Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in biological, biomedical and biotechnology research. By describing the systems in a quantitative manner, their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter.
Such mathematical areas are calculus, probability theory, statistics, linear algebra, graph theory, combinatory, algebraic geometry, dynamical systems; differential equations are now being applied in biology.
Mathematical models are important tools in basic scientific research in many areas of biology, including physiology, ecology, evolution, toxicology, immunology, natural resource management, and conservation biology. These models are equations or formulas that can predict or describe natural occurrences, such as organism behavior patterns or population changes over time. The result obtained from analysis and simulation of system models are used to test and extend biological theory, and to suggest new hypotheses or experiments. Models are also widely used to synthesize available information and provide quantitative answers to practical questions.
Another field of mathematical biology is statistics which is known as biostatistics. Biostatistics deals with the application of statistics to a wide range of topics in biology. It is one of the disciplines of biological sciences concerned with the application of mathematical-statistical theory, principles, and practices to the observation, measurement, and analysis of biological data and phenomena. Biostatistics often involves the design of experiments in medicine, online pharmacy, agriculture, and fishery. It also involves collecting, summarizing, and analyzing the data received from those experiments as well as the decided results.


Annual percentage yield or APY is a measurement that tells you just how much money a deposit or investment with a fixed interest rate will make over a year. It can be a great tool for deciding where best to deposit your money. Of course, you could look at interest rates as absolutes, but that won’t give you an accurate picture of performance over a period, for example, a year.
That’s because of compounding. Supposing you invest an amount of money and receive monthly interest. Provided that you don’t withdraw anything, the interest now forms part of the interest-earning total, so you’re literally earning interest on interest.
Calculating Your Annual Percentage Yield
You can calculate your APY using a formula: APY= (1 + r/n )n – 1
In this formula, r is your interest rate per period and n is the number of periods you want to calculate APY for. But calculating your annual percentage yield can be as easy as using an online calculator.
What’s The Use Of APY?
If you’re choosing between options that pay a fixed interest rate periodically, looking at the percentage interest doesn’t give you the full picture because it doesn’t take compounding into account.
By using APY, you can determine exactly how much interest you will earn on your investment and be better able to choose the most profitable option. For example, a deposit that offers you 5% quarterly and compounds will be better than one that offers you 6% annually.
Since you would like to grow your savings, APY can be compared to inflation so that you know whether your money is gaining or losing value over time. If your money buys less because of inflation, an investment can actually become worth less in real terms even if it’s growing in absolute terms. Remember to factor this in when making decisions.
Fixed interest rate investments are reliable in that you know what to expect from them, and the risk is low. But if using them actually makes your overall worth lower, you will know that it’s time to look at alternatives that might be riskier, but will also have the potential to generate inflation-beating returns. You don’t know for sure till you do the math or use the calculator to understand the numbers.
Make The Most Of Your Cash
Even if you’re already an investor who generates greater value through a balanced stock portfolio, you will need to have some easily available cash in interest-bearing accounts. Placing it where it earns the highest APY helps you to get the most value from it.
At times when stocks and shares are volatile, you might decide to leave some of your investment funds in an interest-bearing account while you wait to see which way markets are headed. 2022’s market volatility, for example, calls for investors to be cautious when redeploying resources. And that, in turn, implies finding a good place to leave your cash until you can decide whether investments are likely to gain or lose value.
Investment is a long-term way to grow your money, but carries risk. Interest-bearing accounts are generally slow growers, and inflation can outpace them, but they’re safe options and may actually gain value provided the APY outperforms inflation. Do the calculations to choose the smartest options.


Yes sir. Installation of ocaml and texvc was among the first things that I tried. I installed and compiled texvc myself and modified $wgtexvc accordingly. It didn't work.
In the interim, I found a work-around. Instead of Math extension, I used SimpleMathJax (MathJax was deprecated a while back). It is definitely not the best Math rendering engine around, but at least for now it works. One downside I found is sometimes SimpleMathJax renders an equation twice.
Meanwhile, I will keep working on the Math extension. I'm sort of a LaTex person.
No matter what, I think Bitnami Mediawiki is an amazing product. It saved a lot of my time. Keep up the good work, guys.


With Clockwyz, you can estimate the amount of time you think it will take to complete a task. When you or a team member log in hours on a task you would be able to compare how long it took to complete the task to how long you thought it would take. You can also plan how long it will take to complete the project, as a whole, by estimating how long each individual task will take.
This makes it easier to project when you would reasonably be able to complete your goal. You can input the estimated time a task will take by clicking on a task and entering your estimation under "Estimated Time".


The 100th day of school is literally the 100th day of the school year. More importantly, it is a great way for elementary school teachers to celebrate the various mathematical concepts that can be taught using the number 100. From the very first day of school, classes begin keeping track of the number of days they’ve been in school in anticipation of the 100th day. It’s this anticipation that is actually the first math lesson, as the days are often marked using coffee stirrers or Popsicle sticks, ten of which become a “ten bundle,” paving the way to counting by tens and ones. (verywellfamily.com)
Today our 1st graders will be dressing up like they are 100 years old! What a fun way to practice and learn counting!


Probability For Dummies
Every so often someone asks a question like, "What's the probability of getting at least one 6 rolling two dice?" A common answer is 1/3, but this is not correct. The thinking here is that there's a 1/6 probability of rolling a 6 on each die, and 1/6 + 1/6 = 1/3. So I thought I'd talk about it.
Let's start with the basics.
The probability of any event occurring ranges from 0 (will not occur) to 1 (will occur). Probabilities can be expressed as a fraction, a decimal value, or a percent (which really just a decimal times 100).
The most common kinds of randomness in games are rolling dice and drawing cards from a shuffled deck. In both of these cases, the results are independent. This is a very important concept.
An independent event occurs without relation to any other event. For example, the results of any die roll are not affected by any past die rolls. This means that if you roll ten 6's in a row, the probability of rolling another 6 is still 1/6.
The other basic assumption is that the randomizing mechanism is fair. In the case of a 6-sided die, it is assumed to be balanced in such a way that the probability of each result is equal. This does not mean that if you roll 1,2,3,4,5 that the next roll will be a 6. It means that over a very large number of rolls, the number of each will be very close. The assumption also holds for cards. A well-shuffled deck implies that each card has an equal chance to be on the top.
To calculate the probability of a single fair event, divide the number of desired outcomes by the number of possible outcomes.
For example, the probability of rolling exactly a 3 on a d6 is 1 (one outcome in question) divided by 6 (six possible outcomes) making 1/6. This is probably intuitive for most people.
The probability of rolling an odd number is 3 (1, 3, 5) divided by 6, which is 3/6 or 1/2. This true of any 3 possible outcomes (rolling an even number, rolling 3 or less, rolling 2 3 or 5, etc.)
Now for more complex stuff.
The probability of multiple independent events ALL occurring is the product of their individual probabilities. However, you must be careful with the exact meaning of what you are looking for.
What is the probability of rolling two 6's on 2d6? The probabilities for each are 1/6. 1/6 * 1/6 = 1/36 (or 2.77%). Easy!
What is the probability of rolling 3 or more (both times) on 2d6? The probabilities for each are 4/6 (or 2/3). 2/3 * 2/3 = 4/9 (or 44.44%).
What is the probability of rolling a 6, then rolling an even number? 1/6 * 1/2 = 1/12 (or 8.33%).
What is the probability of rolling a 1 and a 5? This is the kind of question where you can easily go wrong. Do you mean rolling a 1 first, and rolling a 5 second? If so, it's 1/6 * 1/6 = 1/36 (or 2.77%). If you mean rolling two dice and having a 1-5 appear then what you really mean is rolling a 1 then a 5, or rolling a 5 then a 1. This leads us to:
The probability of one of several independent events occurring is the sum of their individual probabilities.
In the previous example, the probabilities of both 1-5 and 5-1 rolls are 1/36, so the probability of either of them occurring is 1/36 + 1/36 = 2/36 or 1/18 (or 5.55%).
What is the probability of rolling an odd/even pair on 2d6? This is a similar question. The probability of rolling odd then even is 1/2 * 1/2 = 1/4. The same for even-odd. So the total is 1/4 + 1/4 = 1/2 (or 50%).
Now let's kick it up a notch.
What is the probability of rolling 9 or more on 2d6? Rather than iterating all the probabilities for 9's, 10's 11's and 12's, you can do this by making a table:
You can see that 10 of the 36 results are 9 or more. And, as you guessed, the probability is 10/36 or 5/18 (or 27.77%).
What is the probability of rolling doubles (not triples) on 3d6? This one is tricky. Let's break it down by first examining case one: the first two rolls match and third does not. The first roll can be anything (probability 1). The second roll must match the first (probability 1/6). The third roll must not match (probability 5/6, do you see this?). So to total probability for case one is 1 * 1/6 * 5/6 = 5/36.
The second case is the same as the first one, but rearranged: any roll, no match, match first roll. Again we have 1 * 5/6 * 1/6 = 5/36. Likewise with the third case: anything, no match, match second roll = 1 * 5/6 * 1/6 = 5/36.
The final answer is the sum of the three separate cases: 5/36 + 5/36 + 5/36 = 15/36 (41.66%).
What is the probability of rolling three completely different numbers on 3d6? This one should be easy by now. The first number can be anything (1), the second must be anything but the first (5/6), and the third must be anything but the first two (4/6 or 2/3), resulting in 1 * 5/6 * 2/3 = 10/18 or 5/9 (or 55.55%).
Interesting note. Look at the previous two examples. One is for none of the three rolls to match, and other is for two of the rolls to match. The only other possibility is 3-of-a-kind. 1 (36/36) - 5/9 (20/36) - 15/36 = 1/36, which is the answer! The sum of the probabilities of all possible outcomes is 1.
What is the probability of drawing 2 aces from the top of a shuffled standard deck of cards? In this case, the second drawn card is going to be affected by the first drawn card, because the number of cards remaining in the draw pile has been changed. There are 4 aces in the deck and 52 cards. The probability of the first card drawn being an ace is therefore 4/52 or 1/13. Now what's left? There are 3 possible aces out of 51 cards: 3/51 or 1/17. So the answer is 1/13 * 1/17 = 1/221 (or 0.45%).
Now back to the original question. What's the probability of getting at least one 6 rolling two dice? Can you do it? There are two cases: 6-anything, and anything-6. However, notice that these two cases overlap. To make them distinct, we need to change the wording a bit: 6-anything, and not6-6. Case one is 1/6 * 1 = 1/6. Case two is 5/6 * 1/6 = 5/36. The final answer is 1/6 (6/36) + 5/36 = 11/36 (or 30.55%)!


