Explanation: On the over-considering matter, as the, the fresh amounts was 8 and eight. 8 + eight = 15. therefore, the sum was 15.
Explanation: Regarding the significantly more than-considering matter, because the, the newest wide variety are 2 and you can 2. dos + dos = cuatro. therefore the share is actually cuatro.
Explanation: On the more than-given matter, just like the, new amounts are 2 and you may 3. dos + step three = 5. so that the contribution are 5.
Explanation: About more than-offered concern, just like the, brand new numbers is 6 and 3. six + 3 = nine. so that the share are nine.
Explanation: Regarding the significantly more than-considering concern, as the, the new quantity is actually step 3 and you can six. step 3 + 6 = 9. so the sum is actually 9.
Explanation: On the significantly more than-given question, as, the brand new amounts was 6 and you can six. 6 + 6 = several. and so the share was a dozen.
Explanation: Regarding a lot more than-given matter, because the, the fresh new amounts is 6 and 7. six + 7 = 13. so that the share was thirteen.
Explanation: Throughout the over-provided matter, given that, the brand new number was 0 and you will nine. 0 + nine = nine. therefore the share is actually nine.
Explanation: About significantly more than-given question, given that, this new numbers try 0 and eight
Explanation: Regarding the a lot more than-offered matter, while the, the latest wide variety was 5 and 5. 5 + 5 = 10. therefore, the share are ten.
Explanation: On more than-given matter, since the, the latest quantity are 5 and you can six. 5 + 6 = eleven. and so the sum is eleven.
Matter thirteen. Jason has eight puzzles. Quincy provides the exact same amount of puzzles due to the fact Jason. How many puzzles perform he has got completely? _14_ puzzles
Explanation: Throughout the above-considering question, since the, Jason provides 7 puzzles. Quincy contains the exact same number of puzzles because the Jason. seven + eight = fourteen. The number of puzzles he has completely = fourteen.
Matter 14. Establish Mathematics Write or draw to spell it out ways to discover each sum: 6 + seven, 8 + 4, 2 + nine. ____13__________
Explanation: Regarding above-considering matter, since, the new numbers try 6 and you may eight. 6 + seven = 13. so the contribution are 13.
Explanation: Regarding above-given matter, once the, the latest numbers was 8 and you will 4. 8 + cuatro = 12. so the sum is twelve.
Explanation: On a lot more than-offered matter, since the, the newest numbers is 2 and you will nine. 2 + 9 = 11. and so the sum try 11.
Explanation: On the more than-considering matter, as, the fresh new numbers are 8 and you may eight. 8 + 7 = fifteen. and so the contribution is 15.
Explanation: Regarding above-provided matter, because the, the fresh quantity was dos and you may nine. 2 + nine = eleven. therefore, the sum was eleven.
Explanation: Regarding more than-considering matter, as, the brand new numbers try dos and you may nine. dos + nine = 11. and so the share is actually eleven.
Explanation: On the above-offered concern, just like the, the quantity are 276. one hundred more than 276. 276 + a hundred = 376. so the number is 376.
Explanation: About above-provided concern, https://datingranking.net/pl/feabie-recenzja/ as the, the latest amounts is 20, 30, and you will forty. amount by tens. 20, 31, forty, fifty, 60, and you may 70. The new wide variety try fifty, sixty, and 70.
Explanation: In the significantly more than-given concern, due to the fact, this new quantity is 0 and you may eight
Explanation: Regarding the over-considering matter, because, this new wide variety was 5 and you will 5. 5 + 5 = 10. and so the contribution are 10.
Explanation: In the over-offered concern, while the, the brand new quantity is actually 6 and you may 4. six + cuatro = ten. and so the share is 10.
Explanation: On the a lot more than-offered concern, because, the newest quantity try eight and you may step 3. eight + step 3 = 10. so that the sum try ten.


Dynamic Mean - Preserving Spreads and the Agency Problem
Daniele Rinaldo, Ph.D. Candidate in Development EconomicsBrown Bag Lunch
Shekhar Hari Kumar
We consider a problem of finding optimal contracts in a principal/agent model when the observable output is driven by a non-Gaussian, mean-preserving noise source, in which the agent's actions increase the drift of the output process and simultaneously spread probability to the tails of the distribution. We consider a one-period model with exponential utilities for both principal and agent, solve the general problem for risk sharing, moral hazard and adverse selection up to a transcendental equation and discuss the numerical properties of the optimal contract. We fully solve the case of a risk-neutral principal for both moral hazard and adverse selection, and explore the possibility of a continuous time extension.


From the Greek, eight-faced or eight-sided, the octahedron is eight equilateral triangles joined along 12 edges to make six vertices or corners. The shape has four pairs of parallel faces.
The Five Solids, also called the Platonic Solids or the Pythagorean Solids, are directly associated as particles of the elements, are seen as the building blocks of reality. By using geometric figures, Pythagoras (and later Plato) implicitly connects the 5 solids to ideas of harmony and symmetry, and as such offer insights into the nature of the cosmos.
The octahedron symbolizes Air, according the Plato and the other Pythagoreans.


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This term our STEAM focus is engineering and we will be answering the question ‘Can I design and construct a product using my understanding of the engineering process?’.
Through our project, the children will:
- Design and make a product to help teachers “have eyes in the back of their head”.
- Create observational drawings which incorporate tonal contrast and shadow
Through this term’s STEAM project we will be covering the disciplines of Science, Design and Technology, and Art. In our learning journey we will:
- investigate how light travels and interacts with different surfaces (science)
- explore how to create a variety of tones (art)
- use knowledge of how light travels to solve problems (science)
- use tone to create a 3D illusion (art)
- explore the relationship between objects and their shadows (science & art)
- use knowledge of how light travels to solve a problem (science & DT)
- explore techniques for joining materials (DT)
- explore the needs of a consumer (DT)
- use market research to create a design brief (DT)
- create a range of suitable design proposals (DT)
- construct a product to meet a design brief (DT)
- create a product pitch for a target audience (DT)
In our work on area and perimeter, we will learn that:
- Area is the measurement of the surface of a flat item.
- Area is measured in square units, such as square centimetres (cm2) and square metres (m2).
- The area of a rectangle can be calculated using multiplication; the area of a composite rectilinear shape can be found by splitting the shape into smaller rectangles.
In our work on decimal place-value knowledge, multiplication and division, we will learn that:
- To multiply a number by 10/100/1,000, move the digits one/two/three places to the left; to divide a number by 10/100/1,000, move the digits one/two/three places to the right.
- Measures can be converted from one unit to another using knowledge of multiplication and division by 10/100/1,000.
In our work on ×/÷ decimals by whole numbers, we will learn that:
- Decimals (with a whole number of tenths or hundredths) can be multiplied by a whole number by using known multiplication facts and unitising.
- Multiplying by 0.1 is equivalent to dividing by 10; multiplying by 0.01 is equivalent to dividing by 100. Understanding of place value can be used to divide a number by 10/100: when a number is divided by 10, the digits move one place to the right; when a number is divided by 100, the digits move two places to the right.
- To multiply a single-digit number by a decimal fraction with up to two decimal places, convert the decimal fraction to an integer by multiplying by 10 or 100, perform the resulting calculation using an appropriate strategy, then adjust the product by dividing by 10 or 100.
- If the multiplier is less than one, the product is less than the multiplicand; if the multiplier is greater than one, the product is greater than the multiplicand.
- To divide any decimal fraction with up to two decimal places by a single-digit number, convert the decimal fraction to an integer by multiplying by 10 or 100, perform the resulting calculation using an appropriate strategy, then adjust the quotient by dividing by 10 or 100.
In our work on playscripts, we will learn to:
- identify the organisational and presentational devices of a playscriptâ
- infer character through what is said, how it is said and through actionsâ
- vary levels of formality in a playscript for effectâ
- use correctly punctuated stage directions to direct speech and actionâ
- write a scene for a playscript
- perform a scene from a playscript
We will then move on to looking at writing biographies and poems which explore form.
This term in Computing we will learn to create vector drawings. We will learn how to use different drawing tools to help them create images. The children will recognise that images in vector drawings are created using shapes and lines, and each individual element in the drawing is called an object. We will layer their objects and begin grouping and duplicating them to support the creation of more complex pieces of work. We will use Google Drawings to create our vector drawings.
In HRE this term, we will learn:
- that there are different types of relationships and the way we show affection can change depending on the relationships we have; we will think about appropriate ways of showing affection depending on the relationship and the situation.
- That it is against the law for anyone to make harmful changes to our body; we will explore who we can talk to if something worries us and how we can seek advice.
- That friendships can provide us with support; how to deal with loneliness or feeling excluded; resolving differences in the ups and downs of friendships.
- what it means for a friendship online or offline to be safe; how to manage risks and get support.
- about the impact of bullying and the consequences of hurtful behaviours on others.
- strategies to respond to hurtful behaviour experienced or witnessed, offline and online (including teasing, name-calling, bullying, trolling, harassment or the deliberate excluding of others)
This term we will be learning about Jazz in our Music lessons. Over the course of the term, we will focus on two tunes: Three Note Bossa and The Five Note Swing.
Throughout the unit we will:
- Listen and Appraise the two tunes and other supporting tunes.
- Learn about the interrelated dimensions of music through playing instruments and improvising.
- Perform and share
This term in PE, we will be focussing on Gymnastics and Hockey.
In Gymnastics, our unit will take inspiration from the gold medal winning Bulgarian Gymnastic team. We will learn to:
- create an extended sequence of 5 – 7 elements
- add in a piece of equipment to a routine e.g. ball, hoop, ribbon.
- perform a routine to a piece of music
In Hockey this term, we will learn to:
- dribble a hockey ball
- pass and receive the ball the ball standing still or on the move
- shoot the ball into a goal / target
- develop attacking and defending skills for playing a game
- use our skills to play a competitive game
This term in French we will be exploring the topic, “Food, glorious food!”
We will learn to:
- Appreciate stories, songs, poems and rhymes in the language in the context of food.
- Follow a familiar story in French.
- Understand key features and patterns of basic grammar in the context of requesting something to eat.
- Use determiners for identifying quantities in making polite requests.
- Understand key features and patterns of basic grammar in the context of stating preferences about food.
- Use the definite article when generalising.
- Give a preference for or against things.
- Describe people, places, things and actions orally and in writing in the context of describing food by colour.
- Describe the colour(s) of an object by modifying adjectives.
- Describe people, places, things and actions orally and in writing in the context of describing objects by size.
- Begin to place adjectives appropriately before or after the noun they modify.
- Begin to understand that adjective spelling depends on number and gender.
- Engage in conversations; ask and answer questions; express opinions and respond to those of others; seek clarification and help in the context of preparing, eating and talking about food.
- Use a range of grammar structures to practise a set of vocabulary groups.
Across the school this term, we will be exploring the big question “Are people really good or bad?” in our RE assemblies.
In our RE lessons in class, we will be learning to:
- explain Christian beliefs about forgiveness
- explain Hindu beliefs about free will
- explain Hindu beliefs about karma


