In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups.
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry.
Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory.
The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.
The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations.
The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous.
Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations,
A significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X, the factor group G/H is no longer acting on X; but the idea of an abstract group permits one not to worry about this discrepancy.
The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.
An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations m (multiplication) and i (inversion),
are compatible with this structure, that is, they are continuous, smooth or regular (in the sense of algebraic geometry) maps, then G is a topological group, a Lie group, or an algebraic group.
The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G, the geometry and analysis pertaining to G yield important results about Γ. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its finite quotients.
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.
During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical orphysical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups,which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.
This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group G: often, the group operation in G is abstractly given, but via ρ, it corresponds to the multiplication of matrices, which is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts (see Maschke's theorem). These parts, in turn, are much more easily manageable than the whole V (via Schur's lemma).
Given a group G, representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value 1, acting on the L2-space of periodic functions.
A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse, page 3.
Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. 350c69d7ab
Welcome to the group! You can connect with other members, ge...


Overview of exam results
GCSE results 2017
66% of students gained a grade 4 or higher in both English and Maths.
In English 74% of students gained a 4 or higher.
In maths 76% of students gained a 4 or higher.
20% of all GCSE grades were awarded at A*/A.
In Chemistry 66% of grades were at A*/A. In Art 42% of grades were at A*/A.
There were some notable high achievers in this year’s results. Abbey gained the equivalent of 10 A* grades closely followed by Mahil and Robert with 8 A*s and Jenna with 7 grades at A*.
Headteacher Sam Griffin said “These are an impressive set of results for this year group. I am delighted to see so many students doing so well. I would like to congratulate all of students on their achievements and to thank their staff and parents for their ongoing support. We are looking forward to welcoming students back into our Sixth Form at the start of term.”
The latest official DfE performance tables for the school's GCSE results can be found here.
A level results 2018
We are delighted to have received another strong set of A Level results, particularly with over half of all grades at A*, A and B, and a quarter of grades at A*A! These results have enabled the vast majority of our students to progress to the university courses of their choice. Individual successes at the school include 12 students with straight A*A grades (or equivalent), with two of these students being awarded straight A* grades. Maths, Chemistry and Sport students have also performed particularly well, with over 50% of grades in these subjects at A*A or equivalent.


Elementary school children who need a little extra help with their homework have a new resource at the library. Brittini Trytek, a teacher at Holy Name, will be available at the library from 5-8 on Monday, Wednesday and Thursday and all day on Saturday to assist children who need help in getting their homework done.
The children should bring their books and their homework assignment with them to the library. Brittini will help them understand how to do the assignment. She will not do the assignment for the children. With Learning Express, an online database, and other library resources, Brittini and the children can do extra practice in math and reading.
Brittini can help children research material for projects using the books in the library and trusted Internet sources. She will teach the children how to find accurate online information using the Student Research Center, Learning Express, Ebsco Digital Books and Kids.gov.