What is the significance of calculus in everybody’s career?
Calculus is very essential part of study life because it makes a student success in many disciplines like physics, engineering, math, computer science, banking and many more. If a student good in the part of math, then he/she can get good marks in his/her academic career. They choose their aim and then choose accordingly. In business, economics, banking, finance, medicines calculus plays a great role. So it is better to learn calculus from its basic and Adacsoc provide the facility of online tutoring that helps the students to become strong in their week points and also in their poor subjects. The free calculus tutor is also available so that you can ask you doubts from them without any hesitation. Calculus plays an important role in the life of every person.
The features provided by the online calculus courses
- Online calculus helps the students to learn calculus part easily by providing new tips and tricks to solve the problems.
- Free calculus tutor provide the best faculties that you ever find. The tutors are very helpful and cooperative and they do interaction with the children. They behave as they are the friends of the children.
- Tutors provide the best quality of video related to study as a study material and also make them do the homework at home.
- The online tutors are available for 24×7 to address the queries the students asked and give proper solutions as soon as possible.
- The online free calculus tutor gives personal attention to each and every student individually and encourages them to grow their confidence level.
Why an online calculus tutor scores heavily?
Online tutors hire only those tutors who are very well known and reputed persons and goal oriented. While choosing the online tutor a person has to go through various stages and then finally some people are selected and some are not. The selected people are very talented and they have the capability to teach you in an easy manner so that the students will not face any problems. The online tutors are given the instructions to give maximum to the students. Online tutoring helps to save your time and also try to make you confident and increase your communication skills. There should be a strong bond between the student and the teacher so that there will not be the chances of occurring problem.
How we can understand the calculus easily?
The online calculus are either provided free of cost or they charges some nominal fees. The Adacsoc free online tutoring charges very less fee so that it can be affordable by every person and they can enjoy the learning strategy from their online tutoring. The tutors provided by the Adacsoc breaks the complex problems and make it easy to understand and also easy to practice. They provide all the instructions step by step and they first teach all the fundamentals to the students so that they become strong in the basic parts of the calculus.


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What is chemometrics?
Chemometrics means performing calculations on measurements of chemical data. This can be anything from- calculating pH from a measurement of hydrogen ion activity to computing a Fourier transform interpolation of a spectrum.
More recently, the common usage of the word refers to using linear algebra calculation methods, to make either quantitative or qualitative measurements of chemical data, primarily spectra. Nearly all trained spectroscopists have the basic understanding of the concepts necessary to apply these methods. Unfortunately, like all specialty areas of science, chemometrics has a language all its own that makes it difficult for the beginner to understand.
The science of chemometrics gives spectroscopists many different ways to solve the calibration problem for analysis of spectral data. Some are very simple to understand, while others require a strong background in linear algebra. However, they all have one thing in common: they each solve an individual problem but do not address ALL possible problems.
Some methods have the advantage of being simple to understand but may not be very robust for all possible samples. Others are very complex to understand and implement, but give solutions that are very stable and can handle a large variety of “unknowns.”
The key to understanding chemometrics is in not necessarily understanding the mathematics of all of the different methods; it is to know which model to use for a given analytical problem and properly applying it.
Application of Chemometrics
- To explore possible outliers and indicate whether there are patterns or trends in the data
- PCA is an important part of chemometrics and provides the most compact representation of all the variation in a data table
- Exploratory algorithms such as principal component analysis (PCA) are designed to reduce large complex data sets into a series of optimized and interpretable size
- To predict related properties that are easier to measure
- The goal of chemometric regression analysis is to develop a model which correlates the information in the set of known measurements to the desired property. Chemometric algorithms for performing regression include partial least squares (PLS) and principal component regression (PCR)
- Chemometric regression is extensively used in making decisions relating to product quality in the on-line monitoring and process control industry where fast and expensive systems are needed to test
- To assign to predefined categories to samples and predicting an unknown sample as belonging to one of several distinct groups
- A classification model is used to predict a sample’s class based on closest examples. K-nearest neighbour (k-NN) is primarily used in chemometrics
- Chemometrics helps in standardizing data. The classification models are more reliable and include the ability to reveal unusual samples in the data
Various chemometric methods are applied for the analysis of complex data in a number of applications viz.
- Spectroscopic calibrations
- Process modeling for optimization
- Process models for monitoring and fault detection
- Dynamic model identification for process control
- Multivariate statistical process control
- Process analytical instrument standardization
- Analytical instrument design and development
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Understanding Payout rates
Payout rates are statistical results that give players the idea of the probability of winning in a particular game. It could also relate to the probability of losing while pursuing for the big win. This is commonly expressed in percentage; the figure in the casino payout rate designates how much of the total wagered amount in a casino is returned to the players on the average.
Online casino provides monthly reports to an independent testing body like GLI or eCOGRA, which releases a percentage payout report. This can be access through the links at the bottom of the online casino web page or in the help section and/or in the fairness policies. These test results are available publicly and shows the average returns to the players who joined in that specific game type during the previous month.
RTP or % Return to Player
On every particular game there is the RTP which is a useful indicator most commonly seen in online slots, though this can also be found occasionally in other games. They function in the same manner as payout percentages but are calculated differently through a simulation of the games. In the slots it is best to check the variance between the possibility of winning and the chance of losing.
House Edge: Understanding
The terminology house edge refers to games that are affected by a player’s strategy. One example is the blackjack. With RTP, it shows the probability and some indicator for player’s knowledge when to bet and where to place their bets. In these cases it pays to be skillful in the game.


These classes are intended to review and reinforce foundational information that students may be missing in a part or fully virtual environment or may have missed last year.
Classes are appropriate for students who have previously completed the math class or who are curently taking the class.
The first hour will be spent teaching a core concept for the appropriate math class.
The last 30 minutes will be designated for completion of homework, asking questions and/or completing practice problems provided in order to reinforce the concept.
If the tutor identifies a gap in knowledge that is concerning, they will communicate that to the parents and work on a remediation plan so that it can be addressed before the student is overwhelmed.
Location: Vickery Village


Math Attacks – iPhone / iPod game
Number infested meteors are about to hit and destroy the Earth. Is your math good enough to save it?!! Try to save the world in over 400 levels.
Use your mental arithmetic to add, subtract, multiply and divide numbers in order to solve the number that destroys the dangerous meteors.
Math Attacks is a learning game for children age 4 and up, that will have fun practicing their arithmetic skills.
Math Attacks has progressive levels, becoming challenging even for grown ups at more advanced levels.
Math Attacks has 4 different game plays, each of them becomes progressively more challenging:
“Classic” – start with simple addition of two small numbers and by level 100 you get to master addition, subtraction and multiplication of three and more numbers.
“SubZero” – starts also easy and gets more difficult with each level, however it involves also among positive numbers also negative ones.
“Zen” – Never fail scenario. It has the same game play as classic, however meteors do not crush into Earth, therefore the player can take his/her time to solve the equations.
“Blitz” — it is also a progressive a game play, however it starts with many meteors threatening Earth, making the player compute faster and faster to solve equations.
All this arithmetic fun for free, with In-App purchase you can get unlimited levels of entertainment and learning.
The game play is very simple and intuitive, just ONE tap to select meteors for calculations in order to solve the given number. Math Attacks has another approach as most of the game out there, instead of having “2 + 2 =?” it encourages the player to find different ways of calculating 4.
Math Attacks has also Game Center enabled therefore you can see how good are your math skills around the world
Posted on May 3, 2012, in children, game, ios, math and tagged apple, arcade, children, division, explosion, fun, game, ios, iphone, ipod, learning, math, meteors, subtraction, sum. Bookmark the permalink. Leave a comment.


