A Ph.D. dissertation is a significant and rigorous endeavor that requires a great deal of planning, research, and analysis. One of the most critical components of a Ph.D. dissertation is the statistical analysis. This section is crucial as it helps to establish the validity and reliability of the research findings. In this article, we will discuss the importance of statistical analysis in a Ph.D. dissertation, the types of statistical analysis commonly used, and the steps involved in conducting a statistical analysis.
The Importance of Statistical Analysis in a Ph.D. Dissertation
Ph.D. dissertation statistical analysis plays a crucial role in a Ph.D. dissertation as it helps to establish the validity and reliability of the research findings. This is particularly important in the social sciences and other fields where quantitative data is collected. The goal of statistical analysis is to draw inferences from the data, and this process requires the use of appropriate statistical techniques to ensure that the findings are accurate and unbiased. For example, statistical analysis can be used to test hypotheses, establish relationships between variables, and identify patterns in the data. This information can then be used to support or refute the research question, and it can also be used to develop new hypotheses for future research. In addition, statistical analysis allows researchers to make generalizations about the population based on the sample data collected.
Types of Statistical Analysis Commonly Used in Ph.D. Dissertations
There are several types of statistical analysis that are commonly used in Ph.D. dissertations. The most commonly used types of statistical analysis include:
- Descriptive Statistics: This type of statistical analysis is used to summarize the data and describe the characteristics of the sample. Descriptive statistics include measures such as mean, median, standard deviation, and frequency distribution.
- Inferential Statistics: This type of statistical analysis is used to make inferences about the population based on the sample data. Inferential statistics include techniques such as t-tests, ANOVA, and regression analysis.
- Non-parametric Statistics: This type of statistical analysis is used when the data does not meet the assumptions of parametric statistics. Non-parametric statistics include techniques such as chi-square, Wilcoxon rank-sum test, and the Kruskal-Wallis test.
- Multivariate Statistics: This type of statistical analysis is used to examine the relationships between multiple variables. Multivariate statistics include techniques such as factor analysis, principal component analysis, and multiple regression analysis.
Steps Involved in Conducting a Statistical Analysis in a Ph.D. Dissertation
The process of conducting Ph.D. dissertation statistical analysis in a Ph.D. dissertation involves several steps, including:
- Defining the research question: The first step in conducting a statistical analysis is to define the research question and determine what type of analysis is needed to answer the question.
- Collecting the data: Once the research question has been defined, the next step is to collect the data. This may involve conducting surveys, experiments, or other forms of data collection.
- Cleaning and organizing the data: After the data has been collected, it needs to be cleaned and organized. This may involve removing outliers, missing data, and other errors in the data.
- Analyzing the data: Once the data has been cleaned and organized, it can be analyzed using appropriate statistical techniques. This may involve conducting descriptive statistics, inferential statistics, non-parametric statistics, or multivariate statistics, depending on the research question.
- Interpreting the results: After the data has been analyzed, the results need to be interpreted. This involves identifying patterns in the data.
Ph.D. Thesis Analysis Using SPSS – How to Analyze Thesis Data
A Ph.D. thesis is a culmination of a student's research efforts and represents the most advanced level of academic accomplishment. As such, it is important to ensure that the data and analysis presented in a thesis is accurate and statistically sound. One tool that can be used to aid in this process is SPSS, or Statistical Package for the Social Sciences. SPSS is a software package that provides a wide range of statistical analysis tools, including descriptive statistics, inferential statistics, and data visualization. It is commonly used in the social sciences, as well as in other fields such as education, psychology, and business. One of the key advantages of Ph.D. thesis analysis using SPSS for analyzing data is its user-friendly interface. The software is relatively easy to navigate, even for those with limited statistical expertise. This allows students to focus on their research and analysis, rather than struggling with technical issues. Another advantage of using SPSS is its wide range of statistical analysis options. The software can be used to perform a variety of statistical tests, including t-tests, ANOVA, and regression analysis. This flexibility allows students to choose the most appropriate test for their data, rather than being limited by the capabilities of a single software package.
When using SPSS for data analysis in a Ph.D. thesis, it is important to ensure that the statistical tests used are appropriate for the data at hand. This includes ensuring that the data meets the assumptions of the test, such as normality and independence. It is also important to interpret the results of the analysis in light of the research question and the limitations of the data. Data visualization is also an important aspect of data analysis in a Ph.D. thesis. Ph.D. thesis analysis using SPSS provides a wide range of options for creating charts and graphs, including histograms, scatter plots, and line graphs. These visualizations can be used to clearly and effectively communicate the results of the analysis to the reader. In addition to the above, it is important to note that SPSS is highly customizable and flexible software. This means that it can be easily integrated with other tools and software, such as Excel or R, to perform more advanced analysis or visualization. Overall, SPSS is a powerful tool that can aid in the analysis of data in a Ph.D. thesis. Its user-friendly interface, wide range of statistical options, and data visualization capabilities make it a valuable tool for students and researchers. However, it is important to ensure that the statistical tests used are appropriate for the data at hand and that the results are interpreted in light of the research question and limitations of the data. In conclusion, SPSS is a powerful tool that can be used to analyze data in a Ph.D. thesis. Its user-friendly interface, wide range of statistical options, and data visualization capabilities make it a valuable tool for students and researchers. However, it is important to ensure that the statistical tests used are appropriate for the data at hand and that the results are interpreted in light of the research question and limitations of the data. With the help of SPSS, Ph.D. students can effectively analyze their data and present their findings in a clear and convincing manner.


Author: Mark Szymczyk
Last Update: August 20, 2005
Vectors are important in game development, especially in the areas of physics and 3D graphics. You can use vectors to represent a game object’s position in the game world, its velocity, and its acceleration. Vectors are also used to respond to collisions between objects. 3D graphics uses vectors for numerous operations, including lighting calculations, bump mapping, and environment mapping. This article introduces you to vectors and vector math.
Before defining what a vector is, I need to define what a scalar quantity is. A scalar quantity has magnitude only. Examples of scalar quantities are 51, 1.34, and –12.56. A vector quantity has both magnitude and direction. Traveling 35 miles per hour on a motorcycle is a scalar quantity. Traveling 35 miles per hour north is a vector quantity.
Vector quantities use the sign to signify direction. The vector quantities 5 and –5 are the same vector with opposite directions, which you can see in Figure 1.
The most common vectors used in computer programs are 2D and 3D vectors. 2D vectors have two components: x and y. 3D vectors have three components: x, y, and z. I’m going to use 3D vectors for the examples in this article. If you can understand 3D vectors, you’ll have no trouble with 2D vectors.
Figure 1: Two vectors with the same magnitude, but different directions.
Next (Vector Length)


Help children to visualise and understand fraction concepts!
Hands-on learning set provides young learners with a visual representation of fractions
Cubes feature fractions, decimals and percentages so young learners can clearly visualise equivalencies
Colour-coded cubes allow children to manipulate fractions
All Rainbow Fraction products use the same colors illustrate each fraction: Red = 1, Pink = ½ Orange = ⅓, Yellow = ¼ , Green = ⅕, Teal = ⅙, Blue = ⅛, Purple = 1/10 and Black = 1/12.
Set of 51 includes enough cubes of each fraction to create a whole
Great for home use or in the classroom
Product Size: One complete rod is 12 cm high. Box is 20.6 x 18.3 x 5.3 cm


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See you at SheTech 2017 to discover the exciting world of science, technology, engineering, and math.


讲述问题：Fast computing Tukey depth regions of dimensions of p > 2
择要：Given data in R^p, a Tukey tau-trimmed region is the set of all points that have at least Tukey depth tau w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension p > 2. We construct two novel algorithms to compute a Tukey tau-trimmed region, a naive one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the naive one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median. Supplementary material is available online.
现已在中国迷信数学英文版、数学学报中英文版、Journal of Econometrics
(学界公认的计量经济学顶尖期刊，是教诲部承认的12本经济学国际顶级期刊之一)、Journal of Statistical Software(top期刊，2017/2018影响因子22.737)、Journal of Computational and Graphical Statistics、Statistics and Computing等海内外刊物上揭晓任命相干学术论文30篇。刘传授先后掌管国度天然迷信基金区域项目、青年项目、面上项目各1项，掌管中国博士基金一等赞助、稀奇赞助各1项，掌管江西省天然基金青年严重、江西省卓越青年基金各项目1项。同时是Journal of American Statistical Association（JASA）等多个国际期刊的审稿人。


How to calculate the sample size
Many researchers claim to have a representative response when their sample is large enough. Two mistakes can already be found in this vision. First, the sample size is not related to representativeness. And second, the size of the sample is not the problem. The problem that counts is the size of the response.
Now, you might be worried. Maybe you should be.
Another way to calculate the sample size is to calculate an estimated proportion with a confidence interval of 95%. Based on the standard calculations, the result is always a sample size of 384.1 or 385. This view also does not stand in a critical discussion. In many studies, estimating a proportion is not important. It is therefore not a reason to calculate the sample size based on the estimation of a proportion.
The correct way to calculate the sample size is much more complex. And by the way, the right way does not exist.
This paper explains how to deal with the problem of sample size. Read it and discover how you can apply the solution to your research.
Our high quality manuals cost € 5.45 each.


