Fed up with squares?If you have used mtf_generate_rectangle before you will know why it is called mtf_generate_rectangle. A bit over two years ago, when I started working on the first release of MTF Mapper, I ran into a specific problem: I could not find any synthetic images rendered with specific, exactly known point spread functions. This meant that the only way that I could tell if MTF Mapper was working correctly was to compare its output to other slanted edge implementations.
While this is sufficient for some, it did not sit well with me. What if all those other implementations were tested in the same way? If that was the case, then all slanted edge implementations (available on the Internet) could be flawed. Clearly, some means of verifying MTF Mapper independently of other slanted edge algorithm implementations was required.
Thus mtf_generate_rectangle was born. The original implementation relied on sampling a dense grid, but quickly progressed to an importance sampling rendering algorithm tailored to a Gaussian PSF. The Gaussian PSF had some important advantages over other (perhaps more realistic) PSFs: the analytical MTF curve was know, and simple to compute (the MTF of a Gaussian PSF is just another scaled Gaussian). Since the slanted edge algorithm requires a step edge as input, it seemed logical to choose a square as the target shape; this would give us four step edges for the price of one.
Such synthetic images, composed of a black square on a white background, are perfectly suited to the task of testing the slanted edge algorithm. Unfortunately, they are not great for illustrating the visual differences between different PSFs. There are a few well-know target patterns that are found on resolution test charts designed for visual interpretation. The USAF1951 pattern consists of sets of three short bars (see examples later on); the width of these bars are decreased in a geometric progression, and the user is supposed to note the scale at which the bars are no longer clearly distinguishable.
Another popular test pattern is the Siemens star. This pattern comprises circular wedges radiating from the centre of the design. The main advantage of the Siemens star is that resolution (spacing between the edges of the wedges) decreases in a continuous fashion, as opposed to the discrete intervals of the USAF1951 chart. I am not a huge fan of the Siemens star, though, mostly because it is hard to determine the exact point at which the converging bars (wedges) blur into a gray mess. It is far too easy to confuse aliasing with real detail on this type of chart. Nevertheless, other people seem to like this chart.
Lastly, there is the familiar "pinched wedge" pattern (also illustrated later in this post), which contains a set of asymptotically convergent bars. The rate of convergence is much slower than the Siemens star, and a resolution scale usually accompanies the pattern, making it possible to visually measure resolution in a fashion similar to the USAF1951 chart, but with slightly more accuracy. I rather like this design, if only for the fact that the resulting pictures are aesthetically pleasing.
Today I announce the introduction of a fairly powerful new feature in mtf_generate_rectangle: the ability to render arbitrary polygon shapes.
The implementationYou can now pass fairly general polygonal targets using the "--target_poly <filename>" command line option. The format of the file specified using <filename> is straightforward: any number of polygons can be specified, with each polygon defined by an integer <n> denoting the number of vertices, followed by <n> pairs of <x,y> coordinates. I usually separate these components with newline characters, but this is not critical.
The polygons themselves can be convex or concave. In theory, non-trivial self intersections should be supported, but I have not tested this myself yet. There is currently no way to associate multiple contours with a single polygon, thus you cannot specify any polygons containing holes. I work around this by simply splitting the input polygons with a line passing through such a hole: for example, a polygon representing the number "0" can be split down the middle, producing two concave polygons that touch on the split line.
General polygon intersectionsFor a while now I have cheated by relying on the Sutherland-Hodgeman algorithm to compute the intersection between two polygons. Specifically, this operation is required by all the importance sampling algorithms involving a non-degenerate photosite aperture (e.g., "airy-box" and "airy-4dot-olpf" PSF options specified with the "-p" command to mtf_generate_rectangle). This article explains the process in more detail, but the gist is that each "sample" during the rendering process is proportional to the area of the intersection between the target polygon geometry and a polygon representing the photosite aperture (suitably translated). If we assume that the photosite aperture polygon is simply a square (or more general, convex) then we can rely on the Sutherland-Hodgeman algorithm to compute the intersection: we simply "clip" the target polygon with the photosite aperture polygon, and compute the area of the resulting clipped polygon.
Now this is where the cheat comes in: the clipped result produced by the Sutherland-Hodgeman algorithm is only correct if both polygons are convex. If the target polygon (the clipee) is concave, and the clipping polygon is convex, the Sutherland-Hodgeman algorithm may produce degenerate vertices. (see figure 1 below). The cheat that I employed relied on the observation that degenerate sections of a polygon have zero area, thus they have no influence on the sampling process.
|Fig 1: Clipping a concave polygon with the Sutherland-Hodgeman algorithm|
This allowed me to continue using the Sutherland-Hodgeman algorithm, mostly because the algorithm is simple to implement, and very efficient. This efficiency stems from the way in which an actual intersection vertex is only computed if an edge of the clipee polygon is known to cross an edge of the clipper polygon. This is a huge saving, especially if one considers that the photosite aperture polygon (clipper) will typically be either entirely outside the target polygon, or entirely inside the target polygon, except of course near the edges of the target polygon.
All this comes apart when the photosite aperture polygon becomes concave. To solve that problem requires significantly more effort. For a start, a more general polygon clipping algorithm is required. I chose the Greiner-Hormann algorithm, with Kim's extensions to cater for the degenerate cases. In this context, the degenerate cases occur when some of the vertices of the clipee polygon coincide with vertices (or edges) of the clipper polygon. This happens fairly often when constructing a quadtree (more on that later). At any rate, the original Greiner-Hormann algorithm is fairly straightforward to implement, but adding Kim's enhancements for handling the degenerate cases required a substantial amount of effort (read: hours of debugging). The Greiner-Hormann algorithm is quite elegant, and I can highly recommend reading the original paper.
Internally, mtf_generate_rectangle classifies polygons as being either convex or concave. If the photosite aperture is convex, the Sutherland-Hodgeman algorithm is employed during the sampling process, otherwise it will fall back to Kim's version of the Greiner-Hormann algorithm. The performance impact is significant: concave photosite polygons render 20 times more slowly than square photosite polygons when rendering complex scenes. For simpler scenes, the concave photosite polygons will render about four times more slowly than squares; circular (well, 60-sided regular polygons, actually) will render about two times more slowly than squares.
Part of this difference is due to the asymptotic complexity of the two clipping algorithms, expressed in terms of the number of intersection point calculations: the Sutherland-Hodgeman algorithm has a complexity of O(c*n), where "c" is the number of crossing edges, i.e., c << m, where "m" is the number of vertices in the clipee polygon, and "n" is the number of edges in the clipper. The Greiner-Hormann algorithm has a complexity of O(n*m); on top of that, each intersection vertex requires a significant amount of additional processing.
Divide and conquerTo offset some of the additional complexity of allowing arbitrary target polygons to be specified, a quadtree spatial index was introduced. The quadtree does for 2D searches what a binary tree does for linear searches: it reduces the number of operations from O(n) to O(log(n)).
First up, each polygon is wrapped with an axis-aligned bounding box (AABB), which is just an educated-sounding way of saying that the minimum and maximum values of the vertices are recorded for both x and y dimensions of a polygon. This step already offers us a tremendous potential speed-up, because two polygons can only intersect if their bounds overlap. The bounds check is reduced to four comparisons, which can be implemented using short-circuit boolean operators, so non-overlapping bounds can be detected with as little as a single comparison in the best case.
Once each individual polygon has a bounding box, we can start to aggregate them into a scene (internally, mtf_generate_rectangle treats this as a multipolygon with its own bounding box). The quadtree algorithm starts with this global bounding box, and splits it into four quadrants. The bounding box of each quadrant is taken as a clipping polygon, clipping all the polygons to fit exactly inside the quadrant.
After one iteration, we have potentially reduced the number of intersection tests by a factor of four. For example, if we determine (using the bounding boxes) that the photosite aperture polygon falls entirely inside the top-right quadrant, then we only have to process the (clipped) polygons found inside that quadrant. If a quadrant is empty, we can simply skip it; otherwise, we can shrink the bounding box to fit tightly around the remaining clipped polygons (see figure 2 below).
The next logical step is to apply this quadrant subdivision recursively to each of the original quadrants. We can keep on recursively subdividing the quadrants until a manageable number of polygons (or more correctly, a manageable number of polygon edges) is reached in each quadrant subtree. We must balance the cost of further subdivision against the gains of reducing the number of edges in each subdivided quadrant. Every time that we add another level to the quadtree we add four additional bounds checks --- eventually the cost of the bounds checks add up.
|Fig 2: Levels 0 (green), 1 (blue) and 2 (magenta) of the Quadtree decomposition of the scene (light gray)|
|Fig 3: Levels 2 (green), 5 (blue), and 6 (magenta) of the Quadtree decomposition|
If the number of quadtree levels is sufficient, we end up "covering" the target polygons with rectangular tiles (the bounding boxes of the quadrants), providing a coarse approximation of the target polygon shape (see figure 3 above). Every sampling location outside these bounding boxes can be discarded early on, so rendering time is not really influenced by the size of the "background" any more.
If the quadtree is well-balanced, the amount of work (number of actual polygon-polygon intersection tests) can be kept almost constant throughout the entire rendered image, regardless of the number of vertices in the scene. I have confirmed this with some quick-and-dirty tests: halving the number of vertices in a scene (by using a polygon simplification method) has almost no impact on rendering time.
Some examplesEnough talk. Time for some images:
|USAF1951-style chart with 4 levels|
|Siemens star chart|
|Pinch chart (click for full size)|
mtf_generate_rectangle.exe --pixel-pitch 4.73 --aperture 4 -p airy-box -n 0 --target-poly pinch.txt
where "pinch.txt" is the file specifying the polygon geometry (which happens to be in the same folder as mtf_generate_rectangle.exe in my example above). These target polygon geometry files are included in the MTF Mapper package from version 0.4.16 onwards (their names are usaf1951r.txt, siemens.txt, and pinch.txt.)
OLPF demonstrationThe "pinch chart" provides a very clear demonstration of the effects of the Optical Low-Pass Filter (OLPF) found on many DSLRs (actually, most, depending on when you read this).
Before I present some images, first a note about effective chart magnification. Most real-world test charts are printed at a known size, i.e., you can say with confidence that a particular target (say, a USAF1951 pattern) has a known physical size, and thus measures physical resolution expressed in line pairs per millimetre (lp/mm). It is relatively straightforward to extend this to synthetic images generated with mtf_generate_rectangle by carefully scaling your target polygon dimensions. For the time being, though, I prefer to fall back to a pixel-centric view of the universe. In other words, I chose to specify my target polygon geometry in terms of pixel dimensions. This was mostly motivated by my desire to illustrate specific effects (aliasing, etc.) visually. Just keep that in mind: the images I present below are not intended to express resolution in physical units; they are pretty pictures.
With that out of the way, here is a sample of a hypothetical D7000 without an OLPF --- this could be something like the Pentax k5-IIs.
|Hypothetical D7000 without OLPF, f/4|
And here is the same simulated image, but this time using an OLPF, i.e., this should be quite close to the real D7000:
|D7000 (with OLPF), f/4|
I repeated the simulations using an f/8 aperture:
|Hypothetical D7000 without OLPF, f/8|
and again, the D7000 (with OLPF) simulated at f/8:
|D7000 (with OLPF), f/8|
Here is a crop comparing the interesting part of the chart across these four configurations:


