Godel was a mathematician, logician, and philosopher whose theorems provided the foundation for theoretical computer science
Kurt Godel was a mathematician, logician, and philosopher born in April 1906 in what is now Brno, Czech Republic. Godel was a sickly child who suffered from rheumatic fever as a six-year-old. It’s said that Godel’s early illness left him paranoid about things like cleaning eating utensils obsessively, and about the purity of his food. Godel went to Austria in 1924 to study at the University of Vienna, where he earned a doctorate in mathematics before joining the faculty in 1930.
The Completeness Theorem
Godel’s doctoral thesis, Über die Vollständigkeit des Logikkalküls (On the Completeness of the Calculus of Logic), proved the completeness theorem. Godel showed that: “…classical first-order logic, or predicate calculus, is complete in the sense that all of the first-order logical truths can be proved in standard first-order proof systems.”
The Incompleteness Theorem
In 1931, Godel published Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions of Principia Mathematica and Related Systems). The incompleteness theorem showed that: “…in any consistent axiomatic mathematical system there are propositions that cannot be proved or disproved within the system and that the consistency of the axioms themselves cannot be proved.” To put it another way, Godel proved that there are mathematical statements out there which are true, but which can not be proven.
It’s so hard to understand that it must be important. Godel’s work, particularly his Incompleteness Theorem, has been described as one of the most remarkable mathematical or logical achievements of the 20th century. Godel has been described as the founder of theoretical computer science.
Godel and Einstein, intellectual peers
Publication of the incompleteness theorem brought Godel international fame. Godel lectured extensively in the United States, including at Princeton University, where he met Albert Einstein. Godel and Einstein would become and remain friends until Einstein died in 1955.
Einstein and Godel were very different men. In The New Yorker’s February 20, 2005 article, Time Bandits, described them this way:
Whereas Einstein was gregarious and full of laughter, Gödel was solemn, solitary, and pessimistic. Einstein, a passionate amateur violinist, loved Beethoven and Mozart. Gödel’s taste ran in another direction: his favorite movie was Walt Disney’s “Snow White and the Seven Dwarfs,” and when his wife put a pink flamingo in their front yard he pronounced it furchtbar herzig—“awfully charming.” Einstein freely indulged his appetite for heavy German cooking; Gödel subsisted on a valetudinarian’s diet of butter, baby food, and laxatives.
But there was an undeniable kinship, too. Einstein is said to have told people that he went to his office “just to have the privilege of walking home with Kurt Godel.”
In 1951, Godel was awarded the first Albert Einstein Award, along with Julian Schwinger. (The photograph above is of Einstein presenting that award to his colleague.) Godel was awarded the National Medal of Science in mathematics and computer science from President Ford in 1974. The citation for the 1974 prize says: “For laying the foundation for today’s flourishing study of mathematical logic.” An annual award for outstanding papers in theoretical computer science, The Godel Prize, was named after him.
Godel transitioned from mathematics and logic to philosophy in later years. He was particularly interested in the idea of Mathematical Platonism. He argued that “sentences” like 2+2=4 truly described non-physical and non-mental collections of objects that existed outside of time and space in a mathematical realm called “Platonic Heaven.” Godel’s philosophical views, though tied to his accepted mathematical and logical theorems, were not widely accepted.
A fascinating and comprehensive source for all things Godel, including his philosophical work, is the Stanford Encyclopedia of Philosophy.
Godel’s paranoia progressed with age. Godel would not eat unless his wife, Adele, tasted his food first out of certainty he was being poisoned. Adele was hospitalized for a time due to her own illness. As a result, Godel stopped eating and starved to death.
A deeper dive – related reading from the 101:
Legend has it that a mathematician could not write a false theorem with this type of chalk
Occasionally, physics and philosophy seem to meet. Like the time the physicist imagined a cat that was dead and alive… at the same time


We can interpret the action of XOR in a number of different ways, and this helps to shed light on its properties. The most obvious way to interpret it is as its name suggests, ‘exclusive OR’: A ⊕ B is true if and only if precisely one of A and B is true. Another way to think of it is as identifying difference in a pair of bytes: A ⊕ B = ‘the bits where they differ’. This interpretation makes it obvious that A ⊕ A = 0 (byte A does not differ from itself in any bit) and A ⊕ 0 = A (byte A differs from 0 precisely in the bit positions that equal 1) and is also useful when thinking about toggling and encryption later on.
The last, and most powerful, interpretation of XOR is in terms of parity, i.e. whether something is odd or even. For any n bits, A1 ⊕ A2 ⊕ … ⊕ An = 1 if and only if the number of 1s is odd. This can be proved quite easily by induction and use of associativity. It is the crucial observation that leads to many of the properties that follow, including error detection, data protection and adding.
Essentially the combined value x ^ y ‘remembers’ both states, and one state is the key to getting at the other.


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The Rise of Homeschooling and Math Tutoring
In recent years, homeschooling has become an increasingly popular educational choice for families across America. According to the National Home Education Research Institute, homeschooling in the United States has been growing at an average rate of 2 to 8 percent per year, with an estimated 2.5 million students currently being homeschooled. With the benefits of individualized attention, flexible schedules, and personalized curriculum, many homeschooling families seek additional support in the form of math tutoring.
In addition, math has long been considered one of the more challenging subjects to teach, particularly for parents who do not have a strong math background. For this reason, many families turn to dedicated math tutors for assistance. To keep growing your understanding of the topic, don’t miss out on the carefully selected external resource we’ve prepared to complement your reading. ACT/SAT Test Preparation!
The Role of Technology in Math Tutoring
Technology has played an increasingly important role in education over the past decade, and math tutoring for homeschooling families is no exception. Online tutoring platforms such as Khan Academy, Mathnasium, and Chegg offer personalized, one-on-one math instruction to students of all ages and abilities, allowing families to access quality tutoring Learn from this informative document the comfort of their own homes.
Another advantage of online math tutoring is the ability to record and review tutoring sessions. With online platforms, families can easily revisit lessons and concepts covered during previous sessions, enabling students to reinforce their understanding of key math concepts.
The Need for Personalization and Attention to Individual Needs
Despite the growing popularity of online math tutoring, there is still a need for personalized attention and instruction tailored to the individual needs of each student. Many students benefit from the one-on-one attention of a dedicated math tutor, particularly those who struggle with math concepts or who have unique learning styles.
This is especially true for homeschooling families, who rely heavily on personalized attention to ensure that their children are receiving an appropriate education that meets their individual needs and goals. While online platforms can provide quality instruction, they may not be able to offer the same level of personalization as a dedicated math tutor.
The Future of Math Tutoring for Homeschooling Families
Looking forward, it seems likely that math tutoring for homeschooling families will continue to grow and evolve as technology advances and the number of homeschooling families increases. While online platforms will likely play an increasingly important role in math tutoring, there will still be a need for personalized, one-on-one instruction that can be tailored to the unique needs of each student.
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Math tutoring for homeschooling families has become an increasingly important educational choice for families seeking personalized attention and support in math instruction. While online platforms offer numerous advantages, there is still a need for dedicated math tutors who can provide personalized, one-on-one attention tailored to the needs and goals of each student. As technology continues to advance, it seems likely that math tutoring will continue to evolve and adapt to meet the changing needs of homeschooling families. Immerse yourself in the subject with this external content we suggest. ACT/SAT Test Preparation!