Are you looking to level up your forex trading game to maximize your profits? One of the most popular and effective techniques is the use of Fibonacci numbers. This powerful tool can help traders predict market trends and identify the best entry and exit points. Whether you’re a beginner or an experienced trader, mastering the Fibonacci sequence can give you the edge you need to succeed in the ever-evolving world of forex trading.
The Fibonacci numbers, named after an Italian mathematician, are a sequence of numbers generated by adding the two preceding numbers to create the next number in the sequence. The pattern goes 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. Traders use these numbers to plot key levels of support and resistance on a price chart. By identifying these levels, they can make informed decisions about when to buy or sell a currency pair. This strategy works particularly well in volatile markets, where prices tend to move in repetitive cycles.
But how exactly does one use Fibonacci numbers in forex trading? It all starts by identifying the highs and lows of a price chart. From there, traders can use the Fibonacci ratios to identify key levels to watch for a potential breakout. By setting stop-loss orders just below these levels, traders can minimize their risk and increase their chances of success. With practice and persistence, even beginner traders can master this powerful tool and start making more informed and profitable trades.
What are Fibonacci numbers and ratios?
Fibonacci numbers are a series of numbers where each number is the sum of the two preceding numbers. The series starts with 0, 1, 1, 2, 3, 5, 8, 13, and continues infinitely. These numbers were first introduced in the book Liber Abaci by Italian mathematician Leonardo Fibonacci in the 13th century, which described various problems involving arithmetic, algebra, and geometry.
Fibonacci ratios are the values that are derived from the series of Fibonacci numbers. These ratios are used for technical analysis in various financial markets, including forex trading. The most popular ratios are 0.236, 0.382, 0.500, 0.618, and 0.786. The inverse of these ratios, which are 1.618, 2.618, and 4.236, are also commonly used in forex trading.
Fibonacci retracements as a trading tool
One of the most popular ways to use Fibonacci numbers in forex trading is through the use of Fibonacci retracements. These retracements are based on the idea that after a significant price move in one direction, the price will often retrace a certain percentage of that move before continuing in the original direction.
Fibonacci retracements are calculated by taking two extreme points on a price chart, typically a swing high and a swing low, and dividing the vertical distance between them by the key Fibonacci ratios of 23.6%, 38.2%, 50%, 61.8%, and 100%. These ratios are based on the Fibonacci sequence, which states that each number in the sequence is the sum of the two preceding numbers (i.e. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.).
Using Fibonacci retracements in forex trading
- Identifying potential support and resistance levels: By plotting Fibonacci retracements on a price chart, traders can identify potential levels of support and resistance where the price is likely to stall or reverse. For example, the 38.2% and 61.8% retracement levels are often considered key levels to watch, as the price will often test these levels before continuing in the original direction.
- Setting profit targets and stop loss levels: Fibonacci retracements can also be used to set profit targets and stop loss levels. For example, a trader might set a profit target at the 61.8% retracement level, where there is a higher likelihood that the price will stall or reverse. Likewise, a trader might set a stop loss at the 38.2% retracement level, which would indicate that the trade is likely to be unsuccessful.
- Confirming other technical analysis: Fibonacci retracements can also be used to confirm other technical analysis indicators, such as trend lines or moving averages. If a Fibonacci retracement level coincides with a trend line or moving average, it is more likely to serve as a significant support or resistance level.
Fibonacci retracement levels table
Overall, Fibonacci retracements are a powerful tool for forex traders looking to identify potential levels of support and resistance, set profit targets and stop loss levels, and confirm other technical analysis indicators. With the help of these retracements, traders can more effectively navigate the complex world of forex trading and increase their chances of success.
How to Draw Fibonacci Retracement Levels on a Trading Chart
Fibonacci retracement levels are used in forex trading to identify potential support and resistance levels. These levels are based on the Fibonacci sequence of numbers and are widely used by traders to determine entry and exit points for currency trades.
Drawing Fibonacci retracement levels on a trading chart is a simple process that traders can easily learn and apply. Here are the steps to follow:
- Step 1: Identify the trend – Before drawing the Fibonacci retracement levels, you need to identify the trend of the currency pair you are trading. You can do this by analyzing the price movement over a period of time and looking for the direction of the trend.
- Step 2: Select the swing high and swing low – Once you have identified the trend, you need to select the swing high and swing low points. The swing high is the highest point reached by the currency pair during an uptrend, while the swing low is the lowest point reached during a downtrend
- Step 3: Draw the retracement levels – To draw the Fibonacci retracement levels, you need to use a Fibonacci retracement tool. This tool is usually available on most trading platforms. Click on the tool and then click on the swing low and drag it to the swing high point. The tool will automatically draw the retracement levels for you.
The Key Fibonacci Retracement Levels
The retracement levels that traders commonly use are 38.2%, 50%, and 61.8%. Traders also use the 23.6% and 78.6% levels, although they are not as significant as the three key levels. Here is how to interpret the retracement levels:
|38.2%||This is the first retracement level and is considered to be a shallow retracement. If the price retraces to this level, it is a sign that the trend is still strong and is likely to continue in the same direction.|
|50%||This is the second retracement level and is considered to be a moderate retracement. If the price retraces to this level, it is a sign that the trend is likely to continue, but may be losing momentum.|
|61.8%||This is the third and final retracement level and is considered to be a deep retracement. If the price retraces to this level, it is a sign that the trend is losing momentum and may be reversing.|
Tips and Tricks for Using Fibonacci Retracement Levels
Here are some tips and tricks for using Fibonacci retracement levels in forex trading:
- Don’t rely solely on Fibonacci retracement levels. They are just one tool that traders use and should be used in conjunction with other indicators and analysis.
- Use Fibonacci retracement levels in combination with support and resistance levels. This will help you identify potential entry and exit points for your trades.
- Practice drawing Fibonacci retracement levels on a demo trading account to get comfortable with the process before you begin trading with real money.
By following these steps and tips, traders can effectively use Fibonacci retracement levels to identify potential support and resistance levels for their forex trades. Remember to always practice risk management and never risk more than you can afford to lose.
Fibonacci extensions and their use in trading
Fibonacci extensions are another powerful tool in forex trading. They are used to help traders identify potential profit targets on trades. Once a trader has identified a trend, they can use Fibonacci extensions to predict where the price may move in the future.
- First, the trader identifies the swing high and swing low points of the trend.
- Then, they apply Fibonacci extensions levels to the chart.
- The levels indicate where the price may potentially reach, based on the length of the trend.
There are different levels of Fibonacci extensions, including 127.2%, 161.8%, and 261.8%. These levels can be used as potential exit points for traders, as the price may encounter resistance or support at these levels.
Fibonacci extensions can also be used in conjunction with other technical analysis tools, such as trend lines and moving averages, to confirm potential price targets.
Example of Fibonacci extensions in trading
|Date||Swing High||Swing Low||Price Target|
|January 1st||1.2000||1.1000||1.2720 (127.2% extension)|
|January 15th||1.4000||1.2500||1.6170 (161.8% extension)|
|February 1st||1.8000||1.6000||2.0860 (261.8% extension)|
In this example, the trader has identified the swing high and swing low points of three different trends. They have then applied Fibonacci extensions levels to the chart, with the levels indicating where the price may potentially reach. The trader can use these levels as potential profit targets, or as points to exit the trade.
Common Fibonacci levels used by traders
Fibonacci levels are widely used by traders in forex trading to identify potential support and resistance levels. Understanding these levels is crucial for traders to make informed decisions when it comes to placing trades. The most commonly used Fibonacci levels in trading are:
The Number 5
The number 5 is an important Fibonacci level that is used in trading. This level is not directly related to the actual Fibonacci sequence, but it is still important for traders to understand its significance. The number 5 is derived from the relationship between adjacent numbers in the Fibonacci sequence. The ratio of any two adjacent numbers in the Fibonacci sequence approaches the golden ratio of 1.618. When we add two adjacent Fibonacci numbers together, we get the next number in the sequence. For example, 3 + 2 = 5, and 5 + 3 = 8.
Traders use the 5 level as a potential reversal point. When prices break above or below this level, it often indicates a potential reversal in the direction of the trend. This level is also used in combination with other Fibonacci levels to identify potential support and resistance levels. For example, the 50% level is often used in combination with the 5 level to identify a potential reversal point.
In addition to its use in Fibonacci retracements, the number 5 is also important in other aspects of trading. For example, some traders use a 5-period moving average to identify short-term trends in the market. This moving average is calculated by taking the average closing price of the last 5 periods.
Overall, understanding the significance of the number 5 in trading can help traders make more informed decisions when it comes to placing trades. By combining this level with other Fibonacci levels, traders can identify potential support and resistance levels and make more accurate predictions about future price movements.
Identifying support and resistance levels with Fibonacci
One of the most popular ways to use Fibonacci numbers in forex trading is to identify support and resistance levels. Support levels are price points that a currency pair tends to bottom out at, while resistance levels are price points that a currency pair tends to peak at. By identifying these levels, traders can use them as indicators of potential future price movements and make more informed trading decisions.
- To use Fibonacci levels to identify support and resistance, traders first need to locate the most recent high and low points on a chart. These points will be used to draw Fibonacci retracement levels. These levels are drawn by placing horizontal lines at the appropriate Fibonacci ratios.
- Once the retracement levels are drawn, traders can look for support or resistance at these levels. Price action that shows a strong bounce off of a Fibonacci level can suggest that the level is acting as support or resistance.
- Traders should also be on the lookout for multiple Fibonacci levels that occur close to each other. This confluence can provide additional evidence that a particular level is acting as support or resistance.
For example, suppose a currency pair has recently been trending upwards and has reached a high point of 1.4500. The pair then begins to retrace downwards, with a low point of 1.4000. To draw Fibonacci levels, traders would draw a horizontal line between these two points and place horizontal lines at the appropriate Fibonacci ratios, such as 38.2%, 50%, and 61.8%.
If the currency pair bounces strongly off the 38.2% level, traders might expect that level to act as support in the future. Similarly, if the currency pair encounters strong resistance at the 61.8% level, traders might expect that level to act as resistance in the future.
Overall, using Fibonacci numbers to identify support and resistance levels can be a useful tool in forex trading. By paying attention to these levels and the price action surrounding them, traders can gain valuable insights into potential future price movements and make more informed trading decisions.
Fibonacci and Elliot Wave Theory in Forex Trading
Fibonacci retracements are a widely-used technical analysis tool in forex trading. They are based on the sequence of numbers known as the Fibonacci sequence, which is derived from the mathematical discoveries of Leonardo Fibonacci in the 13th century. Elliot Wave Theory, on the other hand, is a technical analysis approach developed by Ralph Nelson Elliot in the 1930s that identifies waves in financial markets.
Using Fibonacci Numbers in Forex Trading
- Fibonacci retracements are used to identify potential levels of support and resistance in a market. Traders use Fibonacci retracements to identify potential entry and exit points for their trades.
- The most common retracement levels are 23.6%, 38.2%, 50%, 61.8%, and 78.6%. These levels are calculated by taking the high and low points of a price movement and applying Fibonacci ratios to determine where the price is likely to retrace to.
- Fibonacci retracements are often used in conjunction with other technical indicators, such as moving averages, to confirm potential entry and exit points.
The Relationship between Fibonacci and Elliot Wave Theory
Elliot Wave Theory identifies trends in the market as a series of impulsive and corrective waves. Fibonacci retracements are often used by Elliot Wave analysts to identify potential levels of support and resistance within these waves.
For example, an Elliot Wave analyst may identify a wave as being in an impulsive phase, and use Fibonacci retracements to identify potential levels of support for buying opportunities.
Fibonacci Retracement Levels
Overall, the use of Fibonacci retracements in forex trading can be a useful tool for identifying potential entry and exit points for traders. When used in conjunction with other technical indicators, such as Elliot Wave Theory, Fibonacci retracements can help traders gain a better understanding of market trends and make more informed trading decisions.
Combining Fibonacci with other technical analysis indicators
Fibonacci retracement is just one of several technical analysis tools used in trading. However, it is commonly used alongside other indicators to confirm a trend, support or resistance level, or possible reversal points. Below are ways on how to combine Fibonacci with other technical analysis indicators:
- Moving Averages: Moving averages can help validate potential Fibonacci retracement levels. For instance, traders may be more inclined to look for retracements near key moving averages; the 50, 100, and 200 SMA. Similarly, if a retracement level coincides with a moving average, this may provide more emphasis on the level.
- Relative Strength Index (RSI): RSI is a momentum indicator used to determine if a currency is overbought or oversold. If the RSI is over 70, the currency may be overbought, and if the RSI is below 30, the currency may be oversold. Fibonacci levels can confirm RSI signals. For example, if the RSI becomes oversold at a Fibonacci retracement level, it may indicate a possible reversal of the downtrend.
- MACD: Moving Average Convergence Divergence (MACD) is a trend-following momentum indicator. This indicator is used to show the relationship between two moving averages. Similar to the RSI, traders combine it with Fibonacci retracements to confirm trend or reversal. If prices pull back to a Fibonacci level and the MACD is showing a bullish crossover, that could signal a potential bullish trend reversal.
Retracement Trading Strategies
There are different trading strategies that traders use to apply Fibonacci retracements alongside other indicators. One strategy is the “Fibonacci Fan” strategy. This involves drawing diagonal trend lines to connect significant price points and form a fan-like pattern.
Another popular Fibonacci strategy is known as the “Fibonacci extension” strategy. This technique uses Fibonacci retracements and extensions to determine price targets. Extensions are calculated by projecting the price level of a retracement from the previous trend wave. A trader can enter a position near a key Fibonacci retracement level and place a stop loss underneath. Then, they can use a Fibonacci extension level as a profit-taking target to exit the trade.
Fibonacci Retracement Levels Table
In conclusion, combining Fibonacci retracements with technical indicators can provide traders with additional confirmation of trend, support and resistance levels, or potential reversals. It is essential to have a solid understanding of both the tools and the market before applying any strategy to ensure positive trading outcomes.
Pros and cons of using Fibonacci in forex trading
Using Fibonacci in forex trading is a popular technique among traders. Fibonacci numbers are a mathematical sequence that appears in many natural phenomena. In forex trading, these numbers are used to identify potential price movements and support/resistance levels.
- The Fibonacci sequence can help traders identify possible price movements based on past trends and patterns.
- It can serve as a guideline for where to enter or exit the market.
- Fibonacci retracements can provide support and resistance levels that traders can use to set stop-loss or take-profit orders.
- It can help traders manage risk by identifying potential entry and exit points.
- The use of Fibonacci requires a certain level of technical analysis knowledge and expertise.
- The accuracy of Fibonacci levels is not guaranteed, and they should not be used as the sole basis for trading decisions.
- The strategy can be subjective and open to interpretation, leading to different outcomes.
- It can be easy to get caught up in the patterns and forget about other important factors that may affect the market.
Despite its advantages and disadvantages, the Fibonacci sequence is still a widely used tool in forex trading. Traders should always exercise caution and use it as part of a comprehensive trading plan that takes into account various factors, such as market conditions, news events and economic data.
One of the key Fibonacci numbers used in forex trading is 9. This number is often used in conjunction with the 38.2% and 61.8% retracement levels to identify potential support and resistance zones. The table below illustrates how the 9 Fibonacci number can be used in trading:
Traders can use the above table to determine potential levels for entry, stop loss and take profit orders based on the 9 Fibonacci number and the corresponding retracement and extension levels. However, it is important to keep in mind that this is not a foolproof method and should be used in conjunction with other strategies and analysis.
Real-life examples of successful Fibonacci trading strategies
There are many trading strategies that incorporate Fibonacci retracements and extensions. Here, we will explore some real-life examples of successful Fibonacci trading strategies.
- Retracement Strategy: This strategy involves identifying a trend, drawing a Fibonacci retracement from swing low to swing high (in an uptrend) or swing high to swing low (in a downtrend), and entering a trade at a retracement level. For example, if the retracement level is 50%, the trader would enter a buy (in an uptrend) or sell (in a downtrend) order at that level. If price bounces off the retracement level and continues in the direction of the trend, the trade is successful.
- Extension Strategy: This strategy involves identifying a trend, drawing a Fibonacci extension from swing low to swing high (in an uptrend) or swing high to swing low (in a downtrend), and placing buy (in an uptrend) or sell (in a downtrend) orders at the extension levels. For example, if the extension levels are 127.2%, 161.8%, and 261.8%, the trader would place buy (in an uptrend) or sell (in a downtrend) orders at these levels. If price reaches these levels and bounces, the trade is successful.
- Fibonacci Fan Strategy: This strategy involves drawing a Fibonacci fan from a swing low to swing high (in an uptrend) or swing high to swing low (in a downtrend). The fan creates diagonal support and resistance levels. Traders can use these levels to enter trades or take profit. For example, if price is in an uptrend and approaching a fan level, traders can enter a buy order at that level. If price bounces off the level and continues in the direction of the trend, the trade is successful.
Overall, Fibonacci retracements and extensions can be powerful tools when used in conjunction with other technical analysis tools and market knowledge. However, traders should always use proper risk management and consider the possibility of false signals or market manipulation.
Below is a table of common Fibonacci retracement and extension levels:
Remember that Fibonacci levels are not always foolproof, and it’s essential to use other technical analysis indicators and market knowledge in combination with Fibonacci tools. When used properly, Fibonacci retracements and extensions can give traders an edge in analyzing market movements and making informed trading decisions.
FAQs about How to Use Fibonacci Numbers in Forex Trading
1. What are Fibonacci numbers in forex trading?
Fibonacci numbers are a series of numbers used by traders in financial markets to predict future price movements.
2. How are Fibonacci numbers used in forex trading?
Fibonacci retracement levels are used by forex traders to identify potential levels of support and resistance in the market. Traders plot these levels on their charts to help them identify potential buy and sell points.
3. What is Fibonacci retracement level?
Fibonacci retracement level is the percentage of a price move that retraces before continuing in the original direction. These levels are drawn from the highest point to the lowest point, or vice versa, in a given price move.
4. Can Fibonacci numbers be used to predict future forex market trends?
Fibonacci numbers and retracement levels cannot predict future market trends with certainty. In fact, traders should always consider other technical and fundamental indicators before they make trading decisions.
5. How do I add a Fibonacci retracement level to my chart?
Most forex trading platforms have built-in Fibonacci tools that you can use to add retracement levels to your chart. Look for the Fibonacci Retracement option in your platform’s charting tools.
6. How do I interpret the Fibonacci retracement levels on my chart?
The retracement levels help traders identify areas of support and resistance in the market. These levels can be used to determine potential entry and exit points, as well as stop-loss and take-profit levels.
7. Are there any downsides to using Fibonacci numbers in forex trading?
Some critics argue that Fibonacci numbers and retracement levels are not based on any sound mathematical or statistical principles. However, many traders have found success using these tools in their trading.
Closing Title: Thanks for Reading Our Guide on How to Use Fibonacci Numbers in Forex Trading
We hope this guide has been helpful in gaining a better understanding of how to use Fibonacci numbers and retracement levels in forex trading. Remember that Fibonacci numbers are just one tool in your trading arsenal, and it should always complement your trading strategy. Thanks for reading and be sure to visit us again for more useful trading tips and information.