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EducationUniversity of West Georgia - Bachelors, Computer Science
ExperienceI earned my undergraduate degree in computer science from the University of West Georgia. I love tutoring students because I see it as an opportunity to build their confidence and show them they are smarter than they believe. As a student, I found I always did better when I would work on topics independently. The process of figuring out solutions made it easier for me to understand and retain. While this made me a difficult student to teach, the experiences helped me become a great tutor because I have the ability to figure out new ways to teach until the student resonates with the lesson and understands the material.
I offer tutoring for a wide variety of subjects, including algebra 1 & 2, calculus 1, elementary math, geometry, middle school math, pre-calculus, essay editing, writing, English grammar and syntax, computer programming, computer science, drawing, Java, arithmetic, nutrition, algebra 3/4, painting, 2D design, photography, technical writing, drawing, web design, pre-algebra, web development, discrete math, English, technology and computer science, and general study skills. I also teach similar AP subjects. I love finding new ways to help my students develop a passion for math. When they are able to move beyond the textbook and into a real world experience, the subject comes alive and their interest and success soars.
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See picture above. Because the circuit is semi-infinite, total impedance of circuit in the box is the same with total impedance of the whole circuit. So we can draw the circuit like picture below.
To make it easier, we should work with complex number.
Hence, the impedance of resistor, inductor, and capacitor are respectively
So the total impedance of the circuit is
And now we need to change the form, so we assume that
Then give the power of two for each sides of the equation.
And now we can get
Solving these equation, and we get
So the total impedance is
Because the real part of impedance cannot be negative, so the total impedance is
So, the magnitude of total impedance is
That's my solution.
If you have any question, correction, or comment, please leave a comment below.


My life is massively busy these days, but that doesn't mean I can't update my site once in a while.
Firstly, last semester (fall 2019) I had only two courses, physics (in which I got a D, sadly) and mathematics (in which I got a B, happily).
Secondly, this semester I have three courses: Linear Algebra (mathematics), Relativity & Astrophysics, and Experiments in Physics Education. Everything is in English, which I'm perfectly happy with.
Thirdly, I didn't get cracking on working on the Books page, but oh well.
Really, this update is a way for me to kick-start myself into working more on the site, and not to let too much time pass between each update. Carry on.


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Becoming a Full Time Developer
A look at my journey to becoming a full time developer.
Tinkercad in Math Class Part 2
Just a bunch of photos of student work.
Tinkercad in math class
How Tinkercad can be used in math class to teach geometric transformations.
TinkerCad Code Blocks
A look at TinkerCad Code Blocks. A great way for students to learn to code...
Create Make Learn 2021
Create Make Learn 2021 - A week long making extravaganza for educators. I was excited...
Using the new Notion API with Gatsby.JS
Quick and dirty intro to using the notion API with Gatsby.JS. Written in a...
NCUJHS Tech Dashboard V2
A quick look at the new version of the NCUJHS Tech Dashboard. This is...
NCUJHS.Tech Guided Tour V1
A guided tour of the NCUJHS.Tech School Dashboard. This is the initial version that...
New Place To Work
A quick look at my new desk in the living room.
The Fantastic Fraction Game
A game I made for my students to practice adding fractions.


Right around the time you get your first basic regression or classification model going, it will at least cross your mind. The vast piles of time series data, coupled with the possibility of retiring young has the irresistible pull of finding an old treasure map in your grandfather’s attic. How can you NOT think about it? Can you use machine learning to predict the market?
I had to at least try. Here’s what I did, and what I learned.
The ApproachThere are plenty of small scales tutorials on the web that are a great place to start. They show you how to pull down the history of a stock, perhaps calculate a few indicators, and feed it to a regression algorithm and try to predict the next day’s value. Alternatively, they use a classifier to predict whether the stock will rise or fall, without predicting a value.