- How much would $1000 be in 50 years?
- What Will 100000 be in 30 years?
- What is the future value of $50000?
- What will 60000 be worth in 20 years?
- What will $1000 be worth in 20 years?
- What stock will double in 2020?
- How much will $600000 be worth in 10 years?
- How much money do I need to invest to make $2000 a month?
- What will 700k be worth in 10 years?
- How much money do I need to invest to make $3000 a month?
- How can I invest 50k wisely?
- Is having 50k saved good?
- Is 50k in savings good?
- What will a million dollars be worth in 40 years?
- How much will $2000 be worth in 20 years?
- What will 50000 be worth in 10 years?
- What will 400k be worth in 10 years?
- What will a dollar be worth in 2040?
How much would $1000 be in 50 years?
Value of $1,000 from 1950 to 2021 $1,000 in 1950 is equivalent in purchasing power to about $11,081.08 today, an increase of $10,081.08 over 71 years.
The dollar had an average inflation rate of 3.45% per year between 1950 and today, producing a cumulative price increase of 1,008.11%..
What Will 100000 be in 30 years?
At the end of 20 years, your savings will have grown to $320,714. You will have earned in $220,714 in interest….Interest Calculator for $100,000.RateAfter 10 YearsAfter 30 Years0.00%100,000100,0000.25%102,528107,7780.50%105,114116,1400.75%107,758125,12753 more rows
What is the future value of $50000?
$50,000 Savings Calculator – Future ValueFuture Value$253,205.82Total Invested$50,000.00
What will 60000 be worth in 20 years?
The first result (Reduced Amount) is $33,220.55, which represents the value of $60,000 in 20 years. The second result (Required Amount) is $108,366.67, which is amount of money that you need in 20 years to match the purchasing power of $60,000.
What will $1000 be worth in 20 years?
After 10 years of adding the inflation-adjusted $1,000 a year, our hypothetical investor would have accumulated $16,187. Not enough to knock anybody’s socks off. But after 20 years of this, the account would be worth $118,874.
What stock will double in 2020?
With that in mind, some stocks that could double in 2020 and fly higher in the back half of the year include: Beyond Meat (NASDAQ:BYND) Nio (NYSE:NIO) Canopy Growth (NYSE:CGC)
How much will $600000 be worth in 10 years?
Investing $600,000. How much will $600,000 be worth in the future?YearValue91,150,343101,236,619111,329,365121,429,06811 more rows
How much money do I need to invest to make $2000 a month?
To cover each month of the year, you need to buy at least 3 different stocks. If each payment is $2000, you’ll need to invest in enough shares to earn $8,000 per year from each company. To estimate how you’ll need to invest per stock, divide $8,000 by 3%, which results in a holding value of $266,667.
What will 700k be worth in 10 years?
Investing $700,000. How much will $700,000 be worth in the future?YearValue10987,419111,021,979121,057,748131,094,76910 more rows
How much money do I need to invest to make $3000 a month?
By this calculation, to get $3,000 a month, you would need to invest around $108,000 in a revenue-generating online business. Here’s how the math works: A business generating $3,000 a month is generating $36,000 a year ($3,000 x 12 months).
How can I invest 50k wisely?
Here are ten ways to invest 50k:Individual Stocks. Individual stocks represent an investment in a single company. … Real Estate. … Individual Bonds. … Mutual Funds. … ETFs. … Invest with a Robo Advisor. … CDs. … Invest in Your Retirement.More items…
Is having 50k saved good?
For most people, $50,000 is more than enough to cover their living expenses for six full months. And since you have the money, I highly recommend you do so. On a different, and equally important note, when you set up an emergency fund, it should be separate from any other savings.
Is 50k in savings good?
Saving $50,000 per year is well ahead of most people, so first off congratulations. Your plan of action should be something like the following: Make an emergency fund. It should be multiple months’ worth of expenses.
What will a million dollars be worth in 40 years?
Time magazine recently estimated that for a millennial with 40 years until retirement, $1 million in savings is not likely sufficient. Taking into account 3% inflation over that time period, it would be worth just $306,000 in today’s dollars.
How much will $2000 be worth in 20 years?
How much will an investment of $2,000 be worth in the future? At the end of 20 years, your savings will have grown to $6,414. You will have earned in $4,414 in interest.
What will 50000 be worth in 10 years?
This calculates what a $50,000 investment will be worth in the future, given the original investment, annual additions, return on investment, and the number of years invested….Investing $50,000. How much will $50,000 be worth in the future?YearValue961,0861062,4601163,8661265,30211 more rows
What will 400k be worth in 10 years?
How much will savings of $400,000 be worth in 10 years if invested at a 8.00% interest rate?…$400,000 at 8% Interest for 10 Years.YearAmount9$799,60210$863,5709 more rows
What will a dollar be worth in 2040?
Future inflation is estimated at 3.00%. When $5 is equivalent to $9.55 over time, that means that the “real value” of a single U.S. dollar decreases over time. In other words, a dollar will pay for fewer items at the store….Buying power of $5 in 2040.YearDollar ValueInflation Rate2040$9.553.00%23 more rows


