|Maxwell's Dream Help: How do I control the calculation?||Back to Overview|
The calculation is started with a push of a button:
During the calculation a process bar is active. The calculation can be stopped by pressing the button again.
At the beginning of the calculation the calculated fields and results are far away from being the correct solution. But with every iteration of the calculation the results are getting better and better. The calculation ends, when a certain level of accuracy is reached.
In the colored graphic not the field strength, but the electrical potential is shown.
It is assumed, that the ground planes have a potential of zero Volts. The signal trace has a high potential. The area between the signal trace and the ground planes has some potential value inbetween.
You find the Calculation Monitor in the upper right part of the window:
The first value is the actual size of the finite elements array. This value is chosen automatically for you.
The second value is the number of iterations processed. For informational purposes only.
The third value is the elapsed time. You may want to look at this value, if you compare the speed of different Macintosh models.
The fourth value QS / QM is a criteria for the remaining errors of the iteration step. This value is used to end the iteration. To understand this value, you have to be aware, that avery electrical field in the surrounding of a metallic conductor induces some electrical charge at the surface of this conductor.
QS is the induced charge at the surface of the signal trace. QM is the induced charge at the surface of both ground planes. In the real world, these charges are balanced, which means that QS and QM have exactly the same value.
In the world of our calculation, in the beginning of the iteration the charges are not balanced at all. To be honest, QS is very large and QM is zero. Of course this is far away from a correct result. But the good news is, that with every iteration step the finite elements array is getting more and more similiar to the real world. The program Maxwell's Dream monitors this "similiarity" by looking at the ratio of QS and QM. If this ratio is equal to -1, the result is perfect. As you can see, the snapshot shown above is somewhere in the beginning of a iteration. The snapshot below is taken after the completion of an iteration:
The fifth value QM / QB is not used to terminate the iteration. It is a proof, whether the calculation in a bounded area has the same result as you would expect from the unlimited and unbounded area of a real triplate. You can find more on this topic in the chapter Tell me about the accuracy!
Of course there is a trade-off between calculation speed and accuracy. In the optimization menu you can choose between three modes:
You may use the Coarse Accuracy, if you want very fast results and an accuracy of about 10% is acceptable. A good example for the Coarse mode is the search for a geometry with an wave impedance in the desired range.
The Normal Accuracy is best used to define the exact geometric parameters for the desired wave impedance.
You can use Fine Accuracy, if you want to verify the results from the Normal Accuray or if you just want to give some work to your new high speed Mac. Be aware, that calculation with Fine Accuracy may take some time on older Macs.
With the default settings the programs enables the usage of an AltiVec unit, whenever such a unit is available. In the optimization menu you can disable the usage of AltiVec. The single purpose of this option is to see, how useful an AltiVec unit is. For finite elements analysis you can expect a speed bump factor of 2...3 by using the AltiVec unit. Test it yourself.
|How do I define the geometry?||Back to Overview||Tell me about the accuracy!|


Building & Room:
1253 S. Halsted Street, Room 215
Clay Johnson joined the Office of Institutional Research in September 2016 and was previously with the University Office of Planning and Budgeting in Urbana. His prior experience includes teaching middle and high school math in Arkansas public schools. Clay has a PhD in Educational Statistics & Research Methods and a graduate certificate in Educational Measurement from the University of Arkansas. He also holds an MA in Mathematics Education from Teachers College, Columbia University.


Department of Mathematics, 6th Floor, Room 6-230
City University of New York
One Bernard Baruch Way
(55 Lexington Ave. at 24th St)
New York, NY 10010
Office: VC 6-294, Phone: (646) 312-4142
55 Lexington – Newman Vertical Campus
- MTH 4500: Introductory Financial Mathematics, KMWA, MW 2:55PM – 4:35PM, VC 10-180.
- MTH 9878: Interest Rate Models, UMA, M 6:05PM – 9:00PM, VC 9-140.
Click here to access the information on current and past courses.
I am interested in probability, statistical mechanics, partial differential equations, combinatorics, and dynamical systems. Click here to get a more detailed description of my research.
- I. Matic, A parallel algorithm for the constrained shortest path problem on lattice graphs, preprint [PDF]
- I. Matic, D. Sivakoff, Excited deterministic walk in a random environment, Electronic Journal of Probability, vol. 20 (2015) no. 44, 1-19, Institute of Mathematical Statistics, ISSN 1083-6489 [PDF]
- I. Matic, Inequalities with determinants of perturbed positive matrices, Linear Algebra and its Applications 449 (2014), 166–174 [PDF]
- I. Matic, J. Nolen, A sublinear variance bound for solutions of a random Hamilton-Jacobi equation, Journal of Statistical Physics, 149 (2012), 342-361 [PDF]
- I. Matic, Large deviations for processes in random environments with jumps, Electronic Journal of Probability, vol. 16 no. 87 (11-23-2011), pp. 2406-2438, Institute of Mathematical Statistics, ISSN 1083-6489 [PDF]
- I. Matic, Homogenizations and large deviations (PhD Thesis), UC Berkeley, 2010 [PDF]
- I. Matic, Parallel algorithm for constrained shortest path problem in C++/OpenCL, open source, available at github.com/maticivan/parallel_constrained_shortest_path
- D. Djukic, V. Jankovic, I. Matic, N. Petrovic, The IMO Compendium 1959-2009 (second edition), Springer 2011
- V. Baltic, D. Djukic, Dj. Krtinic, I. Matic, Practice Problems for High School Math Competitions in Serbia (in Serbian), Serbian Mathematical Society, 2008
- Z. Kadelburg, D. Djukic, M. Lukic, I. Matic, Inequalities (in Serbian), Serbian Mathematical Society, 2003


There's an old saying out there that goes something like, "stats are like bikinis..they show you a lot..but they leave out the most important stuff". That may not be it exactly, but you catch my drift. Basically, stats can tell you some of the story, a lot even, but you need to watch the games to get the whole picture. And that is definitely true. There will always be things that a player does, both good and bad, that statistics cannot measure.
However, thanks to programs like Synergy Sports — which charts every single play of the NBA season — that gap is getting smaller and smaller.
Synergy breaks down each and every play, charting who had the ball, what type of play it was, and whether it finished in a field goal make, field goal miss, free throw attempts, or a turnover. It then comes up with a number of how many points you create (both via field goal and free throws made) per play. They do the same thing defensively, charting who was covering the player when they were shooting, and coming up with a number for how many points each player allows per play.
Offensively, league average is approximately 0.94 points-per-play (ppp) created, while defensively that number is 0.86 ppp (the difference between the two is mostly fast break points that no defender is blamed for. Examples: two or three on one breaks, defense comes up with a steal and finds one of their teammates cherry picking, ect).
With all of this in mind, I thought it would be interesting to see who on the Celtics rates well, and who not-so-well, in terms of these metrics.
To come up with a "Celtics MVP" I came up with the following (embarrassingly simple) formula:
(Offensive ppp created - (defensive ppp allowed x 1.1075)) x 100
You multiply the defensive number by 1.1075 to make up for the fact that the the average number of point created per play is 0.94, while the average number of points allowed is only 0.86. This will allow for an exactly league average player to be given a 0, with anyone in the positive being above average, and anyone in the negatives being below average. Then you multiply that number by 100 just to avoid a bunch of decimals being the players' scores.
Ok, with that in mind, here are the best and worst of the Celtics this season (minimum 200 minutes played..sorry MarShon and Keith).
1. Courtney Lee: (1.03 ppp created) - (0.79 ppp allowed x 1.1075) = +.155 x 100 = +15.5
2. Kris Humphries: (0.97 ppp created) - (0.78 ppp allowed x 1.1075) = +.106 x 100 = +10.6
3. Jeff Green: (1 ppp created) - (0.81 ppp allowed x 1.1075) = +.103 x 100 = +10.3
4. Brandon Bass: (0.95 ppp created) - (0.77 ppp allowed x 1.1075) = +.098 x 100 = +9.8
5. Jordan Crawford: (0.97 ppp created) - (0.84 ppp allowed x 1.1075) = +.040 x 100 = +4.0
6. Jared Sullinger: (0.91 ppp created) - 0.89 ppp created x 1.1075) = -.079 x 100 = -7.9
7. Avery Bradley: (0.88 ppp created) - (0.87 ppp allowed x 1.1075) = -.083 x 100 = -8.3
8. Vitor Faverani: (0.76 ppp created) - (0.86 ppp allowed x 1.1075) = -.192 x 100 = -19.2
9. Gerald Wallace: (0.67 ppp created) - (0.87 ppp allowed x 1.1075) = -.296 x 100 = -29.6
10. Kelly Olynyk: (0.73 ppp created) - (0.95 ppp allowed x 1.1075) = -.322 x 100 = -32.2
11. Phil Pressey: = (0.53 ppp created) - (0.97 ppp allowed x 1.1075) = -.544 x 100 = -54.4
So judging by the numbers, Courtney Lee has been the Celtics most efficient player this season (You hear that opposing GMs? Trade for this man!), with the best offensive number on the team and the 3rd best defensive number. Overall, the Celts have five guys in the positive, and six in the negative, which would seem to make sense for a team currently a few games below .500.
The two biggest surprises are Avery Bradley and Jared Sullinger, but when you really look at their numbers, it becomes less so. Bradley has taken steps forward offensively, but he's still shooting just about league average from the field, and barely gets to the foul line. When you throw in a major step backward on the defensive end (thus far)..his number makes sense. And then you have Sullinger, who has also been just below average defensively, and has also been trying to expand his range this season. While that may end up being an awesome decision in the long-term, his 30% three-point shooting isn't helping his efficiency at the moment.
I think these numbers also shed positive light on Jeff Green and Brandon Bass, who are well into the positive side while being among the team leaders in minutes played. Green has the 3rd highest usage rate on the team, yet is still putting up 1 point per play. Impressive stuff. And then you have Bass, who's solid offensively while posting an elite defensive number for the 3rd consecutive season.
Unfortunately, there is also a dark side to these numbers. Celtics first round pick Kelly Olynyk has been pretty awful on both sides of the floor, to the point where it's not all that difficult to see why the Cs are only 4-10 when he plays this season, compared to 6-4 when he doesn't. The numbers are also ugly for Gerald Wallace, specifically on offense where that 0.67 ppp number is just horrific for someone playing as many minutes as he is. And then there's Phil Pressey. Poor, poor Phil Pressey. The worst defensive number on the team, and the worst offensive number I could find in the entire league. Not an ideal combo.
Again, I'm not saying you can derive a players entire worth from these numbers. Not by a long shot. But I do think there's a ton of value in them. Especially on defense, where no matter how closely you watch, there is no way your eyes can do as good of a job as a system that tracks every single play over the course of an 82 game season. So take the above for what it's worth. Like a girl in a bikini, they tell us a lot. But not quite everything.
Follow Mike on twitter - Mike_Dyer13
For more of my articles, click here Michael Dyer 12/13/2013 02:21:00 PM Tweet Edit


