From “Brain Mysteries” 5-2-2012:
Tape measures. Rulers. Graphs. The gas gauge in your car, and the icon on your favorite digital device showing battery power. The number line and its cousins – notations that map numbers onto space and often represent magnitude – are everywhere. Most adults in industrialized societies are so fluent at using the concept, we hardly think about it. We don’t stop to wonder: Is it “natural”? Is it cultural?
Now, challenging a mainstream scholarly position that the number-line concept is innate, a study suggests it is learned.
Another example of the vastly under appreciated role of learning in how our minds develop. What’s not learned?
Read: Study finds twist to the story of the number line (5/2/2012)
I believe it is both innate and taught. “”These findings suggest that how we think about abstract concepts is even more flexible than previously thought and is profoundly affected by language, culture and environment,” said Nunez.
“Our familiar notions on ‘fundamental’ concepts such as time and number are so deeply ingrained that they feel natural to us, as though they couldn’t be any other way,” added former graduate student Cooperrider. “When confronted with radically different ways of construing experience, we can no longer take for granted our own. Ultimately, no way is more or less ‘natural’ than the Yupno way.”
Example – true story
One night, walking down to the store with my youngest child who was 3 1/2 at the time, she pointed to a star, the brightest star in the sky, and spoke. Back then, my child barely talk, let alone said more than ten words in a conversation, and rarely ever initiated a conversation.
“Look Mommy, there’s heaven, and the number line goes through heaven.” I did not think too much of it, and I put it down to nursery school, but when we arrived home, she wanted to explained further. She grabbed a book, that had the star constellations and planets information and turned to the page where Earth is shown with the star constellations. She pointed to Earth, and told me that Earth is zero, and pointed to a star constellation that she believe is heaven, and said that is where the numbers ended.
By age 5, different place, different view of constellations, but I believed the same star, my 5 year expands her description of the number line. All positive numbers goes through heaven and all negative numbers are returned to Earth. Since Earth is the number zero, zero represents all the positive and negative numbers. She sees the number line as a circular line, instead of a straight line, that goes on forever.
I thought at the time, she should not have any problems with math at school. Was I wrong, the math curriculum being taught today, does not abide by such radical concepts as zero being one composed of negative and positive numbers but still represents nothing. Zilch. Nada. By the end of grade 1, my youngest child could no longer do the simple addition and subtraction done with such ease when she was smaller, and she was hopelessly confused. In her dyslexic mind, the number line was three dimensional that could be manipulated. In the classroom, math was flat and two-dimensional.
By grade 6, home tutoring and re-teaching at home, determined to turned my child into a child that is adept with numbers – I started with the number line. I only had 4 weeks before the unit test, to undo the damage by the fuzzy illogical math curriculum and math instruction. I had my work cut out for me. In the process, we spent a great deal of time on the number zero and the number line itself. It was one of those aha moments I had, remembering my child’s early concept of heaven and the number line. I told her to look at the straight line on the paper as a compressed circle that has also been stretched to where it is no longer a circle but a straight line. From the point, it was easy and math became easy for my child, and the first unit test came back with a 82.
Rather funny, my dyslexic child would relate very well with the natives of Papua New Guinea, but not in the 21st century elementary school math class. “Most adults in industrialized societies are so fluent at using the concept, we hardly think about it. We don’t stop to wonder: Is it “natural”? Is it cultural?”
It could be both. In the industrialized societies they are only fluent in the first and second dimensional thinking, but the cultural aspect won;t allow people to venture out into the third and fourth dimensions.


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.


This game requires 2 or more players. Each player gets one dart to shoot at the number they would like to be. Nobody can have the same number. Once the numbers are selected, the one with the lowest starts the game. Three darts are allowed per turn. The object is to accumulate 5 points in your number to become a killer (like baseball the outer part is worth 2, the inner 1). As killer you can shoot other player’s numbers. When they are knocked down to zero, they are out. The last survivor wins.


Proof complexity is a fast growing field connected to logic, computational complexity, and combinatrorial optimization. In order to fully understand the concepts and their importance, one should go to the roots of this field and follow its history. It started with investigations of the Hilbert school at the beginning of the 20th century, whose aim was to prove consistency and completeness of basic theories used in the foundations of mathematics. After the groundbreaking result of Goedel in the 1930s, the incompleteness of Peano Arithmetic and any theory extending it, the focus turned to the study of fragments of Peano Arithmetic. With the advent of computational complexity theory in the 1970s, connections between weak fragments and complexity classes were discovered. Soon after that provability of certain sentences in weak fragments was connected with the existence of polynomial length propositional proofs of sequences of tautologies expressing these sentences in propositional logic. Another boost for the emerging field of proof complexity came from the realization that several heuristic used in combinatorial optimization could be viewed as propositional proof systems.
In this lecture we will briefly review the history and introduce basic concepts of proof complexity. After that we will demonstrate on a concrete example how one can prove an independence result using propositional proof complexity.


