Math Arcade V1.2
Put away your calculator and give Math Arcade a try! Whether you’re a grade school student learning basic math for the first time or an old timer in need of a refresh, Math Arcade will hone your mental math skills like nothing else!
Math Arcade is a collection of 8 different games with 5 skill levels each. The games vary from simulated flash card learning for addition, subtraction, and multiplication, to full blown long hand addition, subtraction, multiplication, and division. There is even a mind-bending equation builder that will be sure to test even the best mental math wizards!
There is both a practice and an arcade mode for each game. In practice mode, you can try as many problems as you want. In arcade mode, up to 4 people at once can compete solving a fixed number of problems in a race against the clock and accuracy for a shot at the high score table.
At the end of each arcade game attempt, a summary sheet is provided to review your performance. There is even printing support so you can have a permanent record of any summary report.
The demo version of Math Arcade has 2 of the 8 games enabled for your evaluation. If you enjoy this program, please consider purchasing the full version to help support independent software development.
System Requirements: Windows OS, 800×600+ desktop resolution, 300+ MHz CPU recommended
Download the demo version here: MathArcadeDemo.zip (1.04 MB)
Order the full version of Math Arcade V1.2 for $9.95


R. Renée Bembry
Tips for teaching carrying and borrowing
Once these simple straightforward concepts are mastered children will be able to carry and borrow with ease.
CARRYING is a mathematical action you perform when adding two numerals (1, 2, 3, 4, 5, 6, 7, 8, 9, and 0) together in a mathematical problem. Numerals are commonly referred to as numbers (a series of numerals such as 12, 123, and 585). I will address them as such in these instructions. In performing these procedures with a child, you will need a place to sit, a surface to work on (I will call a table), 31 popsicle sticks/pennies/like pieces of candy, et cetera, 2 pencils, at least 2-3 sheets of blank paper, patience, a willingness to teach, and the belief this process may be easier than you think.
Allow the child to get a visual image by counting out twenty-four of the items you’ve chosen to work with, and placing them in a column on the table. Then, have him/her count out seven items and align them below the twenty-four. Ask how much is twenty-four plus seven. The answer, thirty-one should be given.
Have the child write twenty-four plus seven in a columned math problem on a piece of paper. You do the same. The twenty-four should be above the seven. There should be a plus sign to the left of the seven, further out than the number two.
Draw a vertical line between the number two and the number four, down passed the seven, and across the horizontal line (that should be written below the minus sign and the seven). Have the child do the same. After the vertical line crosses the horizontal line, there should be enough space to separate the numbers that are going to be written below the horizontal line. Explain that the tens column is the column the number two is in; and that numbers four and seven are in the ones column.
NOTE: The purpose of the vertical line is to help the child grasp the concept that the numbers are lined up in columns. Make sure your and the child’s numbers are lined up properly.
Tell the child that the numbers in the ones column must be added together before the numbers in the tens column are added together. Ask the child how much you get when you add four to seven. Once the correct answer, eleven, has been given, explain that you are going to “put down” one of the number ones below the seven just below the horizontal line in the ones column, but that you’ll have to “carry” the other number over to the tens column because you cannot have two numbers next to each other in the ones column. Write a number one below the horizontal line. Place the other number one above the number two. Have the child do the same.
Tell the child that the number you carry will always be the numeral to the left of the added numbers. In this case, since the added numbers make eleven, you’re carrying the same numeral you’re going to keep. However, this will not always be the case.
NOTE: A zero may be placed to the left of the seven and just under the number two so the child can get a visual image that there are zero items in the tens place.
Now the carried number one must be added to the two. Ask how much is one plus two. After the correct answer, three, is given, you and the child write a number three below the horizontal line just below the number two.
The problem has been completed. The child can see the answer to the problem, thirty-one, is the same as the answer when he/she added their items together on the table.
Repeat this process until the child is comfortable the carrying concept.
BORROWING is a mathematical action you perform when subtracting high valued numerals from low valued numerals in a mathematical problem.
Allow the child to get a visual image by counting out 31 items (as in carrying above) by placing them in a column on the table. Have him/her count out twenty-four items and align them near the upper area, and align the remaining seven below. Ask which is more, twenty-four, or seven.
Ask how many items will be left in the column of twenty four if you take seven items away. Have child consider the problem and allow him/her to remove seven items from the column of twenty-four. After the child counts the remaining seventeen items in said column, tell him/her that is the number you should have after you subtract seven from twenty-four on paper.
Instruct the child to align four items horizontally on the table. Below the four items, instruct him/her to align seven items horizontally. Ask which column has more. After the correct answer, seven, is established, tell the child to take seven items away from the column of four. This action helps the child understand the concept that you cannot subtract more from less; and gives him/her a chance to think about how to solve the problem.
Write twenty-four minus seven in a columned math problem on a piece of paper. Have the child do the same. Explain that just like with the tangible items, you cannot take seven away from four and that you’ll have to “borrow” from the “tens” column. Explain that the tens column is the column the number two is in on the paper; and that numerals four and seven are in the ones column. Then draw a vertical line as with “carrying” above.
Going back to the tangible items, have the child form two columns of ten items, and a column of four items to the right of the column of ten so that the combined columns symbolize twenty-four items in a problematic structure. There should be a space between the twenty-four items and the four items. The column of seven items should be below the column of four.
Next have him/her “borrow” one of the columns of ten items by placing them closer to the column of four items. Tell the child to count the items. After the correct number, fourteen, has been totaled, ask if he/she can take seven items away from fourteen items and how many will be left. Let him/her perform the process of moving seven items from the ones column. After the questions have been answered and it has been established that seven items will be left, ask why he/she was able to take seven away from fourteen. Child should answer fourteen is more than seven. Transfer this concept to the math problem on paper.
Reiterate the fact that you are going to “borrow” ten from the tens column. Draw a diagonal line through the number two; this is called “crossing out”. Place a one about one half the size of the four just to the upper left of the number four. Have the child do the same. Reiterate the fact that you have “borrowed” ten from the tens place; and ten plus the four that were already there makes fourteen. Make sure the child understands that there are now fourteen invisible items in the ones column where there used to be only four.
Place a small number one next to the crossed out number two. Have the child do the same. Ask how many invisible items are left in the tens column. Although there is only a numeral one in this column now, the correct answer is, there are ten invisible items left-this is the tens column. Numbers in the tens column are in degrees of ten. To reinforce this concept, return the child’s attention to the items on the table. Ask how many tangible items are left in the column that used to be twenty after you borrowed ten of the items and gave them to the four. The answer should be ten.
Ask how much will be left after you take seven from the fourteen. After the correct answer, seven, has been established, you and the child write the answer, seven, below the horizontal line, just below the number four. Now, you and the child bring down the one that is remaining in the tens column and place it next to the seven below the horizontal line on your papers. Tell the child there are zero items to add to the one in the tens column.
NOTE: A “0” may be placed to the left of the seven, just below the initial number in the tens column-in this case, number 2-to help the child understand there are zero tens available to subtract from the one.
These simple tips should make carrying and borrowing a cinch once children master them. Start using two numerals at the “tops” and at the “bottoms” of problems after the child conquers these basics of adding or taking away one numeral. Eventually, the child will be able to add or subtract three, four, and more numerals in a row; and to “add” three, four and more “rows” of numbers.


Since March 14 is also written 3/14, it's become a day to celebrate the number Pi, which starts off 3.14... but what is Pi?
Try this with any circle!
1) Measure the circumference, or distance all around the edge of the circle.
2) Measure the diameter, or distance across the circle (being sure to pass through the center point).
3) Divide the circumference by the diameter.
You should get Pi, or 3.14159... Using long division, you will find that the number keeps going and going and going!
No matter what size circle you measure, the answer to circumference divided by diameter will always be Pi.
Some people have tried to see if the number ever ends. But it has been carried out to 31.4 trillion decimal places and still continues!
Have you ever tried to memorize the numbers of Pi? How far did you get?
It may take a little while to load, but you can see 1,000,000 digits of Pi on one page on the website https://www.piday.org/million/.
Here are the first 100 decimal places (shown in groups of 10):
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679
Scientists use Pi in their calculations whenever circles are involved. Many things are circular, so this number comes in handy. Imagine going to outerspace. Orbits, planets and all kinds of things have circles in them.
Check out places to find Pi in the universe with NASA here: https://nasa.tumblr.com/post/612574998435135488/cosmic-piece-of-pi
For fun challenges at home, try the NASA Pi Day challenge, which gives real-life problems scientists solve using Pi as they explore the universe.
Happy Pi Day!