Plinko is a popular game that is often seen on game shows, online casinos, and other gambling platforms. The game consists of a board with a series of pegs and slots, and players drop a ball or chip from the top of the board and hope that it lands in a slot with a high payout. While the game itself may seem simple, there is a lot of math and probability involved in determining the odds of winning.
In this article, we will explore how online Plinko probabilities and odds are determined.
In Plinko, players drop a chip or ball from the top of the board and watch as it bounces off a series of pegs before landing in a slot at the bottom of the board. The slots are assigned different payouts, with some slots having higher payouts than others.
The placement of the pegs on the board is what makes the game random and unpredictable. As the chip or ball bounces off the pegs, it can change direction and speed in unexpected ways, making it difficult to predict where it will land.
The probability of a chip or ball landing in a particular slot depends on the placement of the pegs on the board and the size of the slots. For example, if a slot is twice as wide as another slot, then the probability of the chip or ball landing in that slot is twice as high.
To determine the probability of winning in Plinko, the first step is to calculate the total number of possible outcomes. In other words, how many different paths can the chip or ball take as it bounces off the pegs? The total number of possible outcomes will depend on the number of pegs on the board and the spacing between them.
Once the total number of possible outcomes has been determined, the next step is to calculate the number of outcomes that result in a win. This will depend on the placement of the slots on the board and their respective payouts.
The probability of the ball landing in a particular slot depends on the layout of the pegs and the number of slots. Let’s consider a simplified version of the game with 5 rows of pegs and 6 slots at the bottom of the board:
Each row has one fewer peg than the row above it, and the pegs are arranged so that the ball has an equal chance of bouncing to the left or right at each peg. When the ball reaches the bottom of the board, it falls into one of six slots labeled A, B, C, D, E, or F:
The probability of the ball landing in each slot can be calculated using the binomial distribution. For example, to calculate the probability of the ball landing in slot A, we can count the number of paths the ball can take that lead to that slot. There are four rows of pegs the ball must pass through to reach slot A, and at each row, it has a 50-50 chance of bouncing to the left or right. So there are 2^4 = 16 possible paths the ball can take to reach slot A. The probability of any one of those paths occurring is (1/2)^4 = 1/16, since the ball has a 50-50 chance of taking either path at each row of pegs. Therefore, the probability of the ball landing in slot A is:
Similarly, we can calculate the probabilities of the ball landing in slots B, C, D, E, and F:
So the probabilities of the ball landing in the six slots are:
These probabilities determine the expected payout for each slot, which is the sum of the product of the probability of landing in that slot and the value of the prize for that slot. For example, if the prizes for the six slots are $1, $10, $100, $1000, $10000, and $100000, then the expected payout for each slot is:
The expected payout for the player is the sum of the expected payouts for each slot, which is:
The odds are calculated by dividing the probability of winning a particular prize by the total number of possible outcomes.
Let’s consider a simplified version of the Plinko game with 5 rows of pegs and 6 slots at the bottom of the board, as shown in my previous response. The probabilities of the ball landing in each slot are:
To calculate the odds of winning each prize, we need to divide each probability by the total number of possible outcomes. In this case, there are 2^5 = 32 possible paths the ball can take through the pegs before landing in one of the six slots.
The odds of winning each prize are:
These odds represent the likelihood of winning each prize on a single play of the game. Note that the odds can vary depending on the layout of the pegs and the number of slots in the game.
To maximize your winnings in online Plinko, it is important to understand the odds of the game and employ certain strategies. Here are some tips to help you increase your chances of winning:
In conclusion, maximizing your winnings in online Plinko requires an understanding of the game’s odds and the use of certain strategies. By choosing a reliable online casino, setting a budget, and using a betting system, you can increase your chances of hitting a big payout. Remember to play for the long term, choose your bets wisely, and most importantly, have fun!


报告题目 (Title)：padic tensor network and emergent Einstein equation（p-adic张量网络和诱导爱因斯坦方程）
报告人 (Speaker)：孔令欣 研究员（复旦大学）
报告时间 (Time)：2021年4月8日(周四) 14:00
We take the tensor network describing explicit p-adic CFT partition functions proposed in , and considered boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree. Moreover, the (perturbative) definition of the graph curvature in the Mathematics literature [2–4] naturally emerges from the consistency requirements of the emergent Einstein equation. This could provide new insights into the understanding of gravitational dynamics potentially encoded in more general tensor networks.


We revisit the polarized L calculus, an abstract-machine-like calculus with mixed evaluation order (i.e. call-by-name and call-by-value) and pattern-matches, and its relation to the λ-calculus. We then show that it is a more symmetric syntax for call-by-push-value. We also introduce a dynamically typed / bi-typed variant of this calculus which completely eliminates clashes (i.e. pattern-matching failures) without relying on any form of typing judgments, and illustrate its usefulness in the study of extensions of the untyped λ-calculus with constructors.


To estimate the amount of wallpaper needed to paper a room. Measure the height of the room. Add on an allowance for trimming. And calculate how many lengths can be cut from one roll; the usual length of a roll is 10 m 33ft.
Multiply the width of the paper. Usually 520 mm by the number of lengths that can be cut from one roll. Ignoring doors and windows, unless they are extremely large, measure the perimeter of the room, divide this figure by the first figure to calculate the number of rolls required. Remember that fractions must be made up to the next whole number.
If the room has different ceiling-heights, such as a stairwell. Measure those areas with the same height and add the totals together.
Preparing paper for pasting
As most modern wallpapers are rolled with the pattern on the outside, problems can arise when pasting the paper. This is because. When the paper is laid on a table for pasting, the ends roll back into the length, which can result in paste getting on to the surface of the pattern.
Deal with this by planning the job in advance, so that all the lengths of paper required are rough cut to size and left lying on the pasting table overnight, with the patterned side downwards. Hold the ends down by loosely looping a length of string round the pasting table legs and over the combined lengths of paper.
Next day. The paper will no longer curl, making it much simpler to paste individual lengths.


The rules for the presentation of abstracts: A4 format, 2.5 cm fields on all sides, single interval, Times New Roman font size 12 pts for main text, volume up to 4 pages. The “header” of the thesis includes the title of the report, information about the authors, contact details. Then follow (necessarily!) the abstract of the report and the keywords. The main text of the report follows. Tables and figures are placed inside the text. At the end of the main text, a support fund is indicated (if available). Then follow (necessarily!) references.
THE PROBLEM OF GENERATION OF SHIFT WAVES IN BOSE GAS WITH A VIBRATING BOUNDARY AND ITS EXACT SOLUTIONS
Bedrikova E. A., Latyshev A.V.
Moscow State Regional University
An analytic solution of the problem of generation of shear waves in a Bose gas in the case of a constant amplitude of the surface oscillation is found. A kinetic equation with boundary conditions is derived. The eigensolution of the equation is found. The properties of the dispersion function are investigated. A general solution of the boundary value problem is formulated in the form of an expansion in eigensolutions.
Keywords: Bose gas, eigensolutions, kinetic equation, boundary value problem
Recently, the problem of the behavior of gas over a moving surface is of great interest. This is due to the rapid development of modern technology.
- Akimova V.A., Latyshev A.V., Yushkanov A.A. Analytic solution of the second Stokes problem on the behavior of a gas over an oscillating surface // Fluid Dynamics. A Journal of Russian Academy of Sciences. 2013. No 1, pp. 125-140.
- Stokes G.G. On the effect of internal friction of fluids on the motion of pendulums. Cambr. Phil. IX, 8 A851), Math, and Phys. Papers III, 1-141, Cambridge, 1901.


• Why does zero factorial (0!) equal 1 ?
Because there is exactly one way to do nothing.
• How (a+b)2 = a2 + 2ab + b2 ?
We can prove it geometrically.
Step 1: Draw a line of length (a + b)
Step 2: Let's draw a square having sides of length (a + b). Area of this square will calculate the value of
This is a square consisting two rectangular and two square parts.
Area of square part 1 = a2
Area of rectangular part 2 = ab
Area of rectangular part 3 = ab
Area of square part 4 = b2
So, Area of square of length (a+b) = (a+b)2 = a2 + 2ab + b2
i.e. (a+b)2 = a2 + 2ab + b2


enter search term and/or author name
Automated container transport system between inland port and terminals
Jianlong Zhang, Petros A. Ioannou, Anastasios Chassiakos
In this article we propose a new concept called automated container transportation system between inland port and terminals (ACTIPOT) which involves the use of automated trucks to transfer containers between an inland port and container terminals....
Fast simulation of overflow probabilities in a queue with Gaussian input
A. B. Dieker, M. Mandjes
In this article, we study a queue fed by a large number n of independent discrete-time Gaussian processes with stationary increments. We consider the many-sources asymptotic regime, that is, the buffer-exceedance threshold B and the...
How heavy-tailed distributions affect simulation-generated time averages
George S. Fishman, Ivo J. B. F. Adan
For statistical inference based on telecommunications network simulation, we examine the effect of a heavy-tailed file-size distribution whose corresponding density follows an inverse power law with exponent α + 1, where the shape...
A discrete event method for wave simulation
This article describes a discrete event interpretation of the finite difference time domain (FDTD) and digital wave guide network (DWN) wave simulation schemes. The discrete event method is formalized using the discrete event system specification...