How Much Math Do You Need for Computer Science?
Mathematics is an essential element in the field of computer science and plays a vital role in various areas within the discipline. Whether you are interested in coding, algorithms, data analysis, or artificial intelligence, a solid understanding of math concepts is crucial for success.
Foundational Math Skills
At the very least, computer science students should have a strong foundation in basic math skills. This includes a solid understanding of arithmetic, algebra, and geometry. These fundamental areas of math provide the building blocks for more advanced concepts, making them essential for any aspiring computer scientist.
Arithmetic (basic calculations like addition, subtraction, multiplication, and division) is essential for understanding the basics of coding and algorithms. Algebra helps students develop problem-solving skills and logical thinking, which are crucial for writing efficient and optimized code. Geometry, on the other hand, is necessary for understanding concepts related to geometric algorithms, computer graphics, and visualization.
Discrete mathematics is an area of math that focuses on countable and distinct elements, and it is highly relevant to computer science. It covers topics such as set theory, logic, graph theory, combinatorics, and probability theory. These concepts help computer scientists understand and analyze algorithms, data structures, and cryptography.
Set theory provides a foundation for understanding data structures like sets, arrays, and trees. Logic is essential for designing and analyzing the correctness of algorithms. Graph theory helps in understanding and optimizing network algorithms and data relationships. Combinatorics plays a role in analyzing the efficiency of searching and sorting algorithms. Probability theory is crucial for machine learning algorithms and artificial intelligence tasks that involve uncertainty and randomness.
Calculus and Continuous Mathematics
While not every computer science student may need to delve deeply into calculus, having a basic understanding of calculus and continuous mathematics can be beneficial, especially for certain areas within computer science.
Calculus helps computer scientists understand concepts such as optimization, approximation, and continuous change over time. It is particularly important in areas like machine learning, data analysis, and computer graphics. A solid foundation in calculus allows computer scientists to develop and optimize algorithms for complex computational problems.
Linear algebra is another important branch of math that is highly applicable to computer science. It deals with systems of linear equations, vectors, and matrices. Linear algebra is widely used in diverse areas such as computer graphics, cryptography, data compression, and machine learning.
Understanding linear algebra allows computer scientists to manipulate and analyze large data sets, apply transformations to graphical objects, optimize algorithms, and solve systems of equations efficiently.
Mathematics forms a strong foundation for computer science and is integral to many areas within the field. While the level of math required may vary depending on the specific area of focus, having a solid understanding of arithmetic, algebra, discrete mathematics, calculus, and linear algebra is crucial for success in computer science.
By mastering these math concepts, computer scientists can develop efficient algorithms, design robust systems, analyze data effectively, and explore the frontiers of artificial intelligence. So, embrace math as you embark on your computer science journey and unlock the immense possibilities it offers.
Foundational Mathematical Concepts
A solid foundation in discrete mathematics, including sets, logic, and algorithms, is essential for tackling computer science problems effectively. Discrete mathematics provides the logical framework and tools required to understand the fundamental principles of computer science and programming.
One of the key components of discrete mathematics is understanding sets. In computer science, sets are used to organize and categorize data. They are particularly useful when dealing with large amounts of information. For example, in a database, sets can be used to represent a collection of records or entities. Understanding concepts such as unions, intersections, and complements of sets is crucial for efficiently manipulating and analyzing data.
Logic is another important aspect of discrete mathematics. It provides the foundation for creating and evaluating logical statements, which are fundamental in computer science. Logic helps identify patterns, make deductions, and reason through complex problems. It allows computer scientists to construct solid and reliable arguments, which are essential for designing algorithms and creating robust software.
Algorithms form the backbone of computer science. These step-by-step procedures are used to solve problems and perform computations. To effectively design and analyze algorithms, a strong understanding of mathematical concepts is necessary. This includes grasping concepts such as efficiency and complexity, which are directly related to the mathematical representation and analysis of algorithms. By understanding these concepts, computer scientists can create efficient and optimized algorithms to solve problems in the most effective manner.
Additionally, a strong foundation in discrete mathematics helps develop critical thinking skills, which are essential in computer science. The ability to break down a problem, analyze its components, and devise a solution is crucial when working with complex algorithms and systems. Discrete mathematics provides the necessary tools and techniques to approach problems logically and systematically.
In summary, a solid understanding of the foundational mathematical concepts in discrete mathematics is crucial for computer scientists. Sets, logic, and algorithms are at the heart of computer science, and a strong foundation in these areas is necessary for tackling complex problems and creating efficient solutions. By developing a solid mathematical foundation, computer scientists can effectively navigate the ever-changing landscape of technology and contribute to advancements in the field.
Calculus and Analysis
A basic understanding of calculus, particularly differential calculus and sequences, is useful for analyzing algorithms and computational complexity. Calculus is a branch of mathematics that deals with the study of change, particularly of functions and their rates of change. It helps in understanding how quantities change over time or space.
In computer science, calculus is particularly important for analyzing algorithms and computational complexity. Algorithms are step-by-step procedures used for solving a problem, and understanding their efficiency is crucial for developing efficient programs. Computational complexity, on the other hand, deals with measuring the resources required to execute an algorithm, such as time and memory.
Differential calculus, which focuses on the concept of derivatives, is especially relevant in computer science. Derivatives measure the instantaneous rate of change of a function and have applications in optimization and approximation algorithms. For example, in machine learning, derivatives are used to optimize the parameters of a model to minimize the error between predicted and actual outcomes.
Sequences, another topic within calculus, are important for understanding the behavior of algorithms. In computer science, algorithms often deal with sequences of data, such as arrays or linked lists. Understanding the properties and behavior of these sequences helps in analyzing and designing efficient algorithms.
Overall, a basic understanding of calculus, particularly differential calculus and sequences, provides essential tools for analyzing algorithms and computational complexity in computer science. It helps in optimizing algorithms and understanding their efficiency in terms of time and memory consumption.
Linear Algebra and Matrices
In computer science, a strong foundation in linear algebra and matrices is essential. These mathematical concepts play a crucial role in various applications such as graphics, machine learning, and cryptography.
Linear algebra deals with vector spaces and linear transformations. It provides the tools and techniques to solve systems of linear equations, analyze geometric transformations, and understand the properties of vectors and matrices. Matrices are a fundamental component of linear algebra, representing linear transformations and assisting in solving complex problems efficiently.
One significant area where linear algebra is utilized in computer science is graphics. Graphics involve creating and manipulating images, animations, and visual effects using computer algorithms. Techniques such as 3D modeling, rendering, and image processing heavily rely on linear transformations and matrix operations. For instance, rotating and scaling objects in computer graphics involve applying linear transformations to vectors, and matrices are used to represent these transformations.
Another important application of linear algebra in computer science is machine learning. Machine learning algorithms train models to make predictions or decisions without being explicitly programmed. Linear algebra is used to represent and manipulate data in these models. For example, in linear regression, a popular machine learning algorithm, linear algebra is used to estimate the coefficients of a linear equation that best fits the given data.
Cryptography, the study of secure communication, also heavily relies on linear algebra. Cryptographic algorithms, like the widely used RSA algorithm, involve modular arithmetic operations that can be represented using matrices. Linear algebra techniques, such as matrix multiplication, inversion, and exponentiation, are fundamental in implementing and analyzing these cryptographic algorithms.
Having a strong understanding of linear algebra and matrices allows computer scientists to develop efficient and optimized algorithms, understand the mathematical foundations of various computer science applications, and tackle complex problems in the field.
Probability and Statistics
Probability and statistics are essential concepts in computer science, as they provide the foundation for understanding and analyzing data. Whether you’re working on data analysis, machine learning, or network modeling, a strong grasp of probability and statistics is crucial.
Probability is the study of chance and uncertainty. It allows us to quantify the likelihood of an event happening. In computer science, probability is used to analyze data and make predictions. For example, in machine learning algorithms, probability is used to determine the likelihood of a certain outcome based on patterns found in the data.
Statistics, on the other hand, involves the collection, analysis, interpretation, presentation, and organization of data. It helps us make sense of the information we gather and draw valuable insights. In computer science, statistics is used to analyze large datasets and make informed decisions.
One area of computer science where probability and statistics are extensively used is data analysis. Data analysis involves sorting, cleaning, and analyzing large sets of data to uncover meaningful patterns and insights. By applying statistical techniques, such as hypothesis testing and regression analysis, computer scientists can draw valuable conclusions from the data.
Machine learning, a subset of artificial intelligence, also heavily relies on probability and statistics. Machine learning algorithms learn from past data and use that knowledge to make predictions or decisions. Probability is used to find the most likely outcome, while statistical models help in training and evaluating machine learning algorithms.
For example, in image recognition tasks, probability and statistics are used to classify images into different categories. The algorithm compares the features of the image with the probabilities of each category to determine the most likely classification. Statistical models are then used to optimize and improve the accuracy of the algorithm.
Network modeling is another area of computer science where probability and statistics come into play. Networks, such as social networks or computer networks, are complex systems with interconnected entities. Probability is used to model the behavior of these networks and predict outcomes.
For example, in network security, probability is used to analyze the likelihood of a cyber-attack happening and the potential impact it could have. By understanding the probabilities, computer scientists can develop strategies to protect the network and mitigate the risks.
In conclusion, probability and statistics are vital for computer science. They provide the necessary tools to analyze data, make predictions, and understand complex systems. Whether you’re working on data analysis, machine learning, or network modeling, a solid understanding of probability and statistics is essential for success.