Binary is a method of representing states, as either
1. These binary digits are called bits. With one bit, we can represent 2 states, which is why binary can also be called Base 2.
To represent more than 2 states, we use groupings of bits called bytes. A byte is a grouping of 8 bits. With 1 bit, we can represent 2 states (
1), and with 2 bits, 4 states (
11). With 3 bits: 8 states, and 4 bits: 16 states.
Binary Permutation Equation
So in Binary, we have an exponential relationship between the number of bits we have (b) to how many states we can represent (s).
s = 2b
Using this equation, we now know that a byte can represent 28, or 256 states. Since binary includes the number 0, this means we can count up to the number 255 using 1 byte.
The number 255 in binary looks like
General Permutation Equation
For different Base systems, this general equation applies, relating the number of bits we have (b) to how many states we can represent (s), to the Base number system (n):
s = nb
Hex is a Base 16 system, so one bit, or character in this case, goes from 0 - 9, then A - F, representing 16 total possibilities. Each character in Hex can represent 4 bits in binary, since 161 == 24. With only 2 slots in Hex, we can represent 162, or 256 states. Sound familiar?
The number 255 in Hex looks like
Base64 is a method of representing binary in ASCII string format, which allows for reliably transmitting data across systems designed to handle only 7 bit data. Each character can represent 64 possible states, which is the equivalent of 6 bits in binary (26 == 64).
The number 255 in Base64 looks like


Modern App-stores have evolved very quickly but the pricing models are very much out-dated and static in the sense that there is no quantitative way to intelligently estimate the price of an app depending upon the demand and usability of the app. In this article I have tried to answer the question ‘at what price should I sell my app?’ I have extended this idea to a dynamic and adaptive model wherein the prices can change depending upon demand. In both the cases I have estimated the total revenue returns. I have used Lagrange multipliers to arrive at the optimization of neural networks.
Before knowing how this all works I will try to explain a few terms that are going to be useful.
Demand (m) : Average number of installs per day of an app. this is the maximum likelihood of a Poisson distribution.
Latent Demand (M): This is the maximum average demand an app can have at any price, easily we can arrive at this from the number of installs its free app can have. This in some sense represents the usefulness or the marketability of an app.
Price (p): That’s a no brainer.
How does it work?
Ever heard of the Duckworth-Lewis method in cricket? Its an estimate of scores in a game of cricket in case the match has to be abandoned or otherwise cut short. It arrives at a function of score and overs using historical data of the game, meaning we can estimate the score at the end of the match given some initial value conditions. We are going to do similar stuff here. We arrive at a function of Demand vs. Price using historical data, and using the Latent Demand that the app currently has we give estimate for a most optimized price where we can maximize the profits. I have used Lagrange multipliers to differentiate the neural networks and arrive at the most maximized price. A back propagation algorithm has been used for arriving at the neural network. A Modified (modified to implement the Lagrange multipliers) Feed forward algorithm has been used to predict the prices.
What is a neural network?
It’s an universal approximation function/algorithm which approximates any function – so long as such a function exists; it is meaningless to approximate a non-existent function – to any amount of accuracy given the input data or parameters, it fits a curve that suits the best to give input data. Just like the Newton Forward Method or Taylor Series Approximation. Intact it is a linear regression approximation of any curve over multiple parallel iterations. Now, did I use the word neuron in defining it? No I didn’t!
Why is there a neuron in its name?
Simply, some one felt like naming it that way (it has nothing to do with neurons or the way the human brain functions).
How does it look like in case of two input parameters (M,m in our case)?
Where P ,M,m stand as defined above and represents the neural net function in one layer with k nodes. This function can be written as below (tanh has been used as the activator function for its mathematical ease in representation in contrast to the sigmoid).
Any function can be approximated using above setup. The problem of back propagation is finding the values of U,V,W vectors that best fit the above function to predict the value of P.
Maximizing profits with forward feed
Let be the price prediction neural network for a given app whose latent demand M is already known.The problem of predicting the right price to maximize the profit is to see that the value of total sales
* price is maximized. i.e. is maximum.
This converts into a Lagrange multipliers problem in two unknowns P and m. The multiplier equations can be written as follows.
Solving the above three equations and eliminating we get
This can be reduced to a mathematical form as
where the weights are already known from the back propagation algorithm. To solve this we can use Newton-Raphson method or Muller Method, though even Bisection Method should be very efficient .
Proof of No guarantee of a primitive and nature of the
We already know that M is the maximum value at which price is 0 and m=0 is another point at price infinity hence. Intuition is that there must be a root for m between m=M and m=0 .
is a monotonically decreasing function; as price decreases the number of installs increases its trivial from basic economics. Hence < 0 and > 0. We can quickly verify that at m=M above equation 4 has a positive value and at m=0 it has a positive value as well. Hence there is no guarantee that there will always exist a root, there are always going to be an even number of solutions, which is understandable as the curve might look like a parabola. At the same time there can be no primitives, this usually happens in the case when an app’s Latent demand is already lower than the market average and any price you put on such an app will not guarantee any maximization of proofs but in all other cases you should find even number of primitives .
Expected Revenue for static and dynamic pricing
In case of a static price the expected price is very straightforward and its (where is the average no.of requests or demand for an app)
In case of dynamic price changes its slightly trickier we need to evaluate it over all of Poisson distribution and arrive at something like this.
I will explain all the algorithms needed for implementing these in python or R in a separate blog.


Pai-gow poker strategy
Pai-Gow Poker strategy look so:
- There is no combination: to leave the senior card on the bottom hand, two subsequent - upward.
- One pair: to leave pair on the bottom hand, two seniors from remained - upward.
- Two pairs:
- To not divide, if at you the sum of points nine and less at king or an Ace;
- To not divide, if at you the sum of points fifteen and less at an Ace;
- Exception: to divide the seven and the two at king and above;
- Exception: to divide kings and the two at an Ace;
- Exception: to divide jacks and the four at an Ace;
- Exception: to not divide the seven and the eight (nine) at an Ace,
- Where points correspond to face value of cards (2 - for deuces, 11 - a jack, 12 - queen, 13 - king, 14 - Ace, that is jacks and give the four the sum of points 4+11=15).
- Three pairs: the senior pair - upward.
- Three: to leave a three below, two senior cards - upward, except for three Aces: two Aces below, one - upward.
- Two three: from the senior three pair - upward.
- Straight and Flush: we leave Straight and Flush if there is no double pair, differently we divide by rules of two pairs.
- Full-house: to divide into a three and the two (if there are two pairs or two three - upward we put the senior pair).
- Four of kind: Four-of-kind from the two up to the six - to not divide; Four-of-kind from the seven up to tens - to divide, if there is no king or above; Four-of-kind from a jack up to king - to divide, if there is no Ace or above; Ace-Four-of-kind - to divide, if there is no other pair.


The aim of the Mathematics curriculum in our Primary School is to strive for enjoyment and excellence in mathematical applications and to provide students with the skills to communicate their mathematical thinking confidently and effectively. Students are given the opportunity to develop the knowledge and skills required to apply mathematics in the world and learn to recognise links between mathematics and other subject areas.
In addition to normal class tests and quizzes, students’ individual mathematical progress is assessed three times per year by MAP Testing. This method shows us as a school where our students are placed in comparison to international schools worldwide and allows teachers to closely supervise and support each student’s development in this subject area throughout the school year by integrating differentiated resources to challenge all students at their personal level.
Our curriculum uses a variety of textbooks and websites, depending on student ability. Emphasis will mostly be on ‘using and applying mathematics to solve problems’. Students undertake numerous problem-solving exercises, investigations, puzzles and games. They will learn to work in a systematic and logical way, appreciating the power and beauty of Mathematics along the way.
All classrooms are equipped with interactive SMARTboards, individual whiteboards, worksheets, books, counting cubes and measuring equipment. In grades 4 & 5 students enjoy weekly ICT Maths lessons using Mathletics software that is designed to improve students’ understanding of core mathematical principles. Within the classroom we see to it that a three tier differentiation process is in operation. This ensures that all students have a positive and profitable experience in their learning.
Differentiated learning as well as nurturing mathematical talents
We have different Math Clubs on offer, where both gifted and talented students as well as students who find maths challenging can get individually targeted support to further progress their inquiring minds. Gifted and talented students take part in both national and international Maths competitions such as Kangaroo and the Maths Olympiade.