PUBLICATIONS BY STUDENTS IN THIS PROGRAM
Here we present some of the papers that the participants in our program have written, submitted, and/or published recently:
1. "Distribution of the exponents of primitive circulant matrices in the first four boxes of Zn", by M.I. Bueno, K. Fang, S. Fuller, and S. Furtado. Involve 5(2012), 187-205.
2. "The solution of the equation AX+X*B=O", by F. De Teran, F. M. Dopico, N. Guillery, D. Montealegre, and N. Reyes. Linear Algebra and its Applications 438(2013), 2817-2860.
3. "The Alexander and Jones Polynomials through representations of rook algebras", by S. Bigelow, E. Ramos, and R. Yi. Journal of Knot Theory and its Ramifications, 21(2012), 18 pp.
4. "Systematic Stochastic Reduction of Inertial Fluid-Structure Interactions subject to Thermal Fluctuations", by P. Atzberger and G. Tabak. SIAM J. Appl. Math., 75(4), (2015), 1884-1914.
5. "Presentation of the Motzkin Monoid", by K. Hatch, M. Ly, and E. Posner. Technical report.
6. "A parametric model for determining consensus priority vectors from fuzzy comparison matrices", by S. Chouinard, E. Dopazo, J. Guisse, and K. Lui. Fuzzy Sets and Systems, vol. 246 (2014), 49-61.
7. "The kernel of the matrix ij (mod n) when n is prime", by M.I. Bueno, S. Furtado, J. Karkoska, K. Mayfield, R. Samalis, and A. Telatovich. Involve, 9-2 (2016), 265-280.
8. " A Decomposition of Parking Functions By Undesired Spaces ," M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, and I. Nicolas. Electr. Journal of Combinatorics, 23 (3) (2016), p3.32.
9. "On the sign characteristic of Hermitian linearizations in DL(P)", by J. Breen, M.I. Bueno, S. Ford, and S. Furtado. To appear in Linear Algebra and its Applications.
10. "Conditioning and backward error of block-symmetric generalized Fiedler pencils", by M.I. Bueno, F. Dopico, S. Furtado, L. Medina. In preparation.
11. "Binary Codes and Period-2 Orbits of Sequential Dynamical Systems", by C. Defant. Submitted to the Electronic Journal of Combinatorics.
12. "Enumerating Periodic Points of Certain Sequential Dynamical Systems", by C. Defant. Submitted.
13. "Using Tiling Techniques to Enumerate Periodic Points," by P. Bartlett and C. Saunders. In preparation.
14. "Periodic Points over the Cycle Graph on Sequential Dynamical Systems," by P. Bartlett and T. Sakal. In preparation.
15. " A unified approach to Fiedler-like pencils via strong block minimal bases pencils", M. I. Bueno, F. M. Dopico, J. Perez, R. Saavedra, and B. Zykoski. Submitted.
The UCSB Mathematics Summer Research Program for Undergraduates is supported by the National Science Foundation.


There is more to our minecraft adventure than meets the eye.
In our MineCraft Club, kids will use MineCraft to build maths and English confidence, develop creative skills and logical problem solving.
Each session will have a mixture of digital and analogue activities that will develop these skills, that will include using iPads.
We will help with development of how to build in Creative Mode.
Set problems that can only be solved when you engage with others
IPADS are provided for everyone
Saturday 22nd September 2pm - 4pm
Light refreshments provided.


Here are solutions that I have written for the 3x3x3 Rubik's Cube. Click to download!
teaches you how to easily solve the cube
teaches you how to solve the cube faster
teaches you how to solve the cube even faster
Information about these pages are below. Download Links:
Here is the layout of the Bandaged Cube. This is for you to know how to "bandage" the pieces. Click on the link above to download the .pdf file so that you can print your own.
On the Maze Cube, the objective is to make a continuous line around the cube. Since I have not memorized the solved position, I have made a 3-D model of the solved position from paper. I made images of the sides on my computer, and have included a link to the .pdf file of it so you can download the file, print it, and make your model.
To see my model, you can follow this link to the video in which it is featured. Click Here!
page last edited 9/22/12


Chemin is a puzzle game whose goal is to place numbered pieces in a board in a continuous ordered sequence taking into account the pieces already in the board.
10 levels with increasing difficulty
250 puzzles (25 puzzles per level)
Boards with 4x4, 5x5, 6x6, 7x7, and 8x8 tiles
Listed as one of
Check a review in this blog gottasolveit.blogspot.com
Available for iPhone, iPad, and Apple TV
Designed & Developed by Enrique Brandariz


from brian2 import arcsin)
Inverse sine, element-wise.
x : array_like
y-coordinate on the unit circle.
out : ndarray, optional
Array of the same shape as
x, in which to store the results. See
doc.ufuncs(Section “Output arguments”) for more details.
angle : ndarray
The inverse sine of each element in
x, in radians and in the closed interval
[-pi/2, pi/2]. If
xis a scalar, a scalar is returned, otherwise an array.
arcsin()is a multivalued function: for each
xthere are infinitely many numbers
zsuch that \(sin(z) = x\). The convention is to return the angle
zwhose real part lies in [-pi/2, pi/2].
For real-valued input data types, arcsin always returns real output. For each value that cannot be expressed as a real number or infinity, it yields
nanand sets the
invalidfloating point error flag.
For complex-valued input,
arcsin()is a complex analytic function that has, by convention, the branch cuts [-inf, -1] and [1, inf] and is continuous from above on the former and from below on the latter.
The inverse sine is also known as
Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 10th printing, New York: Dover, 1964, pp. 79ff. http://www.math.sfu.ca/~cbm/aands/
>>> np.arcsin(1) # pi/2 1.5707963267948966 >>> np.arcsin(-1) # -pi/2 -1.5707963267948966 >>> np.arcsin(0) 0.0


This course includes basic math skills, proportions and ratios, converting from US to metric, APQ, EPQ, APC, EPC, through recipe costing. We removed some algebraic formulas and replace them with visual triangle structures for three variable problems to simplify the math. This is a more holistic approach that makes it easier to understand the formulas, remember how they are applied, and apply them successfully.
In the 1980's, Michael Scott opened and managed three restaurants in New York called The Great American Health Bar. After completing his Master degree in Computer Science he taught and consulted in Japan, New York and Pennsylvania. Michael currently teaches, among other courses, culinary math at the Art Institute of Philadelphia.
This website provides a complete course in culinary math.
Skip site news
Using triangle structures for three variable problems removes the algebra by revealing three formulas in one. This makes problem solving much easier. Wherever we can, we use these structures throughout the course. For an example, see the PDF file for Mark-up and Mark-down formulas attached to this message.


We have finally managed to do a video feature. We have been wanting to do this for a long time, but never really managed to find intelligent things to say. We did a piece of work this week which quite frankly is easier explained in a video than written about.
We looked at all the tax saving equity linked savings schemes that are available to investors in India and used their historical returns & current portfolio composition to identify the best possible fund(s). We used Python & some advanced mathematics to compute a number of risk/return ratios and used a technique called principal components analysis to identify similarities between these different funds. If you are looking to make a tax saving ELSS investment before the end of this financial year, this video should help! Happy watching!
P.S : If you are interested in the full code and the data used in this analysis, you can find it on our Github page right here.