F2 - Rainbow Fish
This week, we have been assessing children on their maths skills. They have to count 1:1 , order numbers to 20 and say 1 more or 1 less. We have also looked at addition and subtraction. We are good at recognising the symbols, but still get a little muddled applying our knowledge, eg for 4—1 = some children put up 4 fingers but instead of bending one down they add another one. Our children are trying very hard to grasp this tricky concept! We will continue to practise to put our fingers back down for takeaways. Maybe you could practise at home too.
Our theme this week is the Rainbow Fish. On the Giraffes’ walkabout we saw lots of changes in the wood! We kept stopping to answer questions on lots of different things to earn Rainbow Fish scales. We talked about what makes us happy; what makes us scared and what we do to make us feel better. What would you do if you saw someone crying? The Giraffes and Zebras had lots of wonderful discussions and the children's kind, caring ideas shone through! We earned lots of beautiful Rainbow Fish scales and used these to decorate our fish and make him beautiful again! Check out our F2 notice board to see some of the lovely things our children have been doing.


In each plane, there are two laboratories. There's a computer in each lab. In one of the computers on a plane, there is a fifth of the formula of the red dust. To be able to make the red dust, you need all 5 parts of the formula. When you have them all, use the data recall on your control panel to show it, add the numbers together, convert them into base 6 with the table below and enter the calculated number into the computer in the air plant. Here's how to get a part of the formula:
Locate a lab.
Open the doors and enter it. Stand in front of the computer and use the computer link function on your control panel.
If there was a part of the formula in the computer in it, this is shown on the display:
If this was the wrong lab, this will be shown instead:
Decimal Base 6 ------- ------ 0 0 1 1 2 2 3 3 4 4 5 5 6 10 7 11 8 12 9 13 10 14 11 15 12 20 13 21 14 22 and so on...An example: Suppose you have collected the entire formula, and it reads 0 5 0 3 1 (in visitor digits). The sum is 0+5+0+3+1=9, which is then converted to 13 using the table above. Then the visitor digits 1 and 3 should be entered into the computer in the air plant.


Last time we talked about taking into account the conversion efficiency of the inverter, circuit losses, and protecting the inverter from voltage spikes and surges. The rated power of the inverter we choose needs to reach about 1.4 times the total power of all the loaded appliances.
For example, the total power of a family’s load appliances is 1000W, then you need to choose a 1400W inverter.
Of course, this 1.4 times is only for ordinary load appliances. If the home has air conditioning, water pumps and other inductive appliances, then the inverter should choose more power.
Because the peak power of inductive appliances at start is very large, often 2-5 times the rated power. Take the middle value 3 times, is the power of the inverter.
5 LED lights :5 x 20W = 100W
1 LCD TV :120W
1 laptop :50W
2 phone chargers :2 15W = 30W
1 air conditioner: 1000W
Calculate the total fixed power (W) of all appliances.
Inverter of appropriate size = (100W+120W+50W+30W) *1.4+1000W*3=3420W


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Read on to learn more about what slope is and some easy ways to When you need to solve a math problem and want to make sure you have the right answer, a calculator can come in handy. Calculators are small computers that can perform a variety of calculations and can solve equations and problems. While th How much are you worth, financially? Many people have no idea what their net worth is, although they often read about the net worth of famous people and rich business owners. Your own net worth is a good number to know, though. It can help Starting a new loan is a very big decision.
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There are fifteen numbers. The median value of this set of numbers is 24. Rahul’s family drove through 7 Solution:. Hence, the median of the gasoline cost is 1.84. There are three states with Se hela listan på gigacalculator.com In this video we go over two examples on how to find the median. Remember that the median is basically the middle value in our data. For more videos visit How To Calculate Median For Continuous Series || Median Of Continuous Series || FCA SIR CLASSESHi, I am Abdul Wahab, Welcome to my Youtube Channel FCA SIR To calculate the sample median in R, use the built-in median () function.
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You can type some contents as shown as below: To find the median, order all of the numbers in the set from least to greatest. If the set is made of an odd number of integers, the median will be the middle number, and if it’s an even number, add the 2 middle numbers together and divide them by 2 to get the median. How to Estimate the Median of a Histogram.
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Compute the median of a 2-d array. Next, let’s work through a slightly more complicated example. Here, we’re going to calculate the median of a 2-dimensional NumPy array. 2017-05-19 · How to calculate median in Excel - formula examples.
Values must be numeric and may be separated by commas, spaces or new-line. You may also copy and paste data into the text box. For example, you have a range data shown as below, and you want to calculate the median of value of a in 2-Jan, you can do as follow steps: 1.
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Jämförelsedata - mera om Median Survival time för referensmedlet sunitinib we are not yet able to calculate the median overall survival rate. a. Make the frequency distribution table with appropiate class interval, frequency, cumulative frequency. b. Calculate the mean, median, mode beskriver dessa medianvärden eller medelvärden inte variation mellan confirmation dates and calculated the median time delay between these two dates. This study investigates two methods of how monthly Chl-a medians can be predicted (i) new monthly regression models from median summer total phosphorus av R Edvinsson · 2021 — An alternative method to estimate the Housing Price Index is based on the unweighted central measure (median, arithmetic average, geometric av ML UCOVNIK · 2011 · Citerat av 170 — Calculation of gestational age was based on the last menstrual period or, when it to also accurately determine the mean measurement-to-delivery-interval.