While taking a disease dynamics course, I thought it would be a good opportunity to learn how to use the
Shiny package in
R and create an interactive interface for some of my problem sets. After a few trial runs with smaller, simpler setups, I have wrapped up the side project (for now). You can see it in action here 1 and you can view the final code on my Git.
At some point, I’d like to make an analogous version of these models using network-based approaches. However, all my work in network models has been done using Python so it might take a while.
If you’re unfamiliar with compartmental models, they are deterministic models that use differential equations to describe the spread of an epidemic through a population. The Wikipedia page on them is a pretty good place to start. The parameters on the page are described below—note that certain parameters are only shown when they are applicable.
- Probability of transmission: the probability that an infectious person will infect a susceptible person at any one contact.
- Average contacts: the number of people an infected person will run into (per week).
- Disease properties:
- No recovery: when checked, if somebody becomes infected, they will be infectious forever—never recovering.
- Duration: how long an infected person remains infected
- Latent period: the time between being infected and being infectious (at which point they are neither susceptible nor infected, but are “exposed”).
- Seasonal fluctuations: a cosine function that emulates seasonal fluctuations in contact rates.
- Vital dynamics:
- Birth and death rate: Self-explanatory. In these model, kept equal to each other.
- Proportion of vaccinated a birth: assumes vaccination occurs immediately at birth (therefore, this is the proportion of new births who never enter a susceptible stage).
- Vaccine effectiveness: probability of a vaccine actually working.