It’s time for another daily prompt! The Daily post asks: Which subject in school did you find impossible to master? Did math give you hives? Did English make you scream? Do tell!
Oh, I’m telling. I didn’t like Math. Math and I were natural enemies. Although, not yet proven, I’m positive I started agonizing about it since the womb. There was a bit of a tension break during Grade 1, but that was probably because I was trying to fit in (skipped Kindergarten). Afterwards, Math showed me no mercy, this was the type of subject that would touch your opened wounds, and laugh until you cried in despair. Grade 2 was the year I knew for sure there was a hell; there had to be, Math pushed my brain through the most torturous methods, fractions. Yes, I was good at them, but I honestly refused to complete the work. I remember staying after school and doing the homework I failed to do (such a Math rebel). My teacher was a great person for allowing me to do that, otherwise I would’ve failed the subject (she probably smelled my immaturity). I sat on the desk while my teacher graded papers, and everytime she looked, I would meet her with a big smile. Inside, I was fuming and plotting my escape; through the window, digging a tunnel, crawling under the desks. In the end, I decided to finish the work.
I loved every other subject, but Math was the enemy that wouldn’t die. I even contemplated the idea of dressing up as Math one Halloween. I dreamed of being able to summon magical powers and make it disappear, but then I was told I would always need to use it, so what the heck, I gave it another try. Big mistake. Math became more challenging and a true school pestilence. Middle school were the years I curled up in a ball during class trying not to go into a psychotic outbreak. Grade 9 was the year I tried to get the work done and out of the way, and that was only because my Math teacher was gorgeous; here’s a Math lesson for you, teen+ridiculously good looking teacher = oh yes, you’d do anything to get praised by Mr. Cutie Pie. However, even Sir Striking McGorgeous couldn’t save me from the evil Math. We had to do more complicated equations and I just broke apart. I got a C+ on my quiz and Mr. Guapo (that’s spanish for ‘handsome’) told me he was disappointed. That was it. Math did it. Math crushed my daydreams. Okay, fine, it was MY fault, but I still want to blame Math for it, so do not ruin my moment :p.
I do not know how I survived the rest of High School Math. I think I just went into a trance every time I entered the classroom, or I somehow managed to Astral-project during classes. Perhaps, I was kidnapped by aliens and they finished my classwork for me. I think at one point, I transformed my Geometry shapes into little monsters. I do not know. But, I have to say that it did help me get awards in Information Technology, so there’s a positive note.
Luckily, I didn’t have to be a master of Math in college (computers did most of the work), so there isn’t much to say about those years. But before those years, Math was an allergic reaction I had no medication for. I am extremely thankful I enjoyed all of my other classes, and I think that balanced my war with Math. Today, Math and I have reached an agreement to be acquaintances. It only comes when I need it to. Will it last?
❤ ❤ ❤ Love, Ellie ❤ ❤ ❤


The infinite geometric series 1 + q + q2 + q3 + …, plays a vital role in almost every department of mathematics. In this post we propose a geometric building of this sequence, from which different components of the sequence’ convergence may be analyzed graphically. Especially, it is actually revealed how the Restrict sum of the collection, 1 /(I — q), varies Along with the quotient q, as well as the behaviour of this sum close to the singular details q = l and q= — one is studied from our building. The construction is based over the substitution nine = cos a (a any genuine angle except 0° and a hundred and eighty°), which transforms the sequence into a trigonometric id.Allow $n$ be an integer and $W_n$ be the Lambert $W$ perform. Permit $log$ denote the natural logarithm making sure that $delta=-W_n(-log2)/log2$. On condition that $a$ and $r$ geometric series are respectively the first term and the continuous ratio of an infinite geometric series, it really is proved the Restrict of convergence on the geometric sequence is $displaystylelim_ntopminftyabig[r^delta-1big]big[r-1big]^-1$ the place $rneq1$. By implementing the geometric collection formulation above, it can be even more proved which the harmonic collection $zeta(1)$ is provided by $zeta(1)=-2big[log2+W_n(-log2)major]$ and as $nrightarrowpminfty$, the value of $zeta(one)$ grows really slowly but surely towards $tildeinfty$, confirming the divergence in the harmonic series.In case you multiply any phrase by this value, you end up getting the worth of another expression.The typical ratio is often positive or negative. It may be a whole number, a fraction, or even an irrational range. Regardless of what worth it’s got, it would be the ratio of any two consecutive conditions while in the Geometric Sequence.
Geometric sequence, in mathematics
An infinite series with the forma + ar + ar2 + ar3+⋯,the place r is known as the common ratio. An easy example is the geometric collection for a = 1 and r = 1/two, or1 + 1/2 + one/four + 1/eight +⋯,which converges into a sum of 2 (or 1 if the initial expression is excluded). The Achilles paradox is really an example of The problem that ancient Greek mathematicians experienced with the concept that an infinite series could make a finite sum. The confusion around infinity didn’t abate until the 18th century, when mathematicians formulated Assessment and also the notion of limits.Graphical illustration of an infinite geometric seriesClearly, the sum of your square’s pieces (1/2, 1/four, 1/8, and so on.) is 1 (sq.). Hence, it can be witnessed that one would be the Restrict of this sequence—that is definitely, the worth to which the partial sums converge.The sum of the primary n terms of a geometrical collection is equivalent to a(one − rn)/(1 − r). If absolutely the value of r is lower than 1, the series converges to a/(1 − r). For every other value of r, the collection diverges.a geometrical sequence is really a sequence with a relentless ratio amongst successive conditions. One example is, the sequence is geometric, for the reason that Each and every successive expression is usually obtained by multiplying the previous phrase by one/two.Geometric sequence are between the simplest samples of infinite sequence with finite sums, Though not all of them have this property. Historically, geometric collection played a significant job inside the early development of calculus, plus they proceed to become central in the research of convergence of series. Geometric collection are employed all through mathematics, and they have got critical apps in physics, engineering, biology, economics, Personal computer science, queueing concept, and finance.
Geometric building with the geometric sequence
The infinite geometric collection one + q + q2 + q3 + …, plays a crucial purpose in virtually every department of mathematics. In this article we recommend a geometric design of the sequence, from which several aspects of the sequence’ convergence might be studied graphically. Particularly, it is actually shown how the limit sum in the sequence, 1 /(I — q), varies Along with the quotient q, plus the behaviour of the sum near the singular details q = l and q= — 1 is analyzed from our design. The construction is predicated around the substitution nine = cos a (a any actual angle except 0° and 180°), which transforms the collection into a trigonometric identification.Allow $n$ be an integer and $W_n$ be the Lambert $W$ purpose. Permit $log$ denote the normal logarithm in order that $delta=-W_n(-log2)/log2$. Given that $a$ and $r$ are respectively the initial expression as well as constant ratio of the infinite geometric series, it can be proved the limit of convergence of your geometric series is $displaystylelim_ntopminftyabig[r^delta-1big]big[r-1big]^-1$ where by $rneq1$. By applying the geometric collection formula higher than, it’s further proved the harmonic collection $zeta(one)$ is supplied by $zeta(one)=-2big[log2+W_n(-log2)huge]$ and as $nrightarrowpminfty$, the worth of $zeta(1)$ grows incredibly slowly toward $tildeinfty$, confirming the divergence of your harmonic collection.In the event you multiply any time period by this price, you end up with the value of the subsequent term.The common ratio is usually favourable or detrimental. It may be a whole selection, a portion, as well as an irrational variety. Regardless of what benefit it’s, it would be the ratio of any two consecutive phrases within the Geometric Sequence.


WPAFB, Ohio (WDTN) – 300 area STEM students participated in National Engineer’s Week at the Air Force Institute of Technology.
The field trip put the students’ love of math and science to work.
“We hope that it’s a lot of fun. We hope that it doesn’t feel like a day of work for them. We hope that they come to this and they can say this is what I want to do every day for the rest of my life,” said Major Christina Rusnock, USAF.
Students spent the day in demonstrations ranging from quadcopters to building Lego robots to building bridges.


Smart Snacks Stack and Count Layer Cake Children Add comments Apr 072017 A stackable mathematical treat! Layer by layer this engaging set encourages gross motor development and early math and counting skills. Sale Price: $ 18.99 Read More and Buy it here! Share this:Click to share on Twitter (Opens in new window)Click to share on Facebook (Opens in new window)Like this:Like Loading... Related Leave a Reply Cancel reply This site uses Akismet to reduce spam. Learn how your comment data is processed.


ACQUIRING THE confidence to approach complex problems strategically
Courses in the mathematics department range from fundamental algebra through Advanced Placement calculus and statistics to post-AP linear algebra, differential equations, and multivariable calculus.
Critical thinking and problem-solving skills are essential at every level of mathematics, and faculty instruct with lessons that emphasize student activity and collaboration. Students gain the confidence to approach complex problems strategically and graduate prepared to be successful in the continued study of mathematics at the college or university level.
“In my AOF math class, I have learned how to apply my quantitative skills to real-life situations.”
- Ryan Anderson ’22
Math courses are a significant part of the online program, and many students take advantage of the opportunity to further their study online in order to advance in the math curriculum during the academic year. Avon’s online courses are taught by AOF faculty.
Post-AP Course of Study
The math curriculum extends beyond the Advanced Placement program to offer courses in Linear Algebra, Differential Equations, and Multivariable Calculus on a rotational basis. In this way, talented math students are able to challenge themselves with topics of study that are typically featured at the collegiate level.
Math Enrichment Hour
Enrichment hour offers a great opportunity for math students to work through additional math problems with each other and with the teacher. The extra repetitions do help. A math student might also use enrichment hour to review an old assessment with his teacher to better understand any mistakes or to refine his approach to studying the material for future assessments. In this way, a student might see enrichment hour as a chance to extend the conversation about his studies beyond the classroom.