Edit: In this blog post I talk sometimes about what I called “creased” points on a curve. My analysis of these points was incorrect. Specifically, I state that there are tangent lines at these points, which is false. Such analysis requires more knowledge of singular points than I have.
I’ve been studying from Erwin Kreyszig’s Differential Geometry. It provides a classical perspective on differential geometry through adventures in 3-dimensional Euclidean space, or when we introduce a Cartesian coordinate system. The second chapter is concerned with the theory of curves in , and it ends with a discussion of so-called Bertrand curves. There aren’t many pictures to go with this discussion, so I tried making some for myself. I found success in the case of planar curves and my experience gave me useful insights into the theory.
Bertrand Curves Defined
Discussing Bertrand curves requires familiarity with some tools in the theory of curves. I’ll present an overview here of the necessary terms and notation from my textbook, assuming experience with vector calculus from the reader. Even if you don’t understand the technicalities, though, you should still be able to make sense of things from the pictures.
A curve captures the notion of something which is traced out by a moving point: strokes of a pencil, paths of stars and planets, the flight of a bumblebee. There are different approaches to create a mathematical model of this loose concept; we’ll use a classical one. Loosely, a curve is a set of points in which can locally be represented by a certain equivalence class of vector-valued functions, each on some interval . We use equivalence classes since, for example, a slow-moving pencil and a fast-moving pencil can draw the same curve and we don’t want to make a distinction there. So for any curve we consider many different functions at once, though in practice it’s often enough to choose just one representative. To apply analytical tools, we require that our functions be continuously differentiable (sometimes several times so). For this reason, the objects we are working with are sometimes called differentiable curves. We also require that the our functions have non-vanishing derivatives so we can make use of tangent vectors. Since I will be writing about planar curves, our point sets will be restricted to a subset of which we identify with .
I will write to mean “a curve which has an allowable representation “. The parameter is understood to come from some interval . I will often abbreviate functions of the parameter by removing , for example . This creates a somewhat convenient confusion between the function and its value at a point.
The unit tangent vector to a curve at some point is the unit vector in the direction of . That is, . Each point of has two opposite-facing unit tangent vectors, depending on the direction in which the curve is traversed by the representation . The unit tangent vector is easy to confuse with the parameter , so I’ll point out here that I’m using a bold font for vectors.
A normal to a curve is a line which intersects at a right angle to the unit tangent vector. If this property holds for the unit tangent vector in one direction then it also holds for the one in the opposite direction, so this definition is independent of the choice of representation. This extends the usual concept of a normal to a line.
Note that in these diagrams I was only able to show normals which lie in the plane of the screen. However, we can also consider normals which pierce the screen just at a point. With this in mind, there is no sense in talking without any further qualification about the normal at a point ; in general we talk about a normal at .
Let be a curve such that is two times continuously differentiable. There is an associated function called curvature. Curvature lives up to its name by increasing at points where the curve resembles a smaller circle and decreasing at points where the curve resembles a larger circle. The curvature of a straight line is zero. When another related function called radius of curvature is defined as . In fact, is the radius of the best circular approximation to at any given point. The radius of curvature of a circle is the radius of that circle.
Let be a curve with non-vanishing curvature. The unit principal normal vector at a point is the unit vector in the direction of . That is, . It happens to be normal to and independent of the choice of representation. The principal normal at a point is the normal which intersects at and agrees with the direction of . The principal normal specifies one out of the many normals discussed before, so we do speak of the principal normal. In the case of planar curves, the unit principal normal vector to a curve is the vector which lies in the plane and points towards the curvature of . Because of this, the principal normal to a planar curve at a point (when it is defined) is the plane normal to the curve at that point.
With these terms out of the way, we come to the definition of Bertrand curves:
Two curves which, at any of their points, have a common principal normal are called Bertrand curves.
That’s a nice definition, but it doesn’t have a clear meaning without inspection. By working with some representations of a pair of Bertrand curves, we can get a better idea of what they’re like.
A More Concrete Understanding
Suppose that and are Bertrand curves, and that the shared parameter is chosen so that the points and always lie on a common principal normal (this is always possible). Then the vector is parallel to . That is, where is a function of . is the signed distance between and . It is positive if the direction from to is the same as the direction of and negative if those directions are opposite. We will see that is a constant with respect to by considering the derivative of . We have . The first argument to this dot product is parallel to as we have already noted. The second is a difference of vectors which are parallel to and , respectively. Since and are both perpendicular to , the arguments to the dot product are perpendicular too, and the expression vanishes.
This exploration leaves us with a nice expression relating the representations of our Bertrand curves: , . This expression has been obtained under the assumption that we already have a pair of Bertrand curves. If instead we begin with just one curve with non-vanishing curvature and define for , then we have a family of vector-valued functions of , each of which corresponds to a set of points which might be a Bertrand curve to . The set of Bertrand curves to is contained in the family of these point sets. We only have set inclusion, though; .
If is a planar curve, then the Frenet-Serret formulas allow us to calculate . When (i.e. when ) is an allowable representation of a curve . We can also calculate , , and . This last equation implies that is a Bertrand curve to . When on the other hand we have , so is not an allowable representation of a curve and its point set is not a Bertrand curve to . In summary: contains the members of which correspond to values of that the function never attains.
This gives us a lot Bertrand curves. We always have so, for example, every negative value of corresponds to a Bertrand curve. I found that studying the family associated to a planar curve is a rewarding exercise, both because of the knowledge to be gained and because of the pretty pictures we see along the way.
Example 1: The Circle
An allowable representation of a circle with radius is . We can calculate , , and . So our family of functions is .
With our discussion about and in mind we consider two cases:
- Suppose . Then is a Bertrand curve to . is a circle, concentric to and with radius .
- Suppose . Then the point set of is the singleton containing the center of .
We see that the Bertrand curves to a circle are the concentric circles.
Circles are an important example in the theory of curves. At the local scale, any curve which has non-vanishing curvature resembles a circle of radius . So by studying the behavior of and over a small interval we can gain insights which will be useful when we inspect the behavior of a more complicated curve and its corresponding family of maybe-Bertrand curves. I’ve made a few diagrams with some small-scale behavior for my notes. In each one I’ve indicated the direction in which and travel, as well as the direction of the vector .
This also motivates a mechanical interpretation of our maybe-Bertrand curves in . We can imagine a carriage running along a track with an arm attached at a right angle whose tip traces a path. We can also imagine an ant crawling on one side of the curve while wearing a hat whose top traces a path.
Example 2: The Parabola
The parabola we’ll be considering is given by the allowable representation . We can calculate and . By inspection, we see that is a right-angle rotation of counter-clockwise. That is, . So our family of functions is .
This time is not constant, so we need to be more careful in our dissection of . is an unbounded concave-upward even function, with a minimum at . I’ll write .
We are interested in solutions to the equation for constant . In light of the description of we now consider three cases.
- Suppose . Then we always have . is a Bertrand curve to .
- Suppose . Then only when . If we have no problems; is a Bertrand curve to . If, on the other hand, then does not represent a Bertrand curve to since . Surprisingly, the point set of is still a curve, despite the vanishing derivative. Notice that the first component of is strictly increasing, hence one-one. This means that we can solve for and use as a new parameter: . Except at the point where , is an equivalent representation to . Hence is continuously differentiable, even at . Since the derivative of is non-vanishing, it is an allowable representation of a curve . This curve has a very interesting property at . It possesses a tangent vector, so there is no cusp or corner. However, we see that as , so and . has undefined curvature when ; one might say that it has as its radius of curvature. I would describe it as looking “creased”. For this reason lacks a principal normal. This is all that stops from being a Bertrand curve to ; if we had instead defined Bertrand curves in terms of plane normals then would be a Bertrand curve to .
- Suppose . Then there are two solutions to the equation . Similar to the previous case, if we have no problems; is a Bertrand curve to . If then does not represent a Bertrand curve to since . The point set described by is not a curve at all since there are cusps at where suddenly reverses direction. As before, at these points.
I’ve created animations of a few lines marching along like our friend the ant. I’ve also included the center of the approximating circle to at the base of the middle marching line. This happens to also be the center of the approximating circle to the curve when it exists. In fact, this property extends to all planar Bertrand curves. The approximating circles to planar Bertrand curves at corresponding points are concentric. Each frame approximately resembles one of the four diagrams of small-scale behavior for concentric circles.
Example 3: The Sinusoid
In this section will represent an arbitrary integer so that we can deal with the periodicity of the curve.
The sinusoid we’ll be considering is given by the allowable representation . We can calculate and . Unlike the previous examples vanishes at some points, whenever to be specific. This means that does not possess a unit principal normal vector at these points. When does possess a unit principal normal vector we can see with some thought that . So our family of functions is , defined everywhere except .
Over the interval , is defined. Further, it is an unbounded concave-upward function which is symmetric about the center of and with a minimum at . I’ll write and . We can see that over this interval the sinusoid is geometrically very similar to the parabola.
We are interested in solutions to the equation for constant . To work with we assume that the interval over which is defined is contained in . We once again consider three cases.
- Suppose . Then we always have . is a Bertrand curve to .
- Suppose . Then the only solution to the equation in is . If we have no problems; is a Bertrand curve to . If then we have a similar situation to the one we had for the parabola in this case. We find that is not an allowable representation of a curve but its point set is a curve . is not a Bertrand curve to because it lacks a unit principal normal vector at . It does have a unit tangent vector at this point, and I would say it looks creased.
- Suppose . Then there are two solutions to the equation in . If If we have no problems; is a Bertrand curve to . Similar to the parabola, we find cusps in the point set of when .
We see that the alternating direction in which curves causes jump discontinuities in , which depends on .
Parallel curves are a generalization of planar Bertrand curves. A curve whose points are at a constant normal distance from another given curve is a parallel curve. We’ll treat this as a definition for parallel curves, although there is a more general definition than this.
Since this new definition doesn’t require the existence of principal normals, the questionable “creased” curves which we have encountered do count as parallel curves. The jump discontinuities created by the waving sinusoid are also resolved by the new definition, though this requires deeper consideration.
If we follow through with a similar analysis to the one we did before, we find ourselves researching a family , where is a right-angle rotation of in a constant direction; our crawling ant stays on one side of the curve. In the first two examples we find no difference between the old and new families of functions since we simply have where the sign is a constant. For , however, this equation cannot hold since the direction of rotation from to changes periodically; we have instead . Further, is defined everywhere in , unlike . We can recover the point set of by taking the union of the point sets of and . This gives us , so for each arc of in which we get one point set in the direction of and another in the direction of . There are still holes in the resulting point set where is not defined, but by taking its closure we end with the point set of as well as that of . Now we see that the jump discontinuities are indeed resolved.
Wikipedia offers this exciting interpretation of parallel curves:
In computer-aided design the preferred term for a parallel curve is offset curve. (In other geometric contexts, the term offset can also refer to translation.) Offset curves are important for example in numerically controlled machining, where they describe for example the shape of the cut made by a round cutting tool of a two-axis machine.
Non-Planar Bertrand Curves
It turns out that Bertrand curves to a given non-planar curve are much rarer than they are for planar curves; most non-planar curves have either one Bertrand pair of none. The only exception to this rule are the helices. Each helix has an infinite family of Bertrand curves, made up of other helices. Although they are infinite, no family contains all helices.
Non-planar Bertrand curves seem a lot weirder to me. My textbook spent all but the first opening paragraph of its Bertrand curve section discussing them, and the discussion is very delicate. I found it much more difficult to come up with explicit equations for non-planar curves which have a Bertrand pair. I tried out a few guesses, but none met the right conditions to even have a Bertrand pair. This speaks to their elusiveness. Because of this I don’t have any more nice pictures. I’m sure it would be interesting to study the family corresponding to a non-planar curve, but I’m just not interested since I won’t find Bertrand curves without a lot more work.
Planar Bertrand curves are neat. I like them. I like them so much that I wrote this blog post about them. Aren’t they cool?
Doing this analysis helped me to practice thinking at a local scale; each of the example functions and are defined over a large domain, but in their analysis we often throw away most of that to focus on local behavior. It was also exciting to try out the techniques of the theory of curves on these curves out in the “wild” after studying about them. Once you’ve learned to use these tools, you can create some very interesting pictures with just some calculation and a graphing calculator
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Our research interests are in quantum information theory, a field that explores the generalization of classical information theory to include quantum phenomena.
Quantum mechanics is the theory that governs the behaviour of atoms, photons and even smaller particles. In recent years, researchers have discovered that one can store, transmit and manipulate information in atoms and photons and — even more excitingly — that quantum mechanics opens up new possibilities for faster computation as well as for secure communication. These discoveries have led to the formation of the interdisciplinary research field of quantum information science, in which physicists, computer scientists, mathematicians and engineers work together in order to build a quantum computer, quantum cryptographic systems or machines capable of quantum teleportation.
It is our goal to develop and solidify the mathematical and physical understanding of the new possibilities emerging from quantum information science. For more details see our Research pages.