The aim of the quantity is to supply a aid for a primary path in arithmetic. The contents are organised to charm particularly to Engineering, Physics and computing device technology scholars, all components within which mathematical instruments play a vital position. easy notions and techniques of differential and vital calculus for services of 1 actual variable are offered in a fashion that elicits severe examining and activates a hands-on method of concrete purposes. The format has a specifically-designed modular nature, permitting the teacher to make versatile didactical offerings whilst making plans an introductory lecture path. The e-book could in reality be hired at 3 degrees of intensity. on the effortless point the scholar is meant to understand the very crucial principles and familiarise with the corresponding key ideas. Proofs to the most effects befit the intermediate point, including numerous feedback and complementary notes improving the treatise. The final, and farthest-reaching, point calls for the extra research of the cloth inside the appendices, which permit the strongly prompted reader to discover additional into the topic. Definitions and houses are offered with large examples to stimulate the training approach. Over 350 solved routines whole the textual content, a minimum of half which advisor the reader to the answer. This re-creation beneficial properties extra fabric with the purpose of matching the widest diversity of academic offerings for a primary process arithmetic.
Read or Download Mathematical Analysis I (UNITEXT) PDF
Similar Mathematics books
Chosen Works of Giuseppe Peano (1973). Kennedy, Hubert C. , ed. and transl. With a biographical caricature and bibliography. London: Allen & Unwin; Toronto: college of Toronto Press.
The information of Fourier have made their means into each department of arithmetic and mathematical physics, from the idea of numbers to quantum mechanics. Fourier sequence and Integrals makes a speciality of the intense energy and adaptability of Fourier's easy sequence and integrals and at the impressive number of purposes during which it's the leader instrument.
Authored by way of a number one identify in arithmetic, this attractive and obviously offered textual content leads the reader throughout the a number of strategies desirous about fixing mathematical difficulties on the Mathematical Olympiad point. overlaying quantity thought, algebra, research, Euclidean geometry, and analytic geometry, fixing Mathematical difficulties contains quite a few workouts and version options all through.
A few books on algorithms are rigorous yet incomplete; others hide lots of fabric yet lack rigor. creation to Algorithms uniquely combines rigor and comprehensiveness. The ebook covers a large variety of algorithms intensive, but makes their layout and research obtainable to all degrees of readers.
Additional info for Mathematical Analysis I (UNITEXT)
Then: a) If f expanding on I, then f′(x) ≥ zero for all x ∈ I. b1) If f′(x) ≥ zero for any x ∈ I, then f is expanding on I; b2) if f′(x) > zero for all x ∈ I, then f is exactly expanding on I. evidence. allow f be the map. consider first f is continuous, for this reason for each x zero ∈ I, the variation quotient , with x ∈ I, x ≠ x zero, is 0. Then f‱(x zero) = zero via definition of by-product. Vice versa, consider f has 0 spinoff on I and allow us to end up that f is continuing on I. this may be resembling challenging Take x 1, x 2 ∈ I and use formulation (6. thirteen) on f. For an appropriate among x 1,x 2, we now have therefore f(x 1) = f(x 2). □ 6. 7 6. 7 Monotone maps within the mild of the consequences on differentiability, we take on the problem of monotonicity. evidence. allow us to turn out declare a). believe f expanding on I and think about an inside aspect x zero of I. For all x ∈ I such that x < x zero, we now have therefore, the adaptation quotient among x zero and x is non-negative. nevertheless, for any x ∈ I with x > x zero. the following too the adaptation quotient among x zero and x is confident or 0. Altogether, Corollary four. three on yields f′(x zero) ≥ zero. As for the potential extremum issues in I, we arrive on the comparable end via contemplating one-sided limits of the variation quotient, that's constantly ≥ zero. Now to the results in elements b). Take f with f′(x) ≥ zero for all x ∈ I. the assumption is to mend issues x 1 < x 2 in I and turn out that f(x 1) ≤ f(x 2). For that we use (6. thirteen) and notice that through assumption. yet because x 2 − x 1 > zero, now we have proving b1). contemplating f such that f′(x) > zero for all x ∈ I as an alternative, (6. thirteen) implies f(x 2) − f(x 1) > zero, accordingly additionally b2) holds. □ the theory asserts that if f is differentiable on I, the next common sense equivalence holds: additionally, The latter implication isn't really reversible: f strictly expanding on I doesn't suggest f′(x) > zero for all x ∈ I. we have now somewhere else saw that f(x) = x three is in all places strictly expanding, regardless of having vanishing spinoff on the foundation. an identical assertion to the above holds if we modify the observe ‘increasing’ with ‘decreasing’ and the symbols ≥, > with ≤, <. Corollary 6. 28 permit f be differentiable on I and x zero an inside severe aspect. If f′(x) ≥ zero on the left of x zero and f′(x) ≤ zero at its correct, then x zero is a greatest element for f. equally, f′(x) ≤ zero on the left, and ≥ zero on the correct of x zero implies x zero is a minimal element. Theorem 6. 27 and Corollary 6. 28 justify the hunt for extrema one of the zeroes of f′, and clarify why the derivative's signal impacts monotonicity periods. instance 6. 29 The map f : ℝ → ℝ, f(x) = xe2x differentiates to f′(x) = (2x + 1)e2x , whence is the only serious element. As f′(x) > zero if and provided that is an absolute minimal. The functionality is exactly reducing on and strictly expanding on . □ 6. eight 6. eight Higher-order derivatives enable f be differentiable round x zero and permit its first by-product f′ be additionally outlined round x zero. Definition 6. 30 If f′ is a differentiable functionality at x zero, one says f is two times differentiable at x zero. The expression is termed moment spinoff of f at x zero.


Neng Fan is an associate professor in the Systems and Industrial Engineering Department at the University of Arizona. He obtained his bachelor’s degree in computational mathematics from Wuhan University in 2004 and master’s degree in applied mathematics from Nankai University in 2007. He obtained his master’s and PhD degrees from the Industrial and Systems Engineering Department at the University of Florida in 2009 and 2011, respectively.
Fan’s research interests include integer programming, stochastic and robust optimization methods, and their applications in data mining, combinatorial optimization, energy systems, water systems, and health care. He has been involved in organizing several conferences in the areas of optimization, smart grids, and health care, and also served as a reviewer for many journals. Currently, he is the associate editor for two journals, Optimization Letters and Energy Systems, published by Springer.
Fan is a member of the Institute for Operations Research and Management Sciences (INFORMS) and the Institute of Industrial Engineering (IIE).
- PhD Industrial and Systems Engineering - University of Florida, 2011
- MS Industrial and Systems Engineering - University of Florida, 2009
- MS Applied Mathematics - Nankai University, 2007
- BS Computational Mathematics - Wuhan University, 2004
Teaching Interests: Optimization, operations research, probability, statistics, data analytics, and machine learning
Research Interests: Methodologies in optimization: integer programming, combinatorial optimization, stochastic programming, robust optimization, multilevel programming, and large-scale optimization; applied operations research: energy systems, water systems, renewable energy integration, interdependent infrastructures, healthcare and sustainable agriculture; data analytics: data mining, machine learning, and high-dimensional data with uncertainties


In 2011, the Trends in International Mathematics and Science Study (TIMSS) showed that South African learners have the lowest performance among all 21 middle-income countries that participated. South Africa’s development as a knowledge economy depends partly on improving the teaching of mathematics and numeracy. Furthermore, South Africa’s extremely high youth unemployment, which is currently at 50%, is closely linked to the quality of schooling; numeracy and mathematics competency in particular. We have looked at the education Stats for 2013 and 2014 as shown on department of education website, and the results show that we still have a long way to go in terms of improving the situation for the future of this country.
Nubian Smarts believes that building a solid math foundation is vital for children to succeed. Hence our interest in creating applications for primary school children. Studies have shown that students with weak basic math skills find the subject increasingly difficult and confusing thus leading them to get poor results. This often leads to math anxiety and the belief that mathematics is hard and not fun, which is how most people feel when asked about their experiences with regards to maths.
So in essence when a child develops a solid math foundation, you notice that the stress caused by poor math skills disappears, and you might notice the child even saying that maths is fun. We at Nubian Smarts believe that this is not an unattainable goal and that the key to improving these statistics is by creating mobile applications that focus on creating a lifelong love of mathematics. These mobile applications are not intended to be a unique educational method but rather to complement the other areas of the educational programme.
Studies conducted have reported that mobile apps are not only engaging, but educational, for children as young as preschool. PBS Kids, in partnership with the US Department of Education, found that the vocabulary of kids’ age’s three to seven who played its Martha Speaks mobile app improved up to 31%. Abilene Christian University conducted research around the same time that found math students who used the iOS app “Statistics 1” saw improvement in their final grades. They were also more motivated to finish lessons on mobile devices than through traditional textbooks and workbooks.
More recently, two studies that separately followed fifth and eighth graders who used tablets for learning in class and at home found that learning experiences improved across the board. 35% of the 8th graders said that they were more interested in their teachers’ lessons or activities when they used their tablet, and the students exceeded teachers’ academic expectations when using the devices.


Are you ready to take a step towards understanding the fundamental difference between feet and square feet? Whether you’re a novice in the world of measurement or simply curious about how these terms are used, this blog post will unravel the mystery for you. From finding the perfect shoe fit to calculating your living space, we’ll explore why it’s essential to distinguish between regular feet and their squared counterpart. Get ready to put your best foot forward as we dive into this fascinating topic!
Feet Vs Square Feet (Comparison Table)
|The term “foot” refers to a unit of measurement equal to 12 inches (30.48 cm) in length.
|The term “square foot” refers to a unit of area measuring equal to one foot by one foot (1 sq ft = 12 in x 12 in = 144 sq in).
|The unit symbol for the foot is “ft”.
|The unit symbol for square feet is “sq ft”.
|The Foot is used to measure lengths, such as heights and distances.
|Square feet are used to measure area, often for rooms or land parcels.
|You cannot directly convert feet to square feet due to the different measurement types.
|To convert from feet to square feet, calculate the area in square inches and then convert it using the appropriate factor.
|Feet are measured in linear units.
|Square feet are measured in terms of area units.
|The term “foot” is derived from the Latin word “pedem,” meaning foot.
|The term “square foot” combines “square” and “foot” to denote an area defined by one-foot sides of a square.
Feet (plural of foot) is the non-SI unit for measuring length. It is used mainly in the Imperial and U.S. customary systems of measurement and is equal to 12 inches or 30.48 centimeters. We often use feet to measure the height of something or the distance between two points.
Examples of Feet
- The height of a basketball hoop is 10 feet.
- The length of a soccer field is 120 feet.
- The depth of an Olympic-size swimming pool is 10 feet.
What is Square Feet?
Square feet, on the other hand, is a unit of measurement used in the imperial and US customary systems to measure area. It is equal to one square foot, which is defined as an area with sides of one foot in length. One square foot is equal to 144 square inches and 0.09290304 square meters.
Examples of Square Feet:
- The average size of a two-bedroom apartment is 1,000 square feet.
- The size of an Olympic-size swimming pool is 7,600 square feet.
- A standard bowling alley lane measures 41 ½ feet long by 12 ½ feet wide, for a total area of 518 ¾ square feet.
Key Differences Between Feet and Square Feet Explained
- Meaning: The term “foot” refers to a unit of measurement that is equal to 12 inches (30.48 cm), while the term “square foot” refers to a unit of area measure equal to one foot by one foot (1 sq ft = 12 in x 12 in = 144 sq in).
- Unit Symbol: The unit symbol for the foot is simply “ft”, while the unit symbol for square feet is “sq ft”.
- Conversion: It is not possible to directly convert from feet into square feet because they are different units of measure; however, it is possible to calculate the area of an object or space given its dimensions in feet and then convert this area into square feet using the appropriate conversion factor.
- Measurement: The measurement of feet is taken in linear units while the measurement of square feet is taken in terms of area.
How to Convert Feet to Square Feet
There are 12 inches in a foot, so to convert feet to square feet, you need to multiply the number of feet by 12. For example, if you have a room that is 10 feet wide and 20 feet long, the area of the room would be 200 square feet. Mathematical derivation is as follows:
Area = Length x Width
Area = 10 feet x 20 feet
Area = 200 square feet
Common Mistakes Made When Measuring with Square Feet
When it comes to measuring square feet, there are a few common mistakes that are made.
First, when measuring square footage, you should always use a tape measure. Measuring with a ruler will not give you an accurate measurement.
Second, when measuring the length and width of a room, make sure to measure from the outside edges of the room.
Third, make sure to include all areas of the room when measuring square footage. This includes closets, hallways, and any other areas that are part of the room. Don’t forget to convert your measurements into square feet. To do this, simply multiply the length by the width.
We hope that this article has helped you understand the fundamental difference between feet and square feet. Knowing the unit of measurement for each can be important when planning projects or taking measurements. It is also a useful tool to know when shopping for items that require precise measurements, such as flooring, appliances, furniture, etc.
In addition to understanding these units of measure, it is important to have the right tools in order to make sure your project turns out perfectly!