1) Addition + Adds two Numbers.
2)Subtraction – Subtract two Numbers.
3)Multiplication * Calculate product of two numbers.
4)Division / Divides two numbers.
5)Modulus % Divides by others.
6)Increment ++ Increments the numbers applied by 1.
7)Decrement — Decrement the numbers applied by 1.
PHP also supports the comparison operators which compares two or more values to determined which values is greater ,lesser , equal or not equal to the other.
Greater than >
Less than <
Greater than equal to >=
Less than equal to <=
Equal to = =
Not equal to ! =
PHP supports the logical operators that you can use to create Boolean expressions.Boolean expressions return either True or False.
OR | |
Assignment and Compound operators:
We can use assignment operators in PHP to assign values to the variables .PHP also supports compound operators that perform the functions of the arithmetic and assignment operators in a single statement.
Operators Implements Description
= a=5 Assigns the value, 5, to the variable ,a
+= a+= Adds the value ,5 , to original value of the variable a and then assigns sum to the variable , a .this is equivalent to a=a+5
-= a-= subtract the value
*= a*= Multiply the value
/= a/=5 Divides the value in the variable.
%= a%=5 Divides the value in the variable


Marketers love to throw around numbers and stats, as if the world rose and set around them. Ever go to the Clickbank marketplace to see some of the top selling products and the insane statistics that they’re claiming as far as conversion? It’s enough to make you sick. Well, I’m going to cut through all the garbage as far as these stats and also clear up some confusion regarding conversion rate versus conversion ratio. It is my hope that when you are finished reading this article, you’ll be able to take all of this with a grain of salt and also understand what these numbers really mean.
The first thing you have to understand is the difference between conversion rate and conversion ratio because many marketers interchange these terms and they are not the same.
Conversion rate is an actual percentage. For example, if somebody says that their conversion rate is 6%, that means that for every 100 people that come to their site, they make 6 sales. You get the percentage by dividing the number of sales by the number of visits and then multiplying by 100. So if you get 70 visits for a test and make 7 sales, you divide 7 by 70 and then multiply by 100. That will give you a 7% conversion rate.
Conversion ratio is presented as X number of sales for every Y number of visits. So a conversion ratio of 4:25 means 4 sales for every 25 visits. If you want to convert this to conversion percentage, then you simply take the number of sales divided by the number of visits and multiply by 100. In this case, the conversion rate would be 16%.
What you need to know about these numbers, when they get thrown at you, is that most marketers will not give you true statistics. They’ll tell you that their conversion rate is 6% but what they don’t tell you is that this is based only on Adwords advertising, or on list promotion. If they were to give you the stats for all their promotions, they would probably turn out to be lower than the stats they are showing. The reason for that is because they always put their best foot forward. So it is important to be aware of this when you decide on a product to promote as an affiliate.
Naturally, marketers are going to want to put their best foot forward when giving you their conversion stats. As long as you understand this, you’ll be better able to make an informed decision when deciding w
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Mastering Excel Sum & Count Formulas Summing and counting values are the most common spreadsheet operations, and Excel's capabilities go far beyond routine SUM and COUNT formulas. Author Tim Hill shows you how to use worksheet functions, array formulas, AutoSum, and other Excel features to summarize an entire range of values, or only values that meet specific conditions. If you're using an older version of Excel that doesn't support the latest worksheet functions, you'll find equivalent formulas that work in Excel 2003 or earlier. You can download the sample workbook to follow along with the author's examples.
• Covers all versions of Excel.
• Display sums and counts without using formulas.
• Master the basics of COUNT, COUNTA, COUNTBLANK, and other counting functions.
• Create conditional counts with COUNTIF and COUNTIFS.
• Calculate the mode for numeric or text values.
• Count unique values in a range.
• Count occurrences of specific text strings.
• Create frequency distributions and histograms.
• Master the basics of the SUM function.
• Use AutoSum to sum values quickly.
• Calculate running totals.
• Sum only the highest or lowest values in a range.
• Eliminate rounding errors in financial calculations.
• Sum every Nth value in a range.
• Create conditional sums with SUMIF and SUMIFS.
• Plenty of tips, tricks, and timesavers.
• Fully cross-referenced, linked, and searchable.
1. Getting Started with Sums & Counts
2. Counting Basics
3. Counting Tricks
4. Frequency Distributions
5. Summing Basics
6. Summing Tricks
About the Author
Tim Hill is a data scientist living in Boulder, Colorado. He holds degrees in mathematics and statistics from Stanford University and the University of Colorado. When he's not crunching numbers, Tim climbs rocks, hikes canyons, and avoids malls.


If you weren’t aware yet, I’m a Middle School Math teacher. I’m also an Apple fanboy, so this post is going to be near and dear to my heart as it covers two of my favorite subjects.
I could write page after page on math apps, but I’m trying to keep this post fairly short. Of course, these are subject to change over time. Here are 5 apps to get you started on your math app journey. I encourage you to do your own research and find out more about each one of these apps and see if they are a good fit for you and your teaching style.
Great iPad Apps for Middle School Math Teachers
MyScript Calculator is an easy, simple and intuitive mathematic demonstration, just write the mathematical expression on the screen then let MyScript technology perform its magic converting symbols and numbers to digital text and delivering the result in real time. With MyScript© Calculator, perform mathematical operations naturally using your handwriting.
MyScript currently recognizes over 200 mathematical symbols and characters, providing a robust toolkit for handwritten mathematical work. Even better, the technology is time-order independent, emphasizing spatial layout of mathematical terms to better reflect user perspective and ensure accuracy.
Factor samurai is a great way to learn times tables. You play as the samurai whose sacred duty is to cut all the numbers down to their prime factors.
You will be thrown one number after the next, cut them up if they can be factored into smaller values, if not, if they are prime, do not cut them.
Prime numbers are numbers that cannot be divided into small whole numbers. Two, three, five, seven, eleven, and thirteen are all examples of prime numbers. Four is not prime because it can be broken down in the whole numbers two times two.
Factor Samurai will reinforce the arithmetic principles of multiplication and division by continually challenging the player to remember which numbers can be factored into smaller values, and which ones can’t.
As the player slices numbers into factors, those factors can be broken down further, until they are split into their prime factors.
Knowing that 63 is 3 times 3 times 7 or that 128 is 2 raised to the seventh power (7 twos!) is how you become a true Factor Samurai!
Factor Samurai is used extensively in the classroom. More than half of the recorded downloads are to public schools.
Operation Math: Code Squad
Take the action to a whole new level, in the fun multi-player math game based on the original Parents’ Choice Award-winning game — Operation Math Code Squad. Join up to three other Base 10 agents to solve the equations and disarm Dr. Odd’s latest device before your time runs out in a challenging game of tick-tock tech guaranteed to keep you on the edge of your iPad.
DragonBox Algebra 12+ is a must-have tool for students so they can earn better grades and gain confidence in algebra and mathematics. It is based on the award winning game DragonBox Algebra 5+, but covers more advanced topics in mathematics and algebra.
I know I’ve left about a million other apps off of this list, so what are your favorite iPad apps for middle school teachers?
[Tweet “Here’s some great iOS math apps for middle school, what’s your favorite?”]


In three months, we transform data analytics and machine learning into a transparent process that every manager can access, develop and improve.
DataVision allows any business user to build machine learning models of any complexity. Building models with the platform is no more difficult than working on a calculator. For data analysts, the system allows setting up automated procedures for hypothesis testing and initial data analysis.
DataVision key features
Access to developed models, factor settings and data is strictly limited by system settings
DataVision does not require users to have specific knowledge in the field of mathematics and machine learning, building a model in the system is no more difficult than building a graph in office software
All input data and characteristics are available in system interfaces including model versioning, statistical characteristics, model risk
The ease of integration
The system supports the ability to upload processing algorithms as python scripts, models developed in the system can be referred to as services
Complete independence from external software and ability to work on unix operating systems
The system is adapted to the solution of modeling problems of any complexity and any types


Extrapolate is to estimate a value or values on the basis
of collected data. For example, a firm may forecast annual sales during the next
five years from historical data of actual sales and sales growth during the most
recent ten years. A process in mathematics used to find the value of a
function outside its tabulated values. This is done as in interpolation by
assuming that over a small range of x the function may be closely approximated
by a polynomial or some other readily computed function. Any of the
interpolation formulas can be used, therefore, and the desired value of x
substituted in them.
A graph is an image that represents data symbolically. A graph is used to present complex information and numerical data in a simple, compact format. Bar graphs, line graphs, scatter graphs, and pictographs are some types of graphs. Graphs then, are visual representations of worksheet data. Graphs often makes it easier to understand the data in a worksheet because users can easily pick out patterns and trends illustrated in the graph that are otherwise difficult to see.