Math is not an easy subject for everyone. In fact many people have a hard time getting the knack over certain math skills, while it comes easy for others. If you are not careful when you are teaching your students math you could easily lose them, and they could get left behind. When their confidence falls they won’t ask questions and then it makes the situation even worse. Rather than let your student get to that point you can give them the confidence they need when they are doing well.
A mathematics award certificate such as this one, can give them a boost in their confidence and shows them that you appreciate the hard work they put into it. Simple and easy to use this certificate can be awarded anytime you need to, even if you had to give the student a little extra attention in their work.
Download options for Award Certificate for Mathematics :
[ct_label type=”info”] You can also add 2nd and 3rd position badges in premium version. [/ct_label]


What is an 'Image Rank' block on OpinionX?
Image Rank is a survey question format on OpinionX which is used to turn a set of images into a ranked list by using a series of head-to-head pair votes. For example:
How are Image Rank scores calculated?
Image Rank on OpinionX is based on the same algorithm as our Pair Rank block, which uses a format called "Win Rate Analysis". For each option, this takes number of wins and divides by its total number of pair comparisons, giving a score between 0-100 (ie. the percentage win rate).
What do the Image Rank exported results look like?
For more information about exports, including examples of exported spreadsheets, please visit the Knowledge Base page on exporting.
Can I change the number of image pairs each participant must vote on?
Yes, you can set your own custom number of pairs for each participant or set your survey to show every pair possible to each participant. Note: only participants who finish your entire survey will be counted on your results page.
How many votes per participant will generate robust results?
There are two formulas you can use to make sure you have a good minimum sample size:
Both are explained below...
Formula 1: Limited pool of participants
This formula tells you how many votes each participant will need to cast so that every possible image pair combination has appeared at least 3 times in total across the whole survey. The formula to calculate this is as follows, where "n" is the number of options you're ranking:
Here's a breakdown of how this formula works:
For example, if you have 16 options to rank and know that you'll only have 30 people to complete the ranking survey, our formula would be (3(16x15)/2)/30 = 12 votes per person. In this case, when deciding how many votes each participant must cast, I should opt for at least 12 votes or more per person.
Formula 2: Large pool of participants
When you know that you've got a good plenty of participants you can engage, you'll want to get as much data per person without making your survey overwhelming. We'll use a slightly different formula:
votes per person
Here's a breakdown of the meaning of this formula:
Whereas before our limiting factor was the number of participants, now our limiting factor is the number of pair votes we will ask each participant to cast.
Let's say we have 40 options we want to rank using the Image Rank format. I usually use 10 votes per participant as my starting point. That formula would then look like: (3(40x39)/2)/10 = 234 participants. If that seems like a lot more than I'll be able to reach, I can try increasing the number of votes to 15, which adds up to 156 participants. As you can see, the number of votes per participant changes your required sample size a lot!
Full Formula Explanation:
Total number of possible pairs is
n(n-1)/2, where n is the number of ranking options.
Multiply the total possible pairs by 3 to calculate your target number of total votes.
Divide your target number of total votes by your minimum estimate of participants you expect will complete the survey.
The answer you get is the minimum number of pair votes you should require from each participant.
Generally, if the formula results in a number below 10, I would just leave the default 10 votes in place as it is a very reasonable amount to ask from each participant.
If your survey has multiple Image Rank questions, you should consider the total number of votes a participant will have to cast across your entire survey. The higher your total number of pair votes, the lower your completion rate will be. Generally, anything above 40 pair votes within a survey is a lot to expect of any participant (unless you provide a strong participant incentive).
If segmentation is going to be an important part of your analysis, you should substitute the "
participants" variable for your estimate of the total number of people you expect to reach from your smallest key segment. Here's an example:
I have 14 ranking options and expect to reach 150 participants, but I plan to segment my results to see the differences between customers on my four different pricing tiers. The segment I'll get the least responses from is Enterprise Customers, but I think I can get at least 16 participants from them. So my formula would be
3(14*13)/2/16= 17 votes per participant.
Can I use multiple Image Rank blocks in my survey?
All OpinionX users can have an unlimited number of Image Rank blocks in their survey, regardless of whether they're a free or paying customer. This apples to all block types on OpinionX.
How many images can I include in an Image Rank question?
There is no limit to the number of images you can include.
Do Image Rank questions work on mobile devices?
Yes. All question types available on OpinionX are designed to be suitable for participants on any web browser -- whether that's a desktop monitor, laptop, tablet or smartphone.
Can I use emojis in my Image Rank captions?
Yes, you can use emojis in the caption sections of Image Ranking options on OpinionX.
I have a question about Image Rank blocks that wasn't answered here!
Send us a message using our live chat or email [email protected] to get a quick answer from the OpinionX team.


to make a pentagon is a piece of cake after reading this post.
You will need a degree ruler (don't know if that's the correct word but look at the pictures)a compass, pen and paper.
It's a big calculation (at least it is to me)to find the radius in a circle wich will give us a pentagon with sides measuring 3 cm.
My darling husband did this job for me, and the radius must be 2.55 cm.(US 1 inch)
Make a circle that big. Use the ruler and mark 90 deg. from mid point. Then mark 36 deg. to each side of the 90 deg. mark on top.
Take your compass and fit it in between the 2-36 deg. marks. Mark this distance a further 3 times at your circle. Draw straight lines through all marks and VOILA you have your pentagon with 3cm sides.
Look at your figure and imagine a house, you could easily lengthen the "walls" skip the lid and you have made yourself a unik vase.
I will write down the calculation formula wich gives the radius of a circle when side length is given in forehand, for those of you who can manage such kind of magic.
R = side lenght / 2 x sin36 deg. (This is Greek to me. :-)
I believe you all can make a hexagon so I don't mention that. If not ask me.


Open Excel 2013. Type a birthdate into a cell in the format MM/DD/YYYY (if in the United States) or in the format DD/MM/YYYY if your country uses that format. Type =DATEDIF(XX, TODAY(), “Y”) then press Enter on your keyboard. Change the “XX” part of the formula to the cell containing the birthdate.
How do you calculate aging from dates?
In some cultures, age is expressed by counting years with or without including the current year. For example, one person who is twenty years old is the same as one person who is in the twenty-first year of his/her life.
How do I calculate age in an entire column in Excel?
How to calculate age in ExcelThis is a pretty simple three-column task. To get started, enter the date of a persons birth into your first cell. Now, enter todays date into the cell next to your first cell. In the third cell, for us its C2, enter the following formula: =DATEDIF(A2, B2, “y”).
How do I calculate average age in Excel?
You can calculate the average age by year or month with array formulas quickly in Excel. Average age by Month: Select a blank cell besides the table, for example Cell F2, enter the formula =SUM((MONTH(B2:B15)=12)*C2:C15)/SUM(IF(MONTH(B2:B15)=12,1)) into it, and press the Ctrl + Shift + Enter keys at the same time.
What is the formula for finding 20 percent of 50?
Percentage Calculator: 50 is what percent of 20? = 250.
What does DD mean in date of birth?
What is the formula to calculate average?
How to Calculate Average. The average of a set of numbers is simply the sum of the numbers divided by the total number of values in the set. For example, suppose we want the average of 24 , 55 , 17 , 87 and 100 . Simply find the sum of the numbers: 24 + 55 + 17 + 87 + 100 = 283 and divide by 5 to get 56.6 .
What is the formula for sample mean?
Calculating sample mean is as simple as adding up the number of items in a sample set and then dividing that sum by the number of items in the sample set. To calculate the sample mean through spreadsheet software and calculators, you can use the formula: x̄ = ( Σ xi ) / n.
What is the formula for population mean?
The formula to find the population mean is: μ = (Σ * X)/ N. where: Σ means “the sum of.” X = all the individual items in the group.
What number is 25% of 50?
Answer: 25% of 50 is 25/2 or 12½ as a fraction and 12.5 as a decimal.
What number is 45% of 50?
Percentage Calculator: What is 45 percent of 50? = 22.5.


The relationship between two things is something that we learn early in life. It's quite an important thing to know about, especially for understanding the world we live in.
However the question of 'are these the same' or 'are these equal' is surprisingly tricky. It might immediately seem rather obvious, but in actual fact there is a surprising amount of complex mathematics to do with equality and how you define it. As it turns out there are quite a few ways of saying that things are the same, leading to a description of all the different ways of comparing things being called an Equivalence relationship.
Equivalence relationships have three key properties:
- Reflexive - That everything is equal to itself.
- Symmetric - That if object A is equal to thing B then thing B must be equal to object A.
- Transitive - That if A equals B and B equals C then A must equal C.
For example consider the following:
Where it starts to get tricky is if there is some variation between the two things being compared. Initially the answer to this might appear to be "no, they aren't equal", but consider the following:
- Yes, it's the same image.
- Yes, apart from one is a bit distorted.
- No, they look different.
This all ties in very deeply with one of the foundations of computer science; the distinction between comparing the concept represented by some data and the data itself. A clearer example of the distinction here would be to use two pictures where the idea of equality is even more fuzzy.
As before the answer to if these two are equal is yes and also no. If the question being asked is "are these the same cat?" then the answer is yes; if the question is instead "are these photos the same?" then the answer is no. Usually it's obvious which of these questions is being asked from the context and generally the question is phrased in the more explicit way.
Computers are no different. Ever since Ada Lovelace realised that numbers could be used to represent ideas, concepts and physical objects there has been the need to distinguish between the two types of equality. It's necessary to explicitly tell the computer if you're asking if the numbers are the same (asking if the photos are the same for the above example) or asking if the thing the numbers represent are the same (asking if it's the same cat).
This distinction between the values and what the values are meant to represent occurs frequently in computer languages. For example in the Java language the operator '==' is used to compare the number representing the object, where as the 'equals()' method is used to compare the concept/item that the object represents.
This distinction between a character/digit of 5 and the number 5 (which you get from adding 2 and 3) might seem a little strange, but consider the phrase "I ate 5 strawberries". It's necessary for there to be some way to represent each character/letter as a number and the method of representation that has become standard has the digit 5 represented by the number 53.
So the next time you're trying to work out if two things are the same, just remember to think what you are wanting to compare.