Date(s) - 01/26/2019
9:30 am - 11:30 am
Concordia Parish Library
When: Saturday, January 26th @ 9:30am
Where: Ferriday Branch
Details: Tips and tricks for scoring higher on the ACT, understanding you ACT score, and finding scholarship opportunities. Featuring an ACT Math Practice Test, guided math practice and strategies for maximizing your time. Instructor Stephen Collins is Louisiana & Mississippi certified in Mathematics, Business Education, Computer Science and Computer Literacy. Call (318) 757-3550 to register. Free and open to the public!


Lots of us read bedtime stories to our kids. But how does math fit in?
Inspired by a listener question along those lines, we invited Laura Overdeck to this week’s episode of Best of Both Worlds. Overdeck is the author of the beloved Bedtime Math series of books. Each book has dozens of short stories that lead to little math problems you can do with your kids. You can also get the problems by signing up for the Bedtime Math organization’s emails or their app. These fun problems are a fantastic way to make math a regular part of life.
Which it should be! There’s some evidence that kids struggled more to keep up with math during the pandemic than they struggled with reading. It’s possible that adults felt more comfortable reading with kids, and incorporated more reading into everyday life, whereas math was more of a school-based subject. But math doesn’t need to be confined to school. With bedtime stories we have fond memories of snuggling up with parents, feeling cozy and loved. The idea of Bedtime Math is to encourage kids to feel the same way about mathematical concepts.
I can say that my kids have really enjoyed reading the Bedtime Math books! So please check them out.
In the Q&A section, we address a listener who wants details on how Sarah is having one on one dinners with her kids. The listener struggles with the idea of using childcare outside of work hours. So we tackle that topic too.
Please give the episode a listen, and as always, we welcome ratings and reviews!


Kepler’s third law in classical mechanics states that the square of the orbital period, , of a planet or particle is proportional to the cube of the semi-major axis of its orbit. The classical equation describing this is
where is the semi-major axis, the Newtonian time, and the mass of the test particle. We are interested in test particles, so .
An interesting question is how this generalizes to test particles moving around a very massive body in general relativity, such as a black hole. The metric in Schwarzschild coordinates that describes this geometry is
where is the mass of the black hole. The question is whether Kepler’s third law applies to the Schwarzschild time, , or the proper time, , of the test particle? To answer this question let us assume that and that the motion is circular, so .
Using our assumptions this simplifies to
From this we see that
and that it is the Schwarzschild time coordinate that is analogous to the Newtonian time in Kepler’s third law.


Journal of Computational Physics 291 (2015) 99?119
Contents lists available at ScienceDirect
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BG 1. flu fo us ou co fin pr so th by pr of ? of * vi ht 00Journal of Computational Physics www.elsevier.com/locate/jcp multiscale fast semi-Lagrangian method for rarefied gas ynamics? iacomo Dimarco a,b,?, Rapha?l Loub?re b, Vittorio Rispoli b epartment of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy niversit? de Toulouse; UPS, INSA, UT1, UTM; CNRS, UMR 5219; Institut de Math?matiques de Toulouse; F-31062 Toulouse, France r t i c l e i n f o a b s t r a c t ticle history: ceived 7 March 2014 ceived in revised form 18 December 2014 cepted 9 February 2015 ailable online 6 March 2015 ywords: netic equations screte velocity models mi-Lagrangian schemes undary conditions main decomposition ultiscale problems
In this paper we consider the extension of the method developed in Dimarco and Loub?re (2013) [22,23] with the aim of facing the numerical resolution of multi-scale problems arising in rarefied gas dynamics. The scope of this work is to consider situations in which the whole domain does not demand the use of a kinetic model everywhere. This is the case of many realistic applications: some regions of the computational domain require a microscopic description, given by a kinetic model while the rest of the domain can be described by a coarser model of fluid type. Our aim is to show how the kinetic scheme developed in the pre-cited articles is perfectly suited for building domain decomposition strategies which make the method more attractive with respect to classical numerical techniques for kinetic equations and multi-scale realistic problems. Several numerical evidences are provided in this work in the two dimensional and three dimensional settings to assess the efficiency of the domain decomposition scheme. ? 2015 Elsevier Inc. All rights reserved.
Kinetic models are intended to give a probabilistic description of particle systems. Concerning the case of gases and ids, this microscopic description is necessary whenever the system is far from its equilibrium state. This is often the case r rarefied gases and plasmas, for which the mean free path between two successive collisions is large enough to forbid the e of macroscopic models based on the hypothesis of local thermodynamical equilibrium . Thus kinetic equations turn t to be very useful in describing many physical phenomena, however their use is limited by the excessive computational st when realistic simulations are considered. On the one side classical deterministic techniques such as finite volume, ite elements or spectral methods [37,39,40,27] are too costly when it is desirable to deal with the full seven dimensional oblem (six dimensional phase?space plus time). Recently it has been shown that the spectral approach  is able to lve, at least partially, this problem. On the other side, probabilistic Monte Carlo techniques [3,10] are often used to tame e curse of dimensionality. However, these methods are affected by many fluctuations which are commonly attenuated time averages for steady state situations while, for unsteady problems, the reduction of the variance is still an open oblem [15,21] with apparently no easy solution. In order to reduce the statistical noise, in practice an enormous number samples is generally used and this leads again to very expensive simulations.
This work was supported by the ANR Blanc project ?BOOST?, by the ANR JCJC project ?ALE INC(ubator) 3D? and by the ?Bando Giovani Ricercatori 2013?
Ferrara University (Italy).
E-mail addresses: firstname.lastname@example.org, email@example.com (G. Dimarco), firstname.lastname@example.org (R. Loub?re), email@example.com (V. Rispoli). tp://dx.doi.org/10.1016/j.jcp.2015.02.031 21-9991/? 2015 Elsevier Inc. All rights reserved. 100 G. Dimarco et al. / Journal of Computational Physics 291 (2015) 99?119
Ki pa se fix lin co of do pr cl is st si
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A m [2 th to
Ki of in a
Se 2. pa 2.1 w
Th of in co
Al diIn a recent paper  the authors have developed a new deterministic method to solve kinetic equations named Fast netic Scheme (FKS). In particular they have designed an efficient technique for the discretization of the linear transport rt of these equations. The proposed method is based on the so-called Discrete Velocity Model (DVM)  and on a mi-Lagrangian approach [13,14]. The DVM is obtained by a direct discretization of the velocity space into a finite set of ed velocities [5,37,39]. As a result of such discretization the original kinetic equation is then represented as a system of ear transport equations plus an interaction term coupling all equations and corresponding to the discretization of the llision operator. The purpose of the method described in [22,23] was to drastically reduce the cost of the transport part the discrete velocity model using a semi-Lagrangian technique, allowing to exactly solve the transport part on the entire main at a negligible cost. Hence, the cost of the solution of the original kinetic equation was almost entirely due to the ojection of the solution onto the grid to compute the collision operator. The resulting scheme shares many analogies with assical semi-Lagrangian methods [28,13,14] and Monte Carlo schemes . However, contrarily to the latter, the scheme as fast as a particle method while providing a fully deterministic numerical solution, not affected by any source of atistical error. The authors have shown that the full six dimensional problem (space and velocity) can be solved on a ngle processor laptop on decent meshes for a simple kinetic equation consisting of transport plus a relaxation operator. owever such scheme still remains very expensive for a single processor machine if space or velocity grids need to be larged to treat more realistic problems. Consequently parallel versions of this scheme have been designed in .


Happy π Day! It’s also Einstein’s birthday. π, a transcendental number, is the ratio of a circle’s circumference to its diameter. It has been calculated to over a trillion digits after the decimal. Babylonian mathematics had a crude approximation. Two millennia ago, Chinese mathematicians had a fairly accurate approximation to seven digits. At about the same time, India was using one to five digits.
Circles are a lot of fun. Using only a compass and straight edge (no need for measuring; the compass holds the radius of the circle), you can draw all the other shapes–triangle, square, pentagon, hexagon, septagon, etc. My favorites are multiples of threes, ie, triangle, hexagon and nonagon, which can all be drawn by a special, secret method. If you connect the centers of circles drawn such that the center of the next circle is placed on the circumference of the previous circle, and then use the first two circles’ center points to define a line… Hmmm… I’d have to show you. I would guess Anaxagoras played with this while he was stuck in prison. He was the first Greek to come up with an good approximation of π during his tenure there.
Mathematics has always fascinated me, especially the very basic roots of geometry. Give a child a pad of plain paper, a (fairly good) compass and a straight edge (not a ruler). Show her a couple of ideas, then let her go with it. You might be amazed at what she can discover on her own.
Here is π to 607 digits. I stopped here because I thought it was interesting that it could have three zeros in a row, then pick up again with a 5.
Copyright ©2014-17 Ramona Ridgewell. All rights reserved.