‘A’ level maths tuition is offered at Bonython Road, covering pure maths topics such as….
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3 edition of Computing methods in applied sciences and engineering found in the catalog.
|Statement||Second International Symposium, December 15-19, 1975 / [organized by] IRIA LABORIA, Institut de recherche d"informatique et d"automatique ; edited by R. Glowinski and J.L. Lions.|
|Series||Lecture notes in physics -- 58|
|Contributions||Glowinski, R., Lions, Jacques Louis., Iria Laboria.|
|The Physical Object|
|Pagination||viii, 593 p. :|
|Number of Pages||593|
8 issues per year. Subscription price. IJCSM is a peer-reviewed international journal that publishes high quality original papers and comprehensive survey articles in all areas of computing science and mathematics, with interfaces to physics, engineering, chemistry, biology, statistics, economics and the social sciences.
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An Essay on the Foundations of Geometry (Geometry) by Bertrand Russell.
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Discover the Fascinating World of Roman numerals with Our Roman numerals chart 1-100. But first, what are Roman numerals?
TABLE OF CONTENTS
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Kevin Wilson Ph.D.
Kevin was a Senior Data Scientist in The Lab @ DC. He played a leading role in envisioning and implementing The Lab’s data analytic standards and protocols, executing (and wherever possible automating) important data analytic tasks, and guiding the use of machine learning, as well as developing entirely novel uses of data that maximize how much we can learn from the District’s vast amount of administrative data.
Kevin brought years of experience in a wide array of data analytic topics, ranging from system and security architecture to advanced statistical techniques to the development of novel predictive models and commercial tools. As the Principal Data Scientist at Knewton-an education technology company-Kevin developed and implemented methods for drawing real time and retrospective insights from massive, complex datasets, and was especially involved in bridging the divide between classic techniques like psychometrics with the modern techniques of machine learning. Kevin also has experience explaining such complex topics to diverse audiences, ranging from small academic seminars to high school students. His work has appeared in such diverse venues as the Journal of Combinatorial Theory and Educational Data Mining, and some of his popular projects have been written up in the Washington Post and Daily Kos.
Kevin earned a B.S. in mathematics from University of Michigan and a Ph.D. in mathematics from Princeton University.


Please use this identifier to cite or link to this item:
|Title:||A novel hybrid representation and control of convective spatially distributed systems|
|Abstract:||In this work we study spatially distributed convective systems described by first order partial differential equations and we show how this systems in a fixed spatial point can be exactly represented by a hybrid system with two commutable ordinary differential equations subsystems, where the first one is dominated by the initial condition, whereas the second one is governed by the boundary condition. Then we define the boundary control problem for spatially distributed convective systems and as a first approach, we propose a controller for one-dimensional systems which solves this problem. Finally, in order to test hybrid representation and controllers, a study case is presented for an enzymatic reactor, where dispersive terms can be neglected. � 2007 World Scientific Publishing Co. Pte. Ltd.|
|Appears in Collections:||Producción científica UdeG|
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Classic Hens Eggs Puzzle Solution - 22 March
A chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days?
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the farmer needs 3 hens to produce 12 eggs in 6 days
This is a classic problem that many people get wrong because they reason that half of a hen cannot lay an egg, and a hen cannot lay half an egg. However, we can get a satisfactory solution by treating this as a purely mathematical problem where the numbers represent averages.
To solve the problem, we first need to find the rate at which the hens lay eggs. The problem can be represented by the following equation, where RATE is the number of eggs produced per hen·day:
1½ hens × 1½ days × RATE = 1½ eggs
We convert this to fractions thus:
3/2 hens × 3/2 days × RATE = 3/2 eggs
Multiplying both sides of the equation by 2/3, we get:
1 hen × 3/2 days × RATE = 1 egg
Multiplying both sides of the equation again by 2/3 and solving for RATE, we get:
RATE = 2/3 eggs per hen·day
Now that we know the rate at which hens lay eggs, we can calculate how many hens (H) can produce 12 eggs in six days using the following equation:
H hens × 6 days × 2/3 eggs per hen·day = 12 eggs
Solving for H, we get:
H = 12 eggs /(6 days × 2/3 eggs per hen·day) = 12/4 = 3 hens
Therefore, the farmer needs 3 hens to produce 12 eggs in 6 days.


Round-off errors arising from the difference between real numbers and their floating-point representation cause the control flow of conditional floating-point statements to deviate from the ideal flow of the real-number computation. This problem, which is called test in- stability, may result in a significant difference between the computation of a floating-point program and the expected output in real arithmetic. In this paper, a formally proven program transformation is proposed to detect and correct the effects of unstable tests. The output of this transformation is a floating-point program that is guaranteed to return either the result of the original floating-point program when it can be assured that both its real and its floating-point flows agree or a warning when these flows may diverge. The proposed approach is illustrated with the transformation of the core computation of a polygon containment algorithm developed at NASA that is used in a geofencing system for unmanned aircraft systems.


Monday 11th January
Can you count how many characters there are in The Enormous Turnip.
Can you represent that number on your fingers and by making marks?
Tuesday 12th January
Wednesday 13th January
Count how many pieces of pasta you used yesterday to make your necklace. If you used different colours count how many pieces of each colour.
Thursday 14th January
Take a walk in your local area. Count how many red cars you see. Do the same with blue cars and black cars.
Can you count how many buses you see? Try counting vans, lorries, bikes and airplanes.
Friday 15th January
Go outdoors and create a pattern using natural resources.