Fannin County High School will host a 9th Grade Parent Night on October 3, 2016 starting at 6:00 p.m. in the cafeteria.
Parents and students will rotate through three 20 minutes sessions including "Being Successful in High School" presented by Mr. Erik Cioffi, Principal of Fannin County High School, which will include tips and advice on how to succeed in high school. Mr. Cioffi will review FLEX period, expectations, grades, balancing school and social life along with opportunities to become involved at school.
CTAE Director, Patti DuBois, and 9th Grade Counselor, Tammy Roberts, will present “College and Career Academy-Pathways” which will focus on Career Pathways and certifications offered at Fannin County High School, the Capstone Project as a graduation requirement, and preparing for post-secondary options during 9th grade year.
The third session will be “How Math Adds Up,” with Fannin County High School Math Teachers, Angela Jones and Kim Kribbs. Mrs. Jones and Ms. Kribbs will present information on how to be successful in math class, the math sequence, and the importance of math in our everyday lives. Also, information will be provided on resources available for students of all levels of mathematic ability.
For additional information, please contact the school at 706-632-2081.


English & Math
All undergraduate students who enroll at LIM College must demonstrate their readiness to learn course material successfully. English and mathematics aptitude is assessed and students are placed in appropriate courses based upon level of aptitude.
Students can demonstrate their readiness by either:
- Submitting ACT or SAT scores for review*
- Completing the LIM College placement test
In addition, college-level English composition and mathematics courses will be reviewed for equivalency and used when determining placement test requirements.
Students who choose not to submit ACT/SAT scores and do not take the LIM College placement test, will automatically be placed in the lowest level developmental courses (Writing Essentials & Mathematics Through Business Applications).
For those who chose to complete the LIM College placement test, study guides are below. It is highly recommended that you review these guides before taking the test.
Additional information, including timeline and instructions, will be communicated to students prior to new student registration.
ACT/SAT Placement Criteria
|ACT Math Score||NEW SAT Math Score||OLD ACT Math Score||Math Course|
|500-540||19-21||Introduction to Algebra|
|ACT Score||NEW SAT Evidence Based||OLD SAT Writing Only Score||Course Placement|
|0-529||0-459||ENGL 0700 - Writing Essentials|
ENGL 1100 –
- International students who submit official ACT/SAT scores will be placed into English and math according to the same ranges posted above. In addition, if an international student transfers equivalent credit in English and mathematics, from a US college/university, they will also be placed accordingly.
- Those international students who do not have the appropriate ACT/SAT scores, and/or do not have equivalent transfer credit, will be required to complete both the English and mathematics placement tests (available through Blackboard).
- Any English or mathematics course completed at an international college/university must be reviewed (via a syllabus review) by the appropriate LIM College department Chair in order to determine transferability and course equivalency.


2 edition of Diophantine analysis and related fields 2010 found in the catalog.
Diophantine analysis and related fields 2010
DARF 2010 (2010 Seikei University)
|Statement||editor, Takao Komatsu ; sponsoring organization, Japan Society for the Promotion of Science|
|Series||AIP conference proceedings -- v. 1264, AIP conference proceedings -- no. 1264.|
|Contributions||Komatsu, Takao, Nihon Gakujutsu Shinkōkai|
|LC Classifications||QA242 .D27 2010|
|The Physical Object|
|Pagination||xi, 151 p. :|
|Number of Pages||151|
|LC Control Number||2010908305|
A. Dujella, Diophantine triples and construction of high-rank elliptic curves over with three non-trivial 2-torsion points, Rocky Mountain J. Math. 30 (), A. Dujella, A note on Diophantine quintuples, Algebraic Number Theory and Diophantine Analysis (F. Halter-Koch, R. F. Tichy, eds.), Walter de Gruyter, Berlin, , pp. The book opens by introducing material usually found in an undergraduate number theory book. The writing style here is, at times, lively and cute. The book then moves into the elementary aspects of Diophantine approximation (a la Dirichet, Kronecker, and Hurwitz), introduces Pade approximation.
Descriptions of areas/courses in number theory. Mathematics Subject Classification, XX Serge Lang's review of Mordell's book Diophantine Equations. Also see Serge Lang's commentary on Mordell and Siegel; A bibliography of papers related to simultaneous diophantine approximation. Free kindle book and epub digitized and proofread by Project Gutenberg.
Diophantine Equations. algebraic number field applied arise Cassels Chapter class number congruence mod cubic curve cubic equation cubic field degree diophantine equation equation 19 equation ax equation f Elliptic Curves: Diophantine Analysis S. Lang Limited preview - Definitely one of the best. As for Diophantine analysis, well that can mean a few things. Lang has a good book on Diophantine approximations. I suspect that going through a book like Ireland and Rosen's will give you a better idea of what it is, specifically, that you're looking for.
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Buy Diophantine Analysis and Related Fields - DARF - (AIP Conference Proceedings / Mathematical and Statistical Physics) on Price: $ Setting the stage for tackling the field's many open problems and conjectures, Diophantine Analysis is an ideal introduction to the fundamentals of this venerable but still dynamic field.
A detailed appendix supplies the necessary background material, more than exercises reinforce the concepts, and engaging historical notes bring the subject to life.4/5(1).
Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, including some of its applications.4/5(1).
Diophantine analysis and related fields DARF, Musashino, Tokyo, Japan, March Starting with linear and quadratic equations, the important connections with Diophantine Approximation are presented and Thue's celebrated results are proved in full detail.
In later chapters more modern issues on heights of algebraic points are dealt with, and applied to a sharp quantitative treatment of the unit equation.
The author’s purpose in writing this book has been to supply the reader with a convenient introduction to Diophantine Analysis. The choice of material has been determined by the end in view. No attempt has been made to include all special results, but a large number of them are to be found both in the text and in the exercises.
Diophantine analysis and asymptotic FLT Whilst class field theory has distinguished historical connections to Fermat's Last Theorem, the subject of Diophantine analysis (Diophantine approximation, linear forms in logarithms) seems to have had little or no influence on the mathematics of the Fermat : Nuno Freitas, Alain Kraus, Samir Siksek.
A polynomial-exponential Diophantine equation related to the sum of consecutive k-th powers Tomohiro Ooto (Tsukuba Univ.) Quadratic approximation of continued fractions with low complexity in F q ((T −1)) Takumi Noda (Nihon Univ.) Confluent hypergeometric type Ramanujan's formulas Yasutsugu Fujita (Nihon Univ.).
Diophantine Analysis and Related Fields held at Raiosha, Hiyoshi Campus, Keio Univ. Yokohama, JAPAN Abstracts of the Talks Thursday 1 March Daniel Duverney (Baggio Engineering School, France) Title: Irrationality of the sums of certain series of reciprocals of Fi-bonacci numbers Abstract: We show how to prove irrationality results by using File Size: 63KB.
cae to Mirzakhani’s Fields Medal . An adequate survey of this area, even if limited to the most important and most recent contributions, would be beyond the scope of this introduction.
The books by Aigner  and Cassels  are ex-cellent references for Markov’s theorem, Bombieri  provides a concise proof,File Size: KB. This lesson is about Diophantine Equations or indeterminate polynomial equations that allows the variables to be integers only (or in some cases fractions).
They have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface or more general object. Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects.
Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, 3/5(1).
Get this from a library. Diophantine analysis and related fields DARF - ; Musashino, Tokyo, Japan, March [Takao Komatsu;] -- "The purpose of the conference was to report recent progress and developments of diophantine aspects of analytic number theory, especially focusing upon the topics in diophantine analysis and related.
Diophantine Analysis and Related Fields ・ Period: From Thursday 1 March to Saturday 3 March, ・ Location: Large Conference Room, Raiosha, Hiyoshi Campus, Keio Univ. (Yokohama, Japan) ・ Organizers: Shigeki Akiyama (Tsukuba Univ.), Masaaki Amou (Gunma Univ.), Noriko Hirata-Kohno (Nihon Univ.), Masanori Katsurada (Keio Univ.), Ryotaro Okazaki.
History of the Theory of Numbers, Volume II: Diophantine Analysis (Dover Books on Mathematics) Paperback – June 7, and many other related subjects. Indexes of authors cited and subjects appear at the end of the book. this set of volumes is a masterpiece and stands alone--very few fields have such a fine authoritative concise Cited by: Buy Diophantine Analysis and Related Fields: DARF / (AIP Conference Proceedings) on FREE SHIPPING on qualified orders.
An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants — including Olympiad and Putnam competitors — as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine.
Abstract. Beginning with the work of Osgood , it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth’s theorem on diophantine by: On and related Diophantine equations Article in Journal of Number Theory (9)– September with 61 Reads How we measure 'reads'.
Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived.
In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields.
An introduction to various Diophantine problems in number theory, with approaches involving analytic methods. The exact subject matter of the course will depend on the participants and general interest, potential topics could include the basic theory of Abelian Varieties, the Lang conjecture, the Manin-Mumford and Andre-Oort conjectures, as well as the .Open Diophantine Problems.
Diophantine approximation on manifolds and Diophantine analysis on function fields. Schikhof also studied this set in his book Author: Michel Waldschmidt.