What is the Difference between Profit Margin and ROI?
When it comes to calculating your profit, two different metrics can guide you: “ROI” (return of investment) and “profit margin“.
- Profit Margin: It is a profitability ratio that measures the amount of net income earned with every dollar of sales generated.
Profit margin can be calculated with the following formula: Profit / Revenue.
Here, the cost and fees should be considered to calculate the correct profit margin.
Let’s make it more clear with an example. You buy a product for $30 and want to sell it for $100. Amazon cuts of 15% fee, which means $15 goes to Amazon. As a result, the profit is $55. (Revenue ($100) – Cost ($30) – Fees ($15))
By calculating the profit, we can find the profit margin: Profit ($55) / Revenue ($100) = 55%
- ROI (Return of Investment): ROI measures the return on investment against the investment cost. It tells how much you gain for a single dollar you spend. The higher ROI, the better.
Return of investment can be calculated by the following formula: Profit / Cost
Based on the previous example, let’s assume the profit is $55, whereas the cost is $20. The ROI will be: Profit ($55) / Cost ($20) = 275%
Don’t be confused with all these numbers and formulas.
Let Eva calculate for you and help you on your way to having better profits.


Operators in Swift can be:
Unary, which requires only one operand or input.(e.g. -a)
Binary, which requires two operands or inputs. (e.g. a+b)
Ternary, which requires three operands. (e.g. ?:).
Swift Assignment Operator
The assignment operator initializes or updates a variable.
var name = "John Doe"
name = "Dave Mathias"
Swift Comparison operators
The comparison operator returns a Boolean true if the statement is true or a Boolean false if the statement is not true.
Swift Arithmetic operators
The arithmetic operators perform the four basic mathematical operations: add, subtract, multiply, divide.
Swift Remainder operator
The remainder operator calculates the remainder if the first operand is divided by the second operand.
var num1 = 20, num2 = 3
num1 % num2 // remainder is 2
Swift Increment, decrement and Compound operators
The increment and decrement operators are shortcuts to increment or decrement a variable by 1.
The compound assignment operators combine an arithmetic operator with an assignment operator.
Swift Logical operators
Logical AND operator returns true if both operands are true, otherwise it returns false.
The logical OR operator returns true if either of the operands is true.
The logical NOT operator inverts a Boolean value.
var bool1 = true, bool2 = false
bool1 && bool2 // false
bool1 || bool2 // true
!bool1 // false
Swift Ternary operator
Ternary operator can be used as a short form for if else conditions.
The ternary conditional operator assigns a value to a variable, based on the condition in first parameter.
boolean expression ? value if true : value if false
var x = 10
var msg = x > 0 ? "positive" : "negative or zero"
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Optimal Control of Markov Processes with Incomplete State
For every stationary Markov process in the first sense, there is a corresponding stationary Markov process in the second sense. The chapter reviews equivalent Markov processes, and proves an important theorem that enables one to judge whether some class of equivalent non-cut-off Markov processes contains a process whose trajectories possess certain previously assigned properties. Hidden Markov models - Traffic modeling and subspace methods Andersson, Sofia LU ( 2002 ) Mark Faculty of Engineering, LTH (1) Faculty of Science (1) 2021-04-24 A Markov process is a random process in which the future is independent of the past, given the present. Thus, Markov processes are the natural stochastic analogs of the deterministic processes described by differential and difference equations.
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for Mathematical SciencesLund University2 Department of Mathematical SciencesNorwegian Title: Lum 8 2016, Author: Lund University, Name: Lum 8 2016, Length: en smärtsam process att lyfta denna problematik på arbetsplatsen, men pollen data: Gaussian Markov random field models for compositional data”. Numerical discretisation of stochastic (partial) differential equations. David Cohen Atomic-scale modelling and simulation of charge transfer process and photodegradation in Organic Photovoltaics Mikael Lund, Lunds universitet Fredrik Ronquist. Introduction to statistical inference. Bayesian phylogenetic inference and Markov chain Monte Carlo simulation.
CV Jennifer Alvén
The following is an example of a process which is not a Markov process. Consider again a switch that has two states and is on at the beginning of the experiment. We again throw a dice every minute. However, this time we ip the switch only if the dice shows a 6 but didn’t show MIT 6.262 Discrete Stochastic Processes, Spring 2011View the complete course: http://ocw.mit.edu/6-262S11Instructor: Robert GallagerLicense: Creative Commons The prototypical Markov random field is the Ising model; indeed, the Markov random field was introduced as the general setting for the Ising model.
CV Jennifer Alvén
Thus, Markov processes are the natural stochastic analogs of the deterministic processes described by differential and difference equations. They form one of the most important classes of random processes. Any (Ft) Markov process is also a Markov process w.r.t. the ﬁltration (FX t) generated by the process. Hence an (FX t) Markov process will be called simply a Markov process. We will see other equivalent forms of the Markov property below. For the moment we just note that (0.1.1) Deﬂnition of a Markov Process † Let (›; F) be a measurable space and T an ordered set.
Lund SLE Research Group
Markov Decision Processes. The Markov Decision Process (MDP) provides a mathematical framework for solving the RL problem. Almost all RL problems can be modeled as an MDP. MDPs are widely used for solving various optimization problems.
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A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). In mathematics, a Markov decision process (MDP) is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision maker. MDPs are useful for studying optimization problems solved via dynamic programming.MDPs were known at least as early as the 1950s; a core 6. Linear continuous Markov processes 7.
Remark on Hull, p. 259: \present value" in the rst line of
Abstract Let Φ t, t ≥ 0 be a Markov process on the state space [ 0, ∞) that is stochastically ordered in its initial state. Examples of such processes include server workloads in queues, birth-and-death processes, storage and insurance risk processes and reflected diffusions. Markov process whose initial distribution is a stationary distribution. 55 2 Related work Lund, Meyn, and Tweedie () establish convergence rates for nonnegative Markov pro-cesses that are stochastically ordered in their initial state, starting from a xed initial state. Examples of such Markov processes include: M/G/1 queues, birth-and-death
A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as " memorylessness ").
Thus, Markov processes are the natural stochastic analogs of the deterministic processes described by differential and difference equations. They form one of the most important classes of random processes 2014-04-20 2005-10-25 2019-02-03 Textbooks: https://amzn.to/2VgimyJhttps://amzn.to/2CHalvxhttps://amzn.to/2Svk11kIn this video, I'll introduce some basic concepts of stochastic processes and Markov Processes And Related Fields. The Journal focuses on mathematical modelling of today's enormous wealth of problems from modern technology, like artificial intelligence, large scale networks, data bases, parallel simulation, computer architectures, etc. For every stationary Markov process in the first sense, there is a corresponding stationary Markov process in the second sense.
A Markov chain is a discrete-time process for which the future behaviour, given the past and the present, only depends on the present and not on the past. A Markov process is the continuous-time version of a Markov chain. Clear, rigorous, and intuitive, Markov Processes provides a bridge from an undergraduate probability course to a course in stochastic processes and also as a reference for those that want to see detailed proofs of the theorems of Markov processes. It contains copious computational examples that motivate and illustrate the theorems.
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Kemister jobb i Lund Lund lediga jobb
Since the characterizing functions of a temporally homogeneous birth-death Markov process are completely determined by the three functions a(n), w + (n) and w-(n), and since if either w + (n) or w-(n) is specified then the other will be completely determined by the normalization condition (6.1-3), then it is clear that a temporally homogeneous birth-death Markov process X(t) is completely Research Portal. Find researchers, research outputs (e.g. publications), projects, infrastructures and units at Lund University For this reason, the initial distribution is often unspecified in the study of Markov processes—if the process is in state \( x \in S \) at a particular time \( s \in T \), then it doesn't really matter how the process got to state \( x \); the process essentially starts over, independently of the past. 2021-04-24 · Markov process, sequence of possibly dependent random variables (x1, x2, x3, …)—identified by increasing values of a parameter, commonly time—with the property that any prediction of the next value of the sequence (xn), knowing the preceding states (x1, x2, …, xn − 1), may be based on the last Lindgren, Georg och Ulla Holst. "Recursive estimation of parameters in Markov-modulated Poisson processes". IEEE Transactions on Communications.
Efficient Monte Carlo simulation of stochastic hybrid systems
† Let K be a collection of subsets of ›. Thus decision-theoretic n-armed bandit problem can be formalised as a Markov decision process. Christos Dimitrakakis (Chalmers) Experiment design, Markov Decision Processes and Reinforcement LearningNovember 10, 2013 6 / 41. Introduction Bernoulli bandits a t r t+1 Figure: The basic bandit process CONTINUOUS-TIME MARKOV CHAINS Problems: •regularity of paths t7→X t. One can show: If Sis locally compact and p s,tFeller, then X t has cadl` ag modiﬁcation (cf.
Almost all RL problems can be modeled as an MDP. MDPs are widely used for solving various optimization problems. In this section, we will understand what an … De nition 2.1 (Markov process). The stochastic process X is a Markov process w.r.t.