I believe only a dedicated follower of mathematics would be able to understand my this piece of work, if not a person who loves having a great deal of knowledge about common issues. Before I start up with anything about the issue that I am writing about, let me just give out a small definition as what this Reimann Hypothesis is so as to help our non-mathematics followers understand my piece. I wouldn’t give out the actual definition of the entitled hypothesis. I would rather go with a more generalised statement.
Named after the Greman mathematician Bernhard Riemann, Reimann hypothesis focuses on how the prime no.s are distributed and how they occur. The Generalised Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2. The Reimann Hypothesis is renowned amidst the mathematicians for being a maths problem dating back to 1859 which has yet not been solved. The hypothesis is so far known to be resisting all the attempts made to prove it which basically means that no one has so far been able to solve this hypothesis.
But today, I have come up with a news where a renowned mathematician claims of having solved it. Let’s have a look as to what this exact news is. Speaking at the Heidelberg Laureate Forum, an event that was organised to create a connection between the experts from the fields of science and maths with the students of the younger generation, retired mathematician Michael Atiyah, claimed about having a solution for the two-century-old problem, renowned as Reimann Hypothesis.
Atiyah during the talk said, “Solve the Riemann hypothesis and you become famous. If you are famous already, you become infamous, Nobody believes any proof of the Riemann hypothesis because it is so difficult. Nobody has proved it, so why should anybody prove it now? Unless, of course, you have a totally new idea.” “It’s so difficult nobody’s proved it, so why should anyone prove it now, unless of course, you have a totally new idea.”
The problem is known to be so difficult that the Clay Mathematics Institute offers a prize of $1 million to anyone who solves it. Though claiming to have solved the same, Atiyah will have to wait for some more time before he claims the prize, i.e., he will have to wait until the solution is published in the worldwide mathematics journal and be accepted by the mathematician community two years after the publication of the solution.
Based on the earlier works of mathematicians Friedrich Hirzebruch and John von Neumann, Atiyah said that the Reimann’s hypothesis of prime numbers’ distribution is correct. Jørgen Veisdal, an economist at the Norwegian University of Science and Technology, being sceptical, spoke to the Magazine Science and said, “It is simply too vague and unspecific.”


– The number of lanes on the road
– The average speed on the road
– The standard deviation of speeds on the road
– The length of the road
– The traffic volume on the road
Based on these factors, the algorithm calculates an expected travel time and a reliability index. The expected travel time is simply the average speed multiplied by the length of the road. The reliability index is a measure of how close the actual travel times are to the expected travel time; a higher reliability index indicates that the travel times are more reliable.
In addition to these factors, FREEVAL-RL also takes into account the type of road (highway, urban, etc.), the time of day, and the day of the week. These factors can all affect travel time reliability.


Published March 21, 2006 by John Wiley & Sons .
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Advanced Engineering Mathematics 10th Edition Binder Ready Version with WileyPLUS Set (Wiley Plus Products) Out of Print--Limited Availability. U.S. 9th Edition Hardcover Book in Like New Condition, Save MONEY and buy the previous edition!/5().
Advanced Engineering Mathematics 9th Edition with Wiley Plus Webct Powerpack Set book. Read 36 reviews from the world's largest community for readers. A /5. Erwin Kreyszig: Advanced Engineering Mathematics 9th Edition with Wiley Plus WebCT Powerpack Set 9th Edition 0 Problems solved: Erwin Kreyszig: Advanced Engineering Mathematics and ODE Architect Companion 8th Edition 0 Problems solved: Erwin Kreyszig: Advanced Engineering Mathematics, 10th Edition International Student Version 10th Edition 10th edition Loose-leaf.
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More than 50% of the homework problems are new, and there are also a number of new sample problems. To help students build necessary visualization and problem-solving skills. Advanced Engineering Mathematics 9th Edition with Math Computer Guide Set (Hardcover) Published September 8th by John Wiley & Sons Hardcover, 0 pagesCited by: How is Chegg Study better than a printed Advanced Engineering Mathematics 8th Edition student solution manual from the bookstore.
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This note is a survey of observational astronomy across the electromagnetic spectrum. Topics covered includes: overview of current. Here is an unordered list of online mathematics books, textbooks, monographs, lecture notes, and other mathematics related documents freely available on the web.
I tried to select only the works in book formats, "real" books that are mainly in PDF format, so many well-known html-based mathematics web pages and online tutorials are left out.
Mathematics books Need help in math. Delve into mathematical models and concepts, limit value or engineering mathematics and find the answers to all your questions. It doesn't need to be that difficult. Our math books are for all study levels.
In addition to expanded explanations, the 10th edition includes new problems, updated figures and examples to help motivate students. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study.
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plus two others characteristics (cumulative frequency and cumulative relative frequency) thus including information about how they are sorted. Cumulative frequency of the i-th variant mi - is a number of values of a variable showing the frequency of variants less or equal the i-th variant.
The first prerequisite for learning the book is a working info of calculus, gained from a standard two, or three semester course sequence or its equal.
Some familiarity with matrices can also be helpful inside the chapters on methods of differential equations. How to Download Elementary Differential Equations, 10th. Description. For courses in college algebra. This package includes MyMathLab ®. Prepare. Practice. Review. Mike Sullivan’s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts.
The Tenth Edition has evolved to meet today’s course needs. With this new edition, Mike Sullivan has Format: On-line Supplement.Download Form "Bookz2" Discription is Advanced Engineering Mathematics -9th Edition By Erwin Kreyszig, Published by John Wiley & Sons Inc.,1, Pages with size tely Free For students DMCA copyright claimed.The title, author or ISBN number can also be used to find information from on-line bookstores such as Amazon, Barnes & Noble or Borders; or from sites like Aaapricesearch, A1Books, BookByte, BookFinder4U, (this one allows comparison of several books in a single search), BuyUsedTextbooks, Campus Books, CampusI, Chambal, Cheap-Textbooks, CheapestTextbooks.


An example of one of my abstractions, created from a photograph. Abstract paintings are traditionally renderings of image mappings made in an artistís head. My abstract works are mostly based on mathematical transformations applied to my photographs. Mathematically speaking, an image is a two dimensional distribution of colors and intensities. A mathematical ď1-to-1 mapping transformationĒ assigns a color and intensity to each destination point based on the color and intensity of a corresponding original image point. That is, the mapping says where each original point goes and how itís properties are changed in the process. It is surprising what you can get starting out with something simple and colorful. I donít remember what this photographic image was to start with. Perhaps a mouse or a moose, a horse or a hearse, a cur or a car. The creativity is in picking a transformation that produces a result that works well as art.


I am an applied mathematician, associate professor of
and a member of the
Center for Infectious Disease Dynamics (CIDD).
I run a theoretical biology research group interested in the description, understanding,
and prediction of the dynamics of biological systems, and the
implications of biology for human communities. This
currently includes work in ecology, epidemiology,
immunology, evolution, medicine, human behavior, and economics.
The university did a write-up that says
a little bit more about how I ended up doing mathematical biology.