2048 x Three v1.0.3 – 2048 x Three inspired number merging tile moving puzzle game with lots of features.
One of the best version of the classic number swiping games for your BlackBerry.
Two games modes
2048, always join the same numbers to merge tiles, 2+2=4, 4+4=8 etc. You can keep playing after 2048
Threes, in this mode you get random tiles with the numbers 1, 2 or 3. The rules to merge the tiles are 1+2 = 3, 3+3=6, 6+6=12 etc.
4 different visual themes
3×3 boards for a quicker, more challenging game
4×4 the classic size
5×5 for a more easy game, use this one
6×6 if you just want to relax, sliding tiles, 6×6 is for you, be prepared for very high scores
File Size: 339 KB
Required: BlackBerry Device with Screen Resolution 640 x 480 (9900, 9930, 9981)


Resource page for the book: Applied Adaptive Statistical Methods
By T. W. O'Gorman
Published by SIAM in 2004.The book contains a detailed description of an adaptive testing method, which is based on permutations of independent variables. It also includes adaptive confidence intervals, and adaptive estimation procedures. To obtain SAS macros or an R function for computing adaptive tests of significance for this book click here.
To obtain SAS macros for computing adaptive confidence intervals, using the approach described in the book, click here.
To obtain SAS macros for computing adaptive estimates click here.
To obtain an executable file for performing variable selection with case-control data click here.
If you are interested in adaptive testing please do not hesitate to contact me at my e-mail address (firstname.lastname@example.org) .


For instance, the calculation ends in ERROR. A few examples of using diff are shown below. Key in the function for which you want to discover the limit.
The Importance of Integral Calculator
There are many ways to fully grasp how convolution works. Currently no pure deduction proofs can be accomplished. For instance, the integrals are very important in probability, although there is not any easy formula for the antiderivative.
A plot of S is provided below. Stirling’s approximation provides the asymptotic formula for large x and big y. The idea of Derivative is at the heart of Calculus and contemporary mathematics.
Convolution lets you establish the answer to more elaborate inputs like the one shown below. It can seem daunting initially, but when you get the hang of it, correlation matrix will be as simple as counting 1-2-3. A function isn’t differentiable at an if there’s any sort of discontinuity at a.
In shell integration, it’s the opposite. The accession of a derivative term, however, does not have any influence on the steady-state error. The integral is caused by the continuous summation of an infinitely high number of infinitesimal terms.
As an example, fewer data points are offered for computing the most suitable derivative close to the conclusion of a test. Since finding derivatives via the limit procedure for the previous section can be quite tedious, however, it’s time to introduce a lot faster method. Apparently, in the event the typical ends up being 19, and the very first unit cost 10, then the price of producing a unit has to be changing as we produce various units.
The calculator can figure out the approximate integral. Bring a calculator you are conversant with. An online integral calculator will find the calculation done for you so you don’t have to do it manually and you can obtain the suitable outcome.
Use relevant our financial tools to define how much money you’re able to earn, in addition to how much and how frequently you need to pay your creditors. We’ve covered a great deal of integral calculator tools online but covering double integral calculator was somewhat challenging for all of us. No registration or sign-in is needed.
I am bewildered on how best to attack the issue of calculating the volume of a slice of an ideal circle cone. When rotated, it will appear similar to our preceding rotation but using a cylinder removed in the center. The diameters of the best side and the base are parallel to one another and are found in the middle of the axis.
Our ordinary value of a function calculator provides you a detailed explanation to locate average value of the given function. The volume is figured in units cubed. It ought to be completely submerged to assess the object whole volume.
Details of Integral Calculator
Differentiate by hand then apply your TI-89 to confirm your result. Needless to say, not one of these are in all sorts of standards, yet. As an example, To comprehend how implicit differentiation works and utilize it effectively it is crucial to recognize that the vital idea is just the chain rule.
Here’s the system we’ll utilize. If you really want to know the results of tuning the individual gains, you’ll have to do more analysis, or might have to perform testing on the true system. Be aware that every one of the tests so far are valid just for positive functions.
We are now able to finish the integration issue. You may wish to identify a number of the crucial functions that you can’t do http://www.mathcalculator.org/integral-calculator/ without. It’s a multi-line display that allows multi-replay functionality to be fast and uncomplicated.
The Most Popular Integral Calculator
Taking everything into consideration, Step-by-step Derivate Calculator is a helpful and user-friendly application designed to present a simple yet efficient means of calculating different derivate expressions while explaining each step. There’s a chance to test the answers. You can examine your answer in graphs and visualizations since it helps any individual to comprehend the things quickly.
There’s a chance to look at the answers. It’s also of wonderful assistance to catch a glimpse of a mistake if you’ve got one. You have to use your mathematical knowledge to specify whether the calculator’s results are reasonable and the way the results may be used to answer a question.
Things You Won’t Like About Integral Calculator and Things You Will
How to fix problems throughout the technique of Lagrange multipliers, A set of absolutely free engineering mathematics lectures in videos. Obviously, getting computers to take care of natural language has turned out to be incredibly tough. Exam’ Before moving from the calculus world, it’s fantastic to have a whole kind of miniature exam.
The Key to Successful Integral Calculator
Producing last second choices aren’t normally the very best option. Even though the layout looks user friendly, it’s still true that you have to have background mathematical understanding about definite triple integrals in order in order to execute the automated calculations. Now, every Engineering Major will need to have a scientific calculator.
Top Choices of Integral Calculator
Recall that second derivatives give information concerning the reversal of slope. Treasury bills of the similar time period could act as a benchmark. So as to recognize what derivative rule applies, you want to understand some derivative rules.
The proportion of the derivatives lets us figure out the limits by the L’Hospital rule. As said earlier, trading in the derivatives market is extremely much like trading in the money sector of the stock markets. Vector derivatives can be combined in various ways, producing sets of identities that are also quite critical in physics.
Top Integral Calculator Choices
If you own a business, knowing this standard statistical data analysis is going to be a terrific aid to boost your services. You might even decide on a background, text and border color which fits with your website. The next example demonstrates the way the use of different tests is more than useful.
Without SegWit, signature data is a fundamental portion of the transaction and must be present to determine the transaction hash. There are lots of cases where you are able to enter absolutely any input x value and put on a valid outcome, and here the domain is thought to be over each of the real numbers. When doing proofs, the amount of feasible cases can occasionally explode dramaticly.