Just how likely is it that you'll hit it big in the lottery? Well, the odds are much worse than the probability of your death as a result of one of these:
- Plane crash: 11 million to one
- Car accident: 106 to one
- Falling out of bed: 2 million to one
- Flesh-eating bacteria: 1 million to one
- Lightning strike: 1.2 million to one
- Dog mauling: 118,776 to one
- Snakebite: 50 million to one
- Shark attack: 3.75 million to one
Turns out swimming with the sharks is safer than petting a dog, driving your car, or sleeping in your bed.
There is one thing that you stand less of a chance Dying from...
Let's say you play for 50 years, that's 2,600 weeks.
The probability of no win over 2,600 weeks is:
(1 − 1/13983816)2600 = 0.999814...
That means the probability of winning (after 50 Years) is: 1 − 0.999814... = 0.000186...
Still only about 0.02%
And you would have spent thousands for that small chance.
If you are healthy you have a bigger chance of winning the lottery than of dying of COVID-19


Representational similarity analysis¶
Visualizing dissimilarity matrices¶
Dissimilarities, based on neural data, behavioural data, and/or model data, can be visualised in various ways.
Compute the similarities in the
vt regions for participant
s01 across the six categories, and load the model similarities for the behavioural ratings. Then visualize the similarities in three ways:
with dissimilarity matrices
with multi-dimensional scaling
Hint: run rsa visualize skl.
Comparing dissimilarity matrices¶
It is easy to compare dissimilarity matrices by computing the Pearson correlation between two flattened upper triangle DSMs using the cosmo corr function. For the next exercise, stack flattened DSMs vertically into a single matrix starting with all of the EV DSMs from every subject then all of the VT DSM. You should have an 16x15 matrix. Then add the v1 model and the behavioral DSMs to make it a 18x15 matrix. Now compute the cross-correlation matrix using cosmo corr. Visualize the cross-correlation matrix with imagesc. Try this with demeaning and without demeaning to compare the results. Finally, use matlabs boxplot function to view the distributions of correlations between neural simiilarities and model/behavioral DSMs.
Hint: run compare dsm skl.
Target dissimilarity matrix searchlight¶
The function cosmo target dsm corr measure implements representational similarity. Use this measure to map where the neural similarity is similar to the behavioural similarity.
It is recommended to center the data using the
Advanced exercise: the cosmo target dsm corr measure function can also run regression on multiple dissimilarity matrices. Use this function to estimate the contribution of the V1 and behavioural model using a searchlight.


The National Research Foundation (NRF) has awarded Witwatersrand University’s Distinguished Professor, Jill Adler, the Lifetime Achiever Award at the 2023 NRF Awards ceremony held on Thursday in Durban.
Introduced to the NRF Awards in 2004, the Lifetime Achiever Award recognises an individual who has demonstrably made extraordinary contributions to the development of science in and for South Africa.
These contributions, according to the NRF, must be of international standard and impact. The NRF is a government mandated research and science development agency.
Adler expressed her gratitude for the special recognition.
“This award for me gives recognition to the importance of educational research and educational research that is grounded in the realities of our mathematics classrooms. It gives meaning to the NRF vision for science and service of society,” she said.
Adler said the NRF has been one of the most important institutions throughout her academic journey.
The Professor was born and raised in Johannesburg and completed a BSc in Mathematics and Psychology at the University of the Witwatersrand in 1972.
She pursued a teaching Diploma at the University of Cape Town in 1973 before taking up a teaching post at Harold Cressy High School in Cape Town.
In 1977, she returned to Johannesburg and joined the SACHED Trust, an educational non-government organisation (NGO) concerned with enriching the quality of education of those disadvantaged in apartheid South Africa.
She first taught mathematics to youth who were out of school following the Soweto uprising in 1976.
“Her work at the NGO over the next 10 years enabled her to further her social justice advocacy, seeking to address the educational inequalities in South Africa by improving mathematics education through the development and evaluation of distance education courses,” the NRF said.
At the same time, she studied for her Master of Education, graduating cum laude in 1985.
It was in 1987 that she moved into academia, lecturing in the Department of Professional Studies at the Johannesburg College of Education for two years before moving to Wits University’s Education Department as lecturer, Head of Department and Professor.
She was also a Chair of Mathematics Education at the South African Research Chairs Initiative (SARChI) between 2010 and 2019.
“Prof Adler completed her PhD in 1996 with a thesis that looked at the dynamics of teaching and learning mathematics in multilingual classrooms, specifically examining secondary teachers’ knowledge in this context,” said the NRF.
Her theoretical and practice-based innovations paved the way for new and transformative approaches in mathematics education, addressing two fundamental research problems.
“The complexities and challenges of teaching and learning mathematics in multilingual classrooms, and the enhancement of professional education for mathematics teachers. The latter came to the fore in the mid-1990s with the demand for ‘upgrading’ the qualifications and teaching quality of secondary mathematics teachers in post-apartheid South Africa.”
Working alongside a dedicated cohort of students, she spearheaded efforts to position mathematics education as a robust research domain within the university, and more broadly in the country.
In 2005, Adler led the renowned Marang Centre for Mathematics and Science Education at Wits University.
The centre quickly gained a reputation for excellence in postgraduate research and development, attracting students from across the African continent.
Between 2010 and 2019, as the DSI-NRF SARChI Research and Development Chair in Mathematics Education at Wits, she directed a large professional development project.
The programme reached over 200 secondary mathematics teachers and many learners across 80 schools in Gauteng and researching its impact together with a large research team of doctoral and postdoctoral fellows.
She is an Honorary Professor at Oxford University and University College London both in the United Kingdom.
She has published more than 60 peer-reviewed articles and four books, co-edited two special journal issues and contributed 50 book chapters, achieving more than 7 000 citations and has published 37 papers that have each been cited at least 37 times.
Over the course of her 45-year career in teaching and academia, she has garnered numerous accolades in recognition of her contributions to education.
Source: South African Government News Agency


Unless you’re a math geek, understanding slot machine math and “volatility indexes” can be pretty daunting. When designing a new game, software providers first need to figure out 3 essential pieces of information :
- The payback percentage or “return to player (RTP)”;
- The hold percentage;
- The volatility index.
The payback is how much you, the player, theoretically keep of the money you play. As we will see, theory and practice are two different things.
Inversely, the hold percentage is the theoretical percentage that we, the casino, should retain to be able to pay our bills.
The volatility index (for example “22.5”) determines the size and frequency of the payouts, as we saw. The bigger the number, the more volatile the game is.
Doing the math
In reality, for an individual such as yourself experiencing a game, the payback percentage will rarely be exactly what’s disclosed by the game developers and shown on the site. Even if you played ten million spins on a slot machine with a theoretical RTP of 92%, it’s more likely you’d fall somewhere between 91.3% and 92.7%
To calculate the actual range of RTP, you need to consider the game’s volatility. Here’s what it looks like:
RTP +/- (volatility index / square root of # of games played on the slot)
So, if you play 1,000 spins on a slot with a theoretical payback (RTP) of 92% and a volatility index of 22.5, you can expect to keep between 20.85% of your money and 163.15%. The more games are played, the more this percentage stabilizes, which is how casinos can make some money, but at the individual level, the experience can vary quite a bit, which is what makes it so interesting!
Ready to give it a shot?