Arithmetic operators are fundamental components of any programming language, and Python is no exception. These operators enable us to perform mathematical operations, such as addition, subtraction, multiplication, and division, in Python programs. In this video, we will explore the different arithmetic operators available in Python and how to use them.
The basic arithmetic operators in Python are the addition, subtraction, multiplication, and division operators represented by the symbols +, -, *, and /, respectively. These operators are used to perform basic mathematical operations and can be applied to integers, floating-point numbers, and complex numbers.
In addition to these operators, Python also provides a floor division operator (//) and a modulus operator (%). The floor division operator returns the integer quotient of the division operation, discarding any remainder, while the modulus operator returns the remainder of the division operation.
Finally, we have the power operator (**), which is used to raise a number to a certain power. This operator takes two operands, the base and the exponent, and returns the result of the base raised to the power of the exponent.
Table of Contents
Introduction to Arithmetic Operators in Python
Arithmetic operators are used in Python to perform mathematical operations on numerical values. Python supports a variety of arithmetic operators, including addition, subtraction, multiplication, division, and more. Let’s explore each of these operators and show you how to use them in your Python programs.
Addition (+) Operator
The addition operator is used to add two values. For example:
a, b = 4, 5 c = a + b print(c)
Subtraction (-) Operator
The subtraction operator is used to subtract one value from another. For example:
a, b = 6, 4 c = a – b print(c)
Multiplication (*) Operator
The multiplication operator is used to multiply two values. For example:
a, b = 7, 8 c = a * b print(c)
Division (/) Operator
The division operator is used to divide one value by another. For example:
a, b = 22, 7 c = a / b print(c)
Floor Division (//) Operator
The floor division operator is used to get the result of the division as an integer value. For example:
c = 22 // 7 print(c)
Modulus (%) Operator
The modulus operator is used to get the remainder of a division. For example:
c = 22 % 7 print(c)
Power (**) Operator
The power operator is used to raise a number to a power. For example:
c = 2 ** 3 print(c)
Underscore (_) in Python
Underscore is not an operator or a function in Python. It is a variable that stores the last printed value. For example:
a, b = 10, 20 print(a + b)
print(b + _)
Round() Function in Python
The round() function is used to round off a decimal value. If the number at the decimal place is greater than or equal to 5, then the number is rounded up; else, it is rounded down. For example:
print(round(17.2)) print(round(17.7)) print(round(17.5)) print(round(12.5))
Output: 17 18 18 12
Arithmetic operators are an essential part of Python programming. With these operators, you can perform mathematical operations on numerical values and manipulate data in your programs.


Do you know numbers in Italian?
If you have just started your language journey, numbers in Italian are one of the very first topics you learn.
Knowing how to deal with numbers is essential to have a basic conversation, be able to tell your age, give your address and telephone number.
Also, if you can understand and use them, going shopping will be a piece of cake!
Especially if you travel to small towns in Italy, being able to use numbers will make you feel at ease in local bars and restaurant where people often can’t speak English fluently.
So let’s go through and see how Italian numbers work: we start with numbers from 0 to 20!
As you can see, numbers up to 20 are all quite different and they don’t follow any patterns: that is why it is necessary to make a little effort to memorize them.
Once you reach 20 it gets much easier!
From now on you can apply a simple rule. Let’s see!
Basically we can just add the first 9 numbers to 20 in order to reach number 30.
Easy isn’t it?
You can apply the same rule after 30, so you just need to remember the tens.
After 100, again it is just about adding other numbers, for example 110 is “centodieci“, or 157 is “centocinquantasette” and so on.
Time to move on!
What happens with thousands?
After 1000 (Mille), you can easily notice what the pattern is: you can form any numbers adding -mila.
How would 134.560 be? Well…centotrentaquattromilacinquecentosessanta!
It might seem too long, but if you start counting from the left it won’t be difficult!
And if you want to use big numbers, you may need 1.000.000, which in Italian is “un milione” or 1.000.000.000, “un miliardo“.
Milione and miliardo become plural when you have, for example, 3.000.000 “tre milioni“, or 5.000.000.000 “cinque miliardi“.
If you are learning Italian in London at Happy Languages, these contents will be covered during the Beginner 1 course, check it out!
Now practice with any numbers you can think of and get more confident!
If you want to learn more join our Italian Evening Beginner 1 course!JOIN NOW
If you enjoy video tutorials, check our Youtube Channel


Support Vector Machine (SVM) is a popular supervised machine learning algorithm used for classification and regression analysis. SVM works by finding the optimal decision boundary or hyperplane that separates different classes in a given dataset. The algorithm tries to maximize the margin between the decision boundary and the closest points of different classes, making the model more robust to noise and able to generalize well on unseen data.
Here are some pros and cons of using SVM:
- SVM is a powerful algorithm that works well on both linear and nonlinear data.
- SVM can handle high-dimensional data, making it useful for complex data analysis.
- SVM has a regularization parameter that helps to avoid overfitting, which is a common problem in machine learning.
- SVM has a kernel trick that allows transforming the input data into higher-dimensional space, where it may be easier to find a linear boundary.
- SVM can be computationally expensive, especially when working with large datasets.
- SVM can be sensitive to the choice of kernel function and other parameters, which can affect the performance of the model.
- SVM doesn’t provide probability estimates, which can be useful in some applications.
- SVM can be difficult to interpret and explain compared to other machine learning algorithms.
Overall, SVM is a powerful and versatile machine learning algorithm that can be used in a variety of applications, but it requires careful parameter tuning and can be computationally expensive.


Don’t have access to a graphing calculator and need to do some math problems? Check out Desmos a free online graphing calculator. It is currently supported on the following Browsers:
•Internet Explorer (9+)
•Safari on Mac or iPad
Desmos is a fully functional, browser-based graphing calculator that graphs your equations as you type and redraws them as you alter your equations. Desmos performs all of the functions you would expect to see in a graphing calculator. It also has a few extras that you would not find in regular graphing calculators, like equation and graph sharing through a Bit.ly link.
Here is a screen shot of what Desmos looks like:


Bob Dorough helped a generation of children learn about math through songs. Dorough has passed away at the age of 94.
Dorough was a jazz musician who was very involved in Schoolhouse Rock. He wrote and performed songs like Three Is The Magic Number and My Hero, Zero.
His biography says in 1971, he received a commission to "set the multiplication tables to music." This lead to Schoolhouse Rock that was seen between Saturday morning cartoons for 12 years.
RIP Bob. You will be missed.
Check out the Morning Breeze podcast with Jack and Carolyn! Listen here:


Thursday, 15 May 2014
This paper derives the equation of motion that is most commonly used to model the dynamics of a body in water, including ships and wave energy converters (WECs). The part of the mathematical argument essential to WECs will be summarised here.
The paper starts off with an interesting bit of history. Up until the 1950s, `classical' sea-keeping research only modelled sinusoidal waves and response. The models had the form of second order differential equations. In the 1950s there was growing interest in investigating response to spectra. Initially, they kept the form of the second order differential equations, and expressed the hydrodynamic added mass and damping as frequency dependent parameters. To run such a time domain model, it would be necessary to choose `the frequency' at which added mass and damping were to be considered. There was growing disquiet about the rigor of this approach, not to mention the fact that the models did not describe experiments very well. As Cummins muses in this paper "The shoe is squeezed on, with no regard for the shape of the foot". He goes on to derive a better fitting model.


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Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. Metropolitan Museum Cleveland Museum of Art. Internet Arcade Console Living Room. Books to Borrow Open Library. Search the Wayback Machine Search icon An illustration of a magnifying glass. Sign up for free Log in. Additional mathematics : pure and applied in SI units Item Preview.
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Technical Details Uploaded on: March Size: Books related to Additional Mathematics ,6th edition. Past Questions related to Additional Mathematics ,6th edition. Vectors, coordinate geometry and statistics assignment questions Department: Science and Technology Year Of exam: school: University of Benin course code: MTH Topics : Vectors, coordinate geometry, statistics Go to Vectors, coordinate geometry and statistics assignment questions past question.


Document Object Model (DOM)
Round the conversion to three decimal places. The equation for converting miles to kilometers is that 1 mile equals 1.609344 kilometers.
If the user does not put in a number, or puts something other than a number in the text field, have the result text say, “please enter a number”.
To really impress, figure out how to make it so that if the user presses the “K” key on the keyboard, the converter switches to converting kilometers to miles. If the user presses the “M” key on the keyboard, it switches back to converting miles to kilometers. The equation you need for converting kilometers to miles is 1 kilometer equals 0.621371192 miles. Again, round to three decimal places.


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I am a PhD Candidate in Machine Learning at Imperial College London with Dr Stefanos Zafeiriou and Prof Michael Bronstein working on Geometric and Bayesian Deep Learning with applications to Computer Vision and network analysis. I am currently a Research Intern at Google AI in Machine Intelligence/Machine Perception for Fall 2018. Previously, I interned in Quantitative Research (Systematic Trading) at JPMorgan Chase & Co in London for the Summer 2018. I have been a research intern at Speechmatics (Cantab Research Ltd.) where I worked on improving the company’s recurrent neural network language models, and a Data Scientist at HarperCollins Publishers in London. I graduated with an MSc in Advanced Computing from Imperial College London, and a Diplôme d’Ingénieur in Applied Mathematics and CS from Ensimag (Grenoble).
See my Curriculum Vitae or my short Resume.
Besides what I do at work, I am interested in a variety of topics in Machine Learning, notably in statistical learning and optimisation.
My preferred applications are in data science, computer vision, and medical research.
Mehdi Bahri (GitHub/LinkedIn/Quora).
Mail: mehdi [dot] b [dot] tn [at] gmail [dot] com