Description: *W2 Candidates Only - Cannot Support C2C*
• Collaborate with data scientists/engineers/stewards on data extraction and data cleaning.
• Design, implement and optimize OR/simulation/machine learning models based on structured and unstructured data.
• Work closely with the software engineering team in the delivery of analytic software Work in a highly interactive, team-oriented environment.
• PhD in Mathematics, Economics, Statistics, Operations Research or a related field with statistics courses/publications/proven expertise (Electrical/Mechanical/Environmental Engineering, Spatial Statistics, Geoinformatics, Environmental Modeling, Engineering, Computer Science, or Computational Biology) or Master's degree plus 3 years of industry experience.
• Proficient in machine learning algorithms and concepts
• Ability to work in a highly interactive, team-oriented environment.
• Comfort in communicating technical content in an easy to understand way
• Proficiency of at least one of Python or R is a must-have and working knowledge of the other
• Proficiency in Spatial and/or Temporal Statistical Modeling
• Proficiency in Machine learning algorithms and concepts (Ensembles, Deep Learning, SVM, etc.)
• Experience working with agricultural/biological scientific data is highly desired
• Drive for translating business problems into research initiatives that deliver business value Creativity in defining challenging exploratory projects
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Alper Orhon Award
Alper Orhon was a faculty member of the Boğaziçi University Department of Economics from 1968 to 1974. He played an instrumental role in the departmental decision to offer Mathematical Statistics and Econometrics, both of which later became requisite courses. Orhon also coordinated the creation of the first electronic database and program library project.
Alper Orhon Econometrics Award is given in commemoration of Alper Orhon, to students in recognition of their achievements in the courses of Mathematical Statistics and Econometrics.
Students who received the award in previous years:
|2021||Nezihat Bengisu Kaynar||3.91|
|2020||Ahmet Erdem Çetintaş||3.95|
|2019||Sadık Hüseyin Gül & Hasan Alperen Tosun||3.93 & 3.68|
|2018||Nazlı Billur Görgülü||3.76|
|2011||Ali Batuhan Barlas||3.25|
|2010||Hüsnü Çağrı Dalgıç||3.78|
|2009||Sina Teoman Ateş||3.72|
|2008||Mustafa Mete Karakaya||3.79|
|2002||Natig Mustafayev (Business Administration Double Major)||3.66|


This is a really cool article on dental plaque preserved on ancient dead people.
But in addition to the aforementioned cool article, here's what you really wanted to know: Calculus is not only a branch of math but another word for dental plaque. Why? Ah, I'm glad you asked. (Prepare for seemingly long non sequitur that's actually totally sequitur. Sequitur sequitur sequitur.)
The first rudimentary "taxi" meter was created by engineer Vitruvius. Actually his full name was (probably) Marcus Vitruvius Pollio, but for some reason everyone in the olden days went by one name. Like Adam.
So anyway, as carts-for-hire moved forward, Vitruvius rigged it so that previously-counted stones would be dropped at a fixed distance via a turning wheel. The number of stones missing from the original lot were counted and the fare derived. Calculus is the mathematical study of change, where change is very difficult to measure.
For example, say you're standing on a cliff and you want to haul up a long and heavy rope and at the same time you want to figure out how much work you did to pick it up. Because this could totally happen.
So anyways, you're on the cliff and you're like, I know! I'll just figure out how much the rope weighs and then figure out how high I lifted it and BAM, have my answer. But if you calculated it this way, your answer would be wrong. Or better said, it it would be imprecise and here's why: As you're pulling the rope up toward you, some of it is being put on the ground behind you. Which means the work your doing minute to minute is decreasing because the rope is becoming lighter as you're pulling up less of it.
Calculus, then, is the process of setting up the equation so that you can check in with the rope at different times. Maybe you do the math by checking in every minute you are pulling the super-long rope up the super-high cliff. So you've got how much the rope weighs when it's THIS LONG over one minute. And then THIS LONG over one minute. And then THIS LONG over one minute. Only, even then your math would be off because since the rope is continuously moving, even when you break down the length of the rope into shorter segments, you're going to be over-estimating the work as the rope was getting even shorter over the period of that minute. So then you have to re-do it checking in at 30 seconds, but then you're still wrong. And so you check in every 10 seconds, and then five and then one -- always ending up with an imprecise answer. Eventually you set up the number of times you check in to infinity and it all works out, so now you know everything you need to know about calculus, except why it shares a name with the stuff on your teeth.
Calculus is another name for stone. A pile of stones decreasing in number over time as they did in the taxi, is similar to how calculus problems are set up and so Calculus was named. Hardened plaque is also "like stone" and, according to the article, it seems to live on infinitely. So, there you go.
This has been another questionably necessary blog post brought to you by yours truly.
P.S. I wrote this because so many people take calculus and are never told, in simple, straight forward language, what they're actually doing. It's super-important to learn the principles of something before trying to learn the details. When are more people going to get this and adjust?
P.P.S. The end.


Rates are used by people daily, like when they work 40 hours per week or make interest annually at a financial institution. You may always seek the advice of your pals. In the event the answer is no, then 24 isn’t a multiple of 5.
Thus, go on and buy your favourite classroom math games and make your classroom an enjoyable place! Explore various choices for assignments Algebra is a subject where students can explore a whole lot of different things and develop different assignments and so forth. Recent studies reveal that a 4-year college education will most likely cost you somewhere in the selection of 96,000 to 200,000 based on the place you attend.
The One Thing to Do for Terms in Mathematics
Two terms are believed to be structurally, literally, or syntactically equal should they correspond to the exact same tree. In the majority of cases, if you find a lowercase letter (e.g. p), it is a statistic. It was initially published in 1945.
The folding line is known as the line of symmetry. The concept is the fact that it is possible to bring the infinite number of derivatives and think of a single finite sum. In case the term is only a normal number, decimal, or fraction, with no letters on it, it is known as a Constant.
Be aware that the order of terms in the last answer doesn’t impact the correctness of the solution. http://www.enjmin.fr/ If you don’t agree to the terms set out in the ToS you might not use the site. It will wind up a great deal easier in case you attempt to understand it in practical terms, rather than theory.
Eventually, you’re likely to need to learn to advertise and sell your work to other folks. Imagine you could choose that football field up, and set it anywhere in the air which you like. You can do life exactly the same way.
Mathematics is used to make the elaborate programming at the core of all computing. MATH 3303 isn’t allowed. Occasionally, even only a mathematics teacher.
The level of a polynomial is the maximum power of the variable in that polynomial, so long as there is just one variable. Truly if you would like to learn calculus, and you currently have a good handle on algebra, then you’ll end up comfortable with this more advanced subject. In the event the monomials contain the precise same variables, and the exact same variables in each monomial are raised to the exact powers then they’re deemed similar monomials.
These projects need deep thought and much effort, perhaps past the capability of one individual. The usage of continuous assessment is frequently an advantage, since in real life you’re tested on the standard of the job you are able to produce, and the ability to do that work quickly and under pressure. Add to locate the perimeters of polygons on such worksheets.
There are 3 powerful change agents which will eventually facilitate and force significant changes in our math education system. The usage of digital technologies in the main mathematics classroom is not an alternative. The evaluation of training programs are certainly the most significant step in the training procedure.
What Does Terms in Mathematics Mean?
Students would then need to work out the patterns and identify what’s common in the factors and the way to check whether a number is a prime and so forth. These pages list the integrals of several unique functions. Knowing the number e on the flip side, was.
The significance of mathematics is two-fold, it’s important in the development of science and two, it’s important in our comprehension of the workings of the universe. A kid’s psychology is extremely delicate. You have to observe how the apparently abstract character of mathematics is just one of the reason behind its power, because it enables the exploitation of analogies between apparently diverse circumstances.


We're comparing the motivation reasons for different subgroups in this section.
We split the entire set of entries into two subsets and then apply Student's t-test
to find out if significant differences exist.
Confidence = 1 minus P-value
Motivation levels are converted to numbers by the following rules: "Weakly motivating" = 1 point, "Motivating" = 2 points, "Strongly motivating" = 3 points, "It discourages me" = -3 points.
Note that some responses are ignored to create the more significant gap between the two groups and to equalize the group sizes.