After introducing the scale and talking about the terminology (heavier than, lighter than, and as heavy as or the same), I followed the lesson plan set forth in the book that came with the kit. The name of the book is Three Bear Family Math Activities: K-3 from Learning Resources. The name of the activity was "Bear Balance." What I liked about this activity is that we used the balancing learned earlier to fill out a chart (another math skill we have been working on) and then using the pattern set forth (using multiples of 2) to predict how many bears will balance out the other side. It's a real-world type comparison that involves more than one math skill. It also included a cute story that the kids had to listen to in order to complete the activity. Since listening comprehension is always something we need to work on with Ben, it was perfect.
The bears and scales are available on Learning Resources' website. I have never bought a set of counting manipulatives because I have found them to be expensive and I have been able to create or use everyday objects instead. However, I might just be convinced enough to buy these manipulatives or at the very least borrow them often. Having three sizes of manipulatives in lots of colors with the ratio between the weight and size being fixed makes them worth it in my mind. There were lots of good activities in the books, but I can already think of lots more that were not in there.


I've been frustrated with my daughter's school lately. Actually, it's been a growing frustration. I remember (decades ago) carrying multiple books home with me in elementary school to do my homework. These kids have copied work sheets - no books to bring home. I've requested to see her books in the past just so I knew what the heck they were learning. How are parents suppose to monitor and help their children with homework if we have no idea of what they are suppose to be learning?
I also thought math is math - you can't change the basics, however, apparently you can. She showed me her worksheet and it was basic multiplication: 135 x 23 for example. I told her, "You know how to do this, is this just a review?"
Nope. They have to solve the problem using the distributive property of multiplication. Now are they just trying to confuse the hell out of these kids or are they preparing them for algebra? Shouldn't they just practice the normal way of solving this problem to master the basics? She's in the 5th grade - I don't foresee algebra until high school.
So, using that example from above this is what you have to do for the problem:
135 --> 100+30+5
x23 --> 20+3
100 x 3 = 300
30 x 3 = 90
5 x 3 = 15
100 x 20 = 2000
20 x 30 = 600
20 x 5 = 100
I asked my daughter if she remembered how to do this problem 'normally' and she did. I guess I'll have to contact the math teacher and find out what this is all about. I bet these teachers think I'm a pain in the ass because I usually ask for materials and writing notes because I want to know what's going on with their classes when I see something strange to me.
So she comes home this past Friday with a failing grade on a science test. I'm surprised because she loves science and is normally a very good student. She's learning basic chemistry and she was able to bring the book home. We reread every chapter this past weekend together, reviewed her old test and I had her take the make up exam on her own. She had one question that she didn't understand. I read it and couldn't answer it, I showed it to my husband and he couldn't either. He said, "This is a bullshit question." We both have multiple degrees, certifications, and he has a graduate degree - together we couldn't answer this question - how is a 5th grader suppose to do it?
So I wrote yet another note to the teacher requesting that she allow my daughter to bring home the book once a week so I can review it with her.
So now I have math & science to do with my daughter, keep an eye on my son's schoolwork, and prepare for my next degree starting in January. If we could add another few hours in the day, I might be able to do this all. Sigh.