Nim - A Devilishly Difficult Game
Ages: 7 and up
Math Ideas: Subtraction, deductive reasoning, mathematical proof
Questions to Ask:
Want to figure out a strategy to always win?
Do you want to go first or second this time?
If you make that move, what do you think I will do?
Years ago, a teacher friend sent me an online version of the game Nim. "Try it out!" they said innocently enough, "See if you can figure out the strategy!"
The game seemed simple enough, basically just a set of three rows of objects. So I gave it a shot, and I lost. And then I lost again, and again. I finally walked away from my computer, a broken man.
In the years since then, I've moved cities and schools, had three kids, and STILL CANNOT BEAT THIS GAME.
In preparation for this newsletter, I finally gave in and looked up the solution. And having seen a lovely explanation from one of my favorite mathematicians, James Tanton, I think I can provide a way to turn this maddening little game into a meaningful mathematical exploration for you and your child.
How to Play
Nim is an ancient game with many variations, but each version is based on the same idea: rows of stones.
The original version I played had three rows of stones, as pictured. But you can have as many rows of stones as you want, and each row can contain as many stones as you like.
You and your opponent take turns choosing a single row of stones and then removing as many stones from that row as you want. You can remove a single stone, a few stones, or even the entire row. The person to take the last stone wins.
You can play an online version of the game with four rows here. The version I played has been lost to the sands of time (Good riddance).
Playing Nim with your Child
Have you played the version above a couple of times? Have you lost every time, just like me? Great, you're ready to explore this game with your child!
I don't usually get this prescriptive with my advice, but I think that Nim provides a great opportunity to show your child one of the greatest problem-solving skills of a mathematician: solving a simpler problem.
Introduce the game to your child, then play a couple of versions of Nim that start with two rows. Then ask them "Do you want to figure out a strategy to always win?" Kids love nothing so much as winning, so hopefully they'll be on board!
Start with a very simple version of Nim: a one-row version of the game. Lay out a row of stones and ask your child "Do you want to go first or second?" Hopefully, they choose to go first and snatch up all the stones! If not, snatch all the stones yourself on your turn, then set up the game again and ask if they want to go first or second.
This version of the game isn't very fun, but it does teach you and your child an important principle of the game: if there is only one row left on your turn, you win!
Now you're ready to play the two-row version of the game. Again, though, you want to start with a simplified version: one stone in each row. Again, ask your child "Do you want to go first or second?"
This time, going first is a guaranteed loser! No matter which stone you choose, your opponent can take the other stone to win. So now you and your child have another piece of strategy about the game: if you can get the stones down to a single stone in each row on your opponent's turn, you win!
One last mini-game before the final reveal: set up a game with two stones in each row and ask your child "Do you want to go first or second?"
Now you can start to discuss all the possible ways the game can go: If you choose to go first, you can either take a single stone from one row, or you can take both stones from one row. But taking both stones is a bad move, since your opponent will just snatch up both stones from the other row and win!
So you have to take a single stone from one row. But wait! Your opponent can take a single stone from the other row, leaving you with one stone in each row, a situation we know is hopeless! So two rows of two stones are equally as hopeless as two rows of one stone. In fact, two equal rows of stones are a guaranteed loser, no matter how many stones are in each row.
And now the strategy becomes clear: On your turn, take stones from the bigger row until both rows are equal. That way, you force your opponent to make the rows unequal on their next turn, and you can make the rows evenly matched again on your turn. Eventually you will winnow down the rows until each row has exactly one stone, and your opponent is doomed!
The beauty of this game is that this strategy works no matter how many stones are in each row. Once you know the goal of making the two rows equal, you can play any variation on the two-row version of Nim. And what if the two rows start out equal? Simple, just ask your opponent to go first! She'll break up the evenness of the rows, and you are back on track to victory.
I think for most kids in elementary school, the two-row version of Nim is perfect. They get the feeling of accomplishment that they know how to beat the game, and they're sure to teach their friends how to play, simply to show off their prowess.
More importantly, though, they have gone through the experience of solving a game by solving simpler versions of a complicated problem. This is a tool that I teach my students to use in middle school, and it's a tool I use as an adult to solve mathematical problems that I've never encountered before. By modeling this process, you are giving your child an opportunity to experience the joy of mathematical discovery.
But wait - we haven't solved the three- and four-row versions of Nim, the ones that drove me crazy for years! That's true. But we have learned something valuable about the structure of Nim, which is that keeping the rows even somehow is a good strategy. In order to see how you can keep three or four rows even, you'll have to watch that lovely video from James Tanton that I mentioned earlier.


So, tomorrow is the great Pi day of the century. At 9:26:53 you will have the first ten digits of Pi with the day/month/year – 3.141592653.
Do you celebrate Pi day? If you don’t, you can use the following info for other days as just math lessons, but if you do there are a lot of cute ideas. One that we did was giving each number 0-9 a color and making a Pi chain. We also did Buffon’s needle (virtual and in real life) and got pretty close to 3.14.
Pi stuff, jewelry, belts, a few crafts, here.
Books – Sir Cumference and the Dragon of Pi, Why Pi?, A History of Pi, Joy of Pi.
Lots of Pi related ideas, crafts and games here.
Vi Hart shares her anti-Pi day message (she is pro-Tau) here.
And don’t forget to eat some pie (whether that it pizza or just pie) tomorrow!


This week in Sign Language & Music we explored numbers 1-10 in sign language. I mentioned that I often use number signs 1-31 in calendar time for many of my sessions. The numbers 1-10 are a great start to learning all of those number signs!
So heres the big question… Do I use signed numbers, or standard finger counting?
This really depends on what your classroom, students, or clients are using. If they have been using and gaining understanding with standard counting then that might be the most appropriate counting measure to use. Many of my classrooms use signed numbers so I use those to reiterate the method that is already being used in the classroom. At the end of the day I am always in search of the most efficient and effective means to the end. Check out this weeks video below, and have a wonderful week!