A simple calculator Android app for runners and bikers to quickly display the relation of distance, time, pace and speed. Changing one of the values will affect any corresponding values. Using the settings the app can quickly answer questions like:
– If I run this distance in that time, what pace is that?
– With this speed, how long would it take me to run a certain distance?
– If I run an hour with this pace, how far will it take me?
It will also give some basic predictions for other distances, based on the current values.
All settings are stored in the app. Translations availaible in english and german.
Download for free in the Google Play Store:
If you use this script, be awesome and donate a few bucks!


Keno is a game of chance. The primary goal of the game is to match as many numbers as possible out of the 20 numbers that will be drawn. Mathematical evaluation show that the chances of getting 1 number alone is one in 86,446 or roughly 1.1%. Getting all 20 numbers is near unimaginable statistically. How then can one decide the best numbers to play Keno utilizing numbers 1 to 80? Statistically, his may be not possible, but listed here are a few ideas that may assist improve the odds of winning in Keno.
Review Earlier Winning Numbers
To be able to see if there is a sample, one can evaluate previous Keno results. This, nevertheless, could also be very tedious, because this means getting tons of Keno results and attempting to investigate if all these numbers make a pattern. Analysis can involve just the day gone by's results, a week's worth of results, or a month's value of results. Those that are really into the game can even get a full year's data for analysis. This can help a player determine if certain numbers at all times make it to the cut, or if there are some numbers that don't make it in any respect, since it's also possible to win in Keno if none of the numbers picked matched the drawn numbers.
Restrict Your Decisions
New players will be easily misled by the fact that picking more numbers increases their chances of getting some winnings. However, what they fail to realize is that choosing more numbers mean increasing the odds of not getting all numbers. Thus, instead of specializing in the best way to get the best numbers to play Keno utilizing numbers 1 to 80, a new player should begin small by choosing just a number of numbers at a time. This increases the possibility that they will be getting 1 or 2 out of the 20 numbers chosen, and can considerably give them small but steady earnings. After getting the texture of the game, they'll increase the stakes by picking more numbers.
Get Professional Advice
Keno is meant to be performed for fun. But when one feels the need to really get into the game and win some big cash, then one can resort to seeking knowledgeable advice. Through the Internet, one can simply check sites that provide tips about how you can win in Keno, or find out how to play the game correctly. At a really reasonable worth, one may even download some software that may assist analyze previous successful results and provides one the best possible combination based on historical data.
The perfect numbers to play Keno using numbers 1 to eighty may be inconceivable to get. What one should realize, although, is that Keno, and different number games which are based on chance get its recognition because of the joys and pleasure that one gets when the numbers are drawn one by one and when one sees all or any of his numbers being drawn. Too much evaluation into the game may make it less thrilling and fun. Ultimately, Keno just isn't just about winning big money, but additionally about having fun.
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Students from the Faculty of Economics of the University of Ljubljana Take Prizes in the Prestigious Tournament in Mathematics
On Thursday, 15 March 2018, a student tournament in mathematics for the Yuri Vega awards was held under the auspices of the Society of Mathematicians, Physicists and Astronomers of Slovenia. 30 students from the Faculty of Economics of the University of Ljubljana attended the tournament.
Yuri Vega’s bronze medals were presented to Damjan Dragovan, Maja Furlan, Lucija Grebenc, Nina Hočevar, Jaka Josipović, Matevž Kovač, Milan Kovačević, Urh Kovačič, Alan Velić and Tadej Zalar.
Alan Velić, a first-year student from the School of Economics and Business, was selected for participation in the national tournament, which will be held at the Faculty of Mathematics and Physics of the University of Ljubljana on 21 April.


Learning, development, and response to instruction often involve changes in the strategies that learners use to solve problems. In this chapter, our focus is on mathematical problem solving in both children and adults. We offer a selective review of research on three classes of factors that may influence processes of strategy change in mathematical problem solving: contextual factors, individual factors, and metacognitive factors. Contextual factors involve information that learners encounter in the learning context, such as feedback about prior strategies and examples of alternative strategies. Individual factors involve the abilities, dispositions, and knowledge that learners bring to the learning context. Metacognitive factors involve knowledge about strategies and factors that affect the application of strategies —including perceptions of problem difficulty, confidence in the strategies one already knows, and judgments about the qualities of alternative strategies. These factors operate both independently and in combination to influence learners’ behavior. Therefore, we argue that scientific progress in understanding strategy change will require comprehensive conceptual models that specify how different factors come together to explain behavior. We discuss several such models, including vulnerability–trigger models, cumulative risk models, and dynamic systems models. Research guided by such models will contribute to greater progress in understanding processes of strategy use and strategy change.


Actor Ranveer Singh has revealed he was a failure in studies, but feels good education is important.
‘Formal education is very important. You should know what you are getting into,’ said Ranveer.
‘The biggest learning comes from experience…No greater teacher than experience. Education arms you well to work with technical knowledge and I think a solid education goes a long way,’ added the 28-year-old.
Talking about himself, he said: ‘I was a failure in math, a complete dud. I failed in math in every single class since third standard onwards.’
He was more inclined towards screenwriting and acting and added: ‘In these type of classes, in which I had a genuine interest, I used to do very well. I topped my film class when I was studying in the Indiana University. But in other stuff like math, science, and accounts, I was a complete failure.’


Okay, maybe someone ELSE can help me with the mathematics of the following problem:
Current outdoor temp is 18C (though the high today was 19C). Thermostat is set to 18C. Window is open just enough to let in a breeze, and I haven't used the stove since yesterday, haven't used the oven in days, and the radiant heater hasn't let off a burble/snap/pop/click/bang in over a month...
Current indoor temp is 26C.
HOW does that add up??????? (And why must I be stubborn and insist that if it's only warm for a month at a time here, I don't need to buy a fan!)


Today's soundtrack is Petra: Unseen Power.
I'm on to the first of two units in the final module of this math course. Today, I'm learning about the fundamental counting principle. It sounds pretty basic; I know how to count. How hard could this be?
The Fundamental Counting Principle lets us determine how many possible combinations there are of a group of choices. To find the total combinations available, we multiply the number of choices in each category by the number of choices in the other categories. For example, if I am going car shopping, I can determine how many choices are available to me. Let's say I go to the Chrysler dealership and they say "You can choose a truck, a car, or a minivan. Each of them comes in red, blue, black, or silver. And each one can be purchased with either automatic or manual transmission." So I know that I have three choices of vehicle, four choice of colour for each one, and two choice of transmission for each one. So I could choose a black truck with manual transmission, a black truck with automatic transmission, a red minivan with manual transmission, a silver car with automatic transmission, etc. Multiplying my choices together (3x4x2), I know that I have 24 unique combinations available to me.
What if we run into a situation where the number of choices changes? For example, if I want to order a sub sandwich with meat and three vegetable toppings, I want three different toppings. I don't want a sub filled with onions, onions, and onions. In those cases, we subtract from a category as we go along. So let's say I'm at a store that offers two kinds of bread, white or whole wheat. There are three meat options: chicken, roast beef, or ham. There are two sauces: ranch and chipotle. And there are five vegetable toppings available: spinach, lettuce, tomato, olives, and onions. So my equation would be 3x2x2x5x4x3, giving me 720 different available combinations of sub sandwich with three unique vegetable toppings.
In summary, the fundamental counting principle: "If one item can be chosen m ways, and for each way another item can be chosen n ways, then both items can be selected m x n ways" (contentconnections.ca).