Direction To Solve
Quantitative Aptitude Questions: Problems on Simple Interest - In this section you can learn and practice Quantitative Aptitude Questions based on "Simple Interest" and improve your skills in order to face the competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc.) with full confidence. You can easily solve all kind of Quantitative Aptitude Questions based on Simple Interest by practicing the objective type exercises given below also get shortcut methods to solve Simple Interest Questions.
If simple interest on a certain sum of money for 4 years at 5% per annum is same as the simple interest on Rs. 560 for 10 years at the rate of 4% per annum then the sum of money is:
The difference between the simple interest received from two different sources on Rs. 1500 for 3 years is Rs. 13.50. The difference between their rate of interest is:
David invested certain amount in three different schemes A, B and C with the rate of interest 10% p.a., 12% p.a. and 15% p.a. respectively. If the total interest accrued in one year was Rs. 3200 and the amount invested in Scheme C was 150% of the amount invested in Scheme A and 240% of the amount invested in Scheme B, what was the amount invested in Scheme B?
A sum of Rs. 1550 was lent partly at 5% and partly at 8% p.a. simple interest. The total interest received after 3 years was Rs. 300. The ratio of the money lent at 5% to that lent at 8% is:
A sum was put a simple interest at a certain rate for 2 years. Had it been put at 3% higher rate, it would have fetched Rs. 72 more. The sum is:
A sum of money becomes 7/6 of itself in 3 years at a certain rate of simple interest. The rate per annum is:
A certain sum is invested for T years. It amounts to Rs. 400 at 10% per annum. But when invested at 4% per annum, it amounts to Rs. 200. Find the time (T)?
If the simple interest on a certain sum of money for 2 years is one – fifth of the sum, then the rate of interest per annum is:
The Sum of money that will produce Rs. 1770, interest in 7 1/2 years at 8% simple interest per annum is:
The simple interest on Rs. 1820 from March 9, 2003 to May 21, 2003 at 7.5% rate will be:


GCSE results: another year of high performance at Oulder Hill
Date published: 22 August 2019
Oulder Hill GCSE results 2019 - Meeraj Akhtar, Shaun Walmsley, James Blackburn and Hasan Khan
Oulder Hill students celebrated picking up incredible GCSE results on Thursday morning (22 August).
Assistant headteacher Lyn Sanderson said: “Oulder Hill Community School is proud to announce another year of high performing results. We are really proud of our students who have worked very hard and achieved highly across all subjects.
“English and maths are once again above national averages and further improvements have been made across the EBACC subjects.
“All students have also achieved well across our wide range of vocational subjects, and we are delighted for them.
“We wish all of our students well for the future.”
The bottom of a grade 7 is comparable to the bottom of the old grade A, whilst the bottom of a new grade 4 is comparable to the bottom of the old grade C. Grade 9 is the equivalent of above an A*.
Top performers this year include Umar Sarwar, Lauren Borg, Mariyah Ramzan, Sarra Malik, Ella Pogson and Lily Mangan.
Umar achieved five grade 9’s in combined science, computer science, history and maths, two grade 8’s, two grade 7’s, plus an A* in further maths.
He plans to complete A-levels in biology, chemistry, maths and law at Bacup and Rawtenstall Grammar School.
Lauren achieved two grade 9’s, four grade 8’s, two grade 7’s and a grade 6 in maths, chemistry, biology, history, physics, English literature, Italian and art, plus an A* in further maths.
She will be studying maths, physics and art at Holy Cross and hopes to pursue a career in architecture or engineering.
Mariyah achieved grade 9’s in biology and maths, three grade 8’s in chemistry, physics and geography and two grade 7’s in English literature and language. She also achieved an A* in Urdu, an A in further maths and distinctions in hospitality and catering.
She will be studying biology, chemistry and maths at Rochdale Sixth Form College before hopefully pursuing a career in medicine.
Sarra will be studying maths, physics and economics at Rochdale Sixth Form College after achieving grade 9’s in combined science and English language, grade 8 in English literature, grade 7 in maths, grade 6 in German, plus a C in further maths. She also achieved distinctions and distinction*’s in hospitality and catering and childcare.
Ella achieved four grade 9’s in biology, maths, physics and geography, two grade 8’s in chemistry and French, one grade 7 in Art, plus two grade 6’s in English literature and language, and an A in further maths. She is going to study A-levels in maths, geography and economics at Rochdale Sixth Form.
Lily achieved one grade 9 in biology, four grade 8’s in chemistry, maths, physics and English language, two grade 7’s in PE and art. She also achieved two grade 6’s in Spanish and English literature, and a B in further maths. She is going to study the subjects she enjoys at A-level – maths, biology and chemistry – at Rochdale Sixth Form College.
Meeraj Akhtar achieved 10 GCSE’s (grades 6-8). She will be studying history, geography and criminology at Bury College. She wants to attend university and work towards a PhD, and hopes to study sustainable management for global warming.
Shaun Walmsley will be studying physics, maths and chemistry at Rochdale Sixth Form College after achieving grades 7 and 8 in these subjects. He has gained nine GCSE’s (grades 4-8) and hopes to read astrophysics in the future.
James Blackburn achieved grade 8s across all sciences, coming away with 10 GCSE’s (grades 4-8 and one A*). He will be doing maths at Rochdale Sixth Form College and is undecided about his other subjects.
Hasan Khan will be studying biology chemistry and maths at Rochdale Sixth Form College after gaining nine GCSE’s (grades 5-8). He aspires to study biochemistry in the future.
Eve Smith achieved nine GCSE’s grade 5-8 and plans to study A-levels in biology and chemistry at Rochdale Sixth Form College. She wishes to pursue a career in medicine.
Jessica Hughes achieved 10 GCSE’s grade 4-6, plus distinctions and merits. She will be studying applied science and psychology at Rochdale Sixth Form College. She thinks she will go to university to study zoology or marine biology.
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We will show you how to write Chemistry formulas in Microsoft Word of Microsoft Office suite.
We all agree that Word is the most useful and used software in the Microsoft Office suite. And in education, Word is used a lot to draft lectures, as well as lesson plans in subjects. If you are simply plain text, then Word editing is quite easy, but if you have to add math or chemistry formulas it can be very complicated if you don’t know it.
If you first add Math or Chemistry formulas to the text you are editing, this article will show you how to present them. As follows.
1. Formula with superscript and subscript
This is the simplest way to write the formula in this article, here we will divide it into 02 cases, including the formula with superscript and the formula with subscript.
a. Formula with exponent (superscript)
Write the formula as you normally would, but when you get to the exponent, use CTRL + SHIFT + =. Then the editing cursor will appear in the exponent position (hat) and you can simply enter the exponent then press CTRL + SHIFT + = again to return to the normal content insert state.
If you do not want to use the shortcut key combination, you can use the Font panel to do it, simply enter the contents of the formula as usual, then highlight the selected number as the exponent then right-click mouse and select Font …
The Font window appears, check the Superscript option in Effects and click OK.
You continue to perform the operation on other exponents if you want.
b. Formula with lower exponent (subscript)
Write the formula as usual, but then go to the exponential part then use CTRL + =. Then the editing cursor will be in the lower position and you just need to enter the exponent and press CTRL + = again to return to the normal content insert state.
If you do not want to use the shortcut key combination, you can use the Font panel to do it, simply enter the formula contents as usual, then highlight the selected number as the lower exponent and then press Right-click and select Font. The Font window appears, check the Subscript option in the Effects section and click OK to finish.
2. Use the Equation tool
Microsoft also did not forget to build a formula insertion tool so that users can choose the templates that match the formula that they are drafting, the tool is named Equation.
If the built-in templates are not the ones you need, you can perform new creation by clicking Equation> Insert New Equation.
Word will then display a list of options that make it easier to manually enter formulas such as Script to write indexes.
Or Basic Math with math symbols …
In addition, if you want to enter formulas faster, you can use the Ink Equation option to write the formula directly and Word will automatically display the content written to you. And you just need to click Insert to insert it.
3. Use MathType tool
MathType (https://store.wiris.com/en/products/downloads/mathtype) is a third-party tool software that supports editing and inserting Math and Chemistry formulas for office workers. The advantage of this software is the ability to support a variety of formulas of Math and Chemistry, as well as ease of use for anyone.
However, MathType is not a free software, but if you regularly work with Math and Chemistry formulas, this software is worth it.
4. Use Chem4Word extension
Chem4Word is an open source extension for Word, which is designed to help users create chemical formulas according to a very professional molecular format. Specific usage is as follows.
After downloading and installing Chem4Word, your Word interface will add a new tab called Chemistry.
When clicking on Chemistry, the list of working tools will be displayed immediately below. You can perform the insertion of molecular charts from Chem4Word’s database through the Open or Web Search options.
If you want to perform molecular chart insertion manually, click on the Draw option and proceed to work with the tools Chem4Word provides.
The above are the four simplest ways to insert the Chemistry formula into Word, hopefully this article will be useful for you.