PHP Data Types
What are PHP Data Types:
Variables can store different types of data. A data type is a set of values and the allowable operations on those values.
PHP Has Eight data types:
PHP String Type
A string is a sequence of characters, like “This is a sample line”.
PHP Integer Data Type
Integers are a subset of the real numbers. They are written without a fraction or a decimal component.
An integer data type is a non-decimal number between -2,147,483,648 and 2,147,483,647.
PHP Float Data Type
Floating-point numbers represent real numbers in computing. Real numbers measure continuous quantities, like weight, height, or speed. Floating-point numbers in PHP can be larger than integers and they can have a decimal point.
<?php $s = "this is string example"; // $s is a string variable echo $s."
"; $i = 10; // $i is an integer variable echo $i."
"; $f = 1.3; // $f is now a float (3.3) echo $f."
"; $concat_i = $i."10 Little Piggies"; // integer echo $concat_i."
"; ?>
Run : http://ideone.com/42n2cm
this is string example
1010 Little Piggies
|Exercises & Assignments|
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|Interview Questions & Answers|
|What are the different data types in PHP?|


April 29, 2022
National Numeracy Day is celebrated on Wednesday 18th May and is all about recognizing the value and presence of numbers in everyday life.
There are lots of activities on Purple Mash that schools can use on the day to celebrate the power of numbers.
Children can use 2Count to create their own pictograms. This could include counting objects in the house or when out on a walk. How many rainbows in windows can children spot?
Children can learn their times tables with 2Simple Table Toons and test themselves on their number knowledge on Monster Multiplication.
With the Multiplication Tables Check in June, there’s no better time to dive into the Multiplication Tables Check tool. Designed to look and feel like the real thing, this Purple Mash activity is the perfect way for Year 4s to prepare.
There are a range of games that children can use to practice and test their number skills and have fun doing it.
From mermaids to rockets, 2Race is a virtual collaborative racing maths game which has a variety of exciting themes. It is designed to make learning numbers enjoyable, while at the same time helping to firmly secure knowledge and improve mental arithmetic. All areas of number are covered for KS1 & 2, with pupils’ responses to questions automatically recorded to allow for comprehensive diagnostic data. Teachers and pupils can play collaborative live racing challenges or alternatively, pupils can play pre-made games against virtual players.
Recently updated Bond Bubbles is a gripping way to reinforce those pesky number bonds. The aim of the game is to use the bubble blower to fire a bubble at a number that will make the number bond being asked. You will find three mysterious new lands waiting to be discovered. Each land has a series of levels, which all need to be completed before you can progress to the next land.
A path through each land is charted on a map for explorers to follow, and they must complete each level to continue along the path in order to complete their journey along each path. And best of all, the game adapts to the player’s ability, making this the perfect game to support all pupils.
Sequence Snake lets pupils apply knowledge of number facts and sharpen their mental arithmetic in a challenging yet fun game. Pupils must strategically feed Sequence Snake by guiding him to the next correct number. With every correct number eaten, Sequence Snake grows by that amount so helping him to not eat himself becomes pretty challenging after a few 9s!
Or why not get children to use this template to create their own Dice game, they could also use 2Quiz to create their own quizzes for others to solve all based around the theme of number.
There are lots of ways to explore, teach and reinforce number in Purple Mash. However you do it, we’d love you to share your numerical adventures with us on our Twitter or Facebook pages.
The Numeracy Day site will be running a virtual online conference all day with a range of sessions both for children and parents. View the timetable here.


A great start to the term, the children are focussed and energetic.
Homework this week is times tables – repetitive I know, but the sooner the children are fluent then their brain can use this information to solve problems, rather than the multiplication being the tricky part.
English homework is adding prefixes to root words. The children should try each one to see which sounds right. We have had lots of practice this week and have found lots of words. Some root words might work with more than one prefix. For example, unfit and misfit.
Have a lovely weekend.


Fractal art and nature
How are fractal art and nature alike?
When someone looks at nature they do not usually think of math, yet math as “algorithm” is everywhere we look. Most do not know of the “fibonacci sequence or fibonacci series” and its relation to the Phi, but this pattern is everywhere.
It is in the flower that we gaze at with awe and is in the cosmos, when we are staring at the stars. How can this be? and what is the fibonacci sequence?
The fibonacci sequence was first recognized being discussed in the 12th century. Leonardo Fibonacci wrote about it in Liber Abaci. He wrote of a simple numerical sequence which is the incredible mathematical relation behind the PHI. What is amazing is that although it is known today as the fibonacci in dedication to the forthcoming knowledge given to the west by Leonardo Fibonacci. We now know that is dates back to the 6 century AD as knowledge known by Indian mathematicians, North Africa and the Mediterranean.
PHI “Φ” = 1.618033988749895, is important because it is an irrational number like “pi” p= 3.14159265358979 but one with unusual mathematical properties. Where pi is a transcendental number, phi is the solution to a quadratic equation.
Now when we take into account the phi and the fibonacci in nature we understand that there is a common pattern platform governing the design of nature and the cosmos.
Our world is created in a beautiful design and she moves in the rhythm of the Golden Ratio. The golden ratio appears in geometry, art, architecture, plants and even our food. We find the golden ratio when we we divide a line into two parts so that;
The whole length is divided by the long part is also equal to the long part divided by the short part.
The Golden Ratio is the greek symbol PHI. You can find the sequence of the golden ratio everywhere. Scientists have even proven that the golden ratio is found on the human body first shown by Leonardo Da Vinci in his art pieces “The Mona Lisa and The Vitruvian Man” and several other pieces that he created.
Throughout history especially during the renaissance time artist and architects who were also alchemists like Da Vinci (1451-1519) made a point of appreciating the golden ratio sequence and it is seen in the structures and art pieces created by them during that time.
All around us the golden ratio sequence or series is reflective within our world. Many people focus on the underlying balance of the fibonacci series and the golden ratio when they design or choose their home by the plants that they admire, and by the art that they choose; although very few realize that it is the pattern of the golden ratio which prompts their passion, admiration and desires.
A modern Dutch artist Piet Mondrian (1872-1944) was a deep believer that math and art were connected. His art pieces of landscapes and abstract art reflected that. He like Da Vinci used art to express his logic, nature and point of view. Mondrian felt any shape is possible to create with basic geometric shapes. The Golden Ratio is one of the basic shapes continuously appearing in his work.
What many do not realize is that fractal art is not new, it just has a new name and is created a little different, but the format is still the same.
Fractal art by Shivasly is the art of our world being given a new expression through the programmer’s knowledge of algorithms which are written to emulate nature and are expressed through the designer’s sense of style, color, scale, proportion, relation, imagination and human expression.
Art Decor by ShivaSly™
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Visit the gallery page and see some of our amazing stylish pieces for you to buy now for your home or business. Each piece will share to your clients what your deepest desires want to portray. You will convey that you and your business believes in being unique, creative and stylish. One look is all it will take.
Here is a wonderful article from Wiki about algorithm which I believe you will like Algorithmic Art
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String Theory and Mathematical Physics
String theory is an interdisciplinary field of fundamental research, with very fruitful interfaces to high energy physics, general relativity, cosmology, statistical physics and mathematics. It currently addresses two distinct and fundamental problems of modern theoretical physics: The unification of all interactions, including gravity, and the physics of strongly interacting quantum field theories.
String Theory and Gravity
String theory is a far reaching generalization of quantum field theory. It is based on the premise that the fundamental constituents of Nature are strings, that is, one-dimensional rather than point-like objects. Elementary particles such as the electron, photon and also the graviton arise as particular vibrational modes of the string. By allowing strings to split and join, one recovers all known fundamental forces, including gravity. On sufficiently large scales, much larger than the size of the string, the stringy version of gravity looks like Einstein's theory of general relativity. The extended nature of strings, however, leads to important deviations from general relativity at small scales. Thereby, string theory succeeds in consistently combining gravity with the principles of quantum physics. Indeed, some important examples show that the stringy version of quantum space-time is somewhat fuzzy or, in more scientific terms, that the quantum theory of strings can be regular even in cases when the space-time background is singular. When applied to cosmological evolution, big-bang singularities or black-hole physics, this smoothing effect of string theory might be able to resolve the conceptual problems that have been raised in the endeavor to understand quantum effects of the gravitational field.
String Theory and Gauge Theory
Gauge theories have been used for several decades to model Nature. In perturbation theory, many interesting quantities can be computed, leading to theoretical predictions of stunning accuracy. Perturbative gauge theory, however, is restricted to a régime in which the couplings between particles can be considered small. This is not always the case. Strongly coupled gauge physics for example is responsible for the observed confinement of quarks. More recently, a strongly coupled quark-gluon plasma has been produced in heavy ion collisions at RHIC. According to a seminal proposal by Juan Maldacena, strongly coupled 4-dimensional (super-symmetric) gauge theories may be described through closed strings in a carefully chosen 5-dimensional background. In fact, equivalences - or dualities in the modern parlance - between gauge and string theories emerge, provided the strings propagate in a 5-dimensional space of constant negative curvature. Such a geometry is called an Anti-deSitter (AdS) space and the duality involving strings in an AdS background became known as the AdS/CFT correspondence. Through this duality, string theory techniques are beginning to give access to strongly coupled gauge physics.
The string theory group at DESY pursues research on a variety of issues, ranging from string theory in Anti-deSitter spaces, through aspects of quantum (stringy) geometry, to problems in 2-dimensional field theories and statistical physics. It profits from a very stimulating research environment at DESY and Hamburg University. This includes groups working in string phenomenology, astroparticle physics, quantum chromodynamics, mathematics and mathematical physics. The interaction is supported by the German Science Foundation (DFG) through a Collaborative Research Center Sonderforschungsbereich 676 on Particles, Strings and the Early Universe. DESY's string theory group is also part of Hamburg's Center for Mathematical Physics, a common initiative with Hamburg University.
Some suggested lecture notes for further reading:
Strings for Quantumchromodynamics, arXiv:0706.1209 [hep-ph].
Strings through the microscope, arXiv:hep-th/0404262.
Lectures on branes in curved backgrounds, arXiv:hep-th/0209241.
Non-compact string backgrounds and non-rational CFT, arXiv:hep-th/0509155.