What do you like about mathematics?
The answer to this question should not be"I love math."
The answer to this question ought to be,"I wish to find out what people love about mathematics." Love's subject has a significance. To appreciate math would be to love math.
You can ask the question to learn what people love about math. They will probably answer this query with words like"challenging,""interesting,""entertaining," and"amusing" There's one more factor that can be utilized write a paper for me to give you a description of why folks love math Although these may be helpful words in giving a description of the facets of math that people like.
The last thing you will likely need to know is what people love about math because that's the point. The issue with the question is that people don't love math the way that the thing has significance for them. When you ask this question,"What do you like about mathematics?" You're asking a question about a specific kind of math.
It is no more significant than asking,"What do you like about summer camp?" The reply to the question is not important, although this question is intended to point out that the query is requesting about a phenomenon of thought and feeling.
In go the same manner, you aren't asking the question,"What do you like about math?" The answer to the question is not important, although you are asking a question about a phenomenon of feeling and thought.
The practice of locating the reply to the question"Whatdo you like about math?" In terms of its history, how it's taught, the people using it, the math itself, and the practice of doing mathematics is crucial. The answer to the question,"What do you enjoy about math?" Must help you see clearly the value of the question.
It is important to realize that when you ask a question of what people love about math, you are asking the proper thing. You are asking a question concerning the humanity behind the custom of mathematics. You're asking a question about a process of human thought and feeling that are targeted at helping individuals to become better people.
Even when you did not like the query, you still have an answer. There is a process of human thought and feeling. You're asking the right question, when you ask a question of what people like about math. As the answer to the question can help you find that what you http://do.institutogmarconi.edu.mx/musterbrche-in-geschftsmodellen-umbeck-tobias-wthrich-prof-dr-hans-a.pdf like about mathematics has to do with human nature You're asking the question.
It's a much more important thing to learn what nature is than it would be to learn how to have a high score. You have to understand what human nature is for you to help you make a decision for what math you want to study. If you like math and want to improve your score on a math test, you want to understand what human nature is so you may choose the right math for you. You should stay in your home if you don't care enough about math to learn exactly what human nature is.
How can you ask the question,"What do you enjoy about mathematics?" The answer to the question is to do with how the human mind functions.


Table of Contents
- What is DataShop?
- Getting Data In
- Accessing DataShop
- Project / Dataset Administration
- Citing DataShop and Datasets
- Filtering Data
- Export: Getting Data Out
- Using Other Tools
- Contact Us
The Model Values report provides details on the results of the AFM (Additive Factor Model) algorithm1, a logistic regression performed over the “error rate” learning curve data. The AFM logistic regression, a standard regression bounded between 0 and 1, attempts to find the best-fit curve for error-rate data, which also ranges between 0 and 1.
The Model Values page presents a quantitative analysis of how well, given the selected knowledge component model, the AFM statistical model fits the data (via AIC, BIC, log likelihood) and how well it might generalize to an independent dataset from the same tutor (via cross validation RMSE).
Using R notation, the AFM model (applied to a modified student-step export file called "ds") can be approximately* represented as:
R> L = length(ds$Anon.Student.Id)
R> success = vector(mode="numeric", length=L)
R> model1.lmer <- lmer(success~knowledge_component+ knowledge_component:opportunity+(1|anon_student_id),data=ds,family=binomial())
success variable must be 0 or 1. The first three R commands simply convert the "First Attempt" values (in the student-step export) of "incorrect" and "hint" to 0, and "correct" to "1".
* The AFM code is different from the R expression above in two ways:
- To reduce over-fitting the data, AFM assumes learning cannot be negative and thus constraints the
"slope" estimates of the
knowledge_component:opportunityparameters to be greater or equal to 0.
- The optimization applies a penalty to estimates of the student parameters (
anon_student_id) for deviating from 0—essentially treating
anon_student_idas a random effect.
AFM is a generalization of the log-linear test model (LLTM)2. It is a specific instance of logistic regression, with student-success (0 or 1) as the dependent variable and with independent variable terms for the student, the KC, and the KC by opportunity interaction. Without the third term (KC by opportunity), AFM is LLTM. If the KC Model is the Unique-Step model (and there is no third term), the model is Item Response Theory, or the Rasch model.
For KC models that contain steps coded with multiple skills, Datashop implements a compensatory sum across all of the KCs of the multi-skilled steps, for both the KC intercept and slope. The sum is "conjunctive" when two (or more) parameter estimates are negative, i.e. performance is predicted to be worse when the KCs are combined than when they are alone. The sum is "disjunctive" when the parameter estimates are positive, i.e. performance is predicted to be better when KCs are combined than when they are alone. If one estimate is positive and one negative, the sum is "compensatory" like an average.
AFM is run for all datasets in DataShop, once for each knowledge component model of a dataset. For large datasets, AFM will not run. "Large", in this case, is a function of the number of transactions, students, and KCs—a dataset with more than 300,000 transactions, 250 students, and 300 KCs will prevent AFM from running successfully. The current workaround for this limitation is to create a smaller dataset with a subset of the data.
Based on the knowledge component model and the observed data, the following AFM model values are calculated:
KC Model Values
- AIC: The Akaike information criterion (AIC) is a measure of the goodness of fit of a statistical model, in this case, the AFM model. It is an operational way of trading off the complexity of the estimated model against how well the model fits the data3. In this way, it penalizes the model based on its complexity (the number of parameters). A lower AIC value is better.
- BIC: The Bayesian information criterion (BIC) is also a measure of goodness of fit of the AFM model. The BIC penalizes free parameters more strongly than does the Akaike information criterion (AIC)4. A lower BIC value is better.
- Log Likelihood: a basic fit parameter used in calculating both AIC and BIC; also referred to as the log likelihood ratio. Unlike AIC and BIC, log likelihood assumes the model includes the right number of parameters; AIC and BIC take into account that the parameters of the model could be wrong both in number and value.
- Number of Parameters: the total number of parameters being fit by the AFM statistical model. This number will vary with the number of knowledge components in the knowledge component model, as well as the number of students for which there is data in the dataset.
- Number of Observations: the total number of observations used by the AFM statistical model.
Cross Validation Values
Cross validation is a technique for assessing how well the results of a statistical model (in this case, AFM for a particular KC model) will generalize to an independent dataset from the same tutor. It's reported as root mean squared error (RMSE). Lower values of RMSE indicate a better fit between the model's predictions and the observed data.
Three types of cross validation are run for each KC model in the dataset. All types are a 3-fold cross validation of the Additive Factor Model's (AFM) error rate predictions.
- Student blocked. With data points grouped by student, the full set of students is divided into 3 groups. 3-fold cross validation is then performed across these 3 groups.
- Item blocked. With data points grouped by step, the full set of steps is divided into 3 groups. 3-fold cross validation is then performed across these 3 groups.
- Unblocked. The full set of data points is divided into 3 groups, irrespective of student or step. 3-fold cross validation is then performed across these 3 groups.
For item blocked cross validation, the system optimizes data division so that all KCs appear in training sets at maximal possibility.
For unblocked cross validation, each student and each KC must appear in at least two observations; otherwise that student or KC are excluded from cross validation. This procedure ensures that all students and all KCs appear in both training and testing sets. However, the dropping of data points affects the following two values:
- Number of Parameters: the total number of parameters used in cross validation. This number can differ from the number of parameters in the AFM model if there are not enough observations for a student or KC. See the cross validation note above.
- Number of Observations: the total number of observations used in cross validation. This number can differ from the number of observations used by the AFM model if there are not enough observations for a student or KC. See the unblocked cross validation note above.
After taking into account the exclusion criteria described above, the following data requirements must be met for cross validation to run:
|Cross Validation Type||Requirement|
|Student blocked||at least 3 students|
|Item blocked||at least 3 steps|
|Unblocked||at least 3 students and 3 KCs|
- Intercept (logit) at Opportunity 1: a parameter representing knowledge component difficulty, where higher values indicate an easier knowledge component and lower values indicate a more difficult KC.
- Intercept (probability) at Opportunity 1: derived from the intercept (logit). A parameter representing knowledge component difficulty, where higher values indicate an easier knowledge component and lower values indicate a more difficult KC.
- Slope: a parameter representing of how quickly students will learn the knowledge component. The larger the KC slope, the faster students learn the knowledge component.
- Intercept: a parameter representing a student's initial knowledge. The higher the student intercept, the more the student initially knew.
To view model values for a different knowledge component model:
- Select a different knowledge component model from the drop-down menu under Knowledge Component Models. The Model Values report will update to show values for that model.
Tip: To learn how to define a new knowledge component model, see KC Models.
To export model values:
- Click the button labeled Export Model Values. You will be prompted to save the exported text file.
Predicted success rate is the probability of the student being correct on the first try (no hint request or incorrect action) on a step.
- Υij = the response of student i on item j
- θi = coefficient for proficiency of student i
- βk = coefficient for difficulty of knowledge component k
- γk = coefficient for the learning rate of knowledge component k
- Τik = the number of practice opportunities student i has on the knowledge component k
- Κ = the total number of knowledge components in the Q-matrix
- The Τik parameter estimate (the number of practice opportunities student i has had on the knowledge component k) is constrained to be greater or equal to 0.
- User proficiency parameters (θi) are fit using a Penalized Maximum Likelihood Estimation method (PMLE) to overcome over fitting. User proficiencies are seeded with normal priors and PMLE penalizes the oversized student parameters in the joint estimation of the student and the skill parameters.
The intuition of this model is that the probability of a student getting a step correct is proportional to the amount of required knowledge the student knows, plus the "easiness" of that knowledge component, plus the amount of learning gained for each practice opportunity.
The term "Additive" comes from the fact that a linear combination of knowledge component parameters determines logit(pij) in the equation. For more information on the Additive Factor Model, see Hao Cen's PhD Thesis5.
1 Cen, H., Koedinger, K. R., and Junker, B. 2007. Is Over Practice Necessary? --Improving Learning Efficiency with the Cognitive Tutor through Educational Data Mining. In Proceeding of the 2007 Conference on Artificial intelligence in Education: Building Technology Rich Learning Contexts that Work R. Luckin, K. R. Koedinger, and J. Greer, Eds. Frontiers in Artificial Intelligence and Applications, vol. 158. IOS Press, Amsterdam, The Netherlands, 511-518. PDF
Cen, H., Koedinger, K., and Junker, B. 2008. Comparing Two IRT Models for Conjunctive Skills. In Proceedings of the 9th international Conference on intelligent Tutoring Systems (Montreal, Canada, June 23 - 27, 2008). B. P. Woolf, E. Aïmeur, R. Nkambou, and S. Lajoie, Eds. Lecture Notes In Computer Science. Springer-Verlag, Berlin, Heidelberg, 796-798. DOI=http://dx.doi.org/10.1007/978-3-540-69132-7_111
2 Wilson, M., & De Boeck, P. (2004). Descriptive and explanatory item response models. In P. De Boeck, & M. Wilson, (Eds.) Explanatory item response models: A generalized linear and nonlinear approach. New York: Springer-Verlag.
3 Akaike information criterion. (2007, February 25). In Wikipedia, The Free Encyclopedia. Retrieved 16:22, March 7, 2007, from http://en.wikipedia.org/w/index.php?title=Akaike_information_criterion&oldid=110898239
4 Bayesian information criterion. (2007, February 19). In Wikipedia, The Free Encyclopedia. Retrieved 16:29, March 7, 2007, from http://en.wikipedia.org/w/index.php?title=Bayesian_information_criterion&oldid=109323430
5 Cen, H. (2009). Generalized Learning Factors Analysis: Improving Cognitive Models with Machine Learning. PDF