Twice per year, the American Mathematical Society (AMS) and the Mathematical Sciences Research Institute (MSRI) jointly sponsor a Congressional briefing.
These briefings provide an opportunity for communicating information to policymakers and, in particular, for the mathematics community to tell compelling stories of how our federal investment in basic research in mathematics and the sciences pays off for American taxpayers and helps our nation maintain its place as the world leader in innovation. Attendees on 28 June 2017 included Congressional staffers and representatives from the NSF Division of Mathematical Sciences, and Senate Minority Leader Charles Schumer and House Minority Leader Nancy Pelosi dropped in. David Eisenbud (MSRI), Karen Saxe (AMS), Anita Benjamin (AMS), and Kirsten Bohl (MSRI) organized this event. Presenter David Donoho has kindly shared these notes for his Congressional briefing. It is notable that the mathematicians had little interest in applications, yet the math inspired original MRI research and applications.
—Karen Saxe, Director, AMS Washington Office, and David Eisenbud, Director, MSRI
MRI scans are crucial tools in modern medicine: 40 million scans are performed yearly in the US. Individual 2-D MR images portray soft tissues that X-ray CT can’t resolve, without the radiation damage that X-rays produce. MRIs are essential in some fields, for example to neurologists seeking to pinpoint brain tumors or study demyelinating diseases and dementia.
MRI technology is remarkably flexible, constantly spawning new applications. Dynamic MRIs allow cardiologists to view movies showing muscular contractions of the beating heart. 3-D head MRIs allow neurosurgeons to meticulously plan life-and-death brain surgeries, in effect to conduct virtual fly-throughs ahead of time.
Traditionally MRIs required lengthy patient immobilization—in some cases hours. Long scan times limit the number of patients who can benefit from MRI, and increase the cost of individual MRIs. Long scan times also make it difficult to serve fidgety children. Ambitious variations of MR imaging—such as dynamic cardiac imaging and 3-D MRI—require far longer scan times than simple 2-D imaging; such long scan times have typically been awkward or even prohibitive. Yet patients with arrhythmias and afib could get better treatment based on dynamic cardiac imaging; patients with aggressive prostate cancer could get much more accurate biopsies under 3-D MRI guidance; and many neurosurgeries could be much safer and more effective if surgeons could plan surgeries with 3-D MRIbased fly-throughs beforehand.
Help is on the way: inspired by federally funded mathematical sciences research, patients everywhere will soon complete traditional clinical MRI scans much more rapidly. Ambitious but previously rarely available MRI applications will go mainstream.
2. Accelerated Imaging by Compressed Sensing
In 2017, the FDA approved two new MRI devices, which dramatically speed up important MRI applications from 8x to 16x. Siemens’ technology (CS Cardiac Cine) allows movies of the beating heart; GE’s technology (HyperSense) allows rapid 3-D imaging, for example of the brain. Both manufacturers say their product is using Compressed Sensing (CS), a technology that first arose in the academic literature of applied mathematics and information theory.
Roughly ten years ago, mathematicians, in-
cluding Emmanuel Can-dès, now at Stanford, Terence Tao of UCLA, and me, put forward theorems showing that compressed sensing could reduce the number of measurements needed to reconstruct certain signals and images. Partly inspired by these new mathematical results, MRI researchers such as Michael Lustig (now at Berkeley) and collaborators in John Pauly’s lab at Stanford began to work intensively on new scanning protocols and algorithms. Compressed-sensing-inspired imaging became in the next few years a leading theme at international conferences of MRI researchers. Convincing clinical evidence soon emerged. Shreyas Vasanawala at Lucille Packard Children’s Hospital (Stanford) and co-authors (including Lustig and Pauly) showed in 2009 that pediatric MRI scan times could be reduced in representative tasks from 8 minutes to 70 seconds, while preserving the diagnostic quality of images. Fidgety children could thus be imaged successfully and comfortably with far less frequent use of sedation. Other researchers demonstrated impressive speedups in dynamic heart imaging in the clinical setting. Manufacturers ultimately became convinced; serious commercial development followed, leading to 2017’s FDA bioequivalence approvals.
Researchers at GE and Siemens tell me that accelerated imaging can be expected to spread broadly to many other MRI settings. The industry welcomes the new approach to accelerate MRI scans, and is seeking to deploy it where possible.
Transforming the MR industry takes time. In each proposed application, the FDA must first certify that accelerated imaging is bioequivalent to traditional imaging: that it really can produce diagnostic quality images in less scan time. Seeking FDA approval demands a rigorous multi-step evaluation taking years. From this viewpoint, it seems stunning that compressed-sensing inspired products are now on the market, only about a decade after the initial academic journal articles. This is certainly a testament to the energy and talents of MRI researchers and developers, and also a sign that the underlying mathematics was solid, reliable, and applicable.
The technology remains to be deployed into hospitals and clinics. More than 5 billion USD in MR scanners are sold annually; service and maintenance costs add billions more. There are tens of thousands of MRI scanners installed in the US currently, and many more worldwide. Improving MRI, by whatever means—mathematical ideas like compressed sensing, or physical ingredients like more powerful magnets—is always a gradual process of getting new equipment into the marketplace or retrofitting existing scanners.
For patients, such improvements can’t come soon enough. My own son is a neurosurgery resident at a large county hospital that serves many indigent and uninsured patients. He treats dramatic injuries to the head, from gunshot wounds, blunt force trauma, and car wrecks; but also less dramatic yet serious problems like aneurysms and brain tumors. In his hospital, which does not yet have the accelerated MRI technology I’ve been speaking about, lengthy delays for MRIs are common; sometimes he must open the patient’s cranium without any MRI at all. In other cases, he only can inspect a few 2-D slices, rather than a full 3-D image.
My son’s comment about accelerating MRIs: “Faster, please.”
3. The Role of Mathematics Research
NSF-funded Mathematical Sciences research, and NIH funding of cognate disciplines played a key role in these developments.
First: how can mathematics be helpful? Don’t the MRI researchers have all the equipment they need to simply do whatever they need, and then see experimentally if it works?
The answer is interesting. Mathematicians build formal models of systems and then derive logical consequences of those formal models. Whatever mathematicians discover about such models is logically certain. In everyday experience, almost anything can be doubted, nothing is as it seems, whatever can go In mathematics, wrong will go wrong. In mathematics, a true statement ment is true, full stop. If a mathematical theorem says something surprising or impressive, you don’t waste time doubting the messenger—you just read the proof and (if you have the background) you will see why the theorem is true.
Mathematical results can be transformative when applied to a phenomenon in the real world that is poorly understood and as a result controversial. And in the case of accelerated scanning, that was the case. Prior to the mathematical work I mentioned above, isolated projects had observed experimentally that in special circumstances, researchers obtained impressive speed improvements over traditional MRI scanning. However, the MRI community as a whole was uncertain about the scope of such isolated results, and the impact of the published experimental evidence was therefore limited.
Mathematical research can go farther and deeper than experiments ever will go. It can give guarantees that certain outcomes will always result or that certain outcomes will never result. When you first hear of such a guarantee, it can rock your world.
Early mathematical results about compressed sensing guaranteed imaging speedups under certain conditions. This got the attention of MRI researchers and inspired some, such as Michael Lustig, to go all in on compressed sensing. Eventually, the mathematical certainty offered by the guarantees and their breadth broke the apparent ‘logjam’ of hesitation and suspicion that would have otherwise greeted MRI. Compressed sensing became a hot topic in MRI research.
Mathematics can also be a floodlight illuminating clearly the poorly understood path ahead. MRI researchers have always wanted to accelerate MRI; they just didn’t see how to do it. Mathematicians proposed new algorithms and gave persuasive guarantees based on illuminating principles.
4. Some Mathematics
I’ll mention very briefly some mathematical ideas that were mobilized to study compressed sensing.
At its heart, compressed sensing (CS) proposes that we take seemingly too few measurements of an object— so from the given data there are many possible reconstructions. CS selects from among the many possible candidates the one minimizing the so-called Manhattan distance. Mathematics guarantees that the optimizing reconstruction (under conditions) is exactly the one we want.
The bridge between imaging and mathematics is produced by mathematical analysis of convex optimization algorithms that shows that the following statements are equivalent.
• Consider 100,000 random measurements of a 1000 by 1000 image. Suppose the underlying image has only 10,000 nonzero wavelet coefficients. With overwhelming probability this image can be reconstructed by minimizing the Manhattan distance of its wavelet coefficients.
• Consider a random 900,000-dimensional linear subspace of 1,000,000-dimensional Euclidean space. The chance that this slices into a certain regular simplicial cone with 10,000-dimensional apex is negligible.
The mathematical heart of the matter is thus the following geometric probability problem. We are in an N-dimensional Euclidean space, N large. We consider a convex cone K with its apex at zero, and sample an M-dimensional random linear subspace L. What is the probability that L intersects K? (See Figure 1.)
Want to add a caption to this image? Click the Settings icon.
Figure 1. The underlying mathematics asks for the probability that a high-dimensional cone intersects a high-dimensional plane.
As it turns out, this probability depends on the cone K, and on the dimensions M and N—let’s denote it P(K; M,N). The central surprise of compressed sensing is that, for the cones K we are interested in, we can have P(K; M,N) essentially zero, even when M <1, for the regions in question.
In the 1960s two papers by federally-funded US university professors invented key tools to attack our problem. Harold Ruben, then at Columbia, generalized Gauss’s formula for spherical triangles in dimension 3 to all higher dimensions, obtaining general formulas for the spherical volume of high-dimensional spherical simplices. Branko Grünbaum at University of Washington formalized the
cone intersection probability P(K; M,N) in the case K>1 and developed some fundamental formulas for P(K; N,M).
Figure 2. The probability that a line intersects a cone is given by the area of the corresponding region on the surface of the sphere. When R is a spherical triangle, the great mathematician Gauss found a beautiful formula for area: simply the sum of the three internal angles, minus π. This formula is easy to understand but conceals a surprising wealth of intellectual meaning.
In the 1990s, German geometric probabilist Rolf Schneider and Russian probabilist Anatoly Vershik showed how to use Ruben’s and Grünbaum’s formulas to compute P(K’; M,N) for some interesting cones K’. In effect, they showed that (Ruben’s generalization of) Gauss’s charming formula was the heart of the matter; everything reduced to the computation of volumes of spherical simplices.
By the mid 2000s several approaches were developed to show that the central miracle of compressed sensing extended over a wide range of combinations of M and N; Candès and Tao (mentioned above) had developed by other means upper bounds on P(K; M,N), establishing that there was a useful such range. NSF-funded postdoc Jared Tanner (now at Oxford) worked with me to apply geometric tools of Schneider, Vershik, and Ruben to the cones K of interest. We developed precise formulas revealing the precise number of measurements needed for exact reconstruction.
Since then, slick machinery was developed by two teams at Caltech, Oymak/Hassibi and Amelunxen/McCoy/Tropp, to get these and many other results.
5. The Role of Federal Funding
The success story of compressed sensing is a testament to federal funding. Federal funding of pure and applied science has endowed the United States with research universities that are the envy of the world. Federal funding of mathematical sciences has led to amazing collections of sophisticated mathematical talent within those great universities, housed in departments of mathematics, applied mathematics and statistics, electrical and systems engineering and computer science, and even biology and chemistry.
These great institutions attract terrifically talented young people from around the world to come here to learn and become part of our technical infrastructure, strengthening the next generation of US science and industry.
Federal funding enabled the breakthroughs of compressed sensing along three paths:
• Federal funding enabled basic research in high-dimensional geometry, which is at the heart of compressed sensing. Harold Ruben and Branko Grünbaum were housed in statistics and mathematics departments at US universities when they did their foundational work.
• Federal funding enabled cross-disciplinary work that identified the key questions for mathematicians to resolve. In the late 1990s, NSF funded a joint project between optimization specialists Stephen Boyd and Michael Saunders and myself at Stanford, which studied the Manhattan metric for important data processing problems. Prior to this project, most scientists used the Crow Flight metric instead of the Manhattan metric. That project and its sequels funded work by Xiaoming Huo (now at The Georgia Technical Institute), Michael Elad (now at Technion), and myself that proved mathematically that the Manhattan metric could pick out the unique correct answer—when the answer we are seeking had a very strong mathematical sparsity property.
• Federal funding enabled focused research to go far deeper into this surprising area and dramatically weaken the required sparsity, for example by using random measurements. Researchers Emmanuel Candès (then at California Institute of Technology) and Terry Tao (University of California—Los Angeles), and myself and many others all were supported in some way by NSF to intensively study random measurements.
This is above all a case where the federal research funding system has worked. NSF support of mathematics and statistics and NIH support of electrical engineering and radiology have really delivered. The research universities, such as Stanford University, California Technical Institute, University of California—Los Angeles, and University of California—Berkeley, to mention only a few, have really delivered. As we have seen, industrial research groups at Siemens and GE have responded enthusiastically to the initial academic research breakthroughs.
Enabling the rapid transition were the great students and facilities available at Stanford University, produced by decades of patient federal support and also visionary campus planning and generous private donations. When Michael Lustig came to campus in the early 2000s as a graduate student of electrical engineering professor John Pauly, his office in Packard was less than a hundred yards away from the statistics department in Sequoia Hall, maybe 100 yards away from a GE-donated high-field MR research scanner, and maybe 300 yards away from a medical MRI research facility at Stanford Medical school. In a few hours, Lustig could literally engage in impromptu conversations about high-dimensional geometry with mathematical scientists, about specific MR pulse sequences with electrical engineers, and about specific possible clinical trials with doctors at Stanford Hospital. This tremendous concentration of resources allowed him in his thesis to produce results that inspired many MRI researchers to pursue compressed sensing-inspired research programs. Today’s Federal Funding model supported the development of such concentrated facilities and
talent over decades of patient investment; the model has delivered.
The cost-benefit ratio of mathematical research has been off-scale. The federal government spends about $250 million per year on mathematics research. Yet in the US there are 40 million MRI scans per year, incurring tens of billions in Medicaid, Medicare, and other federal costs. The financial benefits of the roughly 10-to-1 productivity improvements now being seen in MRI could soon far exceed the annual NSF budget for mathematics research.
ACKNOWLEDGMENT. There are by now literally thousands of papers in some way concerned with compressed sensing. It’s impossible to mention all the contributions that deserve note. I’ve mentioned here a select few that I could tie into the theme of a Congressional Lunch on June 28, 2017.
Thanks to Michael Lustig, PhD (University of California—Berkeley), Edgar Mueller (Siemens Healthineers), Jason Polzin, PhD (GE Global Research), and Shreyas Vasanawala MD (Stanford University), for patient instruction and clarifying explanation about MRI applications. Thanks to Ron Avitzur and Andrew Donoho for help with Pacific Tech Graphing Calculator Scripts used for the above figures and for movies that were used in the briefing. Thanks to Emmanuel Candès (Stanford University) and David Eisenbud (MSRI) for helpful guidance.
 See the first item of “Inside the AMS” in the September 2017 Notices.
 Here we really mean l1 norm. I figured that since Senator Charles Schumer would be present, we really should have a way to connect the proceedings to New York State!
 Actually many other ways have arisen to understand these results, using sophisticated ideas from metric geometry to information theory. I would mention for example work of Roman Vershynin, Ben Recht and collaborators, and of Mihailo Stojnic. But in a brief presentation for Congressional staffers, I couldn’t mention all the great work being done.
 Of course Crow Flight metric means standard Euclidean or l2 distance.
 Lustig credits many other MRI researchers with decisive contributions in bringing CS into MR Imaging. He says: “I would emphasize the contributions of Tobias Block pubmed/17534903 and Ricardo Otazo (with Daniel Sodickson) pubmed/20535813. These guys have taken the clinical translation of Compressed Sensing orders of magnitude forward.” He also mentions key contributions of Zhi-Pei Lian of UIUC, Joshua Trzasko of Mayo Clinic, and of Alexey Samsonov and Julia Velikina of UW Madison. It seems that federal funding was important in each case.