What is math test anxiety?
Math test anxiety most often appears initially as a “mental block” on a particular problem–often one that the student “knew” how to do moments before the test. Worry over the original problem can lead to more generalized worry and negative self talk, which can sabotage the student’s overall test performance.
How do you define math anxiety?
Maths anxiety has been defined by Tobias and Weissbrod (1980) as “the panic, helplessness, paralysis, and mental disorganisation that arises among some people when they are required to solve a mathematical problem” and it is thought to affect a large proportion of the population.
How does anxiety affect the body?
Cardiovascular system. Anxiety disorders can cause rapid heart rate, palpitations, and chest pain. You may also be at an increased risk of high blood pressure and heart disease. If you already have heart disease, anxiety disorders may raise the risk of coronary events.
How do I relieve stress and anxiety?
Try these when you’re feeling anxious or stressed:
- Take a time-out.
- Eat well-balanced meals.
- Limit alcohol and caffeine, which can aggravate anxiety and trigger panic attacks.
- Get enough sleep.
- Exercise daily to help you feel good and maintain your health.
- Take deep breaths.
- Count to 10 slowly.
- Do your best.
Why do you love mathematics?
1. Numbers help us understand the world, and Math helps us understand numbers. The real-life applications of Mathematics are endless. We are surrounded by numbers, equations and algorithms – especially in this age of data science, with huge data sets that can only be understood through statistical models and analysis.
What is it about maths that you enjoy?
Math is my favorite subject because it is easy to do solve the problem. I like using numbers, fractions, decimals and so much more. Math is also used in the real world like telling time, adding, dividing, percentage and even in a future career like engineering, architect, math teacher and lot more.
What is the importance of mathematics in physics?
For physicists, math is a tool used to answer questions. For example, Newton invented calculus to help describe motion. For mathematicians, physics can be a source of inspiration, with theoretical concepts such as general relativity and quantum theory providing an impetus for mathematicians to develop new tools.
How do you mentally prepare for a math test?
Here are some tips on studying for exams.
- Start on Day One. You should always be studying for the next exam.
- Get a Good Night’s Sleep.
- Make a List of Important Concepts/Formulas.
- Rework Homework Problems.
- Rework Book/Notes Examples.
- Look for Identifying Characteristics in Problems.
- Take a Practice Exam.
How can I be good at tests?
During the test:
- Ask for clarification if you don’t understand the directions.
- Answer all the easy questions first.
- Skip those you don’t know.
- Look for concepts embedded in the question.
- Pace yourself.
- Keep yourself balanced by “psyching up and cooling down” during the test.
- Take brief rest periods.
- Check your work.
Why do I love physics so much?
Physics helps you to understand the world around you, and satisfy your curiosity. Studying physics develops your critical thinking and problem-solving skills. Physicists are versatile, which opens a wide range of future careers.
How can I study with anxiety?
Quick tips for managing anxiety
- take a deep breath and say ‘I can do this’
- speak to family, friends or a health professional.
- practice relaxation exercises before you need them.
- focus on the task, not what others might be thinking.
- remember times you have performed well in the past.
How do I stay calm during a math test?
Here’s what you can do to stay calm in the days leading up to and during your test.
- Be prepared.
- Get a good night’s sleep.
- Fuel up.
- Get to class—or the testing site—early .
- Have a positive mental attitude .
- Read carefully.
- Just start.
- Don’t pay attention to what other people are doing.
Why do many students hate and struggle in mathematics?
Lack Of Practice Many students simply don’t spend enough time practising math concepts. Even though students have a thorough understanding of the math lesson, without practice the concepts will be lost on them. This may also stem from a lack of interest in the subject or topic.


The sin value should be Sin a= Opposite/Hypotenuse=CB/CA. transitions In the problems below, we transitions of trignometric function will use the formula for the period P of transitions trigonometric functions of the form y = a sin(bx + c) + d or y = a cos(bx + c) + d and which is given by. The derivatives of the inverse trigonometric functions are given below. In general, we have for all angles :. For the restricted cosine, sine, and tangent functions, how do we define the corresponding arccosine, arcsine, and arctangent functions?
d (cosec x) = –cosec x cot x dx. trigonometric functions values - a table containing the values of trigonometric functions for the most common angles, e. If transitions of trignometric function tables are used instead of a calculator, the same table is used for either process.
In the first quadrant we get only we get positive values for all trigonometric ratios. . Calculate Arcsine, Arccosine, Arctangent, Arccotangent, Arcsecant and Arccosecant for values of x and get answers in degrees, ratians and pi. NCERT Solutions of Chapter 3 Class 11 Trigonometry is available free at teachoo. Example 3: Differentiate Apply transitions of trignometric function the trignometric quotient rule first. Now transitions of trignometric function that we have the above identities, we can prove several other identities, as shown in the following example. Long version: transitions transitions of trignometric function I got a program that's quite heavy on numerics (it's a physics simulation) and that needs to call trigonometric functions, mostly sin and cos, a lot. Definitions of trigonometric and inverse trigonometric functions and links to their properties, plots, common formulas such as sum and different angles, half and multiple angles, power of functions, and their inter relations.
For a given value of a trigonometric function; they produce the length of arc needed to obtain that particular value. When the value of a trigonometric function is undefined, it means that the ratio for transitions of trignometric function that given function. Next Differentiation of Exponential and Logarithmic Functions. In Class Maths syllabus, you will come across a list of trigonometry formulas, based on the functions and ratios such as, sin, cos and tan.
Note: The use of calculators or tables gives only approximate answers. In this chapter, we trignometric will learnWh. Here is a list of the derivatives that you need to know: d (sin x) = cos x dx.
&0183;&32;The trigonometric transitions of trignometric function functions have to do with angles in the coordinate plane. Trigonometric ratios of supplementary angles Trigonometric identities Problems on trigonometric identities Trigonometry heights and distances. So, if we know all the details of one full transitions of trignometric function cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past. Put u = transitions of trignometric function 2 x 4 transitions + 1 and v = sin u. 6 Introduction Integrals involving trigonometric functions are commonplace in engineering mathematics.
Conversion of Trigonometric Functions. y = cos x is an even function. The integral and derivative of \tan(x) is more complicated, but can be determined by studying the derivative and integral of \ln(x). Now w e have to think about the angle of sin for which we get the value √3/2. Find Period of Trigonometric Functions. $ f(x) = a sin(bx + transitions of trignometric function c) + d$ Let’s break down this function. The transitions of trignometric function Trigonometric Functions - The transition from geometric tools to functions, (2.
P Bockstaele, Adrianus Romanus transitions of trignometric function and the trigonometric tables of Georg Joachim Rheticus (Basel, 1992). All the inverse trigonometric functions have derivatives, which are summarized as follows: Example 1: Find f′( x) if f( trignometric x) = cos −1 transitions of trignometric function (5 x). d transitions of trignometric function (cos x) = –sin x dx. &0183;&32;By strengthening results of N. A somewhat transitions of trignometric function more general concept of angle is required for trigonometry than for geometry.
To recall, inverse trigonometric functions are also called “Arc Functions”. This means the graph of the function has the same output at exactly the same place in every cycle. In this work, a family of four new trigonometric Bernstein-type basis functions with four shape parameters is constructed, which form a normalized basis with optimal total positivity. A transitions of trignometric function trigonometric transitions equation contains at least one trigonometric function, and when asked to solve the equation we must find the angle(s) for which it is valid. Together they make up the set of six trigonometric functions. Trigonometric ratios of 270 degree plus theta. If you continue browsing the transitions of trignometric function site, you agree to the use trignometric of cookies on this website. Let s take an example If cos x = 12/13 lies in second quadrant, find value of other five trigonometric functions Since x is in 2nd Quadrant, tan and cos will be negative But sin will be positive Given cos x = 12/13 We know that sin2 x + cos2 x = 1 sin2 x + (-12/13)2 = 1 sin2 x + 144/169 transitions of trignometric function = 1 sin2 x = 1 144/169 sin2 x =/169 sin2 x = 25/169 sin x = (25/169) sin x = 25 / 169.
This angle is generated by the continuous counterclockwise rotation transitions of trignometric function of a line segment about the point V. Graphs for inverse trigonometric functions. Periodic Functions. sin (-π/3) = - √3/2Hence transitions of trignometric function the principal value of θ is -π/3. Trigonometric ratios of complementary angles. transitions of trignometric function The latency calculation depends on the word length trignometric and LatencyStrategy settings. Previous Higher Order Derivatives.
They are unique because the input is an angle measure, and the output is a ratio. A von Braunm&252;hl, Vorlesungen &252;ber Geschichte der Trigonometrie (2 Volumes) (Leipzig,. Is it possible for a periodic function that fails the Horizontal Line Test to have an inverse? sin 90 degrees, trigonometric identities - a transitions of trignometric function list of different, more or less popular, dependencies between various trigonometric functions. They are: The ratio between the length of an opposite side to that of the hypotenuse is known as, the sine function of an angle. It is possible to find the derivative trignometric of trigonometric functions. Example 2: Find y′ if.
This is especially true when modelling waves and alternating current circuits. We transitions of trignometric function had learned Basics of Trigonometry in Class 10. d (tan x) = sec&178;x dx. We call the number the period of the trigonometric functions and, and refer to these functions. &0183;&32;Principles of trigonometry Trigonometric functions. For the right triangle we have seen the basic trigonometric functions. Identifying Trig transitions of trignometric function Ratios: Quick Formative Assessment. Figure %: The signs of the functions in the four quadrants Values of Quadrantal Angles When an angle lies along an axis, the values of the trigonometric functions are either 0, 1, -1, or undefined.
While we trignometric can find the value of any of the trigonometric functions for any value of θ, there are some angles that are more frequently used in trigonometry and worth memorizing. Domain and range of trigonometric functions. The inverse function has the letters 'ARC' in front of it. Inverse Trigonometric Formulas: Trigonometry is a part of geometry, where we learn about the relationships between angles and sides of transitions of trignometric function a right-angled triangle. The trigonometric functions cosine, sine, and tangent satisfy several properties of symmetry that are useful for understanding and evaluating these functions.
Trigonometric Ratios: In mathematics, trigonometry is that branch of study, which helps a student to learn about triangles. 5 Graphs of the Trigonometric Functions In this section, we return to our discussion of transitions of trignometric function the circular (trigonometric) functions as functions of real numbers and pick up where we left o in Sections10. Using the properties of symmetry above, we can show that sine and cosine are special types of functions. trigonometric function (plural trigonometric functions) (trigonometry) Any function of an angle expressed as the ratio of two of transitions of trignometric function the sides of a right triangle that has that angle, or various other functions that subtract 1 from this value or subtract transitions of trignometric function this value from 1 (such as the versed sine). You transitions can check the detailed explanation of all questions of exercises, transitions of trignometric function examples and miscellaneous by clicking on the Exercise link below. The following table summarizes the derivatives of the six trigonometric functions, as well as their chain rule counterparts (that is, the sine, cosine, transitions of trignometric function etc. The formula for some trigonometric functions is given below.
sin (−x) = −sin x. Since you know how to draw a sine function, this will be easy. Don't forget to try our free app - Agile Log, which helps you track your time spent on various projects and tasks, :) Try It transitions Now.
Example: transitions of trignometric function Solve the equation cos θ = 0, for −π ≤ θ ≤ +π. A student should note that triangles are made out of right angles. Lotte N&248;rtoft, februar. Trigonometric Functions Formula. Trigonometric function values transitions for the special angles. We may imagine the unit circle rolled out, in both. The independent variable x is the radian measure.
Sine & Cosine Period Action (1)! Trigonometric Functions 13. Arc functions are used to find the measure of the angle if the value of the trigonometric function is known. Suppose that a student were asked to provide an transitions of trignometric function estimate for the value of the sine of 20˚. transitions of trignometric function $ f(x) = 2 sin(x)$ Examine this function. An angle A with vertex at V, the initial side of which is VP and the terminal side of which is VQ, is indicated in the figure by the solid circular arc. Moreover, is the smallest positive angle for which Equations 1 are true for any angle. We are normally required to find particular values of θ in a given interval.
Short version: I'd transitions like to know whether there are implementations of the transitions of trignometric function standard trigonometric functions that are faster than the ones included in math. 1 transitions du transitions of trignometric function 1—u2 dx 1 du 1—u2 dx 1 du 1+u2 dx 1 du 1+u2 dx du lul u2—1dx du lul u2—1dx' < 1 < 1 lul>l lul>l 3. The range of an inverse function is trignometric defined as the range of values of the inverse function that can attain with the trignometric defined domain of the function. Inverse trigonometric functions, like any other inverse function, are mathematical operators that undo the function's operation.
The transitions of trignometric function values of trigonometric functions can be found through the coordinate values of the intersections on a unit circle. S o we have to choose one of the angles from 0 to -π/2 that is negative angle. Currently I'm simply using the implementations included in math. While COS tells you the cosine of an angle, ARCCOS tells you what angle has a given cosine. We’ll need more than acute angles in the next section where we’ll look at oblique triangles.
&0183;&32;Inverse trigonometric functions provide anti derivatives for a variety of functions that arise in engineering. Vejleder: Carl transitions of trignometric function Winsl&248;w, Institut for Naturfagenes Didaktik. The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.
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StatisticsInstructor: George Morrison Course ID: RE4010
Statistics may be intimidating to the beginner. George Morrison has been a patient teacher over many years to thousands of people, like you, who want to learn statistics in a way that makes the topic approachable, understandable, and relevant. Students and academics use statistics to make sense of a vast amount of data. Business people use statistics to support business decisions. While some examples in this course are about real estate, some are just about generally interesting things. Homburg Academy’s more advanced finance classes relate quantitative analysis to a range of problems in real estate. This course eases you into the concepts, tools, methods, and approaches of statistical analysis. You will come away with knowledge that will give you an edge when you undertake research, or make your business decisions. You will learn how to pull vital information out of raw data by sampling, analyzing, organizing, and displaying your information in ways that best support your findings.
Lecture 1 - Introduction to Statistics
Lecture 2 - Numerical descriptive measures
Lecture 3 - Probability
Lecture 4 - Discrete probability distribution.
Lecture 5 - Continuous probability distribution
Lecture 6 - Sampling distributions
Lecture 7 - Estimation
Lecture 8 - Practical applications
Lecture 9 - Hypotheses testing
Lecture 10 - Estimation and hypothesis testing two populationsRegister for this Course >


