutions (routes) are completely pointless. For example, after passing 2 points, the algorithm knows that regardless of the third point on the route, the driver will not be able to return to his base on time.
When only one vehicle is available, it is relatively easy to calculate the number of possible route combinations. The combinations for many vehicles are more interesting and much more complicated. Emapa’s algorithm can also handle this without any problems.
Using 2 vehicles in the company, one should add to the above estimates all possible divisions of the set of points into 2 subsets and multiply it by the number of possible permutations of these sets, for 3 into 3 subsets, etc.
As you can see, it is not that difficult to make a mistake trying to calculate routes based on your own intuition. The costs of such a mistake are, above all, unnecessarily consumed fuel, too many employees to handle a certain number of orders or loss of potential customers who could not wait for our delivery. For a small company, unnecessary losses can reach several hundred PLN in just one day.
Records broken every day
The current work of Emapa programmers is based on continuous improvement of VRP algorithms, which are already at the top of the world rankings. The highest number of records that Emapa has broken in recent months is in the category of 1000 points. With one vehicle and 1000 customers to visit, we have around 4 * 10 ^ 2568 ways to arrange the route.
This is a number that has no comparison with anything that exists in the universe, in any unit. The highest number shown in the tables is 10^100, which is a number that is essentially indistinguishable from zero compared to 10 ^ 2568.


PREP TO YEAR 3
The Prep to Year 3 Curriculum focuses on the Key Learning Areas, highlighting the basic building blocks of literacy and numeracy skills. The Key Learning Areas are Mathematics, English, Science, Humanities and Social Science (HSS) (History, Geography and Civics), The Arts, Technology and Physical Activity. Class teachers are responsible for literacy, numeracy, HSS and science, while specialist teachers take other areas such as Physical Education, LOTE and The Arts. Private tuition in Speech and Drama, Instrumental Music and Singing is also available at the school.
YEAR 4 – 6
The Year 4 – 6 Curriculum is developed in line with the Key Learning Areas in Years 7 – 10 (English, Mathematics, Science, HSS, LOTE, Technology, The Arts and Health and Physical Activity). Secondary School facilities are available to Senior Primary students and Specialist Secondary Staff oversee classes in some areas including Science. Private tuition in Speech and Drama and Instrumental Music are also available at the School.
YEAR 7 AND 8
In Year 7 and 8 a broad-based education provides the foundation for informed subject choices in Years 9, 10 and beyond. The following curriculum areas form the basis of the broad Middle School curriculum:
Core Subjects: English, Mathematics, Science, Humanities and Social Science and Physical Activity. Students also study the Arts, LOTE and Home Economics.
YEAR 9 AND 10
Along with core subjects of English, Mathematics, Science, Humanities and Social Science and Physical Activity, in Years 9 and 10 students select three electives of particular interest for deeper exploration. They may choose any three from: Agricultural Science, Business, Dance, Drama, Home Economics/Hospitality, Spanish, Music, Digital Technology, Health and Physical Education and Visual Art.
YEAR 11 AND 12 PATHWAYS
In Year 11 and 12 students prepare for post school life by constructing an individual curriculum plan. Each student maintains a minimum of one English and one Mathematics subject and four other options from our 25 senior subjects or alternate pathway choices. Students may choose to study for a school based apprenticeship or traineeship, a TAFE course or may even choose to start their university studies.