Change is a very scary entity for many individuals. It can be terrifying, even debilitating, for us to identify the habits, routines and rituals that are no longer serving our highest good, and then proceed to change them. Why is that?
I believe it’s a lot like the initial introduction to Algebra. Seeing an unknown variable in an equation can be rather intimidating at first. When an equation is consistent, containing a set of numerical values, we can determine the answer using logic and a set of basic mathematical skills such as addition or subtraction to come to a conclusion. We can confidently assume a solution because we’ve had success in previous experiences using novice principles, and the answer is often rather simple- black and white, if you will. In many instances, we can even count on our fingers to check our work.
Conversely, with algebra, we can’t be sure of our success until after we’ve completed the equation. We have to substitute the unknown variable with a determined solution to know whether or not we’ve concluded correctly throughout. Furthermore, when we realize we went wrong somewhere along the way, we find ourselves having to start from the beginning, slowly working through the problem again with more precision, greater attention to detail, and a little more wisdom than when we began the first time.
Likewise, when substituting variables in our lives, we cannot be sure we’ve made the right choice until we’ve implemented the solution and checked our work. We can make educated calculations, use deductive reasoning, and even apply the process of elimination to come to a decision, but until we insert a value in place of our metaphorical x, we just can’t know for certain we took the right steps. To complicate matters further, life often has many variables that may require substitution, with countless steps we can take, and a myriad of outcomes we could potentially achieve. How do we identify which variables need a substitute? How can we be confident we’re taking all the right steps, and doing so in the right order? And how do we backtrack when we’ve found ourselves defeated, with the wrong solution, having to work our way through the equation of life once more?
THIS is where it can get tricky, considering life is significantly multifaceted compared to even the most complex mathematical equations. The most important thing to remember about solving an Algebraic equation is that you must isolate the variable to either side of the equals sign. This is only possible if you divide the problem into smaller equations, with simpler operations, and solve it one step at a time.
Life comes at us very quickly, and it’s rare we have an abundance of time to really map out the path we’re going to take to get where we’re trying to go. So often, we find ourselves starting back at square one, feeling defeated that we didn’t get it right this time. But consider this… you probably know exactly where you went wrong. You probably know precisely which steps of the equation need a little more fine-tuning. Even better yet, it was probably a minor error you overlooked because you were in a hurry. You may find yourself heading back to the drawing board, but don’t underestimate the wisdom you gained by working through it once, as you are much less likely to make the same mistakes this time around. Hell, you’ll probably even see the pitfall coming before you stumble on it! The point is, you just have to start somewhere in order to isolate the variable.
You must break the complexity of your life down into smaller, simpler equations so you can more easily identify which variables you need to substitute to move forward. Life may be a bit complex, but change is the only constant in its’ equation, and If we would’ve avoided Algebraic equations altogether, then we probably wouldn’t have graduated high school. Well, I’m here to inform you that avoiding necessary changes in your life brings a far worse fate.
As for algebra, I was a straight A student in school… but I certainly failed my fair share of quizzes, and only scored moderately on tests. I blamed it on test anxiety, ha! But I always received a final grade of A in Math. Not because I always took the right steps, and not because I spent an unrealistic amount of time or attention on the equations. I succeeded because I worked my ass off, I always asked for extra credit, I (almost) always did my homework, and I never gave up. Apply these habits to implementing change in your life, and I assure you that every step you take will get you closer to the correct solution.


When you calculate a check digit for GTIN-12, or GTIN-13, the GS1 number will consist of 12 and 13 digits respectively. If the number is to be saved in a database or used in a GS1-128 or a GS1 DataMatrix, you need to make the number 14 digits long by filling in the digit 0 in front of the number. The leading zero should not be included in the barcode EAN-13.
Please note that the GS1 numbers you calculate here are neither saved nor registered.
Click here to calculate the check character pair for the GMN (Basic UDI-DI)
To calculate the check digit for one number, type 1 in the field “Ammount of numbers”, or use the calculator on GS1’s Global website, where you also can calculate the check digit for other GS1 keys.


6 edition of A Short Course in Computational Probability and Statistics (Applied Mathematical Sciences) found in the catalog.
July 12, 1977
Written in English
|The Physical Object|
|Number of Pages||155|
Computational Statistics: Stat Contents. 1 Logistics Prerequisites Text Assignments Pick a paper from the projects below or on the website on the course webpage at A very recent reference is the book called: The EM algorithm and Extensions, by G.J. McLachlan and T. Krishnan,(). SB Applied Statistics SB Computational Statistics SB1 Practicals letter / [PDF] Declaration of Authorship Link to University guidance on plagiarism. SB Foundations of Statistical Inference SB Statistical Machine Learning SB Applied Probability SB Statistical Lifetime Models SB Actuarial Science.
This graduate-level textbook is primarily aimed at graduate students of statistics, mathematics, science, and engineering who have had an undergraduate course in statistics, an upper division course in analysis, and some acquaintance with measure theoretic probability. It provides a rigorous presentation of the core of mathematical statistics. Course goals: (partially adapted from the preface of Givens' and Hoeting's book): Computation plays a central role in modern statistics and machine learning. This course aims to cover topics needed to develop a broad working knowledge of modern computational statistics.
Rubin H. Landau, PhD, is a professor in the Department of Physics at Oregon State University in Corvallis. He teaches courses in computational physics, helps direct the Northwest Alliance for Computational Science and Engineering, and has been using computers in theoretical physics research for the past 30 years. The book that I recommend in the syllabus is this book called All of Statistics by Wasserman. Mainly because of the title, I'm guessing it has all of it in it. It's pretty broad. There's actually not that many. It's more of an intro-grad level. But it's not very deep, but you see a lot of the overview. Certainly, what we're going to cover will.
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A Short Course in Computational Probability and Statistics (Applied Mathematical Sciences) by Walter Freiberger Ulf Grenander (Author) ISBN ISBN Why is ISBN important.
ISBN. This bar-code number lets you verify that you're getting exactly the right version or edition of a book. Cited by: ISBN: OCLC Number: Notes: Published in under title: A course in computational probability and statistics.
A course in computational probability and statistics by Walter F. Freiberger, Walter Freiberger, Ulf Grenander,Springer-Verlag edition, in EnglishPages: Integrates the theory and applications of statistics using R A Course in Statistics with R has been written to bridge the gap between theory and applications and explain how mathematical expressions are converted into R programs.
The book has been primarily designed as a useful companion for a Masters student during each semester of the course, but will also help applied. ISBN: OCLC Number: Notes: Title on cover and spine: A short course in computational probability and statistics.
Computational Probability is a collection of papers presented at the Actuarial Research Conference on Computational Probability and related topics, held at Brown University on AugustThis chapter book explores the development of computational techniques in probability and statistics and their application to problems in Edition: 1.
statistics course at the Master’s level at the university, and the course has also become a compulsory course for the Master’s in eScience. Both educations emphasize a computational and data oriented approach to science – in particular the natural sciences. The aim of the notes is to combine the mathematical and theoretical underpinning.
Don't show me this again. Welcome. This is one of over 2, courses on OCW. Find materials for this course in the pages linked along the left. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.
No enrollment or registration. This book explains the fundamental ideas of Bayesian analysis, with a focus on computational methods such as MCMC and available software such as R/R-INLA, OpenBUGS, JAGS, Stan, and BayesX.
It is suitable as a textbook for a first graduate-level course and as a user's guide for researchers and graduate students from beyond s: 1. This book arose out of a number of different contexts, and numerous persons have contributed to its conception and development.
It had its origin in a project initiated jointly with the IBM Cambridge Scien tific Center, particularly with Dr. Rhett Tsao, then of that Center. We are grateful to Mr. Introduction to Statistics: /5: Free: This course is designed to explain the fundamental of statistics.
The course contains four weeks or four modules. Learn More: Statistical Learning (Self-Paced) /5: Free: This is an introductory-level course in supervised learning, with a focus on regression and classification methods.
Learn More: The final stimulus to the book's completion came from an invLtation to teach a course at the IBM European Systems Research Institute at Geneva. We are grateful to Dr.
J.F. Blackburn, Director of the Institute, for his invitation, and to him and his wife Beverley for their hospitality.
Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. The probability of any outcome of a random An Introduction to Basic Statistics and Probability – p. 32/ Expected Value of X The average of the sample means (x’s) when taken over a large number of random samples of size n will.
This book should appeal to researchers in the mathematical sciences with an interest in applied probability and instructors using the book for a special topics course in computational probability taught in a mathematics, statistics, operations research, management science.
e-books in Probability & Statistics category Probability and Statistics: A Course for Physicists and Engineers by Arak M. Mathai, Hans J.
Haubold - De Gruyter Open, This is an introduction to concepts of probability theory, probability distributions relevant in the applied sciences, as well as basics of sampling distributions, estimation and hypothesis testing.
x: Computational Probability and Inference (Massachusetts Institute of Technology/edX):One of two courses/series to teach statistics with a focus of coding up examples in Python. Reviews suggest prior stats experience is needed and that the course is a bit unorganized.
Latest in Statistics and Probability books, ebooks, and academic textbooks from Cambridge University Press, including machine learning, stochastic networks, econometrics, theory and methods. Computational statistics, machine learning and information science; Statistics and probability: general interest This series of short but.
System Upgrade on Fri, Jun 26th, at 5pm (ET) During this period, our website will be offline for less than an hour but the E-commerce and registration of new users may not be available for up to 4 hours. Her research interests include spatial statistics, Bayesian methods, and model selection.
Givens and Hoeting have taught graduate courses on computational statistics for nearly twenty years, and short courses to leading statisticians and scientists around the world. This text is designed for an introductory probability course taken by sophomores, juniors, and seniors in mathematics, the physical and social sciences, engineering, and computer science.
It presents a thorough treatment of probability ideas and techniques necessary for a ﬁrm understanding of the subject. The text can be used. Meaningful use of advanced Bayesian methods requires a good understanding of the fundamentals. This engaging book explains the ideas that underpin the construction and analysis of Bayesian models, with particular focus on computational methods and schemes.
The unique features of the text are the extensive discussion of available software packages combined with a brief but complete and.The course is part of the Data Science for Executives Professional Certificate program. Explore these and other free online statistics courses that cover inferential statistics, descriptive statistics, statistical analysis software tools and much more.
Many courses are .The book can serve as an introduction of the probability theory to engineering students and it supplements the continuous and discrete signals and systems course to provide a practical perspective of signal and noise, which is important for upper level courses such as the classic control theory and communication system design.