This work deals with quantum systems determined by time-dependent Hamilton operators. Family of quantum systems, whose Hamilton operators take form H(t) = H 0 + V (t), where V (t) is perturbation and H 0 is self-adjoint with pure-point spectrum and constant gaps between eigenvalues in spectrum σ(H 0 ). Existing theory dealing with stability of quantum systems with Hamilton operators of above form, where H 0 is self-adjoint with pure-point spectrum and growing or shrinking gaps between eigenvalues in spectrum σ(H 0 ) is given in corresponding chapter. Because of non-applicability of existing theory to the studied cases the author attempts to device a new approach based on term ”mean of Hamilton operator over infinite time interval“. Lemma 7 can be interpreted as quantum variant of ergodic theorem in a very nice form that the author have not encountered before. In the last chapter is devoted to the study of simple example and to the application of devised theory.
time-dependent Hamiltonians, stability of a quantum system


Have you ever heard of the Fibonacci sequence? Or the golden ratio? Or the golden spiral?
These mysterious numbers and shapes are all connected to each other. If you look closely, they can be found in the most unexpected places, creating beautiful and appealing patterns.
Simply put, the Fibonacci sequence is a series of numbers starting with 1 and 1. From there, add the two previous numbers in the sequence to get the next number. This is a type of recursive sequence.
So 1+1=2, 1+2=3, 2+3=5 and so on. This gives you a sequence that looks like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 etc. But what makes this sequence so special and interesting?
The mathematical sequence that governs nature
For starters, Fibonacci numbers can be found in the natural world around us. For example, most flowers will have a number of petals corresponding to the Fibonacci sequence. Irises have three petals while wild roses and buttercups have five petals. That is, of course, until a petal falls off. No wonder rare four-leaf clovers are considered lucky!
If you cut into a piece of fruit, you are likely to find a Fibonacci number there too, in how the sections of the seeds are arranged. Bananas have three parts while apples have five.
The Fibonacci sequence even plays a role in the subtle spirals you can see in the seed head of a sunflower. This is due to something called the golden ratio, the golden ratio or the Greek letter Phi. If you take a number in the sequence above 5, and divide it by the previous number, you get an answer very close to 1.618. The bigger the numbers, the closer you get to 1.618. The golden ratio is an irrational number, and therefore cannot be written as a fractional number. Again, this is a number that can be found in the natural world.
Take the sunflower. To be as effective as possible, its seeds must be packed tightly together without overlapping. Now, if it only grew seeds in a straight line in one direction, it would leave lots of empty space on the flower head. The best way to minimize waste of space is for the seeds to grow in spirals, with each seed growing at a slight angle away from the previous one.
If the degree of turn was a fraction, like 1/4, then it doesn’t matter because after four laps the seed pattern would be back at the beginning again. There would be four rows of seeds, but it’s not much better than one when trying to cover a circular area. The perfect degree of luck must be an irrational number, not easily approximated by a fraction, and the answer is the golden ratio.
The Fibonacci spiral can be found in art, architecture, biology and many more places
Turning numbers into an eye-catching spiral
Another way to think of the golden ratio is as a spiral. This spiral gets wider by a factor of 1.618 each time it makes a quarter turn (90°).
Drawing a perfect golden spiral is quite tricky to do by hand, but just like with the golden ratio, you can get a close approximation with the Fibonacci spiral. This is created by drawing an arc through a series of squares corresponding to the Fibonacci numbers, so 1², 1², 2², 3² etc.
You may have seen these spirals superimposed over famous works of art, as experts try to explain why we find them so aesthetically pleasing. Often the spiral pulls our eye so that the focus of the artwork is in the center of the spiral. Examples include works by Leonardo da Vinci and Salvador Dali.
Listen for the Fibonacci sequence in music
Fibonacci did not actually discover the sequence himself. Born Leonardo Bonacci in 12th century Pisa, Italy, the mathematician traveled extensively around North Africa. There he learned how the Hindu-Arabic numbers 0-9 could be used to make calculations easier than the Roman numerals still used in much of Europe. Fibonacci explained his discoveries in a book called Liber Abaci, published in 1202, which had a section dedicated to the intriguing sequence that would be named after him hundreds of years later.
However, Indian poets and musicians had already been aware of the Fibonacci sequence for centuries, having discovered its implications for rhythm and various combinations of long and short beats.
Mozart used the golden section when writing a number of his piano sonatas. A sonata can be divided into two separate sections, known as the exposition (where the theme is introduced) and the development and recapitulation (where the theme is developed and repeated). In Mozart’s sonatas, the number of bars in the latter part divided by the former is about 1.618, the golden mean.
There is even a style of poetry inspired by the Fibonacci sequence, known as Fib Poetry. Each line of the poem must contain the exact number of syllables corresponding to the Fibonacci sequence. So a six-line poem would have syllable lines of 1, 1, 2, 3, 5 and 8.
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"Finally, I suspect that it is by entering that deep place inside us where our secrets are kept that we come perhaps closer than we do anywhere else to the One who, whether we realize it or not, is of all our secrets the most telling and the most precious we have to tell." Frederick Buechner
Come in! Come in!
"If you are a dreamer, come in. If you are a dreamer, a wisher, a liar, a Hope-er, a Pray-er, a Magic Bean buyer; if you're a pretender, come sit by my fire. For we have some flax-golden tales to spin. Come in! Come in!" -- Shel Silverstein
Thursday, September 04, 2008
How to fail a math test with dignity. Or: "How to keep your sense of humor during a presidential election."
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Number two reminds me of a final in third year calculus -- I misread a question and did a triple integration instead of double integration and got zero points. A male fellow student did the same thing and got extra points for greater difficulty!!
I hasten to add that this was so long ago that I could no longer do even a single integration -- whatever that is.
Gee, I don't think I ever studied "single, double or triple integration" in math class.
Now, poly sci was a completely different story.
This is why I stopped after Calc I!!!!
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Terence is a data enthusiast and professional working with top Canadian tech companies. He received his MBA from Quantic School of Business and Technology and is currently pursuing a master's in computational analytics at Georgia Tech.
You might not notice it in your daily life, but we actually rely on forecasted time series data all the time.
Have you checked tomorrow’s temperature to see if you’ll need a jacket? 🧥
Have you tried to estimate how many rolls of toilet paper you need to buy for the next month? 🧻
How about predicting the future price of your electricity bill? 🔌
All of these are examples of forecasted data in the real world. We discussed the basics in our previous article, How to conduct simple time series forecasting in SQL, but here’s a refresher: All time-series analyses involve predictions based on historical time-stamped data. As useful as it is for our day-to-day planning as individuals, it may not come as a shock that it’s just as useful for businesses, too.
In this article you’ll learn:
Why businesses care about forecasting
What exponential smoothing is
3 types of exponential smoothing
The guiding mathematical model
How to perform simple exponential smoothing in Python
Why businesses care about forecasting
Let’s start with a basic business-world example: Pretend you own a business that sells products on Amazon. Obviously, it takes time for inventory to be produced and shipped, so how can you prepare how much inventory you'll need to order?
There's really only one solution – you have to predict, or forecast, how much inventory you’ll want to produce to match demand. 🔮
Here’s the tricky part: If your company orders too much inventory and doesn’t sell it all, it loses money on the unsold merchandise – but if your company orders too little inventory, then it misses out on potential profits that it could have made if it ordered more.
This optimization problem is what makes time series forecasting so important, and this is why we’re talking about a specific time series modeling method called exponential smoothing.
What is exponential smoothing?
Exponential smoothing is one of the most widely used time series forecasting methods for univariate data, so it’s often considered a peer of (or an alternative to) the popular Box-Jenkins ARIMA class of methods for time series forecasting.
Exponential smoothing is similar to simple moving averages in the sense that it estimates future values based on past observations, but there’s a critical difference: Simple moving averages consider past observations equally, whereas exponential smoothing assigns exponentially decreasing weights over time. This means that exponential smoothing places a bigger emphasis on more recent observations, providing a weighted average.
Three types of exponential smoothing methods
There are three types of exponential smoothing models: simple exponential smoothing, double exponential smoothing, and triple exponential smoothing.
Single (or simple) exponential smoothing is used for time-series data with no seasonality or trend. It requires a single smoothing parameter that controls the rate of influence from historical observations (indicated with a coefficient value between 0 and 1). In this technique, values closer to 1 mean that the model pays little attention to past observations, while smaller values stipulate that more of the history is taken into account during predictions.
Double exponential smoothing is used for time-series data with no seasonality – but with a trend. This builds on the single exponential smoothing technique with an additional smoothing factor to control the influential decay of the change in trend, supporting both linear (additive) and exponential (multiplicative) trends.
Triple exponential smoothing, also known as Holt-Winters exponential smoothing, is used for time-series data with a trend and seasonal pattern. This technique builds on the previous two techniques with a third parameter that controls the influence on the seasonal component.
Basically, each technique advances the parameters from the previous techniques to handle additional factors and provide more accurate predictions. For the rest of this article, though, we’re going to focus on single (otherwise known as simple) exponential smoothing.
Simple exponential smoothing explained
A simple exponential smoothing forecast boils down to the following equation,
St+1 is the predictedvalue for the next time period
St is the most recent predictedvalue
yt is the most recent actual value
a (alpha) is the smoothing factor between 0 and 1
In simple terms, what this equation tells us is that the predicted value for the next period (St+1) is a combination of the most recent actual value (yt) and the most recent predicted value (St).
So, why should you care about the math behind the principle? Understanding this equation gives you a better intuition in setting the alpha, yielding a more accurate prediction. 🎯
As we briefly mentioned, the smoothing factor, alpha, is the only parameter in this equation that controls the level of smoothing. When alpha is closer to 1, more emphasis is placed on the most recent actual value (so, there’s less of a smoothing effect). When alpha is closer to 0, the level of smoothing is stronger and the prediction is less sensitive to recent changes.
Ready to put it to use in Python?
Simple exponential smoothing in Python
For this purpose, let’s pretend it’s January 2017 and you own a rental car agency. You’ve been open since January 1, 2013, so you have historical data on cars rented from January 1, 2013, to January 1, 2017. In this case, you have a dataset that lists the demand for all 1461 days in the 4 years you’ve been open next to its associated date, like so.
After plotting this data, you get a graph that looks something like this, with the demand plotted on the y-axis against the date:
If you wanted to forecast the number of cars that will be rented for the next week (January 2, 2017, to January 8, 2017), you could perform the time series analysis with exponential smoothing using the following steps:
Step 1. Import a method from statsmodel called SimpleExpSmoothing as well as other supporting packages.
Step 2. Create an instance of the class SimpleExpSmoothing (SES).
Step 3. Set your smoothing factor to 0.2 to take into account previous historical data.
Step 4. Fit the model to the data as shown.
Step 5. Forecast the next 7 days.
Putting that all together, your script should look like this:
After running the script and printing the output, your model gives you forecast values based on previous observations, telling you that the predicted number of cars that will be rented each day is ~15 cars.
Based on this result, you could assume that over the next week you’d expect to see an average of 15 cars rented each day, allowing you and your team to prepare for this upcoming demand and keeping your customers happy.
That’s it! Pretty simple, right? Now you understand how simple exponential smoothing works, why it’s useful, and how to implement it in Python.
Want to learn more SQL? Check out these articles next: 👇