One for statement and one reason, so every statement that you make has to have a reason and that's going to be given linear Lecture 1: Introduction and Proofs. We begin by Algebra. Theorem 1. Software for math teachers that creates exactly the worksheets you need in a matter of minutes. Algebraic Properties of Equalty. If we want to disprove a conjecture, The aim of these notes is to provide a proof of the Fundamental Theorem of Algebra using concepts that should be familiar to you from your study of Calculus, and so we begin by providing an explicit formulation.
Introducing Thinkwell Referral Rewards! Now we'll do algebra in the format of the two-column proof. Abstract algebra proofs. It seems like a special case, an optical illusion: with just the right shape, things can be re-arranged. Anyway, see you in the next video. This proof format is a very popular format seen in most high school textbooks. You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. IT is an alias for cs. If we want to disprove a conjecture, A proof is a series of statements, each of which follows logicallyfrom what has gone before.
An important part of writing a proof is giving justifications to show that every step is valid. An existence statement asserts that objects with a given property exist. All reasons used have been showed in previously algebra courses. A basic knowledge of set theory, mathe- matical induction, equivalence relations, and matrices is a must. There is at least one side of our triangle for which the altitude lies "inside" the triangle.
Something like a square has a lot of symmetry, but circle has even more. The building of a proof requires critical thinking, logical reasoning, and disciplined organization. It is very frustrating for your reader if steps in the proof are skipped, so even if it seems simple or obvious you should include it. Language and Proofs in Algebra: An Introduction. This video provides the student with a walkthrough of one or more examples from the concept ".
Contains hints on classroom presentation, each fallacious proof, and a summary of the source of the fallacy. This makes it easier than the other methods. All the intermediate logical steps are supplied, without exception. Introduction; Lines and Angles. View my complete profile Theorems and Proofs A theorem is a mathematical statement that provides a conclusion, provided that a set of speci c assumptions holds.
Example 1. The trouble with this is that, sooner or later, mathematics becomes Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Come to Mathradical.
Emeritus and Other Faculty
There's only one semi-obnoxious step the main one! You want to write proofs in a natural, step-by-step order, like a manual. Understanding a proof can be a daunting task. Algebraic Proofs for the Trachtenberg System. Some important algebra proofs are demonstrated and explained here. Besides being a rst course in linear algebra it is also supposed to be a rst course introducing a student to rigorous proof, formal de nitions in short, to the style of modern theoretical abstract mathematics. Some of the most important geometry proofs are demonstrated here.
Statements are assertions that can be broadly classified under two types: Existence statements and others. One way is that I can show the task card on the document camera and have students choose an answer choice via Plickers. Prove that 6. Creating lessons is easy! Look for other lessons and click on 'view source'. You are given a problem to solve, and sometimes its solution.
The second is aimed at students who know a little bit of linear algebra, but should be useful for anyone learning how to write proofs, since it doesn't really rely on any knowledge of linear algebra. Once a question is marked, it cannot be edited. This insistence on proof is one of the things that sets mathematics apart from other subjects. The properties of equality can be used to justify each step when solving an equation. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra.
It took a long time to figure out that actually, there was a solution here. Additive Inverse of a Number. Proof of Various Limit Properties. Then, students complete each proof by naming the Proof by Contradiction This is an example of proof by contradiction. A direct proof is a sequence of statements which are either givens or deductions from previous statements, and whose last statement is the conclusion to be proved.
The main activity for today's lesson consists of three algebraic proofs. Connections to matrices and relations Figure 1: Mathematical constructions from surveying. The essence of the talks was to present elements of linear algebra as a powerful tool for proving inequalities. Directions: Examine each proof and determine the missing entries.
Writing a proof can be challenging, exhilarating, rewarding, and at times frustrating. Paperback: ISBN The ruler postulate tells us An algebraic proof shows the logical arguments behind an algebraic solution. Section 1. I post about ideas I have found and have in teaching math.
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Read the problem over carefully. A proof is a convincing demonstration that a mathematical statement is necessarily true. Section 7. Except in the simplest of cases, proofs allow for individual thought and development. For example: Let angle A and angle B be linear pairs. In the Algebra world, mathematical induction is the first one you usually learn because it's just a set list of steps you work through.
Lecture 1: Introduction and Proofs. Version 1. Note that at time of writing, the question reads "How do I mathematical proofs i. An algebraic proof uses algebraic properties, including the Distributive Property and the properties of equality. Subtracting 1 from both sides, The Corbettmaths video tutorial on algebraic proof. Remember that in our scratch work, we found that the conclusion was equivalent to a known fact from above. Here are three examples of the axiomatic method.
See if you can figure out in which step the fallacy lies. Before students write out the proofs, I have them brainstorm the process. They can solve the equation in this space and write out the steps needed in an informal way. We've already learned how to solve equations for a variable. A finest proof of this kind I discovered in a book by I.
Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. A mathematics proof establishes the validity of a mathematics statement. All you have to really know is math. In mathematics, a statement is not accepted as valid or correct unless it is accompanied by a proof. This will give you some reference to check if your proofs are correct. When formulated for a polynomial with real coefficients, the theorem states that every such polynomial can be represented as a product of first and second degree terms.
To be published by MIT Press, someday. Proofs and Reasoning 2. Subtraction Property of Equality. When a statement has been proven true, it is considered to be a theorem. Geometric proofs can be written in one of two ways: two columns, or a paragraph. Here is an alternate proof.
Activities and Resources
One of our goals in this class is to make precise the idea of symmetry, which is important in math, other parts of science, and art. Proof: Assume that x is even neg of concl. Prove and this completes the proof. It's a bit of math inception, where we try to understand the verification step, not simply verify the verification step.
The following properties are true for any real numbers a, b, and c. Hardcover: ISBN External links etc The proof illustrates a standard approach in inequality proofs involving the basic axioms: Convert inequality statements to equations and work with the equations.
How long did she run Rather than staring at a dry proof and trying to understand it directly, get a rough intuition ADEPT method and then see if the proof makes sense. So, like a good story, a proof has a beginning, a middle and an end. Algebraic Proofs Other proofs may be algebraic or combine algebra and geometry on the cartesian coordinate plane.
This activity provides a good review of basic math principals and the structure of mathematical proofs. A pattern of reaoning is a true assumption if it always lead to a true conclusion. This can occasionally be a difficult process, because the same statement can be proven using Directions: Examine each proof and determine the missing entries. A mathematical proof is a series of logical statements supported by theorems and definitions that prove the truth of another mathematical statement. Even more important is the ability to read and understand mathematical proofs.
Certain methods and facts are indispensible. You can use it as a main text, as a supplement, or for independent study. Algebra proofs.
Their improper use results in unclear and even incorrect arguments. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Is to make the formal proof argument of why this is true. However, since it is easier to leave steps out when writing a paragraph proof, we'll learn the two-column method. In the words of Paul Halmos: "Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs.
Given any precise logical statement, a proof of that statement is a sequence of logically correct steps which shows that the statement is true. It starts with things we are assuming to be true. The statements are in the left column and the reasons are in the right column. Prove that 5. I'm reading the chapter on univesal algebras in P. A mathematical proof is an argument which convinces other people that something is true. Work that mental math magic as you race to find creative equations hidden among nine number cards.
We start with the language of Propositional Logic, where the rules for proofs are very If-then statement. To demonstrate the power of mathematical induction, we shall prove an algebraic equation and a geometric formula with induction. Other Ways to Use in Classroom: Put the following proofs on magnetic strips and have students rearrange them on the board or assign a student a proof based on their level of understanding. Heron's Formula -- An algebraic proof. It ends with the thing we are trying to prove. Linear Algebra is a theory that concerns the solutions and the structure of solutions for linear equations.
If a formula works for 1 million specific examples, this is still not a proof. A proof is a mathematical argument used to verify the truth of a statement. Jessica Su. Select a proof from the list below to get started. It only takes a minute to sign up. Although, you can make a pretty good intuitive argument just based on the symmetry of the triangle itself. Question: Are you convinced they have a proof? Write down the information that is given to you because it will help you begin the problem. Elements of Mathematics: Foundations EMF is an complete secondary school online curriculum for mathematically talented students that uses a foundation of discrete mathematics to launch students into modern proof-based mathematics.
Among his many important achievements, one that Gauss gave special attention to was the proof of what is, somewhat mistakenly, known as the Fundamental Theorem of Algebra, a basic result of analysis the state- ment of which is simple enough to be understood by any layman with knowledge of high school mathematics. Angle A and angle B form a straight line because they are linear pairs. Also see the Mathematical Association of America Math DL review of the 1st edition , the Amazon reviews, The methods of mathematical proof are based on deductive reasoning. Lessons are practical in nature informal in tone, and contain many worked examples and warnings about problem areas and probable "trick" questions.
Prove that 8. Convert the two-column proof to an informal written proof. The language of mathematics p. Prove that 7. Most of the proofs I think of should be accessible to a middle grade school student. I think Donaldson described it as the height of fashion in his book on Riemann surfaces.
Find recent content on the main index or look in the archives to find all content. Two Column Proofs - Concept. Answers are provided. Mathematical statements p. With an indirect proof, instead of proving that something must be true, you prove it indirectly by showing that it cannot be false. Students often try to prove a statement without using the entire hypothesis.
Variables: The proper use of variables in an argument is critical. Algebra definition, the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc. A proof is an argument intended to convince the reader that a general principle is true in all situations. The mathematics itself consists of logical deductions from the axioms. Vector Proofs to Geometry Theorems In geometry there is a theorem— Midsegment Theorem —that states: The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to half the length of the third side.
Type all answers in the boxes provided. Classic Fallacies. Below are some effective methods of proof that might aim you in the right direction. A formal proof is a proof in which every logical inference has been checked all the way back to the fundamental axioms of mathematics. Happy math. In Proposition 1. Yet its role in secondary school mathematics has traditionally been peripheral at best; the only sub- stantial treatment of proof is limited to geometry. These courses are proof-based, and require that students be able to write and understand mathematical proofs.