GEO 5210 – Seismology I L14-1
Lecture 14 – Seismic Inversion Part I
1. A brief introduction to Inverse Theory
Inverse theory in seismology has many applications. Some of the most important are:
1) Earthquake location
2) Determining 1D velocity and density structure of the Earth
3) Seismic tomography – i.e., 3D velocity variations.
Many problems in the physical sciences can be written in the form of a linear equation:
Where we define:
A = model/data kernel. This provides the relationship between model parameters and data.
m = model vector. Contains the unknown model parameters.
d = data vector. Contains our observations.
Example: Straight line fitting
A simple example would be temperature measurements (Ti) made as a function of depth (zi) in the Earth.
Looking at these data we see what looks to be a linear increase of temperature with depth. So, let’s try and describe our ith temperature measurement as follows:
That is, we want to find the equation of a straight line that best fits our observations. We can write this in matrix form:
GEO 5210 – Seismology I L14-2
Where our data vector (d) contains our temperature measurements (T1, T2, …, T5), our model vector (m) contains the unknowns (our slope – b and intercept – a), and the model kernel (A) shows the relationship we are assuming between our data and unknowns.
Just multiplying out the first term of our matrix above we can see that this represents the relationship we are looking for:
If the system above was evenly determined (i.e., we had the same number of equations and unknowns) then we could easily find the solution of 𝐀𝐀𝐀𝐀=𝐝𝐝 by just solving for m. That is:
We can see why this is called inverse theory, because to find our solutions (our model parameters) we have to find the inverse of the model kernel.
But, our example problem had 2 unknowns and 5 observations. This is often the case that we encounter, i.e., we usually have more observations than unknowns – this is referred to as an overdetermined system.
Recall that a matrix A is invertible if there exists a matrix B such that: AB = BA = I. Non-square matrices are not invertible. Hence, in our example above we have a 5 × 2 matrix, which is not square and not directly invertible.
Traditionally, what is done is to look for the solution that provides the best fit in the least squares sense. Consider the next image:
GEO 5210 – Seismology I L14-3
Here we show our observed temperature readings (red circles) and a possible line fit to these observations. The black cross shows the predicted (model) value for this line at each depth. What we calculate is the residual. Where we define the residual = observed value – model value.
The least-squares solution provides the best-fit where we define the best-fit as the line for which the sum of the squared residuals has its least value. That is, we seek to minimize:
Where ri is the ith residual.
We set up our equations as:
The solution by least squares can be determined from multiplying each side of the above equation by the transpose of A: 𝐀𝐀𝑡𝑡 (see your linear algebra text for proof).
Now we solve for m:
Note that in our example A is a 5 × 2 matrix. So, At is a 2 × 5 matrix. So, multiplying 𝐀𝐀𝑡𝑡𝐀𝐀 we get a 2 × 2 matrix, which is square, and possibly invertible.
Underdetermined systems are also possible. That is, we have more unknowns than equations. If we try the least squares solution we find that 𝐀𝐀𝑡𝑡𝐀𝐀 is singular (i.e., it has no inverse). There are an infinite number of solutions. That is, if we try to fit a straight line to a single data point, any line will fit through that data point:
GEO 5210 – Seismology I L14-4
This is unfortunate because sometimes parts of problems we are trying to solve may be underdetermined.
The traditional approach to handling this is to look for the simplest solution. That is, we search for the model values that have the shortest length in the least squares sense. Specifically we seek to minimize:
In this case, the solution is:
To make this clear, let’s take a look at an example. Suppose we have just one measurement:
T1 = 5° at z1 = 10 m.
So, we set up our matrix equation as:
Solving 𝐦 =𝐀𝐀𝑡𝑡(𝐀𝐀𝐀𝐀𝑡𝑡)−1𝐝𝐝 we get:
GEO 5210 – Seismology I L14-5
⇒𝑎𝑎=5101=0.0495 (the intercept)
⇒𝑏𝑏=50101=0.495 (the slope)
And our solution length is:
So, we get a straight line passing through our point that looks like:
But is this solution really of any interest to us?
The problem in geophysics is that we often encounter cases where parts of our problem are over-determined and other parts are under-determined. So, typically we try to seek a compromise between minimizing the prediction error (the residuals) and the solution length.
Damped Least Squares
The damped least squares solution is:
Where λ is a damping factor.
We try to choose a value of λ that minimizes both the prediction error and solution length.
• If λ2 is large the emphasis is on minimizing the solution length.
GEO 5210 – Seismology I L14-6
The effect of a large damping factor is to produce a smooth model (i.e., we have a small model length) but not necessarily a great fit to the data.
• If λ2 is small the emphasis is on minimizing the residuals (prediction error).
The effect of a small damping factor is have very small residuals but a very rough model (and a very large solution length).
We typically try to find a balance between these two end-member cases. For example, we can try a range of values for λ and produce a trade-off curve as shown below. Here we try to find the optimal solution as a balance between model roughness and model misfit. Calculating model roughness can be done in a variety of ways, for example by looking at the 2nd derivative of the model values.
GEO 5210 – Seismology I L14-7
But, one must always be cautious of these solutions. The optimal model doesn’t necessarily imply it is really the Earth we are looking at.
The figure below is an example of the effect of damping in a seismic tomography model. This is from the S-wave tomography model S20RTS by Jeroen Ritsema. It shows a cross-section through the Earth (cutting through the central Pacific LLSVP). Here we can see the effect that choosing a different damping parameter can have on the resulting model. On the left we have a heavily damped model which gives us a very smooth model. Whereas on the right we have less damping that results in a much rougher model and stronger perturbations in velocity.
2. Tau-P Inversion
Previously we noted that travel-time vs. distance or ray parameter vs. distance plots do not produce singularly valued functions. To get around this limitation we generate a new function called a Tau-p or τ(p) curve.
GEO 5210 – Seismology I L14-8
Setting up the inverse problem:
Recall from a previous lecture that we defined Tau(p) as follows:
𝜏𝜏(𝑝𝑝)=2(𝑢𝑢2−𝑝𝑝2)12 𝑑𝑑𝑑 𝑧𝑧𝑝𝑝0
We can write this instead for a stack of discrete layers:
That is, we have measured τ at a number of discrete ray parameters (p1, p2, p3, …, pm). And we decide to parameterize our Earth model as a series of n-layers, each with thickness h:
Let’s think about this problem in the sense of straight-line fitting segments of our travel-time vs. distance curve:
GEO 5210 – Seismology I L14-9
That is, what if we approximate the real travel-time curve (blue), by a series of straight line segments (red segments). Each one of these segments has a ray parameter p, and let’s assume that the slowness is u. This is strictly only true if we are looking at refracted arrivals (headwaves – see the aside at the end of this lecture), but is a good first approximation in any case, and is true in the limit that our straight line segment lengths go to zero.
Let’s consider a few values of this equation.
j = 1
Recall that in our setup we assume that u1 = p1, u2 = p2, etc. Hence our first term (𝑢𝑢12−𝑝𝑝12)=0. Since we have increasing velocity with depth, that means that u2 < u1. And u1 = p1, so that u2 < p1. Thus, our second term (𝑢𝑢22−𝑝𝑝12)12 is undefined (hence, the condition that 𝑢𝑢𝑖𝑖>𝑝𝑝𝑗𝑗).
So, 𝜏𝜏(𝑝𝑝1)=0. This makes sense since our first straight-line segment passes through the origin.
j = 2
The second term is zero here, so:
j = 3