Learn about SCIKIT library and how to import it along with other libraries and data. Learn to create important indicators for the algorithm.
Learn about Hyper-parameters and Cross-validation for data pre-processing. Learn to create datasets, standardization and how to handle missing data. Learn to train and test your data.
Learn Regression in detail. Learn about errors and residuals in a regression model and how to predict them. Understand the Cost Function and Gradient Descent algorithm to minimize the cost function. Finally, learn about Multivariate Linear Regression and code with respect to linear regression.
Learn the concept of predicting error and how to identify these errors in any Machine Learning algorithm. Learn about underfitting and overfitting the data and ways to get a good fit. Understand the concept of Regularization using lambda parameter.
Learn how to modify the predictions made by the regression to account for market conditions. Learn how to get actual market high and low predictions from raw predictions.
Learn the concept of classification and how to map input into a discrete category. Learn four types of classifier algorithms, which are K-Nearest Neighbor, Random Forest, Artificial Neural Network, and Naïve Bayes Classification. Learn various indicators such as RSI, SMA, Correlation co-efficient, Parabolic SAR and Average directional index.
Learn the concept of Binary Classification to predict the market direction. Learn the mathematical functions like Sigmoid and hyperbolic tangent to construct a binary classifier. Learn how to implement binary classification in financial market to predict market movement.
Understand the concept of Multiclass classification and how it is different from binary classification. Learn to classify datasets into more than one class using ‘One vs All’ algorithm. Learn how to categorize the data based on numeric encoding of categories followed by an explanation on ‘one hot encoding’. Learn the probability function and performance measures in ML and working of ‘Softmax’ function.
Learn the concept of Hyperplane, Support Vector, and Margin. Learn to how to choose the best hyperplane by maximizing the margin and the mathematics behind it. Learn about classification of non-linear data using kernel and understand different parameters such as C & Gamma and their effects on SVM algorithm.
Learn to build your own trading strategy based on the concepts learned earlier. Learn to properly import libraries, data and create necessary indicators. Learn to compare the strategy’s performance with market data. Learn to implement/modify the given strategy.


Rounding Numbers in Ruby
This article will show you how to round numbers up and down using Ruby.
Published on:August 5, 2013
The goal of this article is to demonstrate how to round numbers up or down in Ruby. The examples below can be run in a regular ruby prompt (you can start one by typing 'irb' from the terminal.) These examples work on Ruby on Rails models, views, or controllers as well.
The first thing we will go over is rounding to the nearest decimal place. This is accomplished using the
round method. To round to a whole number, simply call round without any arguments. For example:
You can also round to a decimal by passing the number of decimals you wish to keep as an argument. For example:
1.2591.round # returns 1 1.6345.round # returns 2
1.259.round(2) # returns 1.26 1.312.round(1) # returns 1.3
Great! However, what if we always need to round up or down? Ruby also provides a number of functions to accomplish this.
To round down we use the
floor function. The floor function. See the following example:
1.954.floor # returns 1
What if we want to round down to a specific decimal? We can simply pass in the number of decimals as a parameter:
1.954.floor(2) # returns 1.95
Similarly, we can round up using the .ceil method:
1.323.ceil # returns 2 1.323.ceil(2) #returns 1.33
Great! That's it for rounding numbers. Thanks for reading!


RSA: Encryption, Signatures and Blind Signatures
RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission.Wikipedia
In this blog post I will try to explain the RSA cryptosystem using simple mathematical principles. In order to read and understand this article, you don’t need any advanced maths knowledge – in fact, my sister, who recently finished the 5th grade, should understand the concepts explained here (if you are reading this, it means that she already did).
0. Modular arithmetic
Trust me, it sounds harder than it actually is. Each operation has one additional element named a modulo. This element is basically a number that dictates the maximum result. Let’s say we want to calculate 3 * 8 modulo 9. The mathematical notation is at follows:
3 * 8 (mod 9)
Simple, right? In order to get the result, we do the following:
- First, we forget about the modulo and just calculate the result of the mathematical expression. In our case, 3 * 8 = 24.
- Then, we repeatedly subtract the modulo from the result until the resulting number is smaller than the modulo. In our case, we subtract 9 two times from 24 to get 24 – 9 – 9 = 6, which is smaller than 9. So, 6 is the result.
Let’s take another example. 4 * 16 (mod 5):
- 4 * 16 is 64
- 64 = 60 + 4 = 12 * 5 + 4, so we subtract 5 twelve times from 64 to get 4.
- 4 is the result.
Also, a good example from Wikipedia:
Also, note that instead of an equals sign, we used a congruence sign (it has 3 parallel lines instead of 2).
0.2 Modular inverse
Remember the multiplicative inverse? If we have a number, say 2, we say that 0.5 = 1/2 is its multiplicative inverse because 2 * 0.5 = 1. In general, for any number x, 1/x is its multiplicative inverse because x * 1/x is always 1.
In modular arithmetic, we only work with integers, so the modular inverse of 2 can’t be 0.5. Also, we need a modulo when calculating it (you will see why in a moment). Let’s take an example: we want to calculate the modular inverse of 2 mod 5. So, we are searching for x such that 2 * x = 1 (mod 5). In this situation, x can be 3, 8, 13, 18. The multiplicative inverse is the smallest possible number (every other number can be reduced to the first one if it’s modulo 5).
1. RSA Key Generation
RSA is a public-key cryptographic algorithm, which means that it uses 2 keys: a public and a private one. During key generation, we link the private key to the public one (again, you’ll see how in a moment). This are the actual steps:
- First, choose 2 prime numbers p and q. For this example, I’ll use p = 7 and q = 11.
- Then, calculate n = p * q and phi(n) = (p – 1) * (q – 1). phi(n) is called a totient function. In our example, n = 7 * 11 = 77 and phi(n) = 6 * 10 = 60.
- Next, choose e such that 1 < e < phi(n). There are a lot of rules on how to choose e in order to not make the key weaker, and in real applications e is either 3 or 65537. In this example, we’ll go with e = 7.
- Now, calculate d as the modular inverse of e mod phi. I know this is unexpected, and the explanation is a bit too complicated for 5th graders, but just trust me that this is what you need to do. In our example, d = 43.
- The public key is the pair (n, e) = (77, 7)
- The private key is the pair (n, d) = (77, 43)
OK. Now we have the public and the private key. Before I teach you how to encrypt and decrypt data, let’s see some conclusions:
- p and q, the prime numbers do not appear in either key and should be deleted immediately after key generation. In fact, these 2 prime numbers link the public key to the private one.
- If an attacker successfully factorizes n (computes p and q), he can calculate the private key. Therefore, the strength of RSA is based on the factoring problem (finding the prime factors of a number as fast as possible). Note that, as our n gets bigger, it becomes exponentially harder to find p and q.
2. RSA Encryption and Decryption
In the above equation, x is any number smaller than n, and N, e, d are previously calculated (or chosen). This equation is the fundamental principle of RSA. This is a bit harder to prove, so I’ll let it up to the reader to demonstrate it. To encrypt a number m (which we will call a message), we use the following function:
enc(m) = m^e mod n
We just raise m to the power of e modulo n. Pretty simple, right? Not that we only need the public key (e, n) in order to encrypt a message. The decryption process is even simpler:
dec(x) = x^d mod n
This time, we will need the private key d to decrypt the message. This means that anyone with our public key can encrypt messages, but only those who have the private key can read them. The demonstration that this algorithm works is as follows:
dec(enc(x)) = dec(x ^ e) mod n = (x ^ e) ^ d mod n = x ^ (ed) mod n
Does the last equation ring a bell? If it doesn’t, read this part of the article again. Now, let’s take an example:
- Let’s say we want to encrypt m = 42. We’ll use our previously calculated keys (n=77, e=7, and d=43).
- First, we encrypt the message. The result of 42 ^ 7 mod 77 is 70. This means that 70 is the encrypted value of 42.
- Then, let’s try to decrypt 70 using our private key. Calculating 70 ^ 43 mod 77, we get, you guessed it, 42.
3. RSA Signatures
It can also tell us another thing: we can change e with d (our public key with our private one) and RSA will still work. But why would anyone need this, you may ask. Well, let me introduce you to the concept of digital signatures.
Let’s say that our hacker, Y, wants to lend $500 from a bank. Before giving out that loan, the bank tells Y to sign an agreement that he will return the money and pay interest. Basically, digital signatures work like real signatures. How do they achieve this? Well, the signer uses his private key to calculate a signature. When someone wants to verify the signature, he/she uses the signer’s public key. In RSA, the sign and verify functions are very easy to define:
s = sign(m, e, d) = m ^ e mod n
verify(m, s, e, n): Is m equal to s ^ e mod n ?
In the above functions, m is the message, (e, n) is the public key, (d, n) is the private key and s is the signature. Let’s take an example:
- We will use the key we already generated (public – (77, 7) and private – (77, 43)).
- We want to sign the message m=13
- The signature s is calculated as m ^ d mod n = 13 ^ 43 mod 77, which is 41.
- If we want to verify the signature, we use the formula s ^ e mod n = 41 ^ 7 mod 77, which is, you guessed it, 13. Notice how we only used the signer’s public key to verify his signature,
4. RSA Blind Signatures
Blind signatures are a very interesting property of RSA. Let’s say you want our hacker, Y, to sign something, but you don’t want Y to find out what he actually signed. It turns out that we can take a message and mask it. We send the masked message to the signer and get a signature from which we can calculate the original message’s signature. The procedure is as follows:
- You have the message m , and the signer’s public key (e, n).
- Choose r such that 1 < r < n and gcd(r, n) = 1 ( for those of you that don’t know what gcd is, simply choose a prime number that is not equal to p nor q).
- Calculate a number called the blinding factor (we will note this with b). The formula is b = r ^ e (mod n)
- Compute the masked message m’ = m * b (mod n)
- Send m’ to the signer. As he doesn’t know b, he can’t calculate the original message. After he signs the masked message, you’ll get s’ = the maked message’s signature.
- To get s, the original message’s signature, use the following formula: s = s’ * r ^ -1 (mod n) (Where r ^ -1 is the modular multiplicative inverse of r mod n)
Again, let’s take an example:
- We want our signer to sign a message m = 13. We know his public key is (e, n) = (7, 77) (this time, we will pretend we don’t know the private key).
- We choose r = 57 (I swear I chose it randomly!).
- Calculate b = 57 ^ 7 mod 77, so b = 29
- Our masked message m’ is calculated as 13 * 29 mod 77, so m’ = 69.
- We get s’ = 27. You should know how I got this value.
- The modular inverse of 57 mod 77 is 50 because 50 * 57 = 1 (mod 77). Knowing this, we can calculate s = 50 * 27 (mod 77), getting 41. This is, in fact, the signature of the number 13.
RSA is one of the most used public-key cryptosystems. I gave my best to describe it in this article using no advanced maths knowedge and no programming. If you want to learn more about this topic, there are lots of articles online. If you have any questions, feel free to message me on Twitter or Reddit.
Until next time, hack the world.