Recently I have been learning more about topics like data analysis and machine learning. For example via various online courses. More broadly, I am interested in how tools like these can be used to solve real problems.
If you have a project where I can help, or something you would like to discuss, please be in touch.
My background is in mathematics and in research. Doctor of science (technology) from 2007, Helsinki University of Technology (currently Aalto University). My old academic homepage including links to my publications can be found here.
This homepage is build using Jekyll by adapting the Vida theme. Opinions expressed here are my own.


Maybe we should not predict the years of Abalone — Part 1
Both Linear Regression and the Classification by Logistic Regression and Decision Tree
We can start from downloading the ‘Abalone Data Set’ from Kaggle with a csv format. You can actually find the original data set with a txt format, and the description in UCI (University of California, Irving) Machine Learning Repository. Now, we can code.
Here are the libraries that we use.
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import LinearRegression, LogisticRegression
from sklearn.metrics import classification_report, confusion_matrix
from sklearn.tree import DecisionTreeClassifier
First, we call the data by Pandas to process it.
pd.set_option("display.width", 1000)# Your local address that holds the data
df = pd.read_excel("C:/.../abalone.xlsx")
# I converted the csv formatted data into xlsx format to actually see the data first, before coding. # You can also type...
# df = pd.read_csv("C:/.../abalone.csv")
Let’s explore the data…
# First 5 rows with the titles of columns
# Number of rows, columns
# Properties of the columns (index, name, number of values that are not missing, type)
# Some statistics about the columns
# Columns with number of unique values
and the output seems to be like this.
We check the correlation matrix to see the correlations between the variables.
sns.heatmap(df.corr(), cmap='Greens', annot=True)
plt.title("Correlation between the Variables")
Here is the preprocessing part that we should do. In this part, we should convert the ‘Sex’ column, which is categorical, into discrete numeric values, the type that machine can understand. For detail, you can check the video.
sex = pd.get_dummies(df["Sex"])
df = pd.concat([df, sex], axis=1)
df = df.drop("Sex", axis=1)
And the new data look like…
As you can guess from the title of this article, we predict the years of trees. In this data set, ‘Rings’ column, which refers to the years, becomes the target variable (objective field, the dependent variable y). Then, we plot the ‘Rings’.
# Split the objective field y (dependent target variable) from the predicting (independent) variables Xs
y = df["Rings"].copy()
X = df.drop("Rings", axis=1).copy()
# Plot y
It does not seem balanced, but let’s move on.
We split the data set into 70% the training, and 30% the test for the modelling. Then, we should scale the independent variables to make the effect of them to be normalized. We determine ‘StandardScaler’ as a scaler, and transform the scaled variables into data frame format.
X_train, X_test, y_train, y_test = train_test_split(X, y, train_size=0.7, random_state=1)
# Scale Xs
scaler = StandardScaler()
X_train = pd.DataFrame(scaler.transform(X_train), columns=X.columns)
X_test = pd.DataFrame(scaler.transform(X_test), columns=X.columns)
First we build a linear regression model to predict ‘Rings’.
# 1) Linear Regression Model
model_1 = LinearRegression()
print("R2 of the linear regression model is: ", model_1.score(X_test, y_test))
The output is…
The model is not so good because the R2 is not high enough.
Now, we jump into classification for ‘Rings’ variable. First, we built a logistic regression model. After the confusion matrix, we check classification report which gives us the precision, recall, f1-score, support, and the accuracy as well. To understand the metrics, you can check the video.
# 2) Logistic Regression Model
model_2 = LogisticRegression()
cm = confusion_matrix(y_test, prediction_2)
fig = plt.figure(figsize=(10,8))
ax = sns.heatmap(cm, square=True, annot=True, cbar=False)
plt.title("Confusion Matrix coming from the Logistic Regression Model")
The scores do not look good.
Secondly, we try a decision tree model, and check the same things as we did in the logistic regression model.
# 3) Decision Tree Model
model_3 = DecisionTreeClassifier(max_depth=4, random_state=42)
prediction_3 = model_3.predict(X_test)
cm = confusion_matrix(y_test, prediction_3)
fig = plt.figure(figsize=(10,8))
ax = sns.heatmap(cm, square=True, annot=True, cmap='Blues', cbar=False)
plt.title("Confusion Matrix coming from the Decision Tree Model")
Unfortunately, we failed in the decision tree model, too.
Frankly, we applied 3 models to predict the year of a tree. All failed, but why? There can be some reasons. Maybe these independent variables are not quailified to predict the ‘Rings’ variable.
However, we won’t give up! :) Since we know the target variable is unbalanced, we will do apply undersampling and oversampling techniques to obtain balanced dependent variable. See you, in Part 2.