Just considering the non-zero terms:
So, it should be obvious by now that we can write this in the form of a matrix:
GEO 5210 – Seismology I L14-10
Which is a matrix equation of the form:
Which can be inverted for the layer thicknesses (h).
3. 1D Seismic Velocity Models
The first 1-D models of Earth’s seismic velocity structure were set up by Jeffreys and Bullen. Their 1940 JB model consisted of 7 layers labeled A through G as follows:
Deepest Depth (km)
The layer names A through G are no longer used. But, it is interesting to note that early on the lowermost part of the D layer was observed to be different, hence the D layer was split into the D’ and Dʺ layers. Dʺ is still used for the lowermost part of the mantle. Another interesting feature was the F-layer, or the Core transition zone. This was dropped in future models, but now researchers are putting forth further evidence that there is a transition region from the outer- to inner- core, and hence the F-layer is making a comeback.
More recent developments are:
1981 – PREM (Preliminary Reference Earth Model) – by Dziewonski and Anderson.
Note that these models typically invert for seismic velocities as polynomial functions in different bands of the Earth as opposed to a stack of layers as we discussed above.
GEO 5210 – Seismology I L14-11
1991 – IASP91 (International Association of Seismology and the Physics of the Earth’s Interior) – by Kennett and Engdahl.
This model primarily improves the seismic velocities at the mantle transition zone.
1995 – AK135 – by Kennett, Engdahl, and Buland.
The main difference in model AK135 from IASP91 is in the velocities near the boundary of the inner core.
The next figure shows the difference between the JB40 model and the IASP91 model. As can be seen the variations aren’t that dramatic.
GEO 5210 – Seismology I L14-12
From Stein & Wysession.
The only other planetary body for which we have 1D seismic velocity profiles is the moon. Recall that we installed seismometers on the moon during the Apollo missions:
Apollo 12 Alsep (Apollo Lunar Surface Experiments Package)
GEO 5210 – Seismology I L14-13
Several competing models currently exist. But, one of the best models is by Nakamura 1983. Note that these models do not resolve structure down to the lunar core.
Model Nakamura 1983. Plot by Yao Yao (green – S-wave velocity; red – P-wave velocity)
4. Density Structure
Models like PREM also include density as a function of depth. How is this determined? The answer can be seen by looking at the Adams-Williamson equation.
Assume that the Earth is made of infinitesimally thin spherical shells with uniform properties.
We know at depth that pressure (P) is given by:
g = gravitational acceleration.
ρ = density
h = depth
If we consider a change in pressure ΔP with increasing depth Δh:
GEO 5210 – Seismology I L14-14
And, in the limit Δℎ⟶0
That is the change in pressure with increasing depth is +ρg. Going the other way, with increasing radius:
⇒𝑑𝑑𝑑 𝑑𝑑𝑑 =−𝑔 (𝑟𝑟)𝜌𝜌(𝑟𝑟)
That is, the pressure decreases with increasing radius.
We also know that the acceleration of gravity at radius r is (just equate Newton’s second law with Newton’s law of universal gravitation):
Mr = Mass within radius r.
⇒𝑑𝑑𝑑 𝑑𝑑𝑑 =−𝐺𝐺𝑀𝑀𝑟𝑟𝑟𝑟2𝜌𝜌(𝑟𝑟)
From the chain rule:
𝑑𝑑𝑑 𝑑𝑑𝑑 =𝑑𝑑𝑑 𝑑𝑑𝑑 𝑑𝑑𝑑 𝑑𝑑𝑑
⇒𝑑𝑑𝑑 𝑑𝑑𝑑 =−𝐺𝐺𝑀𝑀𝑟𝑟𝜌𝜌(𝑟𝑟)𝑟𝑟2𝑑𝑑𝑑 𝑑𝑑𝑑 (Eqn 1)
Recall that bulk modulus (k) (or Incompressibility) is a materials ability to resist being compressed under a pressure equal in all directions.
Bulk modulus = k VVP−ΔΔ=,
Where P is the Pressure (Force/Area) applied to a volume of material given by V. Graphically, if we apply an equal Pressure (P) on all sides…
GEO 5210 – Seismology I L14-15
Then the box deforms as follows…
Where the dashed lines show the volume of the original, uncompressed, box.
For small volumes:
𝑘𝑘=𝑑𝑑𝑑 𝑑𝑑𝑑𝑑−𝑉 ; (𝐸𝐸𝐸𝐸𝐸𝐸 2)
Noting that we have a minus sign so that with increasing pressure we get a smaller volume.
Since we know that 𝜌𝜌=𝑚𝑚/𝑉 we can write:
Carrying out the derivative:
⇒𝑑𝑑𝑑 𝑑𝑑𝑑𝑑=−𝑚𝑚𝑉 2
And substituting back in 𝜌𝜌=𝑚𝑚/𝑉 :
GEO 5210 – Seismology I L14-16
⇒𝑑𝑑𝑑𝑑𝑉 =−𝑑𝑑𝑑 𝜌𝜌
Now, if we substitute this into Eqn 2:
⇒𝑘𝑘=𝑑𝑑𝑑 𝑑𝑑𝑑𝑑−𝑉 =𝑑𝑑𝑑 𝑑𝑑𝑑 𝜌𝜌=𝜌𝜌𝑑𝑑𝑑 𝑑𝑑𝑑
⇒𝑑𝑑𝑑 𝑑𝑑𝑑 =𝜌𝜌𝑘𝑘
And substituting this back into Eqn 1:
⇒𝑑𝑑𝑑 𝑑𝑑𝑑 =−𝐺𝐺𝑀𝑀𝑟𝑟𝜌𝜌(𝑟𝑟)𝑟𝑟2𝑑𝑑𝑑 𝑑𝑑𝑑 =−𝐺𝐺𝑀𝑀𝑟𝑟𝜌𝜌(𝑟𝑟)𝑟𝑟2 𝜌𝜌(𝑟𝑟)𝑘𝑘
Now, if we recall our seismic wave speeds:
P-wave velocity = 𝛼𝛼=𝑘𝑘+43𝜇𝜇𝜌𝜌
S-wave velocity = 𝛽𝛽=𝜇𝜇𝜌𝜌
And substituting back into our previous equation for the rate of change of density with respect to radius:
GEO 5210 – Seismology I L14-17
⇒𝑑𝑑𝑑 𝑑𝑑𝑑 = −𝐺𝐺𝑀𝑀𝑟𝑟𝜌𝜌(𝑟𝑟)𝑟𝑟2 𝜌𝜌(𝑟𝑟)𝑘𝑘=−𝐺𝐺𝑀𝑀𝑟𝑟𝜌𝜌(𝑟𝑟)𝑟𝑟21𝛼𝛼2−43𝛽𝛽2
If we let 𝜑𝜑=𝛼𝛼2−43𝛽𝛽2, then we arrive at the Adams-Williamson equation:
𝑑𝑑𝑑 𝑑𝑑𝑑 = −𝐺𝐺𝑀𝑀𝑟𝑟𝜌𝜌(𝑟𝑟)𝑟𝑟2𝜑𝜑
This equation gives us the density gradient with respect to radius. Unfortunately, density is also on the RHS of this equation, which is what we are interested in solving for. We already know the seismic velocities however in great detail (see previous lecture!). So, how do we solve this problem?
What we do is an iterative approach. That is, we start at the Earth’s surface at a known density:
𝜌𝜌(6371 𝑘𝑘𝑘 )=3300𝑘𝑘𝑘 𝑚𝑚3⁄
We know the Mass of the Earth (Mr) and the seismic wave speeds, so we can now calculate the gradient 𝑑𝑑𝑑 𝑑𝑑𝑑 . Hence, with this gradient we can linearly extrapolate the density to a new smaller radius:
Now, with this new density, we can calculate a new Mr at the new radius, and repeat. Hence, it is an iterative process. Remember, we also have important constraints that must be satisfied:
• Total Mass of the Earth: ME = 5.974×1024 kg.
• Mass at each successive level: 𝑀𝑀𝑟𝑟=𝑀𝑀𝐸𝐸−4𝜋𝜋∫𝜌𝜌(𝑟𝑟)𝑟𝑟2𝑑𝑑𝑑 𝑅𝑅𝐸𝐸𝑟𝑟
• Moment of inertia constant: 𝐼𝐼𝑚𝑚𝑎𝑎2=0.3307
There are a couple problems with direct implementation of this approach that also need to be accounted for.
GEO 5210 – Seismology I L14-18
• This assumes adiabatic compression. But, convection is happening in the mantle so an extra term accounting for convection must be included.
• This approach also assumes that the composition is constant throughout the Earth. This is readily accounted for by introducing density jumps at compositional boundaries.
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PartitionReturnPlaywright Ira Hauptman Producer Indra's Net Theater [Website] Description Shiinavasa Ramanujan said that his mathematical discoveries were given to him in his dreams by a Hindu Goddess. Where ever they came from, they continue to astound mathematicians with their fluidity and depth today, a hundred years after his death. This play is a fantasy based on the life of this legendary Indian clerk who came up with mathematical theorems that rivaled those of any mathematician in history. And with a little dance, and a visit from a mathematical ghost…it can’t be such a bad night.
Running Time 1:50 Performers Alan Coyne [Hardy]
Heren Patel [Ramanujan]
Avanthika Srinavasan [Namagiri]
Marco Aponte [Fermat]
David Boyll [Billington] Creative Staff Bruce Coughran [Director]
Robert "Bo" Golden [Scenic Designer]
Lisa Claybaugh [Costume Designer]
Haley Miller [Lighting Designer]
Robert "Bo" Golden [Sound Designer]
Berkeley City Club
2315 Durant Avenue, Berkeley, CA 94704 Box Office: (415) 613-9210
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The tower defense genre is immensely popular on iOS and has produced some smash hits like Plants vs. Zombies and Fieldrunners. This tutorial walks you through the process of creating a tower defense game step by step. You'll learn game mechanics, art style and how to use Unity Prefabs. Best of all, it's in 3D.
So, you've created a cool 2D game and it's fun to play. But hard to aim. How can you help your players aim? The answer is auto-aiming and this article demonstrates a simple approach that you can use in your own games.
Game development can involve a lot of math and this tutorial by Nick Lockwood is an excellent review of that math. After helping you refresh your trigonometry, it will explain how you can use the math to calculate acceleration, velocity, collisions and smooth out rotations. This is worth a read and is definitely more fun than high school math class. Anybody? Bueller?