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The books listed in this site can be downloaded for free. The books are mostly in Portable Data File (PDF), but there are some in epub format. Feel free to download the books. If you can, please also donate a small amount for this site to continue its operations. Engineering Mathematics with Examples and Applications provides a compact and concise primer in the field, starting with the foundations, and then gradually developing to the advanced level of mathematics that is necessary for all engineering disciplines. Therefore, this book's aim is to help undergraduates rapidly develop the fundamental knowledge of engineering mathematics.
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I am still trying to Author: Ali Kayaspor. Madison College Textbook for College Mathematics Revised Fall of Edition. Authored by various members of the Mathematics Department of Madison Area Technical College.
How to use this workbook. Each chapter consists of text plus worked examples. These are followed by Exercises labeled as Your Size: 2MB.
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Your Contact Information. Emeritus Professor Associate Professor Dr. Mrs. Math is good since it provides a lot of lessons to learn A lot of mathematicians form formulas for us to use during our real life. That is my inspection.
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A note on distances
In order to calculate distances between members we use the longitude and latitude of each member and apply some maths.
While this is normally fairly accurate, it should be noted that the earth is not a perfect sphere, and as such, the distance between members needs to be considered as approximate.
The distances are also calculating a direct route from point A to point B on the globe (an as the crow flies approach) and does not take into account the route you may have to actually travel which could add extra miles to the distance.
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Auto Fibonacci retracement indicator Guide for MT4/MT5
Auto Fibonacci retracement is an MT4 trading indicator in the technical analysis . It aims to determine S/R levels based on precise retracements. To explain, this retracement indicator for MT5 stipulates that the price would retrace by a predictable amount using Fibonacci numbers. Then, the same price will continue its original movement. So, the Fibonacci indicator was invented to anticipate retracement grades on Tradingview charts. This study would demonstrate the importance of the Fibonacci retracement strategy in the FX exchange.
Fibonacci Retracement Indicator FAQ
What is the Fibonacci retracement indicator ?
Auto Fibonacci retracement indicator is based on Fibonacci number sequences. Discovered by the Italian mathematician Leonardo Bonacci, the number sequence is said to be observed in nature. To clarify, it is calculated by adding the number and its predecessor together (0, 1, 1, 2, 3, 5, 8). Chartists believe that the natural aspect of this mathematical tool can also be applied in technical analysis. Besides, they anticipate satisfactory trading results using it. The Fibonacci retracement goal is to give future areas of support and resistance.
To explain, any asset experiences a price retracement in the opposite direction near these grades. For further explanation, when the current price of the instrument is trading upwards, a correction can happen in the opposite direction. This correction is referred to as retracement. In fact, a downward move may cease near retracement grades. In other words, after reaching one of four retracement grades of: 23.6 percent, 38.2 percent, 61.8 percent, and 76.4 percent.
How do you draw Fibonacci retracements ?
Many retracement levels result from the computation of this analytic tool. In fact, each level represents support in the case of an upward move. Contrarily, resistance in the case of a downward move. For instance, we will consider the 23.6 percent level in the computation process.
Case of an upside moveIn an upside move, prices are going up. So at some point, the correction will happen downwards. To clarify, the Fibonacci MT4 indicator predicted that the downer retracement level would be equal to:
- Predicted Support = [ Highest point – ( Upside high – Upside low ) × 23.6 % ]
Case of an downside moveOn a downside move, the price of a security is decreasing. This means that at some point, the prices will start to go up before going down again. The Fibonacci MT5 indicator anticipated that the upper retracement grade would be equal to:
- Predicted Resistance = [ Highest point – ( Downside high -Downside low ) × 23.6 % ]
How to trade with this Indicator ?
It is recommended to use the Fibonacci retracement with another tendency indicator. Because the resulting trading system is more reliable than depending solely on retracements. Also, trendlines could be very useful in this case. So, these are a few steps to trade retracements accurately. Firstly, the forex trader needs to identify a strong move zone. That is to say, a trending upward or downward direction. Secondly, a price correction will happen in the opposite direction. Thirdly, the currency trader will select the Fibonacci Tradingview indicator and choose the trendline. This line should link the lowest point to the highest point. Lastly, four stations of S&R will take place on the screen.