Introduction to Quantum Computing for Machine Learning
|The Bloch Sphere
|QML to Solve Linear Algebra Problems
|Quantum Principal Component Analysis
|Quantum Support Vector Machines
|Quantum Deep Learning
As an outcome of decades of hard work quantum computing has finally found its way with the advancement of technology to realize quantum computers. There has been interest from many big firms like IBM, Microsoft, Google, etc. as they know about the serious number-crunching abilities of quantum computers. Until a few years ago quantum computers were only in theory until serious breakthroughs were made in the field of hardware required to make these machines a reality. It is often considered that the famous physicist Richard Feynman was one of the earliest conceptualizers of the theory of quantum computers. The idea of quantum computers is important to model the vast intricacies of nature as nature is built on top of quantum mechanics which cannot be modeled by classical computing techniques we use.
One of the most important properties of quantum computers is their ability to crunch numbers at a really high rate. For example, take a book in a library and mark an X on one of its pages. Now keep this book among 1 million other books in the library. Now you use a classical computer using 1 and 0 bits as well as a quantum computer using qubits to find that page. It'll take moments for the quantum computer to find that page whereas the classical computer will have to go through all the pages to find that X. This happens because of the interesting properties of quantum mechanics that we'll discuss in this article. We'll also discuss how quantum learning techniques have been applied to machine learning to give birth to a whole new field called quantum machine learning. Let's get started!
Basics of Quantum Computing
The major difference between quantum computers and traditional computers is that the traditional computer works on bits of data that are binary, or Boolean, with only two possible values: 0 or 1. On the other hand, quantum computers use something called "qubits". Qubits act like atoms in quantum computers. There are many tools in quantum physics that describe the interaction between different atoms. An atom can act as both particle and wave with the waveform having the capability of storing enormous amounts of data compared to a particle (or bit).
A quantum computer uses these qubits to supply information and communicate. The most important property of qubits is that they are encoded with quantum information in both states 0 and 1 instead of classical bits which can only take values 0 or 1. This means a qubit can be in multiple places at the same time due to superposition and hold an enormous amount of data. have a look at the following figure to understand how qubits can be distributed.
If all the things discussed above went above your head then don't worry. Now we'll discuss some basics of quantum computing and then we'll get back to more high-level concepts.
This is a commonly used notation to write equations in quantum mechanics and quantum physics. This is a must-know for everyone because it's commonly used in research papers and blogs related to quantum computing.
The notation uses angle brackets < > and vertical bar | to construct "bras" and "kets". A "ket" looks like this : |v>. It denotes a vector mathematically, in a complex space v. In physical terms it represents the state of a quantum system.
A bra looks like: <f| . In mathematical terms, it denotes a linear function f: V → C which maps each vector in V to a number in complex plane C. A linear function acting on a vector is written as (<f| act on a vector |V>):
<f|v> ⍷ C
Wave functions and other quantum states can be represented in the complex vector space using the Bra-ket notation.
Now let's revisit the concept of "qubits". You know that that quantum computer uses them instead of using the standard bits which take only two values i.e. 0 and 1. For a formal definition, qubit stands for quantum binary digit. Qubits can have multiple states at the same time with the value ranging between 0 and 1. Take the example of a coin toss with heads being 0 and tails being 1. When you toss the coin you don't know what will be the outcome until the coin lands. In other words, the outcome will exist in two different states simultaneously. This is the idea of superposition which is one of the core characteristics of quantum mechanics.
Superposition states that Qubits can exist in a combination of two states simultaneously. Another analogy to explain superposition can be when a high and low note from a musical instrument is played at the same time. This sound is neither high nor low note but a combination of two. But there's a catch - in a system following quantum mechanics you cannot directly observe superposition. If you try to measure a state of superposition that was indeterminate till a point becomes determinate as soon as you measure it. This gives a false idea that there is no superposition because you have already managed to determine a state of superposition separately.
Another fundamental principle of quantum mechanics is entanglement. To understand this we'll take another analogy of a game of wishbone. In this game, you both pull on one side of the wishbone to break it into two pieces but never look at each other's pieces to compare which one is bigger. Now in a quantum sense, as soon as you and your friend have your pieces, two things are possible: you have the biggest piece and your friend has the smaller one or vice-versa. You can also think of it as the wishbone pieces that have not yet decided which one is smaller or which one is bigger. Each wishbone piece is in superposition and their states are indeterminate. Now you need to go far away from each other and see which piece you have. As soon as you see your piece is larger it immediately determines the state of the piece your friend has.
Both the pieces were entangled but as soon as you see which piece you have your friends' piece is compelled into becoming the complementary piece. In other words, as soon as you decide to see which piece is yours the other piece loses its superposition. In a similar fashion, qubits are considered entangled if you can't describe the state of one qubit without describing the state of another- their states cannot be separated.
The Bloch Sphere
Now let's talk about The Bloch Sphere which is the mathematical representation of a qubit. It uses a two-dimensional vector with a normal length of one to represent the state of a qubit. There are two elements of this vector: a real number α and a complex number β. Have a look at the following figure and the equations mentioned to have a clearer idea of how the state of a qubit is represented:
Another important phenomenon is quantum decoherence. This happens when qubits superimpose and unwanted collapses happen randomly and naturally because of the noise in the system. This leads to errors in computation. Take the case where we think a qubit is in a superposition when it isn't and we do an operation on it, we'll get a completely different answer in this scenario. Quantum decoherence is caused by the contact of quantum systems with the environment and this is the reason that they need to be separated from the environment. When qubits interact with the environment information leaks out from that and information from the environment leaks into them. The information that leaks out could have been important for some future calculations but the information that leaks in is random noise.
Quantum Coherence helps the quantum computer to process information in a way that classical computers cannot. A quantum algorithm performs a stepwise procedure to solve a problem, such as searching a database. It can outperform the best-known classical algorithms. This phenomenon is called the Quantum Speedup.
Quantum Machine Learning(QML) with Algorithms
Quantum computers use various methods that are derived from classical machine learning to solve problems pertaining to data science. Following are some of those techniques/algorithms:
Quantum Machine Learning to Solve Linear Algebra Problems
A wide range of machine learning algorithms consist of matrix operations in a high dimensional vector space. In quantum computing, the quantum state of the qubits is a vector in 2d - dimensional complex vector space. A lot of matrix transformations happen in this space. Quantum Computers can solve common linear algebraic problems such as the Fourier Transformation, finding eigenvectors and eigenvalues, and solving linear sets of equations over 2a- dimensional vector spaces in time that is polynomial in a (and exponentially faster than classical computers due to the Quantum Speedup). One of the examples is the Harrow, Hassidim, and Lloyd (HHL) algorithm. Following is the circuit diagram for the same algorithm:
Quantum Principal Component Analysis
Principal Component Analysis (PCA) is used for dimensionality reduction and is one of the most popular algorithms in the field of classical machine learning. Dimensionality reduction comes at the cost of accuracy but if done correctly it makes the machine learning task much more comfortable because it's convenient to deal with a smaller dataset.
Take the dataset which has only 10 attributes, now it'll be easier to apply PCA to this dataset and get really good results. But when we have millions of attributes it becomes difficult to apply PCA because the importance of one attribute when compared to another in quantifying the overall variance of the dataset is very difficult to predict.
Another issue with classical techniques is that they are not good at calculating eigenvectors and eigenvalues when the dimensions of the dataset are very large. Quantum computers solve this issue by using Quantum Random Access Memory (QRAM) to choose a data vector at random. It maps that vector into a quantum state using qubits. The summarized vector that we get after Quantum Principal Component Analysis has logarithmic qubits. The chosen random vector forms a dense matrix. This matrix is called the covariance matrix.
By repeatedly sampling the data and using a trick called density matrix exponentiation, combined with the quantum phase estimation algorithm (which calculates the eigenvectors and eigenvalues of the matrices), we can take the quantum version of any data vector and decompose it into its principal components. Both the computational complexity and time complexity are thus reduced exponentially. Following is a diagram of the quantum circuit to perform Principal Component Analysis:
Quantum Support Vector Machines
Support Vector Machine is a classical machine learning algorithm. It classifies the linearly separable data into two different classes but if the data is not linearly separable then it's superimposed to a higher dimension and the dimensionality keeps on increasing until the data becomes linearly separable.
There's an issue with continuously increasing the dimensionality of the data when it comes to classical computers as after a particular limit these computers don't have enough processing power to process high dimensional data. This is where quantum computing comes in. Quantum computers can perform calculations for support vector machines at an exponentially faster rate using the principles of superposition and entanglement. Following is a diagram for the quantum circuit to perform SVM:
Quantum Deep Learning
Quantum computing can be combined with deep learning to reduce the time required to train a neural network. As you know, when the size of a neural network increases the time and the computational power required to train it increases. The solution to this is to mimic classic deep learning algorithms on an actual quantum computer.
Quantum computers are designed in such a way that the hardware can mimic a neural network instead of the software used in classical computers. Here, a qubit acts as a neuron that constitutes the basic unit of a neural network. Thus quantum computers that have a sufficient number of qubits can be used for deep learning with performance that easily surpasses that of a deep learning model on a classical computer. Following is an example of how a deep learning network using quantum computers will look like:
In this article, we covered different aspects of quantum computing starting from basics to how it can be used for machine learning. Quantum computing is a growing field with many more discoveries to be made. The rise of quantum computing will surely change the landscape of machine learning and how the algorithms are implemented using classical computing techniques. Happy learning!