Research article of American Journal of Educational Research and Reviews
Improving Learning Effectiveness- Some Aspects Shouldn’t Be Ignored in Mathematics Teaching
Columbus Humanities Arts and Technology Academy, ACCEL Schools, Ohio, USA, 43229
There are abundant educational resources in the USA, but the performance of math education in primary and secondary schools is not so well, and has lagged behind some Asian countries including developing countries. Among many reasons, the crucial reasons are how teachers teach, how students learn, and the level of school teaching management. This paper focuses on how teachers teach in five aspects, particularly emphasizes on how to improve students’ retention and understanding of knowledge, and to enhance learning and thinking abilities. It points out that teachers should constantly remind students to maintain basic facts, create assignments for students by combining regular methods with the strategies of interleaved and spaced practice, build connections among knowledge, pay attention to improve students’ ability of problem-solving, and use some complementary approaches to generate students’ interest, participation, memory and concentration in math. And also it mentions that self-discipline, persistence and confidence are significant in prompting learning efficiency. In addition, some suggestions are put forward for school education management in supporting teaching and learning.
Keywords: memory, understanding, practice, procedure, connection, problem-solving
How to cite this article:
Yuqing Yan. Improving Learning Effectiveness- Some Aspects Shouldn’t Be Ignored in Mathematics Teaching. American Journal of Educational Research and Reviews, 2020,5:74. DOI: 10.28933/ajerr-2020-07-1505
1. Glenda Anthony, Margaret Walshaw. Effective Pedagogy in Mathematics. Educational Practices Series-19. UNESCO International Bureau of Education. 2009. Pages 32.
2. Barbara Oakley. A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)M. Tarcher Perigee. 2014.
3. Rohrer, D., Dedrick, R. F., Hartwig, M. K., & Cheung, C.-N. (2020). A randomized controlled trial of interleaved mathematics practice. Journal of Educational Psychology, 112, 40-52.
4. Rohrer, D., Dedrick, R. F., & Stershic, S. (2015). Interleaved practice improves mathematics learning. Journal of Educational Psychology, 107(3), 900-908.
5. Sean H. K. Kang. Spaced Repetition Promotes Efficient and Effective Learning: Policy Implications for Instruction. Policy Insights from the Behavioral and Brain Sciences 2016, 3(1) 12-19.
6. Jensen, B., Hunter, A., Sonnemann, J., & Burns, T. (2012). Catching up: Learning from the best school systems in East Asia. Grattan Institute.
7. Nisbet S . Teaching mathematics for understanding J. Steadfast Press, 2010. pp: 1-12.
8. Doug Rohrer, Robert F. Dedrick, etl al.. A Randomized Controlled Trial of Interleaved Mathematics Practice J. Journal of Educational Psychology, 112(1), 40–52.
9. Rohrer D , Taylor K . The effects of overlearning and distributed practise on the retention of mathematics knowledge J. 2006, 20(9):1209-1224..
10. Stylianides A J , Stylianides G J . Learning Mathematics with Understanding: A Critical Consideration of the Learning Principle in the Principles and Standards for School Mathematics J. Montana Mathematics Enthusiast, 2007. 14(1), pp: 103-114.
11. Richard Rusczyk. The Pros and Cons of Math Competitions. SEPTEMBER 15, 2016. https://artofproblemsolving.com/news/articles/pros-cons-math-competitions
12. Gerard W. Kelly. Shortcut mathM. Dover Publications, April 1984.
13. Hermann Ebbinghaus (1885), Translated by Henry A. Ruger & Clara E. Bussenius (1913). Memory: A Contribution to Experimental Psychology. https://psychclassics.yorku.ca/Ebbinghaus/index.htm
14. Stina Söderqvist and Sissela Bergman Nutley. Working Memory Training is Associated with Long Term Attainments in Math and Reading. Frontiers in Psychology, 2015, 6(118):1711.
15. Gathercole S E , Lamont E , Alloway T P . Working Memory in the Classroom. Educational Research & Reviews, 2006, 1(5):382-385.
16. Alexandra Piacenza. 10 Mnemonic Devices That Every Elementary Schooler Will Remember. December 24, 2014. https://www.noodle.com/articles/10-mnemonic-devic-es-that-every-elementary-schooler-will-remember
17. Nidhi Suman. Use of Mnemonics for Teaching Mathematics at the Primary Level J. The International Journal of Indian Psychology. 2016: 3(2),pp: 51-57.
18. Kathy DeLashmutt. A Study of the Role of Mnemonics in Learning Mathematics. July, 2007. https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1006&context=mathmidsummative
19. BARRY SCHWARTZ. Attention Must Be Paid! Schools need to teach students to maintain attention, not cater to short-attention spans. SEPT 23, 2013 2:40 PM. https://slate.com/ human-interest/2013/09/paying-attention-is-a-skill-schools-need-to-teach-it.html
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This work and its PDF file(s) are licensed under a Creative Commons Attribution 4.0 International License.


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Or calculate the spread between 2 interest rates, a and b, by using the formula a - b.
Use the assigned data series variables (a, b, c, etc.) together with operators ( , -, *, /, ^, etc.), parentheses , and constants (1, 1.5, 2, etc.) to create your own formula (e.g., 1/a, a-b, (a b)/2, (a/(a b c))*100).
In Arcata the population under the age of 15 was 28.1%.
Those age 15-24 made up 22.8% of Arcata’s population. Those age 40-54 made up 16% of Arcata’s population.
As noted above, you may add other data series to this line before entering a formula.
In 1850, Douglas Ottinger and Hans Buhne entered the bay, naming it Humboldt in honor of the great naturalist and world explorer, Baron Alexander von Humboldt.
Yet just a few miles inland the temperatures may be up to 25 degrees warmer in the summer and fall.
Winter high temperatures average in the low 40s to mid-50s, with lows in the mid-30s to lower 40s.“Normal” aged workers (age 25-39) made up 23.9% of the population.Older working age (age 40-54) made up 19.4% of the population.Learn more, including about available controls: Cookies Policy.For example, invert an exchange rate by using formula 1/a, where “a” refers to the first FRED data series added to this line.), originally Union Town or Union, is a city adjacent to the Arcata Bay (northern) portion of Humboldt Bay in Humboldt County, California, United States.


maximum likelihood estimation Maximum Likelihood Estimation and Nonlinear Least Squares in Stata Christopher F Baum Faculty Micro Resource Center Boston College July 2007 Christopher F Baum .
Useful Stata Commands (for Stata versions 13 & 14) Kenneth . Useful Stata Commands .. W3.. Maximum Likelihood Estimation for Custom Models .
Maximum Likelihood Estimation in STATA and MATA Econ626 Maximum Likelihood Estimation in STATA and MATA Econ626 .. Need i-th contribution to likelihood function as input .. 14 / 16.
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Maximum Likelihood Estimation - University of Notre Dame The maximum likelihood estimate is that value of the parameter that .. Maximum likelihood estimation is generally more complicated than .
C:paperseeML EstimationpapersMaximum Likelihood Estimation . Maximum Likelihood Estimation of Utility Functions Using Stata by Glenn W.. Harrison May 2008 Working Paper 06-12, Department of Economics, College of Business .
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3-IN-1 SET: Sorting . Stacking . Number & Shape Recognizing in one Wooden Board. Never get bored with this first wood puzzle game for toddler 3 years and older. Enjoy sorting the colors of the rings . counting the pieces . stacking blocks . naming the colors & putting the counter rings on the wooden pegs.
EARLY DEVELOPMENT: A great creative toy that improves kids' cognitive ability. Fun sticks for counting. It is good for developing early sorting skills . teaching colors and numbers . promoting hand-eye coordination . fine motor skills . problem solving skills and encouraging creativity.
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Unleash your Financial Potential: Mathematics Course for Finance and Business
Discover the key to success in the financial world with our specialized course: "Mathematics for Finance and Business". Designed for those seeking to understand and master essential mathematical foundations in the context of finance and business, this course will take you from basic concepts to practical applications that will transform your ability to make informed and strategic decisions.
This course goes beyond formulas and equations, offering a practical perspective on how mathematics plays a crucial role in the financial and business world. From calculating interest rates to evaluating investments, each lesson is designed to provide a solid foundation in mathematics applied to real-world financial situations.
Unit 1: Linear Functions
1.1. Conceptualization of linear models and graphical representation of linear equations. Application in performance calculation through simple composition. Exploration of the concept of slope and elasticity.
1.2. Exploration of Simultaneous Equation Models. Analysis of the structural and reduced form of a model. Characterization of balance, addressing aspects such as existence, uniqueness and stability. Applications in a Supply and Demand model of a good.
Unit 2: Manipulation of Matrices and Vectors
2.1. Detailed study of Matrices and Vectors, addressing operations such as addition, multiplication and the resolution of models through matrix inversion. Applications in determining macroeconomic balance and the fiscal multiplier.
2.2. Non-arbitration principle. Approach to the principle of non-arbitrage and graphical representation of a vector. Application in asset valuation through the creation of a replica portfolio.
Unit 3: Exploration of Nonlinear Functions
3.1. Breakdown of Polynomials. Identification of characteristics of polynomials and their graphic interpretation. Study of root determination. Applications in project decision making, including Present Value and Internal Rate of Return.
3.2. Analysis of Exponential and Logarithmic Equations. Manipulation of exponents, properties of the exponential function, and exploration of the concavity and convexity of a function. Applications in the calculus of continuous interest and decision making under uncertainty, including Jensen's inequality.
Unit 4: Focus on Financial Mathematics
4.1. Money Market Instruments. Study of types of instruments, rates of return and discount rates, considering the frequency of composition and the Day Count. Application in calculating the price of a Cete and comparison of returns using the Effective Rate.
4.2. Bonds and Annuities. Derivation of formulas for perpetuities, bonds and annuities. Application in the valuation of a mortgage and a corporate bond.
Unit 5: Exploration of Infinitesimal Calculus
5.1. Derivatives: Maximum and Minimum. Study of the rules for the derivation of functions and determination of first and second order conditions. Application in determining the optimal amount of production for a company.
5.2. Calculation and Optimization. Exploration of optimization with constraints, Lagrange multipliers and first-order conditions. Application in the construction of a stock portfolio with minimal variance.
What You Will Learn:
1. Mathematical Foundations:
- Key concepts of algebra and calculus applied to financial and business problems.
- Use of essential mathematical functions in financial analysis.
2. Interconnection with Finance:
- How mathematics is intrinsically related to financial valuation and business decision making.
- Interpretation of numerical data to promote effective financial strategies.
3. Practical Applications:
- Solving real-world financial problems using mathematical tools.
- Implementation of mathematical models for financial projections and risk analysis.
4. Risk and Return Evaluation:
- Risk and return calculations in investments.
- Use of mathematical statistics to make informed decisions about investment portfolios.
Practical and Applied Approach:
- Direct application of mathematical concepts to financial and business scenarios.
- Guided by experts in financial mathematics and professionals in the sector.
- 24/7 access to course content to learn at your own pace.
- Upon completing the course, you will receive a certification that validates your skills in mathematics applied to finance and business.
Get ready to overcome any mathematical challenge in the financial and business field. This course not only improves your analytical skills, but also equips you with the tools necessary to thrive in dynamic business environments. Join us and unleash your financial potential with the "Mathematics for Finance and Business" course. Your journey to success starts here!