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Eureka Math (also known as Engage New York) connects math to the real world in ways that build student confidence—while helping students achieve true understanding lesson by lesson.
It’s not enough for students to know the process for solving a problem; they need to understand why that process works so they can use it anytime. Eureka Math builds students’ knowledge logically and thoroughly to help them achieve deep understanding. While this approach is unfamiliar to those of us who grew up memorizing mathematical facts and formulas, it has been proven to be extremely successful and essential. The following resources will provide support for both parents and students as we continue to shift our teaching practices and incorporate different strategies that help children conceptualize the math they are doing.
How to Access Eureka Math Homework HelpFollow the the step-by-step directions below to access a site that provides video explanations of homework problems by grade level.
1. Click here to access the Math Homework Help Page. The screenshot shows you the page you will see. Once you are on this page, click on your child’s grade level.
After you click on your child’s grade level, the following page will appear. Now, select the module your child is working on.
3. Once you select the module, the following page will appear. You will now have access to the actual homework sheet and a video that explains how to do different types of problems found on the homework sheet for every lesson.
NOTE: You can also access a math newsletter for every topic. These newsletters are a terrific resource for both parents and students!
Eureka Math Parent Support Page
Click here to access a parent support page created by the authors of Eureka Math.


The Solaris clock turns telling time into a creative activity. Children can build their own clock using the Solaris components. It is a unique, multipurpose clock. There are several interchangeable faces - Arabic numerals, Roman numerals, divided into minutes and a sundial. These clock faces are commonly found in everyday life. Introducing the concept of time as a function of the sun and the rotation of the earth opens to the children an entirely new perception of time concepts and their relationships. The supplied compass is needed to orient the clock to North and introduces the concept of direction on the planet's surface (also known as "magnetic" North). The many applications of this truly unique material are a valuable addition to any classroom. Includes: one clock (37 x 31 x 10 cm), with a storage compartment for the clock faces, four interchangeable clock faces in different colors (diameter 27 cm), two removable clock hands, a sundial pointer, a compass, a whiteboard marker and instructions.
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A forex swap rate is a rollover interest for holding positions overnight in foreign exchange trading. Swap rates are released weekly by the financial institutions we work with and are calculated based on risk-management analysis and market conditions. Each currency pair has its own swap rate and is measured on a standard size of 1.0 lots (100,000 base units).
The latest swap rates on our trading platforms. Swap rates posted on our platforms are subject to change based upon currency interest rate and market volatility.
Trading 1 lot of EUR/USD (short) with an account denominated in EUR
Swap per lot for long position
Swap per lot for short position
Swap fee: (1*40.72) = $40.72
For forex, the Swap Calculator works as follows:
Swap = (Lots * Swap Rate * Number of Nights)


How is the plan upgrade or downgrade cost calculated?
At Scalenut, we understand that your business may need changes over time, and you may wish to upgrade or downgrade your plan accordingly. To provide flexibility and ensure a fair cost calculation, we use a Pro Rata Basis approach when determining the cost of plan upgrades or downgrades. This article aims to explain how the plan upgrade or downgrade cost is calculated, so you can make informed decisions about your subscription.
Plan Upgrade or Downgrade Calculation:
When you decide to switch from one plan to another, the cost is calculated based on the Pro Rata Basis. Let's break down the calculation process using an example:
Determine the Daily Rate:
To calculate the cost, we first determine the daily rate of your current plan. For instance, if you are on the Growth plan (Monthly) priced at $79, we divide the monthly cost by the number of days in the month. Assuming there are 31 days in the month, the daily rate would be $79/31 = $2.55.
Calculate the Consumed Amount:
Next, we calculate the amount you have consumed based on the number of days you have used the current plan. Suppose you decide to upgrade to the Pro plan (Annual) after using the Growth plan for 10 days. Multiply the daily rate ($2.55) by the number of days used (10), resulting in $2.55 * 10 = $25.5. Therefore, you have consumed a total of $25.5 from the initial $79.
Adjust the Plan Cost:
To determine the final cost of the new plan, we subtract the consumed amount from the total cost of the desired plan. The Pro plan (Annual) is priced at $1072.8 with a 40% discount. Subtracting the unconsumed amount ($53.5) from the total plan cost, we get $1072.8 - $53.5 = $1019.3
Final Cost After Upgrade or Downgrade:
After 10 days of using the Growth plan, the cost of upgrading to the Pro plan (Annual) would be $1019.3. This amount reflects the remaining cost of the new plan after deducting the consumed amount from the initial price.
We aim to provide transparent and fair pricing when it comes to plan upgrades or downgrades. Our Pro Rata Basis calculation ensures that you are charged proportionately based on the number of days you have used your current plan. By understanding how the cost is calculated, you can make informed decisions about upgrading or downgrading your subscription.