Non-classical logics – such as modal logics, conditional logics, intuitionistic logic, description logics, temporal logics, linear logic, dynamic logic, fuzzy logic, paraconsistent logic, relevance logic – have many applications in AI, Computer Science, Philosophy, Linguistics and Mathematics. Hence, the automation of proof search in these logics is a crucial task.
Aims and Scope
The ARQNL workshop aims at fostering the development of proof calculi, automated theorem proving systems and model finders for all sorts of quantified non-classical logics. The workshop will provide a forum for researchers to present and discuss recent developments in this area. The contributions may range from theory to system descriptions and implementations. Contributions may also outline relevant applications, describe problem formalizations, example problems, and benchmarks. We welcome contributions from computer scientists, linguists, philosophers, and mathematicians.
Topics of the ARQNL workshop will cover all aspects related to the mechanization and automation of quantified non-classical logics, including but not limited to:
- Proof theory, semantics, meta theory, and cut-elimination
- Proof search calculi, including sequent calculi, tableau calculi, connection calculi, resolution calculi, and instance-based calculi
- Modal logic, conditional logic, intuitionistic logic, description logic, temporal logic, linear logic, multivalued logic, dynamic logic, fuzzy logic, paraconsistent logic, relevance logic, free logic, and natural logic
- Techniques, strategies and heuristics to deal with first-order or higher-order quantification
- Implementation of theorem provers and experimental evaluations
- Problem libraries and benchmarking for theorem provers
- Applications, formalizations, and example problems
- User interfaces, proof representation, and syntax issues
ARQNL 2016 will be part of the International Joint Conference on Automated Reasoning IJCAR 2016 in Coimbra, Portugal.
23 July 2014, Vienna, Austria
(part of IJCAR 2014).
Proceedings: ARQNL 2014. Automated Reasoning in Quantified Non-Classical Logics. EPiC Series in Computing, Volume 33, EasyChair, 2015.


The Smooth and the Striated
Sierpensky’s sponge: more than a surface, less than a volume. The law according to which this cube was hollowed can be understood intuitively at a glance. Each square hole is surrounded by eight holes a third its size. And so on, endlessly. The illustrator could not represent the infinity of holes of decreasing size beyond the fourth degree, but it is plain to see that this cube is in the end of infinitely hollow. Its total volume approaches zero, while the total lateral surface of the hollowings infinitely grows. This space has dimension of 2.7268. It therefore lies between a surface (with a dimension of 2) and a volume (with a dimension of 3). ‘Sierpensky’s rug’ is one face of the cube; the hollowings are then squares and the dimension of the ‘surface’ is 1.2618.