The course is the first part of 2-year of IB Diploma Program (Mathematics Studies) and it addresses to students with various mathematical backgrounds. It is designed to build confidence and encourage the students who do not anticipate a need for mathematics in their future studies. The course includes one project work (a Maths exploration) that allows students to ask their own questions about mathematics and to take responsibility for a part of their own course of studies in mathematics.
1) Core topics: Numbers & Algebra; Logic & Sets; Probabilities & Statistics.
2) DP Internal Assessment: Maths Project.


6.2 Trying out the model
Now that we have a complete model, we are ready to try it out on some study data. As we’ve emphasised many times before in this book, when using a real data set, it is essential to look carefully at the data set to make sure that it is complete, correct and has the form that you expect. Remember that many common machine learning problems are caused by problems with data (such as those listed in section 2.5). A good way to check your data set is to construct visualisations that let you to see at a glance what it looks like. In this case, we need to create visualisations of the test results for each child, allergen and time point. However, this study data set contains private medical data and so we cannot share the data publicly in this book, even in the form of a visualisation. The most important thing that we learned from doing this visualisation is that there are a lot of test results missing from the data set.
When there are missing data, it is always worth analysing to understand why they are missing. In Figure 6.6, we plot the number of test results in the data set (whether positive or negative) for each age and type of test.
You can see several different patterns of missing data in Figure 6.6. First, the plot shows that there are ages and test types that have no data for particular allergens. For example, peanut has no results at all for ages 1 and 3, and only IgE results at age 5. Mould has no IgE results at all, and no skin test results at age 1. Second, there is a lot more missing data at early ages, particularly age 1. Third, the plot shows that there is a lot more missing data overall for IgE tests than skin tests. We need to take into consideration the effect of all these missing data points.
Working with missing data
Missing data can introduce bias into the posterior distributions computed by running inference on a model, leading to incorrect or misleading results. Whether or not this effect will occur, and how big the bias will be, depends on why the data points are missing in the first place. In statistics, it is common to consider three kinds of missingness, which are referred to using the following (quite confusing!) terms:
- missing completely at random (MCAR) – where the missing data points occur entirely at random. In other words, the fact that the data is missing is independent of the value of the missing data point (the test result that would have been given had the test actually happened).
When data is MCAR, the remaining, non-missing, data points are effectively just a random subset of the overall data set. In this case, the posterior distributions computed by probabilistic inference will be unbiased by the missing data. Unfortunately, in reality, missing data is rarely missing completely at random. However, it may be an acceptable approximation to assume that it is – in which case, this assumption should be made with full understanding of the possibility of introducing biases.
- missing at random (MAR) – where the missingness is not random, but where other known data values fully account for the fact that the data is missing. For example, suppose that boys are more likely to refuse an IgE test than girls. Considering the fact that boys are more likely to have allergies, this would introduce a bias in our results, since the missing tests would be more likely to be positive than the non-missing tests.
When data is MAR, it is possible to correct for the bias, at least to some extent, by changing the model appropriately to account for why the data is missing. This extension requires creating a new variable in the model for each data point, which is true if the data point is missing and false otherwise, and then building a suitable sub model to explain this new variable. For example, if boys are more likely than girls to skip an IgE test, then to correct for bias we would need to extend our model to represent this effect, such as by adding a new gender variable connected to the missingness variable. We would also need to allow this gender variable to affect the probability of sensitization in an appropriate way. The degree to which this approach correct the bias introduced by missing data, depends on how good the model of missingness is. As ever, a better model will give better results.
- missing not at random (MNAR) – where the missingness is not either MCAR or MAR. In this case, the fact that a data point is missing depends on the value of that data point. For example, this would occur if children with lots of allergies were more likely to skip a skin prick test because of concerns about the discomfort involved in having a positive test. Or such children might be more used to medical interventions and so may be less likely to skip a blood test due to fear of needles.
When data is MNAR, it is not possible to correct for the bias without making modelling assumptions about the nature of the bias (which could be dangerous as there would be no data to verify such assumptions). One possible approach would be to try and collect additional information relevant to why the data is missing, in the hope that this would now make it missing at random (MAR).
For our study, we need to find out why the various patterns of missing data arose. Consulting again with the MAAS team, we find that:
- The clinicians chose to omit mould tests at age 1, since this is a rare allergy and there was a desire to minimise the number of tests performed on babies. Similarly, a decision was made half way through the study to add in peanut tests.
- The reduced number of tests at age 1 are due to manpower limitations as the study was ramped up – not all children could be brought in for testing by age 1.
- The greater number of missing IgE tests are due to children not wanting to give blood, or parents not wanting babies or young children to have blood taken.
For 1, we know why the data is missing - because the clinicians chose not to do certain tests. Such data can be assumed to be missing completely at random, since the choice of which tests to perform at each age was made independently of any test results. For 2, the study team chose whether to invite a child in for testing by age 1 and so could choose in a way that was not influenced by the child’s allergies (such as, at random). So again, we could assume such data to be missing completely at random. For 3, we might be more concerned, as now it is a decision of the child or the parents that is influencing whether the test is performed. This is more likely to be affected by the child’s allergies, as we discussed above, and so it is possible that such missing data is not missing completely at random. For the sake of simplicity, we will assume that it is – this is such an important assumption that we should record it:
- Missing test results are missing completely at random.
Having made this assumption, we should bear in mind that our inference results may contain biases. One reassuring point is that where we do not have an IgE test result, we often have a skin test result. This means that we still have some information about the underlying sensitization state even when an IgE test result is missing, which is likely to diminish any bias caused by its missingness.
There is another impact of missing data. Even when missing data is not introducing bias, if there is a lot of missing data it can lead to uncertainty, in the form of very broad posterior distributions. For example, at several time points we have no data for mould or peanut and so the gain/retain probabilities for those ages would be very uncertain, and so have broad posterior distributions. When included in results plots, such broad distributions can distract from the remaining meaningful results. To keep our plots as clear as possible, we will simply drop the mould and peanut allergens from our data set and consider only the remaining six allergens.
Some initial results
Having decided to treat our missing data as missing completely at random, we are now in a position to apply expectation propagation to our model and get some results. Where we have a missing data point, we simply do not observe the value of the corresponding random variable.
Having run our inference algorithm, the first posterior distributions we will look at are those for probSkinIfSens, probSkinIfNotSens, probIgeIfSens and probIgeIfNotSens. These posteriors are beta distributions, which we can summarise using a mean plus or minus a value indicating the width of the beta distribution, as shown in Table 6.3.
The results in Table 6.3 show that the two types of test are complementary: the skin prick test has a very low false positive rate (<1%) but as a result has a reduced true positive rate ( 79%); in contrast, the IgE test has a high true positive rate ( 93%) but as a result has a higher false positive rate ( 4%). The complementary nature of the two tests show why they are both used together – each test brings additional information about the underlying sensitization state of the child. During inference, our model will automatically take these true and false positive rates into account when inferring the sensitization state at each time point, so it will gain the advantage of the strengths of both tests, whilst being able to minimise the weaknesses.
Next let’s look at the inferred probabilities of initially having, gaining and retaining sensitization for each allergen. Figure 6.7a shows the probability of initially having a sensitization (age 1) and then the probability of gaining sensitization (ages 3, 5, 8). Similarly, Figure 6.7b shows the probability of retaining sensitization since the previous time point (for ages 3, 5 and 8 only). Since each probability has a beta posterior distribution, the charts show the uncertainty associated with the probability values, using the lower and upper quartiles of each beta distribution.
Looking at Figure 6.7a and Figure 6.7b together, we can see that different allergens have different patterns of onset and loss of sensitization. For example, there is a high initial probability of sensitivity to egg but, after that, a very low probability of gaining sensitivity. Egg also has the lowest probability of retaining sensitization, meaning that children tend to have egg sensitivity very early in life and then rapidly lose it. As another example, mite and pollen have very low initial probabilities of sensitization, but then very high probabilities of gaining sensitization by age 3. Following sensitization to mite or pollen, the probability of retaining that sensitization is very high. In other words, children who gain sensitization to mite or pollen are most likely to do so between ages 1 and 3 and will then likely retain that sensitization (at least to age 8). Cat and dog have similar patterns of gain and loss to each other, but both have a higher initial probability of sensitization and a lower peak than mite and pollen. Milk shows the lowest probabilities of sensitization, meaning that it is a rare allergy in this cohort of children. As a result, the probability of retaining a milk sensitization is more uncertain, since it is learned from relatively few children. This uncertainty is shown by the broad shaded region for milk in Figure 6.7b.
Another way of visualizaing these results, is to look at the inferred sensitizations. We have inferred the posterior probability of each child having a sensitization to each allergen at each time point. We can then count the number of children who are more likely to be sensitized than not sensitized (that is, where the probability is >50%). Plotting this count of sensitizations for each allergen and age gives Figure 6.8.
Figure 6.8 shows the patterns of gaining and losing sensitization in a different way, by showing the count of sensitized children. The chart shows that egg allergies start off common and disappear over time. The chart also shows that mite and pollen allergies start between ages 1 and 3 and the total number of allergic children only increases with age. In many ways, this chart is easier to read than the line charts of Figure 6.7a and Figure 6.7b because it looks directly at the counts of sensitizations rather than at changes in sensitizations. Also, all the information appears on one chart rather than two. For this reason, we will use this kind of chart to present results as we evolve the model later in the chapter.
To summarize, we have built a model that can learn about the patterns of gaining and losing allergic sensitization. The patterns that we have found apply to the entire cohort of children – effectively they are patterns for the population as a whole. What the model does not tell us is whether there are groups of children within the cohort that have different patterns of allergic sensitization, which might give rise to different diseases. By looking at all children together, this information is lost. Reviewing our assumptions, the problematic assumption is this one:
- The probabilities relating to initially having, gaining or retaining sensitization to a particular allergen are the same for all children.
We’d really like to change the assumption to allow children to be in different groups, where each group of children can have different patterns of sensitization. Let’s call these groups ‘sensitization classes’. The assumption would then be:
- The probabilities relating to initially having, gaining or retaining sensitization to a particular allergen are the same for all children in each sensitization class.
The problem is that we do not know which child is in which sensitization class. We need a model that can represent alternative processes for gaining and losing sensitization, and which can determine which process took place for each individual child. In other words, we need to be able to compare alternative models for each child’s data and determine which is likely to be the one that gave rise to the data. To achieve this will require some new tools for modelling and inference, which we will introduce in the next section.
missing dataIn a data set, a missing data point is one where no value is available for a variable in an observation. The reason for the value being missing is important and can affect the validity of probabilistic inference using the remaining non-missing values. See section 6.2.
missing at randomWhere missing data points do not occur at random, but where other known data values fully account for the fact that the data is missing.
missing not at randomWhere missing data is neither missing completely at random (MCAR) not missing at random (MAR). In this case, the fact that a data point is missing depends on the value of that data point. Where data is missing not at random, it is very difficult to avoid biases in the results of inference.