Exemplos. O exemplo a seguir usa Log10 o método para retornar o logaritmo de base 10 para os valores selecionados. The following example uses the Log10 method to return the base 10. This handy pack will assist you with this objective from the Year 2 Maths curriculum. Help children understand the numbers to 100 with cardinal representations like Base 10 and Numicon using the PowerPoint. Use the accompanying worksheet to draw Base 10 representations of numbers to 100. Base Ten Blocks Instructions Click on buttons to make blocks appear. Show problem to challenge yourself to match the correct number of blocks with a written number. You can adjust the number of columns to make the problem easier or more challenging. Drag your finger to create a box to connect ten blocks to make a larger block. And nearly every civilization has used base-ten math probably for the simple reason that we have ten fingers. If instead we lived in a cartoon world, where we would have only four fingers on each hand. The base-ten "two" 2 10 is written in binary as 10 2. A "three" in base two is actually "1 two and 1 one", so it is written as 11 2.
For example, the most common base used today is the decimal system. Because "dec" means 10, it uses the 10 digits from 0 to 9. Most people think that we most often use base 10 because we have 10 fingers. A base is usually a whole number bigger than 1, although non-integer bases.
Principe de la numération à base 10, notre manière classique de compter. Ce que disait Aristote de -384 à -322 Aristote prolonge son propos en énumérant la manière de représenter sur les doigts tous les nombres de 1 à 9 000.
There has been an acceleration in the war over the timelines involving artificial base 10 architecture that supports the holographic projections of Artificial Timelines used by the NAA to generate splits and inversions throughout the earth’s dimensional fields. What is Base 10? - Base 10 refers to the numbering system in common use around the world, that uses decimal numbers. Base 10 is the basis of a place value number system and is also called the decimal system or denary system. 09/01/2019 · If you understand the everyday decimal base 10 number system, then you already understand the ternary, base 3 counting and numbering system. You just don’t know you know yet. The complete lesson immediately follows the short semantics note about "ternary" versus "trinary". Ternary is. A função Math.log10 retorna o logaritmo de base 10 de um número,. O logaritmo de base 10 de um número. Caso o número seja negativo, NaN é retornado. Descrição. Caso o valor de x seja menor que 0, então o retorno será sempre NaN.
exchanging base-10 blocks Materials base-10 blocks: 2 flats, 30 longs, 40 cubes 2 dice Players 2 Base-10 Trading How to Play 1. Each player starts with 1 flat on the mat. 2. Take turns. On your turn, roll the dice. Return that many cubes to the bank. Make exchanges when needed. 3.. 02/04/2009 · Base-Ten Number System By: Miriam Coyle The National Council of Teachers of Mathematics states that “In prekindergarten through grade 2 all students should use multiple models to develop initial understandings of place value and the base-ten number system” p. 78. It also expects that “In grades 3-5 all students should. Exercice de maths mathématiques "Bases 10,2,3,8,12,16" créé par anonyme avec le générateur de tests - créez votre propre test ! Voir les statistiques de réussite de ce test de maths mathématiques Merci de vous connecter au club pour sauvegarder votre résultat. 1. Amazon's Choice for base 10 math. hand2mind Blue Plastic Base Ten Blocks, The Starter Kit for Elementary Math Manipulatives, Ages 8-11, Master the fundamentals of Place Value & Regrouping Set of 161 4.8 out of 5 stars 20. $26.95 $ 26. 95 $33.95 $33.95. Get it as soon as Tue, Sep 24. The logarithm base 10 that is b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e that is b ≈ 2.718 as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative.
Cube en re-plastique rouge encastrable de 10 x 10 x 10 cm. Cube des milliers illustrant l'espace 1000. Ce cube est compatible avec les centaines, dizaines et unités qui sont vendus séparément. Fabriqué en Allemagne en re-plastic plastique recyclé. Matériel éducatif de qualité pour les maths et base 10. Learning Resources offers a wide selection of Base Ten Blocks and Manipulatives Sets to help kids master math and counting. Shop by age and grade here. Base 10 blocks worksheet on regrouping. In these grade 1 math worksheets, students are given a number of blocks, regroup them into tens and ones and write the total as a two-digit number. This exercise gives practice in counting and emphasizes the meaning of the digits in a two-digit number. Free math worksheets from K5 Learning. This set includes a free printable Base 10 blocks. There is a hundreds board, Tens, and Single Unit blocks. Print on colored cardstock so children can easily identify the size of the unit. Mad Maths MinutesMad Maths Minutes Mad Maths Minutes Mad Maths Minutes 2-digit Base 10 Set A 2-digit Base 10 Set B.
The formula is 1 8 = 1 10 and 10 8 = 8 10. Everything else can be derived from that. If you have a sequence of base 8 digits you want to convert to a base 10 number, process them from left to right, keeping a total you initialize at zero. The following example uses the Log10 method to return the base 10 logarithm for selected values. using System; public class Examplepublic static void Main. "The base 10 log of 0 is 1.", _ number, Math.Log10number Next End Sub End Module ' The example displays the following output: ' The base 10 log of -1 is NaN. ' The.
Differentiated worksheets for addition using Base 10 materials. Includes a blank sheet for children to create their own additions with Base 10 jottings. Ideal for Year 2 or LKS2. Place Value with Base Ten Blocks MAB / Dienes Blocks Base 10 blocks are also called Dienes blocks, Multibase Arithmetic blocks MAB blocks or Place Value blocks.The four types of base ten blocks are named units, rods, flats and cubes. The unit cubes are worth 1, the long rods are worth 10, the flats areContinue reading "Place Value with. BaseTen. Number Pieces helps students develop a deeper understanding of place value while building their computation skills with multi-digit numbers. Students can: Use number pieces to represent multi-digit numbers, regroup, add, subtract, multiply, and divide. Use edge pieces to display the dimensions of arrays created with number pieces.
SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. It includes unlimited math lessons on number counting, addition, subtraction. for instance, the number 978345162 is formed using the base 10 numerals. Fun Facts. Base ten numerals form the basis for our counting system and thus.
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Have you ever wondered is there a way to compute without ever revealing the inputs? For example, you can use a mobile phone to navigate, without the server knowing where you actually are!
It is truly intriguing and has occurred to me several times. This problem, while easy to understand on the surface, turns out harder than I expected at first. I believe it's been studied extensively by crypto experts, but I know little about crypto so I don't even know the keyword to search with. I'll appreciate it if somebody could tell me something about this...
Nevertheless, I will describe my attempt below.
Intuitively: How to compute (evaluate a function) without the computer knowning the inputs?
One way to achieve this is to simply encrypt the input with a (public) key, and different users shall have different keys, so that the computer wouldn't have the chance to ever reveal the plaintext inputs. (I'd like to point out here, this is just one way I came up with upon, on which the rest of this article is based, and there might be other ways that I didn't think of.)
Simply put, given a function $f(x)$, an encryption algorithm $E(x)$, and a decryption $D(x)$, we would like to find a function $F(x)$, satisfying $D(F(E(x))) = f(x)$, where $E(x)$ and $D(x)$ depend on the user (the input provider). Since $E(x)$ and $D(x)$ should adopt the same encryption scheme, $D(E(x)) = x$ holds. Note that I intentionlly leave some details undefined.
It's easy to show that, if we could find a standardized procedure to find such a function $F$ for all $D(x)$ and $E(x)$, we would have solved this problem perfectly. However, this task is too terrifying, isn't it? I would be completely satisfied if I could find a solution when a specific encryption scheme is given.
Firstly, let's think about the trivial cases: what if $E(x)$ and $D(x)$ do absolutely nothing?
$$E(F(D(x))) = F(x) = f(x)$$
Not bad, no?
Similarly, if $f$ does nothing:
$$E(F(D(x))) = f(x) = x$$
We can just take $F(x) = x$.
1-bit, XOR, Not
Simple cases are not really simple, as you can build a Turing machine if it contains sufficiently powerful operations. I would like to know if we can build an adder with this "secure computation" thing. Let's start with NOT.
First consider only the symmetric XOR encryption, and let the variable $x$ to be boolean. The key is also boolean.
The original NOT function is defined as $f(x) = 1-x$, which negates the boolean variable. Simple truth-table computations show that, well, it's actually feasible! We can take $F(x) = x$.
The same reasoning can be applied to any $n$-bit vector easily.
2-bit, XOR, And
Now we've got NOT, let's see if we can solve for AND.
Here comes the first problem: how to represent a binary function? Easy, make the function take one argument which is a 2-bit vector, whose result is boolean. Unfortunately, this naive approach will get in the way of encryption, as encryption takes 2 bits as key, while the output only needs 1 bit, breaking the symmetry. But worry not, since both the key and the 1-bit vec can be padded to 2 bits, there is no problem.
Different from the 1-bit case, this time the truth table is much bigger. I didn't bother to check by hand, and just verified the assertion with Z3, which happens to support binary functions.
from z3 import * x1, x2 = Bools('x1 x2') k1, k2 = Bools('k1 k2') F = Function('F', BoolSort(), BoolSort(), BoolSort()) s = Solver() s.add(ForAll(x1, ForAll(x2, ForAll(k1, ForAll(k2, Xor(F(Xor(x1, k1), Xor(x2, k2)), k2) == And(x1, x2)))))) if s.check() == sat: print(s.model())
Since we pad consistently, the decryption only uses
k2. What if we
use asymmetrical XOR, where decryption uses an arbitrary function that
k2 together? Well, see for yourself, still no
solution. What if we relax the condition, and only uses 1-bit key?
While I strongly doubt the security and usefulness of such a crypto
system, unfortunately, no solution found. Code:
from z3 import * x1, x2 = Bools('x1 x2') k = Bool('k') F = Function('F', BoolSort(), BoolSort(), BoolSort()) s = Solver() s.add(ForAll(k, ForAll(x1, ForAll(x2, Xor(F(Xor(x1, k), Xor(x2, k)), k) == And(x1, x2))))) if s.check() == sat: print(s.model()) else: print('infeasible')
AND is just too powerful, I guess. :-/ Or maybe the universe just forbids us doing such kind of "secure computation"...
From the constraint $E(F(D(x))) = f(x)$, we can conclude $F$ can preserve some information from $D$, and the information about private key is cancelled out by $E$.
I worked on the maths (omitted) when $f(x) = x + c$, but I don't know how to cancel out the private bits.
Z3 to rescue
Z3 is a sophisticated SMT solver, which can solve large models in seconds. Z3 comes with its own Sexp-based DSL, but it's very quirky, so I used the Python binding. Well, it still sucks. Hey, maybe you could design a better DSL!
Generally speaking, we don't have to deal with arbitrarily large numbers, so I drove Z3 to check if there are any solutions when we only consider n-bit vector and don't specify a crypto (let z3 figure it out instead).
from z3 import * def check(sz, f): x, k = BitVecs('x k', sz) k1, k2 = BitVecs('k1 k2', sz) F = Function('F', BitVecSort(sz), BitVecSort(sz)) E = Function('E', BitVecSort(sz), BitVecSort(sz), BitVecSort(sz)) D = Function('D', BitVecSort(sz), BitVecSort(sz), BitVecSort(sz)) s = Solver() s.add(ForAll(x, ForAll(k, And(D(k, E(k, x)) == x, # (1) E(k, F(D(k, x))) == f(x))))) # (2) s.add(Not(ForAll(k1, ForAll(k2, ForAll(x, E(k1, x) == E(k2, x)))))) # (3) if s.check() == sat: print(s.model()) check(1, lambda x: 1-x) # f(x) = Not(x)
- Decrypting a encrypted message will return you the plaintext.
- What "secure computation" is all about.
- Rule out trivial cases, where
Ddon't depend on the key at all.
(Constraint 3 tells us coding z3 is not as easy as it seems. Z3 will happily accept your formulae, but the validity remains unchecked for your specific scenarios. You just don't get it right easily. Actually, there were some bugs in my first attempt, and they weren't discovered until I started writing this article. Z3 is indeed a great listener...)
Believe it or not, this piece of code can automatically solve the
problem at hand, giving you a solution when one is found, or reporting
unsat when no solution is possible. However, it can take centuries to
sz is large.
Running the script above gives back the simple case we analyzed
D(k,x) = E(k,x) = k^x, and
f(x) = not x. Magic, right?
There are two problems with this approach.
- The crypto is not flexible, as it comes "bundled" with the function
we want to evaluate. This could lead z3 to finding an
Dthat encodes the computation in itself with
Fbeing a no-op. Absolutely against our will.
- The crypto is symmetrical. To make it asymmetrical, we must write down the relation between a public key and a private key, in the language of first-order logic.
This is an (unfinished) attempt, on which I spent about two hours of
my spare time. There are still hundreds of problems yet to be solved
or even proposed. Like, we know
F must preserve information about
D, but how much? Which class of functions can have solutions?
What about RSA? What about AND? Maybe our universe just forbids us doing such kind of computation, but there might be other ways around, I don't know.
I believe this topic has been extensively studied. Garbled circuits seem to be relevant, but I haven't understood the Wikipedia page on that subject so I can't say for sure...