Professor Cochran's research interests include statistical methods, statistical learning, inference, sample based and Bayesian optimization, and applications of statistics and operations research to real problems from a wide variety of areas including medicine, social & economic development, politics/elections, sports, finance, economics, engineering, marketing, accounting, and education. He has published over 40 articles in peer-reviewed academic journals and seven textbooks in statistics, analytics, and operations research. Professor Cochran has served as a consultant for many companies, government agencies, and NPOs, and he was a founding co-chair of Statistics without Borders and a member of the founding committee for INFORMS Pro Bono Analytics initiative. In 2008 Dr. Cochran received the INFORMS Prize for the Teaching of OR/MS Practice, in 2010 he received the Mu Sigma Rho Statistical Education Award, and in 2011 he was named a Fellow of the American Statistical Association. In 2014 he received the American Statistical Association’s Founders Award, in 2015 he received the Karl E. Peace Award for outstanding statistical contributions for the betterment of society, in 2017 he received the American Statistical Association’s Waller Distinguished Teaching Career Award and was named a Fellow of INFORMS, and in 2018 he received the INFORMS President's Award.


Add It On - Jennifer Fixman. Anchor charts give kids something to refer back to as they work on new concepts and skills. Create a game with a deck of cards, but remove the face cards (king, queen, ace, etc.) Addition and subtraction are the most basic things of Mathematics. Now group the tens, 50 and 20, and add them separately (50 + 20 = 70) and group the ones, 4 and 5, and add them separately (4 + 5 = 9). Helping your kids understand the rudiments of math can often be a difficult and tiring process, but there's no way to avoid it - these are things they just have to know if they're going to be successful.
This hands-on addition activity is a great math center idea for students. Let's Learn Our Addition Facts 1 - Jack Hartmann. How to teach your child addition and subtraction? Browse christmas tree crafts for addition and subtraction resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources.
A worksheet which asks children to add and subtract thousands, hundreds, tens and units from a set of numbers. Say "3 pieces together with 6 pieces is 9." Write 3 + 6 = 9 and ask the students to write it down.
By using dice students can still see both parts (in the pips!) For example, if you roll a 4 and a 5, you would place 4 bricks in your first ten frame and 5 in your second. For the other parts of this series dealing with bigger number ranges, please refer to the following links: Strategies for Teaching Addition and Subtraction - Part 1: Numbers to 10 (this post) Though we do not have the provision of negative numbers in this, however kids at kindergarten age do not require that. Shade 1 of the 2 pieces. Work up to adding and subtracting larger numbers with the objects. Stage 1 - addition and subtraction. Give each bowler 2 turns. Mantras for subtraction. The practice with addition should make the subtraction concept a bit easier. Write 6 + 2 = 8 or the board and ask the students to write it down. Let them bowl and ask them how many pins are left standing. Play "flip flop" match game to look for sets of equations that follow the commutative property. Step 3: Practice those facts until they're mastered. Have students put two groups of manipulatives together to model addition. When they understand how numbers break apart into ones, tens and hundreds, they can use that to help them when they are subtracting.
This approach is appropriate for a wide range of students. 10 Silly Monkeys - Jack Hartmann. Have the child count the beads one at a time, aloud if necessary. They will teach "fact families," a term for using the inverse relationship between addition and subtraction to solve problems. 2 = 1 ten and 2 ones, then 1 ten goes to the tens column and 2 ones stay in its column. Addition & subtraction in Year 1 (age 5-6) In Year 1, your child will be expected to be able to read, write, and understand mathematical ideas using addition (+), subtraction (-) and equals (=) signs. Teaching your students to add and subtract integers can be tricky.
An open number line is where students have make jumps in order to represent adding or subtracting tens and ones. Teaching Addition and Subtraction KS2: A Guide For Primary School Teachers From Year 3 To Year 6. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. The number line would be blue with black numbers 1-20. We began each lesson by showing pupils the problem on the board and asking them what they noticed; by prompting them to comment simply on what they could . I Can Say My Number Pairs 10 - Jack Hartmann.
This is the most intuitive method.
They also start learning regrouping, or "borrowing.". However, in the subtraction strategy, I have students use sticky notes also to mark the numbers throughout the process. Make Teaching Addition and Subtraction a Breeze with these Done-for-You Lessons.
The hands-on method described here is great for . Week Two: Putting Together, Taking Apart, Mix.
2nd Grade Math - Focus on addition and subtraction within 20, Understanding the place values of ones and tens 3digit addition and subtraction . There are five major types of word problems to focus on in second grade: adding to, taking from, putting together, taking apart and comparison problems.
Add and Step On Ones Rap - Music with Mar. Ten frames are all about understanding the value of numbers. Once your child has learned one specific addition strategy for one specific group of facts, she'll still need some practice before she's able to use the strategy fluently. The addition will involve regrouping (trading) when two digits in the column make more than ten. When it's time to introduce addition and subtraction to young students, it's helpful to start by modeling with manipulatives. 18 June 2021. To add or subtract unlike fractions, the first step is to make denominators the same so that numerators can be added just like we do for like fractions. The first step is to introduce doubles to the students. Plastic math cubes, counting bears, mini-erasers, or even marshmallows and chocolate candies. Teaching your students to add and subtract integers can be tricky. Have them respond with the next number. Adding Doubles- Put 'Em to Bed- for Google Slides (TM) and PowerPoint (TM)
You can use anything! that they are manipulating. Songs for Teaching Addition (Including Skip-Counting Songs) 4 Plus Song - Alan Stern. We talk more about putting numbers together and taking them apart. Friends of 10 - Jack Hartmann. 3rd grade. (one side of the sorting sheet) The first key understanding kids need to develop is place value understanding. By fourth grade: Kids typically understand how to add and subtract multi-digit numbers . Ask the students how many oranges Jordan ate. 2. To teach kids addition and subtraction teaching methods, give them a ball. Write down "-" between the "8" and the "3." Ask the students for the solution to "8-3." Write down "=" followed by a "5." Ten Frames: A ten frame is a 2 x 5 array that is filled with counters to teach children subitizing (seeing a number), combinations of 10, patterns and adding/subtracting. Now ask the child to count out the objects to match the number on the flashcard.
So the second turn would be. The Addition and Subtraction Strategies.
This includes: making and using number bonds to 10 and then to 20. adding and subtracting one-digit and two-digit numbers to 20, including 0. As the child fills the 10 frame they start to "see" the value of different numbers, and can begin to add and subtract numbers less than, equal to and greater than 10. We like this subtraction anchor chart for all the details it provides. share to twitter Grade 2 math worksheets on subtracting a 2-digit number from a 2-digit number with borrowing Teaching Materials Branching 4 Branching 4. 2. 9 addition and subtraction facts within 20 for Google Slides (TM) 2. Activities. Then, I count out 5 more cubes but break them into two parts and we talk about it. simple Method to Introduce a Child to subtraction with the Abacus. When introducing addition to young children, you should explain it as you take two sets of quantities, add one set to another, and count to see how much they make together. The practice with addition should make the subtraction concept a bit easier. First and the foremost is to convey what 'subtraction' means. In this post, we examine how we differentiated learning practically for struggling learners in KS1 when teaching the chapter in the Maths No Problem!
Shortcuts to Learning Most 1st grade teachers employ little tricks to help kids master addition and subtraction. Share on Pinterest. 3. When children understand the relationship between addition and subtraction they will find it easier to master their subtraction facts. Differentiate the Numbers - Give students just the right numbers so that they can read the problem without getting bogged down . Example 4: Give 4 cookies to your child. They also subtract single-digit numbers and 10s. . The act of subtraction is to take one large . We can do this by introducing children to the think addition strategy - "I know that 3 + 4 is 7 so 7 - 4 is 3.". Addition and Subtraction Number Line. Make pins from 2 liter soda bottles.
If you want to a step-by-step approach to teaching addition and subtraction, so that your students really get it, then you need the 3rd Grade Addition and Subtraction within 1,000 Simplified Math Curriculum Unit. 1. The Split Strategy (AKA: Partial Sums, Expanded Form, Decomposition): In this strategy, students are breaking apart a number based on its place value and then adding or subtracting each individual place to find the answer. Wondering how to teach addition and subtraction computation in a kindergarten, first, and second grade classroom?!
Now add the tens total and the ones total (70 + 9 = 79). How I Teach Addition Fact Strategies. 2.
Taking a step away from concrete materials, dice are a great representative tool. When introducing addition to young children, the key vocabulary words to use are "together", "add", and "count". Take turns rolling the dice and showing those numbers in the two ten frames.
Use this lesson to teach your students how to determine if they should use addition, subtraction, or both operations to solve a word problem by following four simple steps. Step 2: Add or Subtract the numerators (keeping the denominator the same) Step 3: Simplify the fraction. Use everyday objects and flashcards. Here is how you can teach addition using physical objects and flashcards: Step 1: Present a flashcard with any number between 1 and 10 written on it. I cannot say this enough times, before we ask children to learn how to add or subtract, we need to make sure they have an understanding of number sense. The reason we teach our students all of these addition strategies is the desire to develop a more robust number sense. Make sure your kids know the concepts of addition and subtraction, the addition subtraction facts, and place value before you start. Next, ask the child to tell you how many beads are set on the abacus. Say "3 pieces together with 6 pieces is 9." Write 3 + 6 = 9 and ask the students to write it down. In this basic m. 7 Plus Song - Alan Stern. There are even digital manipulatives, like base-ten blocks, place value charts, and more! Adding and Subtracting. For a quick and easy game, grab your two ten frames and two dice. Written by: Susan.
Only once the teacher is happy with the concrete and . Once each individual place is found, then the expanded form is returned back to . It's an amazing activity to start learning about subtraction and addition. Vertical addition with regrouping. The teacher uses learners prior knowledge and the concrete apparatus to revise number concepts before dealing with addition and subtraction. Week One: Adding To, Taking From, Mix. We do not place more than 9 in a column.
But the learning process can be made easier if . I'm going to share a few of my favorite addition strategy songs below. So, have your child focus on just that specific set of facts for several days. I Like To Make 10 - Harry Kindergarten. The concept of zero pairs does not come easily to many students, so it really helps to have awesome, engaging lessons prepared when introducing this topic! 3- Addition Strategies: Ten Frames. Kids love using fancy objects and what better way to teach them math than by using objects that they love.
With games specific to the skill, students can play and practice computation at the same time. Teaching children about number fact families is the perfect way to show this link. textbook on addition and subtraction. The Addition and Subtraction Strategies. For example, 7 + 8 is 1 more than 7 + 7, so 7 + 8 is 1 more than 14. 9 pins, 4 hit: 9 - 4 = 5.
Preschoolers can learn a lot about addition and subtraction from their daily lives. In addition to dynamic subtraction or "take away", Children's Mathematics describes two other meanings to subtraction. Now ask him to share 2 of them with you or a sibling.
Also, a sorting page with two different sides is helpful. While helpful when working with equations with unknowns in all areas, it's imperative that students have a solid understanding of .
Young children generally enjoy counting and once they have mastered basic numbers from one to 20, they have the tools to do simple addition and subtraction. In this video, I share some computation ac. In this number sentence, the 7 is called the minuend. For students to build a solid understanding of addition and subtraction, they need to understand what each one represents. This will help them in learning to count easily while playing with their friends. Math. First, have the child set all of the beads on the counting tool, like shown in the photo. Say "1" out loud to your child. This is the first of a 4-part series of discussion on teaching Addition and Subtraction, for numbers to 10. Check out the NEW Math Game we made at https://www.MageMath.com/ It is a full video game called Mage Math that helps kids build confidence in math while ha. Students take turns bowling and writing the subtraction equation. Once each individual place is found, then the expanded form is returned back to . Invite students to physically take away manipulatives when subtracting. Add At a Circus - Music with Mar. Check out the NEW Math Game we made at https://www.MageMath.com/ It is a full video game called Mage Math that helps kids build confidence in math while ha.