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Herzing AC 216 A machine is purchased January 1 at a cost of $59000
STRAIGHT-LINE, DECLINING-BALANCE, SUM-OF-THE-YEARS’-DIGITS, AND MACRS METHODS A machine is purchased January 1 at a cost of $59,000. It is expected to serve for eight years and have a salvage value of $3,000. REQUIRED
1. Prepare a schedule showing depreciation for each year and the book value at the end of each year using the following methods
(b) Double-declining-balance (round to two decimal places)
(c) Sum-of-the-years’-digits (round to two decimal places)
2. Assuming a seven-year class of property, compute MACRS depreciation expense for each year of the asset’s life.


The current topic is not aimed to Discrete Mathematics only, but to all the Mathematics and to all the Logic processes, when we are dealing with it. It’s little bit theory we need to deal with in order to understand the world of proofs – how to construct a solid proof of a specific problem, how to be sure it is a solid and “bullet proof” and what are the most famous approaches when it comes to proofing.
I. Deductive Proof – Modus Ponens
This Proof involves a lot of deduction chains – Modus Ponens. In short – If X, then Y (I am going to publish later this year a great tutorial of logic processes and logic when it comes to mathematics and we are going to inspect with great detail all the kinds of logic relations between 2 or more states). For example:
Fact A:You are reading this sentence right now.
Fact B:All who read this sentence right now and recognize the words has at least mediocre knowledge of English.
Fact C (or conclusion): You have at least mediocre knowledge of English
In this example we gave deduction with 3 statements involved, facts A and B, and the conclusion C. In short – If A and B, therefore C. If we built our proof using such relations or deductions, you can be absolutely sure your statement is correct (all you need to make sure is that your facts are correct).
II. Formal Proof
The major problem of the previous method of proofing – in most cases it is too long and it needs extra tools in order to proof the different facts involved. The Formal Proof is more likely to be used by you in future and the Formal Proof is the tool you will greatly appreciate. A good example of Formal Proof – imagine you have two functions F and G, both depends on one variable only – x. Therefore you have F(x) and G(x).
You have that F(X)>G(X). If you proof that when you increase X, the difference F(X)-G(X) will be increasing too – you have just created one good Formal Proof. You can say – hey, let’s substitute fact A = “X increases” and fact B = “F(X)-G(X) increases“. We can state that if A, then B – so there is no difference between the previous way of proof and the current one. In the previous method we defined A and B as facts, something we are sure it happens. In this case we didn’t check them, are they correct or not? So we can wrap the whole process of proofing with Modus Ponens, but I recommend the states to be transformed to facts using Formal Proof.
I know it’s little perplexed and some of you will find it annoying, but believe me – when we are dealing with hard theorems and lemmas, such details can save you!
III. IFF – if and only if
When we look through the Modus Ponens example, we said it is constructed by deductive fashion:
If A, then B.
What if we proof that If B, then A? Then we will have two rules (proofs):
If A, then B.
If B, then A.
In this case we add the two statements and relate them with the abbreviation IFF.
You can right down A IFF B (which means A happens only when B happens and vice versa).
For example (maybe some philosophers will disproof that statement, but I recall you it is just an example using our daily routines): My eyes are open IFF I look.
You can observe the statement, if my eyes are open, I look (Modus Ponens part I).
If I look, my eyes are open (Modus Ponens part II). Both of those deductive proofs are right, therefore we can join them in the statement My eyes are open IFF I look.
IV. Set related proofs
When we start dealing with Set Theory in the future tutorials, we will often apply this method. What if we need to compare two sets and proof they are equal? We should proof those two characteristics of the sets(let’s define the sets as A and B):
1. Proof that every element of A, belongs to B .
2. Proof that every element of B, belongs to A.
In other words, following the math notations:
\forall a \in A => a \in B \\
\forall b \in B => b \in A \\
I should add – here the symbol “=>” defines the logic link IF,THEN (simple deduction definition, not a symbol for logical operator). We are going to experience lot of such proofs in the future.
V. Converse and Contrapositive – Differences between the both
Imagine you have the statement -> A and B = TRUE
(I should add – this statement is equal to TRUE, only when both A and B are true)
By the definition “converse” we apply NOT(A and B)
Following De Morgan’s laws, when we are applying a NOT to a statement, we should not only change the sub-statements (A and B), but also change the operator (in our case - and). So what this statement means? NOT(A and B) = NOT(A) NOT(and) NOT(B) = NOT(A) or NOT(B). You should pay attention, that the opposite of the operator AND is OR (I will explain that in future, when we start the Set Theory tutorial). So let’s underline:
NOT(A and B) = NOT(A) or NOT(B)
*example of converse*
Now what about contrapositive? Imagine we apply 0 to the statement if A is false, and we apply 1 to the statement if A is true. We have four cases:
If we calculate the four cases, we will see the statement (A and B) is true, only when both the statements are true. If (A is true and B is true), then (A and B) is true. What if we put a NOT operator in front of the two statements?
If NOT(A is true and B is true), then NOT(A and B) is true
The Set Theory is tricky one and you can be easily mislead that you cracked the hidden information, but an inexperienced eye will just scratch the surface. The information here is not notated correctly for an experienced eye or someone with respectful background on this field, but I am just outlining some basic concepts. As I promised, we will later dig in the Set Theory field (and I promise you really, really amusement experience).
You have the statement (All x are good). You need to proof that. In order to do that by applying contradiction, you need to think like pessimist. Hey, you say to yourself, this can’t be true. I am sure All x are bad! So you accept this statement and start making deduction steps or formal steps in order to “unwrap” this statement and the sequences of it. You go further and further using Modus Ponens, and voila – you rich a false statement!
You have the statement (All x are good). You need to proof or disproof it? Well, if you can find such x, which is bad – voila, the proof is one. The statement is wrong, no further steps required!
Because this paragraph is the shortest, I want to add another important detail. We are talking about proofing statements, but how to define a proofed statement? Axiom? Lemma? Theorem? Observation? What is the difference between them? Here is the answer:
1. Axiom – some statement granted for true ( the basic, the fundamentals of mathematics)
2. Lemma – a helping local statement, which is not interesting by itself, but it is a major step in order to retrieve a bigger statement (Observation or Theorem)
3. Observation – a big statement, which is based on finite count of states. For example, if we proof that All the people in the world (planet Earth) are X, from some X, this is not a theorem. It is an observation, because the statement can be applied to finite number of elements (in our case – all the people in the world – is finite count – around 7 billions).
4. Theorem – a big statement, which is based on infinite count of states. For example – all the numbers are X (for some X). The numbers count is infinite, therefore this is a theorem, not an observation!
VIII. Proof by induction
Here comes by mine opinion, the most interesting kind of proof – proof by induction. Let’s go through this proof by an example:
Proof that 2*n is an even number, if n is a positive integer.
Proof by induction
Step 1: The smallest possible n is equal to 0. Therefore, 2*n equals 2*0 = 0, and 0 is an even number, therefore the statement holds!
Step 2: Let’s take for granted, that for some n equals to k, the statement holds too!
Step 3: Can you proof the statement holds, for n equal to k+1? In other words, we should proof that 2(k+1) is an even number. 2(k+1) = 2*k + 2, which is an even number if 2*k is an even number ( even number + even number = even number). But 2*k is an even number (from step 2), therefore 2(k+1) is an even number and statement holds!
Before going to the next chapter, we will take a short break here and I want to challenge you with this math problem:
Can you proof by induction, that all the natural numbers, equal or greater than 8, can be represented by sum of threes and fives only? In other words:
n \ge 8\\
n = 3*x + 5*y
Please pay attention, that proofing this statement by induction is a must!
You can publish your answer as a comment below.