Author: Kevin Mekulu
- Monte Carlo simulation is one of the most important algorithms in quantitative finance
- Monte Carlo simulation can be utilized as an alternative tool to price options ( the most popular option pricing model is based on the Black-Scholes-Merton formula)
How Does Monte Carlo Simulation Work?
Before demonstrating the implementation of the Monte Carlo algorithm, it’s important to fully comprehend the science behind it. Simply put, Monte Carlo simulation generates a series of random variables that have similar properties to the risk factors which the simulation is trying to simulate.
The simulation produces a large number of possible outcomes along with their probabilities. In summary, it’s used to simulate realistic scenarios (stock prices, option prices, probabilities…).
Note: Monte Carlo simulations can get computationally expensive and slow depending on the number of generated scenarios.
Next, I will demonstrate how we can leverage Monte Carlo simulation to price a European call option and implement its algorithm in Python.
Pricing a European Call Option Using Monte Carlo Simulation
Let’s start by looking at the famous Black-Scholes-Merton formula (1973):
Equation 3–1: Black-Scholes-Merton Stochastic Differential Equation (SDE)
S(t) = Stock price at time t
r = Risk free rate
σ = Volatility
Z(t) = Brownian motion
Our goal is to solve the equation above to obtain an explicit formula for S(t).
We utilized Euler Discretization Scheme to solve the stochastic equation above. The solution is given by the expression:
Equation 3–2: Euler Discretization of SDE
Let’s apply the logarithm function to equation 3–2 above which will allow a faster implementation in Python (the vectorization process using the numpy package in Python would easily ingest the log version of the solution above).
Equation 3–3: Euler Discretization of SDE (log version)
Monte Carlo Implementation in Python
We will utilize the numpy package and its vectorization properties to make the program more compact, easier to read, maintain and faster to execute. The source code below is available here.
# Monte Carlo valuation of European call options with NumPy (log version)
from numpy import *
from time import time
# star import for shorter code
t0 = time()
S0 = 100.; K = 105.; T = 1.0; r = 0.05; sigma = 0.2
M = 50; dt = T / M; I = 250000
# Simulating I paths with M time steps
S = S0 * exp(cumsum((r – 0.5 * sigma ** 2) * dt
+ sigma * math.sqrt(dt)
* random.standard_normal((M + 1, I)), axis=0))
# sum instead of cumsum would also do
# if only the final values are of interest
S = S0
# Calculating the Monte Carlo estimator
C0 = math.exp(-r * T) * sum(maximum(S[-1] – K, 0)) / I
# Results output
tnp2 = time() – t0
print(‘The European Option Value is: ‘, C0) # The European Option Value is: 8.165807966259603
print(‘The Execution Time is: ‘,tnp2) # The Execution Time is: 0.9024488925933838
The graph below displays a plot of the first 10 simulated paths. Those simulated paths represent different outcomes for the price of the underlying asset (index level).
import matplotlib.pyplot as plt
Figure 3–1: Simulated index levels by time steps
Next, let’s investigate the frequency of the simulated index levels at the end of the simulation period.
plt.rcParams["figure.figsize"] = (15,8)
Figure 3–2: Histogram of all simulated end-of-period index values
Next, let’s look at the histogram of all simulated end-of-period option values.
import numpy as np
plt.rcParams["figure.figsize"] = (15,8)
plt.hist(np.maximum(S[-1] - K, 0), bins=50)
plt.xlabel('option inner value')
Figure 3–3: Histogram of all simulated option values
We notice something very interesting in the latter plot. The European call option expires worthless the majority of the time. This might be useful information for an astute options trader! But that’s a discussion for another day.
Hilpisch, Y. (2015). Python for Finance: Analyze Big Financial Data
Originally posted here