Audio Domains: Time and Frequency
Audio data can be represented in one of two domains: the time domain and the frequency domain. Editing audio in a stereo audio editor (like Audacity) or DAW (like Pro Tools) typically takes place in the time domain. When you look at a standard representation of an audio signal in an editor you are looking at amplitude values at various times. A waveform display is actually an x-y (Cartesian) graph. Amplitude is represented on the y-axis, and time is represented on the x-axis, but without any negative values for x because audio always moves forward in time as a physical thing.
You can also describe the spectrum of a sound at any given moment. The spectrum view shows the amplitudes of frequencies present in a sound. This view is also an x-y graph, with amplitude values again being represented in the y-axis, but in this case frequency values are represented on the x-axis. While it is possible to have negative frequencies and negative amplitudes, simple spectrum views will only show positive y and x values.
To understand the concept of domains, you must understand that the domain is the parameter represented on the x axis. The standard waveform view has time represented on the x-axis, and therefore is a time-domain representation of sound. The spectrum view has frequency represented on the x-axis, and therefore is a frequency-domain representation of sound.
In a time-domain representation, frequency information is omitted from the graphed parameters. We may be able to deduce frequency information from the graph by counting zero amplitude crossings, but the parameter itself is not part of the graph parameters. With frequency-domain representation you do not have any information about time.
Converting Domains – The Fourier Transform
To convert from a time domain representation of sound to a frequency domain representation, you use a process called the Fourier Transform. To reverse the process and convert from the frequency domain to the time domain you use an Inverse Fourier Transform.


All children can learn math! They learn math by exploring, investigating, observing and making connections with their lives and their needs. Math is a tool for engineering, technology, science and understanding their world. Math takes the problem and helps find, explain and make sense of the solution.
We each learn to understand mathematics through Virginia's Process Goals.
- Become mathematical Problem Solvers who are able to recognize and create problems from real-world data and situations, and then apply appropriate strategies to determine acceptable solutions.
- Communicate their thinking and reasoning using the language of mathematics. Students learn to describe and defend their ideas with precision and conviction, while at the same time, learn to question and debate with others.
- Reason mathematically in order to predict, test, evaluate and justify their thinking. Number sense, proportional and spatial reasoning are essential.
- Make Connections and build upon prior knowledge to relate concepts and procedures from other disciplines in their education and their lives.
- Represent and describe their ideas, generalizations, and relationships using physical, visual, symbolic, verbal and contextual representations.
Learning Loss and Mathematics
The VDOE has launched a new initiative to help support learners in K-12 with learning loss due to the pandemic. Parents may apply for scholarships that will help pay for tutoring services. Please visit the VDOE website to learn more. UPDATE! VDOE has received the maximum number of applicants. Please check back periodically to check for a new window of applications.
FCPS1 understands that there are many children in our community that have suffered learning loss in mathematics from the pandemic. Part of our Algebra Readiness Funds from the VDOE are utilized for scholarships to Mathnasium for 8-12 weeks. There, the students work directly with math support in a one-to-one environment. We will be posting the newest round of scholarship applications for this fall soon!


The algorithm is based on the mathematical principles of the golden ratio and the Fibonacci sequence.
The algorithm begins by creating an initial shape, a square. The square, like the foundation of a building, serves as the basis for creating new shapes. The algorithm then uses the golden ratio to create new shapes that are in proportion to the initial shape. The Fibonacci sequence, with its specific pattern, moves the shapes through a sequence of iterations.
Published February 05 - 17, 2022 on versum/teia


4th October 2010
1st March 2010
One of the biggest problems when working on a game (or any project that is constantly changing) is trying to predict the future. From the start, we knew we wanted to have multiple maps with different layouts in each; however, upon implementation, many unknowns appeared. Will we ever change gravity on a map? What about the map radius? How about friction, lighting, run speed, jump height… the list continues ad nauseum.
As a developer, I’ve got to constantly ask these questions (and often times, come up with my own answers). While working on the map editor, one question came up that had never really been addressed before: Will our towers have different radii? The official answer: “I dunno’.” The programmer’s answer: “They can.”
Warning: The following post contains math. To make up for it, I’ve also included screenshots with captions consisting of mostly smaller words.
Edited: The final bit of math is now actual math and not made up wrong math.


Financial Calculators. Our financial templates are designed with the user in mind. Although intuitive to use, each calculator also contains helpful information within the cell comments, and additional resources and references are provided on the individual download pages.
Professional financial calculators for Microsoft Excel by Spreadsheet123 including Loan Calculators, Mortgage Calculators, Savings Calculators, Investment.
Any financial institutions or banks usually calculate it to determine your mortgage affordability. Calculating your debt-to-income ratio is easy, open up an excel spreadsheet, put and sum up all of your bank or financial institution debts in one column, then put and sum up all of your income in other column, and divide the sum of your debt to.
To make things easier, this Excel loan payment calculator lets you select the payment frequency from a.
Mutual Fund SIP Calculator: What will you get by investing a certain amount of money every month in Mutual Fund SIP in 10 years? If you are a first-time investor, this may be the biggest question on.
Financial Calculator (FC 200V) – Very handy and built in formulae will definitely help us save time in many situations. The CMPD function is the one that is mainly used in the financial calculator as far as the cfp/ cwm examination is concerned. The ease with which one can use this is unmatched with any other instrument.
This page lists the built-in excel financial functions. These functions perform many of the common financial calculations, such as the calculation of yield, interest.
A compilation of free financial calculators involving mortgages, loans, investments, debt, credit cards, retirement, and more, each with related information to gain in-depth knowledge. Also, explore a large collection of other calculators covering math, fitness, health, and many more.
The aim of this free investment/financial calculator and tutorial is to introduce these two concepts and at the same time use Microsoft Excel to develop financial .
Commercial Loan Vs Residential Loan King said that growth in commercial loans and other consumer loans should offset the residential mortgages that M&T has been running off since it acquired hudson city Bancorp in 2015. Credit quality.How To Get A 2 Million Dollar Loan
Create your own personal financial plan with this free excel-based financial planning template. Ever since I made my first retirement calculator, I have had several requests to make an integrated financial plan creation tool. The only reason I took this long to make it is sheer boredom. I thought.


This use case is described based on the 8/5 calendar, which is shown below.
Choose this calendar from the drop-down list directly in the add-on:
Now the calculation for all issues will be according to the chosen calendar.
When you choose Days, the calculation occurs according to the hours in a calendar day (24h):
8h / 24h = 0.33d
So, the task was in active statuses for 0.33d (during the working day).
If you need to get data based ONLY on the working hours (for example, 8 hours), you need to choose the Business Decimal Days. It turns out that 8 working hours is your whole working day. So, the calculation occurs in such a way: 8h/8h = 1d.


It’s hard to imagine a scrappy start-up entering a prestigious business plan competition, working hard to advance, winning the competition — and then walking away from a $70,000 investment prize. But that’s exactly what Virtual Nerd did after winning the 2008-9 Olin Cup competition at Washington University in St. Louis.
An online, video-based math and sciences tutoring company, Virtual Nerd viewed the competition as a kind of “entrepreneurial boot camp,” said the co-founders, Josh Salcman, an interactive designer, and Leo Shmuylovich, an M.D.-Ph.D student in physics and cardiology at Washington University. They said they did not care how much the prize was worth.
On Feb. 5, 2009, when they won, they were ecstatic, but they felt they were unprepared to negotiate the terms of a convertible note, the form of the investment prize offered at the Olin Cup and many other competitions.