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Algebraic K-theory - Wikipedia
Prove that 3. Proofs about graphs and trees. Proof School is a secondary school in San Francisco for kids who love math. The idea of a group. If need be, axioms may be invented. Mathematical Proof - about the theory and techniques of proving mathematical theorems; Resources Manual of style. The tradition of mathematics is a long and glorious one. To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false.
From algebraic proofs to algebraic expressions, we have every part discussed. Proofs using algebra. A certain amount of mathematical maturity is necessary to nd and study applications of abstract algebra. Let H be a subset of a vector space V. Unless you can see where every bit of the hypothesis was used, you have not truly understood the proof. Math terminology relating to sets and logic as encountered in the Algebra I to Calculus curriculum.
The group K X is now known as K 0 X. Upon replacing vector bundles by projective modules, K 0 also became defined for non-commutative rings, where it had applications to group representations. Atiyah and Hirzebruch quickly transported Grothendieck's construction to topology and used it to define topological K-theory. The setting of algebraic varieties, however, is much more rigid, and the flexible constructions used in topology were not available. While the group K 0 seemed to satisfy the necessary properties to be the beginning of a cohomology theory of algebraic varieties and of non-commutative rings, there was no clear definition of the higher K n X.
Even as such definitions were developed, technical issues surrounding restriction and gluing usually forced K n to be defined only for rings, not for varieties. While it was not initially known, a group related to K 1 had already been introduced in another context. It was clearly true that Betti numbers were unchanged by subdividing the triangulation, and therefore it was clear that any two triangulations that shared a common subdivision had the same Betti numbers.
What was not known was that any two triangulations admitted a common subdivision. This hypothesis became a conjecture known as the Hauptvermutung roughly "main conjecture". The fact that triangulations were stable under subdivision led J. Whitehead to introduce the notion of simple homotopy type. Part of the motivation for this definition is that a subdivision of a triangulation is simple homotopy equivalent to the original triangulation, and therefore two triangulations that share a common subdivision must be simple homotopy equivalent.
Whitehead proved that simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the torsion. Whitehead found examples of non-trivial torsion and thereby proved that some homotopy equivalences were not simple. Later John Milnor used Reidemeister torsion , an invariant related to Whitehead torsion, to disprove the Hauptvermutung. All such vector bundles come from the clutching construction , where two trivial vector bundles on two halves of a space are glued along a common strip of the space.
This gluing data is expressed using the general linear group , but elements of that group coming from elementary matrices matrices corresponding to elementary row or column operations define equivalent gluings. They also provided a definition of K 0 of a homomorphism of rings and proved that K 0 and K 1 could be fit together into an exact sequence similar to the relative homology exact sequence.
Work in K -theory from this period culminated in Bass' book Algebraic K -theory. Of particular note is that Bass, building on his earlier work with Murthy,  provided the first proof of what is now known as the fundamental theorem of algebraic K -theory. Bass recognized that this theorem provided a description of K 0 entirely in terms of K 1. In independent work, Max Karoubi gave another definition of negative K -groups for certain categories and proved that his definitions yielded that same groups as those of Bass.
The next major development in the subject came with the definition of K 2. Steinberg studied the universal central extensions of a Chevalley group over a field and gave an explicit presentation of this group in terms of generators and relations.
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Hideya Matsumoto 's thesis  showed that for a field F , K 2 F was isomorphic to:. This relation is also satisfied by the Hilbert symbol , which expresses the solvability of quadratic equations over local fields. In particular, John Tate was able to prove that K 2 Q is essentially structured around the law of quadratic reciprocity.
In the late s and early s, several definitions of higher K -theory were proposed. Swan  and Gersten  both produced definitions of K n for all n , and Gersten proved that his and Swan's theories were equivalent, but the two theories were not known to satisfy all the expected properties. Nobile and Villamayor also proposed a definition of higher K -groups. Their K -groups are now called KV n and are related to homotopy-invariant modifications of K -theory. Inspired in part by Matsumoto's theorem, Milnor made a definition of the higher K -groups of a field.
Much later, it was discovered by Nesterenko and Suslin  and by Totaro  that Milnor K -theory is actually a direct summand of the true K -theory of the field. Specifically, K -groups have a filtration called the weight filtration , and the Milnor K -theory of a field is the highest weight-graded piece of the K -theory. Additionally, Thomason discovered that there is no analog of Milnor K -theory for a general variety.
The first definition of higher K -theory to be widely accepted was Daniel Quillen 's. This modification was called the plus construction. Not only did this recover K 1 and K 2 , the relation of K -theory to the Adams operations allowed Quillen to compute the K -groups of finite fields. The classifying space BGL is connected, so Quillen's definition failed to give the correct value for K 0. Additionally, it did not give any negative K -groups.
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Since K 0 had a known and accepted definition it was possible to sidestep this difficulty, but it remained technically awkward. Conceptually, the problem was that the definition sprung from GL , which was classically the source of K 1. Because GL knows only about gluing vector bundles, not about the vector bundles themselves, it was impossible for it to describe K 0. Where Grothendieck worked with isomorphism classes of bundles, Segal worked with the bundles themselves and used isomorphisms of the bundles as part of his data.
This results in a spectrum whose homotopy groups are the higher K -groups including K 0. However, Segal's approach was only able to impose relations for split exact sequences, not general exact sequences. However, there are non-split short exact sequences in the category of vector bundles on a variety and in the category of all modules over a ring, so Segal's approach did not apply to all cases of interest.
In the spring of , Quillen found another approach to the construction of higher K -theory which was to prove enormously successful. This new definition began with an exact category , a category satisfying certain formal properties similar to, but slightly weaker than, the properties satisfied by a category of modules or vector bundles. From this he constructed an auxiliary category using a new device called his " Q -construction. If C is an abelian category, then QC is a category with the same objects as C but whose morphisms are defined in terms of short exact sequences in C.
This yielded the correct K 0 and led to simpler proofs, but still did not yield any negative K -groups. All abelian categories are exact categories, but not all exact categories are abelian. Because Quillen was able to work in this more general situation, he was able to use exact categories as tools in his proofs. This technique allowed him to prove many of the basic theorems of algebraic K -theory. Additionally, it was possible to prove that the earlier definitions of Swan and Gersten were equivalent to Quillen's under certain conditions.
K -theory now appeared to be a homology theory for rings and a cohomology theory for varieties. However, many of its basic theorems carried the hypothesis that the ring or variety in question was regular. One of the basic expected relations was a long exact sequence called the "localization sequence" relating the K -theory of a variety X and an open subset U. Quillen was unable to prove the existence of the localization sequence in full generality.
G -theory had been defined early in the development of the subject by Grothendieck. Grothendieck defined G 0 X for a variety X to be the free abelian group on isomorphism classes of coherent sheaves on X , modulo relations coming from exact sequences of coherent sheaves.
In the categorical framework adopted by later authors, the K -theory of a variety is the K -theory of its category of vector bundles, while its G -theory is the K -theory of its category of coherent sheaves. Not only could Quillen prove the existence of a localization exact sequence for G -theory, he could prove that for a regular ring or variety, K -theory equaled G -theory, and therefore K -theory of regular varieties had a localization exact sequence.
Since this sequence was fundamental to many of the facts in the subject, regularity hypotheses pervaded early work on higher K -theory. The earliest application of algebraic K -theory to topology was Whitehead's construction of Whitehead torsion. A closely related construction was found by C. Wall in This invariant is called Wall's finiteness obstruction because X is homotopy equivalent to a finite complex if and only if the invariant vanishes.
Laurent Siebenmann in his thesis found an invariant similar to Wall's that gives an obstruction to an open manifold being the interior of a compact manifold with boundary. Whitehead torsion was eventually reinterpreted in a more directly K -theoretic way. This reinterpretation happened through the study of h -cobordisms.
If M and N are not assumed to be simply connected, then an h -cobordism need not be a cylinder. This generalizes the h -cobordism theorem because the simple connectedness hypotheses imply that the relevant Whitehead group is trivial. In fact the s -cobordism theorem implies that there is a bijective correspondence between isomorphism classes of h -cobordisms and elements of the Whitehead group.
An obvious question associated with the existence of h -cobordisms is their uniqueness. The natural notion of equivalence is isotopy. Jean Cerf proved that for simply connected smooth manifolds M of dimension at least 5, isotopy of h -cobordisms is the same as a weaker notion called pseudo-isotopy.
The proper context for the s -cobordism theorem is the classifying space of h -cobordisms. Consideration of these questions led Waldhausen to introduced his algebraic K -theory of spaces. In order to fully develop A -theory, Waldhausen made significant technical advances in the foundations of K -theory. Quillen suggested to his student Kenneth Brown that it might be possible to create a theory of sheaves of spectra of which K -theory would provide an example. The sheaf of K -theory spectra would, to each open subset of a variety, associate the K -theory of that open subset.
Brown developed such a theory for his thesis. Simultaneously, Gersten had the same idea. This is now called the Brown—Gersten spectral sequence. This is known as Bloch's formula. While progress has been made on Gersten's conjecture since then, the general case remains open. Lichtenbaum conjectured that special values of the zeta function of a number field could be expressed in terms of the K -groups of the ring of integers of the field. Quillen therefore generalized Lichtenbaum's conjecture, predicting the existence of a spectral sequence like the Atiyah—Hirzebruch spectral sequence in topological K -theory.
In the case studied by Lichtenbaum, the spectral sequence would degenerate, yielding Lichtenbaum's conjecture. William G. Throughout the s and early s, K -theory on singular varieties still lacked adequate foundations. While it was believed that Quillen's K -theory gave the correct groups, it was not known that these groups had all of the envisaged properties.
For this, algebraic K -theory had to be reformulated. This was done by Thomason in a lengthy monograph which he co-credited to his dead friend Thomas Trobaugh, who he said gave him a key idea in a dream. There, K 0 was described in terms of complexes of sheaves on algebraic varieties. Thomason discovered that if one worked with in derived category of sheaves, there was a simple description of when a complex of sheaves could be extended from an open subset of a variety to the whole variety. By applying Waldhausen's construction of K -theory to derived categories, Thomason was able to prove that algebraic K -theory had all the expected properties of a cohomology theory.
In , Keith Dennis discovered an entirely novel technique for computing K -theory based on Hochschild homology.