The California Mathematics Framework
In many of the discussions about changes in the math program in Palo Alto
schools, you may have heard references to the "Mathematics Framework." If
you are not familiar with that term, this article will provide a little
background on what the framework is and what it says.
What is the California Mathematics Framework?
The California Department of Education publishes a series of documents called
frameworks for each major subject area -- English, math, science, history,
etc. These documents describe the vision of the State Board of Education for
teaching these subjects in California public schools. The frameworks are
advisory documents -- there is no legal requirement that schools must follow
the frameworks. Similarly, there is no funding that is dependent upon a
district adhering to them.
The document dealing with mathematics, entitled Mathematics Framework for
California Public Schools, was last published in 1992, and is a major
revision of the previous edition, published in 1985. In many ways, the
framework concentrates more on how to teach mathematics than on what to teach.
When discussing issues of curriculum content, the framework makes frequent
reference to a document published by the National Council of Teachers of
Mathematics (NCTM) entitled Curriculum and Evaluation Standards for School
The Palo Alto school district has made the decision to follow all of the
framework documents, including the Mathematics Framework. Many of the
changes that have parents so concerned are exactly the changes being pressed
by the Mathematics Framework.
What does the Mathematics Framework say?
The framework analyzes mathematics education by dividing it into eight
"strands" which run through all grades levels: functions, algebra, geometry,
statistics and probability, discrete mathematics, measurement, number, logic
and language. Along with these strands, and viewed as tying them together,
the framework speaks of ten "unifying ideas" which vary from grade to grade.
These include such concepts as proportional relationships, patterns, multiple
The real thrust of the framework, however, is in its recommendation for
changes in the way that mathematics is taught. Some of its main
- It pushes for a major change in emphasis. This includes a de-emphasis on
practicing basic skills and an increase in "questioning", "writing",
"discussion", and "cooperative work".
- It declares that "calculators are the 'electronic pencils' of today's world
.. costing much less than students' textbooks," and recommends that four
function calculators be issued to all kindergartners. Once the students have
these calculators, there should be a decrease in rote memorization, so that
students in 2nd and 3rd grade can "learn the addition and subtraction facts
and some of the multiplication and division facts through their many
encounters with numbers."
- It strongly discourages tracking, counseling that schools should "Maximize
time spent by students in heterogeneous groups. Minimize time spent in
tracked or special-ability groups."
- It promotes a changed role for the teacher, saying that the new approach
"requires teachers to do more coaching and facilitating and less telling..."
- It discourages testing, and promotes portfolios, "authentic assessment,"
and "holistic scoring rubrics" in their place.
How do we know that these changes will work?
To be blunt, we don't. As the Framework itself states, "the ideas in this
Framework have not been implemented all together on a wide scale". There is
not even agreement on how to measure whether the Frameworks are succeeding or
not. Proponents of the framework argue that since we are de- emphasizing
memorization of basic skills, we should de-emphasize testing them. Critics of
the plan feel strongly that fluency with basic math skills is important and
must be tested.
What is the relationship of the framework to the education reform movement?
By any measure, the framework is a direct product of the education reform
movement. Along with embracing all of the reform movement themes, the
framework has a sweeping revolutionary tone in its call to throw out the old
and bring in the new. In its very first sentence it bluntly states:
"Publication of the 1992 Mathematics Framework for California Public Schools
answers a call ... to change what mathematics we teach, how we teach it, and
to whom." It then goes on to state "This Framework is devoted to change,"
and "The time for small scale experiments is past. We know what is
In addition, many of the changes advocated by the frameworks seem to follow a
much broader social agenda than that needed just to improve mathematics
education. "Mathematics instruction can be connected to society by a
conscious choice of contexts for problems that illuminate the mathematics side
of social issues." There is no mention of who decides on the "correct" answer
to these social math problems.
Finally, many of the recommendations given in the Framework seems to address
problems that are not pertinent to Palo Alto. "Progression through the
mathematics curriculum is an unfulfilled promise for the majority of
California's students." However, over 95% of all Palo Alto students complete
four years of high schools math. For 7th through 12th grade, it states that
"shallow courses, which emphasize repetition and low-level skills, must be
replaced...." However, there are no such courses in Palo Alto.
How does the CLAS test relate to the frameworks?
There is direct relationship between the mathematics framework and the now
defunct CLAS test. In simple terms, the CLAS test measures what the
framework teaches. Many reformers lament that falling test scores are not
meaningful because we are using old tests that don't measure all the good
things that the framework is supposed to teach. However, everyone realizes
that assessment drives curriculum, and therefore, the reform movement needed
a new test. This test would de-emphasize assessment of basic skills, and
emphasize issues dealing with mathematical "empowerment." The result was the
Can I get a copy of the Mathematics Framework to see for myself?
The frameworks are published in 8.5 x 11 softcover format by the California
State Department of Education. The Mathematics Framework runs just over 200
pages and may be ordered by calling 1-800- 995-4099. The cost is $6.75 plus
shipping. Credit card orders are accepted. Delivery is generally within 3
days to a week. (Ordering a copy of the frameworks by phone is one area where
Sacramento actually seems to work well.)


One of our readers asked,
I came from a background of small print projects. Now I’m tasked to find photography art for five doors, each 12 x 14 inches. When it comes to finding images, I don’t know where to start. What would be the best size? Best resolution?
For a start, there’s plenty of royalty-free images that you can find from Pexels, Pixabay, or Unsplash. Most photographs from these sources would have good lighting as well. Hence you just need to pick the ones which has enough pixels for your given print project.
Since our dear reader was looking for images to place on doors, I’m going to assume that people would view these images no closer than at an arm’s length. This would be about two feet or 24 inches.
Why does the distance matter? Image resolution is not just how many dots to put in a given inch (the dot per inch or pixel per inch metric), but also how far people would view them. The further the viewing distance, the less pixel density would be required such that people who view them doesn’t perceive the image as blurry, blocky, or otherwise be able to identify the individual pixels (dots) in the image.
What you need is a function that estimates the lower bound pixel density given an expected viewing distance. In other words, a formula that converts a distance in inches into a minimum dots-per-inch density value.
q = f(x)
- x — the expected distance between the viewer and the image, in inches.
- q — the minimum image resolution, in dots-per-inch or pixels-per-inch.
To derive the formula, first let’s draw a schematic of the problem. The image below shows a human eye (representing the viewer) that is looking at an image of height y at distance x. Now the angle that the eye makes between the bottom of the image and its top is ⍺. Let’s keep things simple and two dimensional for now, because making calculations for the width of the image would be exactly the same way, only on a different axis.
We know that magazines are usually printed at 300 dpi (dots per inch). This resolution is effective when a person with normal eyesight is viewing it on a standard reading distance, which is one foot (12 inches). This is probably also the reason why the first retina-display iPhone debuted at that approximate pixel density, which is 326 ppi (pixels per inch). Similarly a retina-display MacBook Pro has only 220 ppi but looks just as good because of the longer viewing distance. Of course, some people has better eyesights and can appreciate higher resolution images whereas some others can’t tell the difference between iPhone 4 and iPhone 3G (before retina display) screens. But just take the middle ground for simplicity’s sake.
Now let’s plug in those standard numbers into our model. Let’s take an image of 1 inch high which is viewed at a distance of 12 inches. The image contains 300 pixels, which is the minimum required to “look good”.
Then we need to find angle ⍺ that the eye makes between looking at the bottom and the top edge of the image. This would be useful to calculate the pixel per degree value later on. Using knowledge from high-school trigonometry, we get…
tan(⍺) = y/x tan(⍺) = 1 inch / 12 inch ⍺ = arctan(1/12) ⍺ ≅ 4.76°
We’ll define p as the pixel-per-degree ratio as follows:
p = q / ⍺
- p — the pixel-per-degree ratio
- q — the number of pixels (or printed dots).
- ⍺ — the angle which contains those pixels.
Our 1-inch image has the bare minimum of q=300 pixels (dots), making it a 300dpi image. Let’s plug those numbers in.
p = q / ⍺ p = 300 pixels / 4.76° p ≅ 62.98 pixels/degree
That means that for every degree of eye movement there need to be at least
p ≅ 62.98 pixels between them so that the eye can’t differentiate the individual pixels.
Now we want an arbitrary distance x to calculate how many pixels q that our 1-inch image can look acceptable. We solve the two equations above for the value of q.
p = q / ⍺ q = p * ⍺ ⍺ = arctan(y/x) q = p * arctan(y/x)
We already know the following values derived from our standard model of 1 inch image at a resolution of 300 pixels per inch viewed at 12 inch distance.
y = 1 inch p = 62.98 pixels / degree
Plug those numbers in and you get the formula to calculate the minimum resolution given an expected viewing distance. This is the formula that we’ve been looking for, the function to convert a viewing distance to a minimum pixel density value.
q = 62.98 * arctan(1 / x)
- q — the minimum resolution in pixels or dots per inch.
- x — the expected distance between the image and the viewer.
Then let’s go back to our dear reader’s initial problem. He/she would need to find images to fill 12 * 14 inch surfaces on doors. We’re assuming people won’t observe these doors closer than an arm’s length (or otherwise risking getting hit by the door if someone else is opening it from the other side), thus the viewing distance is about 24 inches (two feet).
q = 62.98 * arctan(1/24) ≅ 150 pixels per inch.
Given these we can calculate the minimum pixel dimensions of the images for the doors:
width = 12 inches * 150 pixels per inch = 1800 pixels height = 14 inches * 150 pixels per inch = 2100 pixels
Fortunately most images in Pexels, Pixabay, or Unsplash has more pixels than the minimum dimensions above and thus would look good to be printed on doors.
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|Our group brings tools from nonlinear dynamics, statistics, and decision-sciences to study problems in monsoon dynamics, climate dynamics, and climate change policy.
- Nonlinear climate dynamics:   An important research theme in our group is understanding monsoons from a dynamical systems perspective, using techniques in nonlinear dynamics, model-reduction, and time-series analysis.
- Statistical methods in climate:  We work on development and application of statistical methods, with ongoing work in spatial statistics, renewable energy integration, and learning directed graphs from data to understand climate teleconnections.
- Climate change economics and policy:  Related to global warming, we have studied origins of path independence between cumulative CO2 emissions and global warming and comparing mitigation of different climate forcers having different atmospheric lifetimes. We have integrated simple climate modeling and economics to study mitigation, and are collaborating on economics of climate change.


Beat Facebook is an application designed to calculate an approximate timeframe of when a given website will reach the current traffic levels of Facebook. This app uses historical data of website’s past growth (using Amazon’s Alexa data) and applies the exact patterns in several different geometrical and arithmetical algorithms and projections. Furthermore, this app is the most optimistic prediction app you will find - it assumes that any given website will repeat the past growth repeatedly until it beats Facebook. In reality only a few web sites, would be able to pull this off. Please use it as an entertaining tool only and do not make investment decisions based on projections of this app.