By Ishmael Sallieu Koroma
The Acting Deputy Registrar in the College of Medicine, Brima Bah, has said about one hundred and fourteen (114) students at College of Medicine and Allied Health Sciences (COMAHS) have been withdrawn from pre-med one and two in the medical course.
Brima Bah made this statement today on SLBC’s Morning Coffee Program where he disclosed that the withdrawn students in the college of Medicine will not be able to continue their Programme at the college.
‘’Those that have c4 and better in all the science subjects in physics, science, math, English and Biology they will go to pre- med two . That means they are able to go straight to year 1 medicine,’’ he said, adding that for those in Pre-med 1if they pass they will be going over to pre- med 2.
The Acting Registrar said the criteria to pass from pre-, med 1 to pre –med 2 is that the student should pass all four his or her subjects including math’s, physics, Biology and chemistry adding that the students should passed all the subjects with not less than fifty percent in each of the subjects and at the end of the day the University is looking for an average of 55%minimum to be promoted.
He further said if the student doesn’t meet the requirement of 55% but have 50% he or she will have the opportunity to repeat in pre-med 1 adding that if the students do not meet of the 50 % in any of the four subjects , the students will be not be able to repeat again .
‘’For pre-med 2, it’s little bit different, the four core subjects which are Biology, Chemistry, math’s and physics .First of all the very first criterion is the students should pass them with not less than 50%, that means you have cleared, but if you failed one of them with 40% or better you will have the opportunity to sit a reference provided the student making 170% average on the three other subjects’’
Mr. Bah further explained that if the student in pre – med 2 failed two subjects of the four the subjects but has an average of 120% of the two other subjects would have an opportunity again to sit to a re-sit exams adding that if at all the students does not fall in these two situations then the students will not be able to re-sit any exams in the pre-med 2 stating that its part of the regulation at the College of Medicine.
When asked whether it is the highest number of withdrawn students at COMAHS, the acting deputy Registrar replied ’’well I didn’t want to say yes, but I think the number is very high really ‘’.
The deputy Registrar said they at college of medicine were there to train doctors and medical people adding that they want to have people that are fit to go through the course and at the end of the day they will be able to live to the job market.
He said all the students affected can meet the authorities in the college and they will have the opportunity for an explanation on their status and grades for their pre –med in the medical course.
The over one hundred withdrawn students from COMAHS have brewed up serious debate about the means through which senior secondary School students acquired their results as most of them cannot defend their colourful results when they are admitted into the university system with some civil society activists advocating for a blanket entrance exams on all faculties into the University Of Sierra Leone.


Renewable Technologies CoE Manager
Eskom Holdings SOC Ltd
Senzo is a registered Professional Natural Scientist, a Godly family man, currently serving as a Renewables Technologies Manager and a Chief Statistician in Research Testing and Development, Eskom Holdings SoC. Ltd. He has over 9 leadership experience years in the fields of Academia, Finance and Energy. He holds a PhD in Mathematics from University of KwaZulu-Natal (UKZN) and further holds 2 MSc degrees: Applied Mathematics (cum laude) from UKZN and Financial Mathematics from Edinburgh & Herriot-Watt Universities in United Kingdom –Scotland. His research interests lie in three main areas, videlicet: Symmetries in Mathematical Modelling, Quantitative Risk Management and Energy Profile Modelling with applications in Renewable Energy.


Personalised Arithmetic Chains With Names Made of Wooden Beads for Learning First Arithmetic Skills
- Custom-made Arithmetic Chains From the Online Configurator
- Beautiful Design Ideas for Individualised Counting Chains
- Individualised Arithmetic Chains as a Gift Idea for Starting School
Personalised arithmetic chains with names are not only pretty, but useful as well. With the colourful, individualised learning chain, children can playfully learn about the number range up to 20. So design your very own custom-made counting chain by using the online key fob configurator.
Custom-made Arithmetic Chains From the Online Configurator
Use our configurator to design your very own personal arithmetic chain with lots of joyful anticipation and creativity. Our large assortment of high-quality wooden beads invites you to design with ease. First choose a key ring. Would you like a classic round key ring or a key ring in the shape of a heart, a star or a flower? Then you can create the bead strands. You can drag and drop the individual beads onto the string.
Interesting questions and answers about arithmetic chains
An arithmetic chain is useful because it promotes children’s mathematical imagination. By using the colourful bead chain, they learn counting, addition and subtraction. In addition, the chain trains children’s fine motor skills and basic understanding of numbers.
You can usually use an arithmetic chain for children of pre-school age as well as early primary school age. These are children between the ages of about 3 and 7. At this stage of life, children learn basic mathematical concepts: for example, counting, an understanding of quantities, and simple addition and subtraction. An arithmetic chain can therefore provide a playful way to develop and encourage these mathematical skills.
To ensure that the child has a lot of fun with the chain, you can use their favourite colours for the design. You can also select motif beads that the child likes – for example, an elephant bead if elephants are the child’s favourite animal. It is important that you use at least two colours for the counting strand. That is because that way you can divide the number range of 20 beads into 4 x 5 beads by colour – making it easier for the children to visually understand the number range.
You can decorate the strand presenting the child’s first name with a cute motif bead. The string with the wooden beads for counting consists of 4 x 5 beads of the same colour. After receiving your order, our experienced team will make your individualised counting chain for you by hand. In the process, we leave a piece of the string free on the bead strand for counting. That allows the child to push the beads back and forth.
Beautiful Design Ideas for Individualised Counting Chains
Thanks to our large assortment of craft materials, you have many possibilities to design the personalised arithmetic chain of your dreams. Firstly, you can customise the beads’ colours as well as the string’s colour with just two clicks. For example, choose from bold shades like red and orange or delicate pastel colours like light pink, baby blue and lemon. If you use the child’s favourite colours, they will certainly have a positive feeling about the counting chain from the very first moment.
The beads’ shape offers you beautiful possibilities as well. For instance, there are many cute motif beads in the configurator. It’s great if you choose motif beads that suit the child: Is the child perhaps fond of dinosaurs? Do they love sweets like cupcakes? Or does the child like cars? The choice is yours! The strand with the wooden beads for counting practice can either be equipped with classic round beads or, for example, with playful grooved beads. The grooves make the beads particularly easy to grasp.
Individualised Arithmetic Chains as a Gift Idea for Starting School
A customised arithmetic chain with name is a fashionable accessory and a practical learning chain at the same time. Children can use the arithmetic chain to learn and practise counting, addition and subtraction. The key ring makes the colourful counting chain flexible to attach – for example to a pencil case, school bag or accompanying a bunch of keys. Thus, the wooden bead chain is always with them as an everyday companion. Therefore, the child can occupy themselves meaningfully – for example, during the short breaks between school hours or while waiting for the bus. All those advantages make an arithmetic chain a wonderful first-school present.
More lovely gifts for primary school pupils are, for example, the following learning chains: With a personalised year chain, the children can learn about the yearly cycle. A combination of an individualised ABC chain with an arithmetic chain combines two learning contents in one wooden bead accessory. But of course you can also give a cute personalised ABC chain with a name as a gift individually.
Or would you prefer a gift in the form of a key fob that will appeal to both children and adults? Then why not give a classic, personalised key fob as a gift? A particularly extraordinary gift idea is a pretty, personalised bag charm. The wooden bead charm can decorate a diaper bag, a school bag or a handbag, for example.