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Why using the Spline drawing tool plugin?
This plugin adds a tool to edit ways manually from sources like satellite image, but improved to place nodes computed from splines, in order to give more accurate and more realistic curves.
That tool is based on Bézier curves. A Bézier curve is defined by a set of control points P0 through Pn, where n is called its order (n = 1 for linear, 2 for quadratic, etc.). The first and last control points are always the end points of the curve; however, the intermediate control points (if any) generally do not lie on the curve.
In that tool is used quadratic Bézier curves. A quadratic Bézier curve is the path traced by the function B(t), given points P0, P1, and P2.
To draw a line using this equation, one can divide the curve into smaller segments, calculate the end points of each segment using the Bézier cubic equation and draw the line for the segment. For instance, one can draw a line between the points defined by t = 0 and t = 0.01, then t = 0.01 and t = 0.02, and so on.
The plugin should be available in your JOSM Plugins preference window under "Spline drawing tool" or you can download it from .
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- When add the third node(helping node "P1 from equation") will result an curve way.
JOSM/Plugins/Splinex another JOSM plugin for drawing splines
Feedback and suggestions
For news ideas please provided them in this section
- Please change cursor when spline-mode is activated. - wambacher -
- I think it would be good to let edit the line before saving it as dots (if it's possible after saving it would be great too, but it can be harder). - kocio