What is SheTech Explorer Day?
SheTech Explorer Day is a hands-on tech conference for high school girls 9th through 12th grade. This day-long event will include hands-on workshops in science, technology, engineering, and math (STEM). Our ultimate goal is to expose girls to technology in a fun atmosphere as well as meet STEM role models to learn more about opportunities in those fields. Girls who attend will have access to internerships, receive a SheTech certificate, receive pathway information and become part of a community to help girls have opportunities in STEM.


How to Solve the Grand Staircase Tower Door Puzzle in Hogwarts Legacy
Grand Staircase Tower Door puzzle, Hogwarts Legacy is an open-world RPG game in the Harry Potter world.
You can explore familiar and new locations, discover magical beasts, and customize your character.
You can also craft potions, learn spell casting, upgrade talents, and become a master wizard.
As a student, you hold the key to an ancient secret that can tear the world apart.
Battle dark wizards, make allies and choose the fate of the wizarding world.
In this quick guide, you’ll learn how to solve the Grand Staircase Tower Door puzzle in Hogwarts Legacy.
How to solve the Grand Staircase Tower Door puzzle in Hogwarts Legacy
To solve the Grand Staircase Tower Door puzzle in Hogwarts Legacy, you need to decipher the symbols on the door and the writing on the blackboard.
The symbols on the door represent the numbers 0 to 9 from the left to right.
The writing on the blackboard is a math equation where the three smaller circles need to add up to get the bigger circle.
In this puzzle, all three smaller circles need to add up to 23 and 19 respectively.
Correct answer: The lizard and the squid.
Here is the solution to the puzzle (expand the screenshots above):
? = 6 = The lizard.
?? = 7 = The squid.
After you’ve set both symbols to the lizard and squid respectively, you can open the door.
The chest includes a Capricious Vest School Uniform and a Gaphorn Statue..


Practice Test for NET (8th) Like Tweet Pin it Share Share Email Sum of 10 terms of the series where rth term is 3r+1 is 650 175 270 325 Continue >> Common ratio of G.P 1/a , 1/b , 1/c is sq-rt(a/c) - sq-rt(a/c) sq-rt(c/a) a & b Continue >> Product of n geometric mean between 1 and 2 is 2 sq-rt(2) ( sq-rt(2) )^n (2)^n/(n+1) Continue >> A.M of two positive numbers exceeds their positive G.M by 1/2 numbers are 9/2 , 16/2 2 , 16 9 , 16 1 , 4 Continue >> Series 4+2+1+1/2 + 1/4 + 1/8 ------ converges to 8 4 6 7 Continue >> 0.259259259......... = 258/999 259/999 257/999 260/999 Continue >> Interval of convergence for series 1 + 2x +4x^2 + ........ is ] -1/2 , 1/2 [ [ -1/2 , 1/2 ] [ 0 , 1/2 ] ] 0 , 1/2 [ Continue >> If A,G,H are arithmetic, Geometric and Harmonic mean between two numbers a and b then A^2=G*H G^2=A*H G^2=(G*H)^2 H^2=G*A Continue >> How many necklaces can be made of 2n+1 beads of different colour? n! 2n!/2! n!/2! (2n+1)!/2! Continue >> Partial sum of series 1+2+4+ ------ 2^(n-1) is 2^n 2^n + 1 2^n - 1 2^(n+1) Continue >> Graphs of trigonometric functions have Smooth curves Line segments Sharp corners Breaking point Continue >> For the line 4x+5y=3 , point (-2,4) lies Above on Below None Continue >> Condition for lines represented by ax2+2hxy+by2=0 to be coincident is h2=ab h2+ab=0 a+b=0 h=ab Continue >> The resistance of a 60 watt bulb in a 120 volt line is 20 ohm 0.5 ohm 240 ohm 2 ohm Continue >> The resistivity of a 1 m length of wire of iron as compared to the wire of same material of double the length is Same Smaller Greater Much greater Continue >> The terminal potential difference of a cell is greater than its emf, when it is on open circuit Being charged Being discharged both charged and discharged Continue >> When the ohmmeter measures the infinite resistance, its pointer lies at Centre of scale Left end of scale Right end of scale None of these Continue >> One Tesla is equal to 10^8 gauss 10^-4 gauss 10^-7 gauss 10^4 gauss Continue >> A current flows in a conductor from east to west. The direction of the magnetic field at a point above the conductor is towards East North west South Continue >> A dynamo converts Electrical energy into Mechanical energy Magnetic energy into Electrical energy Mechanical energy into Electrical energy None of these Continue >> When two inductors L1 and L2 are connected in parallel, the equivalent inductance is Less than both L1 and L2 L1+L2 Between L1 and L2 None of these Continue >> The armature current in a DC motor is max when the motor has Just been switched off Picked up max speed intermediate speed just started Continue >> Lenz's law don't violate the principle of conservation of mass energy Charge Momentum Continue >> Impedance is the combined effect of Resistance and Reactance Resistance and Inductance Capacitance and Inductance None of these Continue >> The min and max values of power factor in an AC circuit are respectively 1 & 1.5 1 & 2 0.1 & 1 0 & 1 Continue >> A gas which burn with blue flame CO2 N2 CO NO Continue >> What is the nature of solution of Borax Alkaline Acidic Neutral Corrosive Continue >> Sodium silicates dissolve in water the resulting solution is Acidic Basic Amphoteric Neutral Continue >> Chemical composition of fool's gold is FeS Fe2S3 FeS2 HgS Continue >> which compound is used in photography NaCl AgCl AgI AgBr Continue >> Which of the following is the strongest Acid HClO4 HClO3 HClO2 HClO Continue >> Max temperature on the open hearth process is 900 C 1200 C 1400 C 1600 C Continue >> ALACRITY : APATHETIC :: compliance : deft temerity : timid despotism : arrogant candor : bungling Continue >> DORY : VAN dairy : cow fish : vehicle freighter : caisson runners : wheels Continue >> SPIKE : TACK :: bullet : wound knife : cut arrow : bow spear : dart Continue >> TERMAGANT : SHREW virago : harpy anteater : mouse supporter : nag single : married Continue >> Synonym of Ameliorate Increase decrease Improve Amble Continue >> Synonym of Barterer trader shopkeeper leader Businessman Continue >> Antonym of fulminate clamour murmur Vain Efficacious Continue >> Antonym of salacious lustful lecherous succeed frigid Continue >> PLAY AGAIN !