Point to Tangents on a Circle Construction
Geogebra is an interactive geometry, algebra, statistics and calculus application for learning and teaching at all levels. Gnuplot is a portable, multi-platform, command-line driven graphing utility. R is a free software environment for statistical computing and graphics.... GeoGebra Math Apps Get our free online math tools for graphing, geometry, 3D, and more!
Tangents Tool GeoGebra Manual
Secant to Tangent Applet. This applet is designed to build the student's visual intuition about the line tangent to a curve at a point being the limit of the secant lines through that point.... Triangle Construction. Author: hjelm. Overview. Here is a construction of two triangles. One is an equilateral and one is an isosceles triangle. We can see that the properties hold, making the triangles stay as equilateral and isosceles. The dynamic nature is important to show the properties of both triangles. Triangles in GeoGebra. Try moving the vertices around on both triangles to see how
How to draw a tangent to a curve at any point on curve on
I want a quick way to draw the in-circle of many triangles. If drawing a circle, has an option to use three tangents, that would help a lot. Other ways could be to write a macro inside GeoGebra to draw the angle bisector for each apex and find the cross point. how to cook eggs benedict in the oven a) Draw in the tangent to the function at the point as accurately as possible. b) Work out the gradient of the tangent at that point by reading off values on your axes.
GeoGebra Geometry projectmaths.ie
7. To construct point A on function f, select the New Point tool and click graph of the function. 8. To construct a line tangent to f and passing through A, select the Tangents … how to create 3d text effect in coreldraw From B draw a circle through A using the Circle with Centre through a Point tool 5. Find one intersection C of the two circles using the Intersect Two Objects tool .
How long can it take?
Make Math Move with Sliders and Animation in Geogebra
- GeoGebra Tutorial 17 – Functions Tangent Lines and
- To Work With Circles Autodesk Knowledge Network
- GeoGebra Tutorial Drawings
- Chordal Points with equal tangents to two circles – GeoGebra
How To Draw Tangents On Geogebra
- To draw a tangent select point B and the circumference of the circle respectively two tangents will be constructed from the point to the circle. 12. Plot the points : A(4,-3) and B(-2,5), join them and find equation of the line.
- Draw a picture with GeoGebra Use the mouse and various tools from the Tool Bar to draw figures on the drawing pad (e.g. square, rectangle, house, tree,…). To access the tools, you must click on the small triangle to open the tools, then select the tool you want to use. Don’t forget to read the toolbar help if you don’t know how to use a tool. 7. An Introduction To GeoGebra Things to try
- GeoGebra3D. Create a three dimensional Geometry and Graphics View in GeoGebra that is easy to use with the mouse. This view will allow the creation and interactive manipulation of 3D geometrical objects like points, lines, polygons, spheres, and polyhedrons as well as function plots of the form f(x,y).The 3D View should both be usable in the
- Now the problem reduces to finding a circle tangent to two given circles and a given point. Inverting at that point causes the desired circle to become a tangent line to two given circles, which is now easy.


This is a math riddle often asked in Portugal to kids, specially those who are good at math. There are two versions of the same riggle, but if you can do one, you will most likely be able to do the other - I leave you both, just in case:
A hawk flies by a flock of doves and says:
– Hellow hundred doves.
– We are not an hundred doves – the doves replies – but us, as much as us and a quarter of us together, and you hawk, makes a hundred.
So, how many are the doves in the flock?
In the second version, the doves reply, instead:
We are not one hundred doves, but us, as much as us, half od us, a quarter of us and you, hawk, a hundred would be.
So, can you find out how many doves are in each of the flocks?
And, obviously, solve them using only your head, no pen and papel or electronics.


Title：Simultaneous Correspondences and Motion Estimation in Geometric Computer Vision
In this talk, I will introduce our recent efforts of estimating correspondences and motion simultaneously in geometric computer vision. Particularly I will discuss two of our ongoing research projects on this topic in details.
Motion segmentation can be addressed as a subspace clustering problem, assuming that the trajectories of interest points are known. However, establishing point correspondences is in itself a challenging task. Existing approaches tackle the correspondence estimation and motion segmentation problems separately. We introduce an approach to performing motion segmentation without any prior knowledge of point correspondences. We formulate this problem in terms of Partial Permutation Matrices (PPMs) and aim to match feature descriptors while simultaneously encouraging point locations to satisfy subspace constraints. This lets us handle outliers in both point locations and feature appearance. The resulting optimization problem can be solved via the Alternating Direction Method of Multipliers (ADMM), where each subproblem has an efficient solution. In non-rigid structure from motion, given point correspondences across multiple images, it has been shown that 3D non-rigid structure can be recovered through factorization techniques. We present a unified framework to simultaneously solve for point correspondences and non-rigid structure by using the PPMs, aiming at establishing point correspondences and enforcing low-rank constraint in the deformable shape. Our new formulation can handle outliers and missing data elegantly.
Yuchao Dai is currently an ARC DECRA Fellow with the Research School of Engineering at the Australian National University, Canberra. He received the B.E. degree, M.E degree and Ph.D. degree all in signal and information processing under supervision of Prof. Mingyi He from Northwestern Polytechnical University, Xi’an, China, in 2005, 2008 and 2012, respectively. He was a visiting student at ANU from Oct. 2008 to Oct. 2009 under the supervision of Prof. Richard Hartley and Dr. Hongdong Li with the support of the China Scholarship Council. His research interests include structure from motion, multi-view geometry, human-computer interaction, compressive sensing and optimization. He has published papers in both top-ranked journals and prestigious conferences such as IEEE TPAMI, IJCV, ICCV, CVPR and ECCV. He won the best paper award in CVPR 2012 (the first one in mainland China). His recent work aims at developing dense non-rigid structure recovery from monocular video sequences, with a view to support analysis and understanding of complex dynamic scene.
戴玉超博士现为澳大利亚国立大学工程研究院ARC DECRA学者。师从西北工业大学何明一教授乐彩网app下载，他分别于2005、2008和2012年获得信号与信息处理学科学士、硕士和博士学位。2008年至2009年受国家留学基金委资助赴澳大利亚国立大学联合培养乐彩网app下载，对方导师为几何计算机视觉的奠基者Richard Hartley教授。他的研究方向包括结构与运动恢复乐彩网app下载、多视角几何、人机交互乐彩网app下载、压缩感知和最优化等乐彩网app下载。他先后在计算机视觉领域的顶级期刊和会议如IEEE模式分析与机器智能（TPAMI）、国际计算机视觉期刊（IJCV）乐彩网app下载、国际计算机视觉大会（ICCV）、IEEE计算机视觉与模式识别会议（CVPR）和欧洲计算机视觉会议（ECCV）等发表论文多篇。他与何明一教授和澳大利亚国立大学Hongdong Li副教授合作完成在非刚性结构与运动恢复方面的研究工作获得CVPR 2012最佳论文奖（大陆高校28年来首次获得该奖项）乐彩网app下载。近期他的研究工作致力于通过单目视频序列进行复杂动态场景的分析和理解乐彩网app下载。


Is it more practical to invest in your own espresso machine? This redditor does the math
Is it actually cheaper to but an espresso machine and buy imported coffee than to purchase a latte from your favorite coffee shop everyday?
It seems so because a penny pinching redditor has done the math and it seems that buying your own espresso machine could actually be the more practical choice.
According to the redditor, an espresso machine costs an average of roughly $239. Half a kilo of Italian coffee costs around $27, assuming this is the average consumption for a month. Multiply this by 12 (for the 12 months of the year) and you get $239.
That translates to an average of $563 for the first year but after the first year, you can already do away with the machine’s cost and limit it only to the coffee. He also added milk and syrup costs after some commenters noticed it was missing from his computation but after adding $121 for the average annual cost, its still a big difference from the daily visit to the cafe.
Meanwhile, a daily purchase of Latte at an average of $4.75 multiplied by 365 days in a year easily eats at $1710 a year. However, if you value your convenience over practicality, there’s nothing wrong with going for maintaining your daily routine. No judgement. (ANGEL ONG)