The Golden Mean and Fibonacci Numbers
By Geoffrey Dobson
The Golden Mean is a number that mathematicians have given the name Phi (?). It is a never ending (irrational) number and its value is approximately 1.618. It was first defined by Euclid in 300BC, and is regarded as the ideal ratio of the lengths of sides in designs. Euclid defined the Golden Mean as dividing a line into two segments so that the ratio of the lengths of the two segments is the same as the ratio of the total length of the line to the length of the longest segment. You can check for yourselves that (1 + 1.618)/1.618 is equal to 1.618. I will give the proof later – it involves a bit of algebra! I will also show how to construct a “Golden Rectangle” having the “ideal” shape with its sides in the ratio 1.618:1.
Ancient Greek architects and sculptors made extensive use of the Golden Mean in the design of buildings such as the Parthenon and also in the proportions of the sculptures in it.
Twentieth century mathematicians coined the name Phi in honour of the Ancient Greek sculptor Phidias who designed many of these works. Since then the Golden Mean has been used by artists and architects as an important criterion in design to give pleasing proportions. Many artists divide their paintings according to the Golden Mean to place the focus of the design in the best place. Although the Greeks argued that the Golden Rectangle was the most pleasing shape, this is not really recognised today. Most paintings and photographs have side ratios closer to 1.5:1. A modern wide screen television has a picture with the sides in the ratio 1.8 to 1. These ratios are all approximately the same as the Golden Mean but not exactly so. However during the 20th century the famous architect Le Corbusier did use the Golden Mean extensively when designing his buildings.
If you cut a square off the Golden Rectangle, the remaining rectangle is also “Golden”. You can go on cutting off smaller and smaller squares in this way till you get to a point called “The Eye of God”. If you start at the Eye of God and connect successive corners of the squares you generate a curve called a “logarithmic spiral”. I will show later that these curves are very important in nature.
The Eye of God is the focal point of the Golden Rectangle. It is approximately one third of the way in and one third of the way up. Many artists use this as a place to put the main feature of their painting.
Leonard of Pisa (aka Leonardo Fibonacci) invented the Fibonacci Series in 1202. It was based on the birth rate of a population of rabbits giving the number of rabbits in succeeding generations. Fibonacci stated the problem as follows:
“A man puts a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits are produced from that pair in a year, if it is supposed that every month each pair begets a new pair? It is also assumed that each new pair of rabbits does not reproduce until the second month”
We start with one pair that in month one gives birth to a new pair so we have two pairs.
In month 2 the first pair produces another pair but the second does not so we have 3 pairs.
In month 3 the first two pairs each produce another pair so we have 5 pairs.
In month 4 the first 3 pairs give birth so we have 8 pairs.
In this way we can construct the series 1 2 3 5 8 13 21 34 and so on.
Each succeeding number is the sum of the two previous numbers.
I will show later by another bit of algebra that as you move up the series the ratio of each term to the preceding term becomes constant, and, is the same as the Golden Mean. At the time this was regarded as somewhat magical and you might wonder what rabbits have got to do with mathematics.
In the nineteenth and twentieth centuries mathematicians and other scientists began to realise that the Golden Mean and the Fibonacci Series crop up in all sorts of things, ranging from the structure of plants and flowers to the structure of galaxies. The Golden Mean is now regarded as one of the most important constants in science ranking alongside numbers like Pi (? – the ratio of the circumference of a circle to its diameter).
In nearly all flowers the petals are arranged as rings of Fibonacci numbers. For example lilies have 3, buttercups 5, delphiniums 8, marigolds 13, asters 21 and daisies 34. Plant leaves are arranged in a spiral around the stem so as not to shade each other. It turns out that the most efficient way of doing this is to arrange the spirals as Fibonacci numbers. In beech and hazel twigs the spacing is 1/3 of a turn between leaves. Apple and apricot twigs have 2/5 of a turn, while pear and willow twigs have 3/8 of a turn. In each case spacing is the ratio of two alternate numbers in the Fibonacci Series.
The explanation of the spiral is that the growth of a stem occurs at the tip. Infant buds are formed at intervals around the tip. As the stem grows longer the buds grow in a spiral around the stem. In 1979 a mathematician named H. Vogel used a computer simulation to show that the most efficient way that leaves can be arranged round a stem is at a preferred angle of 137.5°, which he named the “Golden Angle”. It turns out that the Golden Angle is the same as dividing a circle (360°) by Phi. Rose petals grow in a spiral with each pair of petals separated by the Golden Angle. If you try any angle other than 137.5° the pattern of packing is not so efficient.
I mentioned earlier that if you divide a Golden Rectangle into successively smaller squares, you form a logarithmic spiral. This shape is a spiral in which, as you go round it, successive loops have diameters increasing in proportion to the Golden Mean. As the spirals get bigger the basic shape does not change. It looks the same whatever magnification you use to see it.
Nature loves logarithmic spirals, and many natural objects including the seeds on a sunflower, snail shells and even spiral galaxies form in logarithmic spirals. Whirlpools and hurricanes are other examples of the logarithmic spiral. Artists use logarithmic spirals when drawing things like clouds and turbulent rivers. In 2000 the biologist Vance Tucker showed that predator birds like falcons attack their prey by flying in logarithmic spirals. He showed that this happens because with their eyes fixed in the sides of their heads, the spiral is the fastest way to close in on the prey without turning their heads.
As a snail grows wider and longer inside its shell, it needs to keep building bigger and bigger chambers in which to live, but its shape does not change. Hence the shell grows in a logarithmic spiral as the perfect shape to accommodate the growing snail. Other molluscs grow in the same way. Ammonite fossils can be formed as very large logarithmic spirals.
In 1991 two French physicists made an experiment showing that the Golden Angle and the logarithmic spiral arise naturally from the dynamics of growth without divine intervention. They put a round dish of silicone oil in a magnetic field that was strongest at the outside of the circle. They dripped drops of a magnetic liquid into the centre of the dish at equal intervals. The successive drops repel each other and also move outwards at increasing speeds towards the stronger magnetic field. In this way the drops form a logarithmic spiral, with successive drops separated by the Golden Angle. Physical systems always settle into a state of lowest possible energy. The Golden Angle in a log arithmic spiral represents the minimum energy positions for the mutually repelling drops. Buds do the same thing as they and the stem grow. A spiral galaxy grows outwards from a central point in the same way.
Doing the algebra according to Euclid we get
(1+x)/x = x
This simplifies to the quadratic equation
x² – x – 1 = 0
which solves to
x = (1 + ?5) / 2
i.e. x = 1.618
For later numbers in the series we get
….., n, xn, x²n, ……
where x is the ratio from one term to the next
by definition each term is the sum of the two previous terms so
x²n = n + xn
which simplifies to
x² – x – 1 = 0
which is the same as Euclid’s equation
The Golden rectangle
From the equation we can see that a golden rectangle has the sides in the ratio
(1 + ?5) to 2
You have drawn a golden rectangle


You are probably about to begin your second exposure to linear algebra. Unlike your first brush with the subject, which probably emphasized Euclidean spaces and matrices, we will focus on abstract vector spaces and linear maps. These terms will be defined later, so don't worry if you don't know what they mean. This book starts from the beginning of the subject, assuming no knowledge of linear algebra. The key point is that you are about to immerse yourself in serious mathematics, with an emphasis on your attaining a deep understanding of the definitions, theorems, and proofs.
You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast. When you encounter the phrase as you should verify, you should indeed do the verification, which will usually require some writing on your part. When steps are left out, you need to supply the missing pieces. You should ponder and internalize each definition. For each theorem, you should seek examples to show why each hypothesis is necessary.


When God commanded Israel in the Bible, He used the 3rd prime (the number 3) extensively. God used this 3rd prime to create the heavens and the earth as illustrated in the book “How Yahweh Numbered Creation”. The 3rd prime is a master prime that God has used throughout the Bible. When the Biblical examples of the number 3 are viewed from a numerical perspective the 2+1 sequence of 3 is revealed. The Biblical examples of the 2+1 sequence are arranged in two parts. One part contains the sequential examples of the 2+1 sequence. In these example, the action of the event are in sequence.
In the non-sequential examples of the 2+1 sequence, the action is not sequential.. In these examples, God delays the +1 part of the sequence to some later time. In these examples, the number three is delayed for a dramatic highlight.


What is The CRM?
As a result, through a public-private collaboration with the University of Sevilla, we have recently developed the Crazy4media RTB Revenue Maximizer.
This system optimises price floors for Video, Display, and Native online advertising inventory sold via RTB with a view to maximising publisher revenues.
How does it work?
Is an algorithm designed to calculate the optimum price floor in Blind Second-Price Auctions. To achieve this, the algorithm uses advanced technologies to converge to the optimal solution in a very short number of stages, grouping thousands of iterations during each stage. The system can also infer the probability distribution of price bids even from reports which only return the values of the first and second highest bids.
The algorithm creates a nonlinear transcendental equation in a web of more than one million calculation points in a dimensionless price range, which gives it extreme precision. After resolving each point with the Newton-Raphson method using a logarithmic form, it rapidly converges on solutions in the real half-plane. From this inference of the joint probability distribution of bids, the algorithm generates a confidence interval in which the optimal price floor for revenue maximisation is discovered within a determined range of probability.
Watch in live
We launched our beta tests in january 2016 and the results of that testing phase are as followed:
We are now able to offer this solution to all our Video publisher partners, and will shortly be able to do the same for our native and display publisher partners.


Multiplication Chart Printables
Welcome to the ultimate resource for multiplication charts online. Download these vibrant PDF multiplication charts to assist your child in developing proficiency and boosting their confidence in mastering the multiplication facts. These printable charts are absolutely free and readily accessible.
Each chart is available in both black and white and colored templates, simply choose the template that suits your preference, download it in PDF format, and print it out
What is a Multiplication Chart
A multiplication chart, multiplication table, or multiplication grid, is a visual tool that displays the products of multiplication for numbers in a specific range. It is typically organized in a grid format, with the numbers being multiplied listed along the top and left side. The intersection of a row and column represents the product of the corresponding numbers. For example, in a 10x10 multiplication chart, the intersection of row 4 and column 6 would display the product of 4 multiplied by 6, which is 24. Multiplication charts are commonly used as a reference tool to help children and students learn and memorize multiplication facts more easily. They provide a visual representation of the multiplication table, aiding in the understanding and mastery of multiplication operations.
How to use a multiplication chart
Start by printing out one of our free multiplication charts pdf templates. Have your child identify the two numbers they want to multiply, such as 4 and 5. Then, guide them to locate the row labeled with 4 and the column labeled with 5 on the chart. The point where the row and column intersect will display the product, which in this case is 20. Encourage your child to practice finding products for different number combinations using the chart. As they become more comfortable, they can gradually transition to solving multiplication problems without the chart, building their confidence and mastery of multiplication facts along the way.