2 edition of Higher mathematics for engineers and physicists found in the catalog.
Higher mathematics for engineers and physicists
Ivan Stephen Sokolnikoff
|Statement||by Ivan S. Sokolnikoff and Elizabeth S. Sokolnikoff.|
Elements of Higher Mathematics Chapter 1. Mechanics Chapter For such a situation this book is ideal. Table of Elementary Integrals
Posted on April 17, by damitr The book in this post is one of the best introductory books for Higher Mathematics and it has been written by two exceptionally talented people, namely, Yakov B. It presents clear definitions and the rationale for such definitions. Curves in Polar Coordinates Volume of a Solid of Revolution
Functions and Graphs Chapter 2. It is a shame you cannot find a math book written in the west that is as clear and lucid. I highly recommend this book. The results of some of the studies are then applied to the theory of vibration by applying the Lagrange method for formulating equations of motion, after the formula establishing the energies and dissipation functions are completed. About this book Introduction This textbook offers an accessible and highly approved approach which is characterized by the combination of the textbook with a detailed study guide available online at our repository extras. Multiplication of Absolutely Convergent Series
A new variorum edition of Shakespeare.
The Western Tradition
Working together for multiple use
An account of the triumphant death of F. S., a converted prostitute, who died in April, 1763, aged 26 years
Black American heritage
community school of social action
Research in public finance in relation to agriculture
Rights and liabilities of law enforcement officers
Art and livelihood.
An answer to depression
The pilgrims progress
The ranks of the black order
Furthermore in case of learning difficulties he or she is given supplementary explanations and in case of individual needs supplementary exercises and applications. Physics students at the graduate level will benefit handsomely.
Such integrations frequently involve not always in immediately re cognizable form elliptic functions and elliptic integrals. It needs to be in print. The PDE chapter covers all the major topics heat equation, circular membrane, blabla. Tests for Convergence Vector calculus: Two chapters, pages.
Cauchy's Integral Test for Convergence It is a good reference though. Basic Properties of Infinite Series Deeper knowledge of series divergence and the like may not be a required course.
Chapter 1 is devoted to complex functions. So picking and choosing becomes a need for shorter courses. Infinite Series Few math books that Higher mathematics for engineers and physicists book at basic underlying math, derivatives and integrals go as far as this one in higher math, and as compactly, densely and succinctly.
Satzer, MAA Review "The majority of applied mathematical fields presently require so much specialization that mathematics often takes a back seat to the particular Higher mathematics for engineers and physicists book of study. Further Remarks on Definite Integrals Properties of Indefinite Integrals They tend to be more confusing Some Additional Topics Chapter The numerous books written on elliptic integrals, while of great value to the student or mathematician, are not especially suitable for the scientist whose primary objective is the ready evaluation of the integrals that occur in his practical problems.
However it is just a single chapter with 13 sections. Preliminary Concepts Arfken and Weber do, poorly. I think a lot of math methods courses would not use this section.
I wish that kind of book had been available when I was a student.Higher Mathematics for Beginning Physicists & Engineers by Ya B Zeldovich, I M Yaglom, B Kila Zel'dovich starting at $ Higher Mathematics for Beginning Physicists & Engineers has 1 available editions to buy at Half Price Books Marketplace.
Nov 14, · Higher Mathematics For Engineers And Physicists Item Preview remove-circle Higher Mathematics For Engineers And Physicists by Sokolnikoff,Ivan S.; Sokolnikoff,Elizabeth S NATURAL SCIENCES, Mathematics, Fundamental and general consideration of mathematics Publisher McGraw Hill Book Company Inc.
Collection universallibrary Contributor. May 09, · A Course of Higher Mathematics, Volume II: Advanced Calculus covers the theory of functions of real variable in advanced calculus. This volume is divided into seven chapters and begins with a full discussion of the solution of ordinary differential equations with many applications to the treatment of physical problems.
This topic is followed by an account of the properties of multiple.The author, Walter Appel, is a renowned mathematics pdf hailing from one of the best schools of France’s prestigious Grandes écoles, where he has taught some of his country’s leading scientists and engineers.
In this unique book, oriented specifically toward physicists, Appel shows graduate students and researchers the vital benefits.Due to the rapid expansion of the frontiers of physics and engineering, the demand for higher-level mathematics is increasing yearly. This book is designed to provide accessible knowledge of higher-level mathematics demanded in contemporary physics and engineering.A Course of Higher Mathematics, Volume V focuses on the theory ebook integration and elements ebook functional analysis.
This volume is suitable for engineers, physicists, and students of pure mathematics. Show less. A Course of Higher Mathematics, Volume V focuses on the theory of integration and elements of functional analysis.
This book is.


Pattern-tastic Treasure Hunt
Can you spot the odd one out in this nature-inspired treasure hunt? Turn the tabs to discover 7 kinds of patterns and 70 first words to see and say.
This visually stunning board book will also help your little one work on developmental skills such as:
- Fine Motor skills (turning the page using tabs)
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- Cognitive skills (identifying patterns and categories, recognizing the odd one out)
Early STEM alert: Pattern identification is KEY for success in math
Perfectly suited for babies and toddlers.


of exponential functions and their new fundamental properties, with Applications to
Midwest Book Reviews (Oregon,
... very highly
recommended reading for advanced mathematics students and academia is
Yuri Shestopaloff's 152-page treatise, "Sums of Exponential Functions
and their new Fundamental Properties, with Applications to Natural
Professor Dr. B. N.
Chichkov, Head of Nanotechnology Department, Laser Zentrum,
The book studies an interesting and practical area of adequate
description of natural phenomena by mathematical models. In the first
chapter the author “plows” unexpectedly fertile soil, with regard to
practical applications in physics, biology, and cosmology, of seemingly
pure classical mathematical constructs. He considers exponential and
logarithmic functions as the most adequate representatives of general
growth, destruction and transitional processes, ...
PhD in Elecrtrophysics
The book is
dedicated to the insufficiently investigated field of mathematical
modeling of complex natural phenomena such as boundary processes. It is
shown that for reasons not completely known, identical mathematics, in
particular, sums of exponential functions, describe many different
phenomena. ... more
V. Agranat, Ph.D in
mathematical methods, modeling of thermodynamical processes
The author discovered fundamental properties
of exponentials functions in the classical area of mathematics studied
for centuries. What else to say?
his findings by convincing mathematical proofs. In particular, he
considers growth of biological cells; he discovers also very interesting
properties of transitional electrical signals.
These are certainly high level and valuable
scientific results on their own, but the author goes to the next level
of generalization and suggests that all these natural phenomena are
closely and inherently interconnected through the boundary functions,
separating the areas of mathematical convergence and divergence of sums
of infinite series. The author provides mathematical support of this
hypothesis. However, this important problem, which he discovers and
formulates, has to be studied on a larger scale. If his finding is true,
then it will have significant implications in many areas of science and
technology, because in this case the introduced boundary functions
become reference points for many natural processes, and this is the
result of a fundamental importance.
Considering relationship of Fermat’s Last
Theorem and physical reality, the author, beside the mathematical
considerations from the area of number theory, includes some
philosophical considerations. This part is certainly thought provoking
and interesting; some new concepts, such as the space rarefaction, can
be used as a foundation for the following research. Overall, it is more
an illustration how the pure mathematical concepts are connected to the
physical world, and how the mathematical modeling should be approached
to create models that are adequate to natural phenomena.
The second part of the book introduces and
proves a Theorem about properties of sums of exponential functions, in
particular “one time oscillation property”. The author switches to
strict mathematical proof of his Theorem, and this is a mathematically
challenging part of the book that requires laborious efforts to get
through. On the other hand, its understanding does not need knowledge of
advanced mathematical subjects, and we agree with the author that
familiarity with calculus is a sufficient prerequisite. Some new
mathematical concepts, which the author developed, have a broader
meaning for calculus in general; in particular, the notion of pair
functions and associated operations certainly deserve close attention of
With regard to the Theorem itself, we think
that it is very important in all aspects. Judging from the mathematical
perspective, it is amazing that the author managed to discover such
fundamental properties of exponential functions that were studied for
centuries by an army of mathematicians. With regard to possible
practical applications of the Theorem, they are numerous, and the
Theorem can be used beneficially virtually in any area of science due to
the adequacy of models based on exponential functions. The author
himself provides several examples of such applications in the form of
corollaries, which compose the last part of the book.
Our impression is that the first part of the
book should evolve into a separate volume in the future editions,
because it appeals to a wide audience of scientists from different
areas, while the second part should be of great interest to
mathematicians. Overall, the book presents high level, and in many
instances fundamental scientific discoveries that certainly have a great
value for science, as well as for different technological and social
disciplines. The general public may be also interested in the content
presented in the first part of this book.
It is well known that the exponential
function and its inverse, the logarithmic function, describe a large
number of Nature’s phenomena. In the first chapter of the book, several
examples of such phenomena in different areas of science and technology
are given. In particular, exponential and logarithmic processes are used
to describe a cell’s replication and growth process.
In order to understand
the importance of the aforementioned example, it is necessary to note
that the discussion deals with the processes of evolutionary development
of any kind of animal populations, beginning from the simplest amoebas
and ending with humans. Here, it is also worthwhile to mention the basic
psychophysical law of the perception of information - the Weber-Fechner
law (“Sensation is proportional to the logarithm of excitation”), which
relates to different systems (detectors) of perception, which plays an
important role in the vital activity of humans and animals.
example examined in the book that deals with the description of
electrical processes of charging and discharging of a capacitor, without
which not one device that generates electromagnetic waves (these are
radio, television, mobile telephones and the like) can exist, gives an
idea about the important role of these functions in inanimate nature.
are also considered in relation to Fermat's Last Theorem. This touches
on another very interesting subject where these functions appear in an
unexpected manner, in such a way that they can be used for the
interpretation of complex natural phenomena.
speaking, a “deficiency” of the first chapter of the book is the fact
that the author did not decide to make this chapter into an independent
book. The series of many interesting questions, which the author
considers here, can be of serious independent interest for many readers.
For the author – a professional mathematician and physicist, purely
scientific problems have a higher priority: namely, the general theorem
discovered and proven by him, which has to do with the special features
of sums of exponential functions of a real variable. To this theorem and
to its consequences the author devotes the second chapter of the book.
In order to see the significance of this theorem, it suffices to say
that it makes possible the description and analysis of the majority of
the processes mentioned in the first part of the book, as well as many
It remains only to add that the book is well published and reads easily,
which is also important for the reader.