Upward move exampleThe trendline on the following chart shows a bullish move. Here the pullback move has attained only the 23.6% Fibo grade. Consequently, forex parties seem optimistic regarding the currency pair evolution. Because they are making active buy positions. Thus, chartists can gauge the prevailing of forex players using Fibo grades. In this example, will use the 23.6 percent level. That means that the predicted S following the price evolution will be at the 1.2026 level. And in case of prices kept going down and breaking the Fibonacci line, we move to a 38.2 percent level and so on.
- Projected S = [ 1.20884 – (1.20884 – 1.172) x 23.6 % ] = 1.2000205
A crypto trader should buy the asset when the corrective downward retracement starts. And since we calculated the upcoming supporting levels, the trader can sell the cryptocurrency back at a higher price after the retracement.
The investor profit in this case would be equal to the difference between the buy and sell price of the chosen asset, multiplied by the quantity acquired. When used correctly, the Fibonacci retracement indicator can assist crypto traders to generate a lot of profits on MT4 charts.
Downward move example
As a start, the forex trader has to find a clear downward line. Clearly, this line should be linking the highest price level to the lowest price level during the selected period.
Then, using the 38.2 percent Fibonacci retracement level, it is possible for the currency trader to calculate the predicted R level for the chosen percentage as follows :
- Projected R = [ 1.14639 – ( 1.14639 – 1.1376 ) x 38.2 % ] = 1.14303
Similarly, the forex trader should sell the asset at the highest point, before the price starts going downwards. Next, buy it back at a cheaper price before the retracement ends.
The investor gain is the difference between the selling price before it falls, and the buy price just after the retracement.
Here the reversal activity stopped to expand near the 38.2% Fibo level. Consequently, this may signal that the downside move would sustain over time. Thus, there is a high probability that prices would resume falling down afterward.
Download Auto Fibonacci retracement indicator for MT4
Auto Fibonacci retracement indicator MT4 belongs to retracement indicators in MetaTrader 4. A lot of forex traders use it because of its simplicity and applicability to any trading instrument. Moreover, this MT4 indicator is easy to implement in the MetaTrader 4 platform using Fibonacci grades. Yet, the detection of an obvious direction is recommended prior to applying this indicator for more precise predictions on the forex market. Indeed it enables traders to plot support lines, determine resistance levels, and set orders.
Download Auto Fibonacci retracement indicator for MT5Auto Fibonacci retracement indicator MT5 is a forex indicator used in MetaTrader 5 to detect the Fibonacci retracements quickly on your price chart. However, this MT5 indicator can be used by newbies and professional traders as it is very easy. The auto Fibonacci retracement indicator for MT5 is preinstalled in the MetaTrader 5 terminal (picture below). Additionally, the MT5 indicator can be found in Insert/Object and then Fibonacci retracement. This screenshot shows two cases in crypto markets. One in an upward movement. And the other in a downward move.
In conclusion, Fibonacci retracement is an effective Tradingview indicator. In fact, it arrives to give an approximation of prospective supporting or resisting grades during a retracement. Most importantly, a forex trending market upwards or downwards must always be the reference before thinking to implement a Fibonacci retracement strategy.
Fibonacci Retracements are:
- Short-term grades that occur after strong upside or downside moves.
- Generated by the software after linking the highest point with the lowest one.
- Horizontal lines indicating where the price could face support or resistance may occur.
- Composed of 4 retracement levels indicating when the price cannot go below or above.


Grades 1-8 Summer Assignments
Students in grades 1-8 have Reading and Math summer assignments. The summer assignments are for incoming grade levels. New students in grades 5-8 should work on grade level IXL math and language arts lessons. Any questions concerning summer reading contact Mrs. Pietsch at firstname.lastname@example.org.
Any questions accessing IXL contact Mrs. Greer at email@example.com.
Combined Summer Math Work for Incoming 5th – 8th Graders (including advanced math students)


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By David P. Landau,Kurt Binder
Read or Download A Guide to Monte Carlo Simulations in Statistical Physics PDF
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Additional info for A Guide to Monte Carlo Simulations in Statistical Physics
A Guide to Monte Carlo Simulations in Statistical Physics by David P. Landau,Kurt Binder


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This frog-themed count and graph activity is a great way for preschoolers to practice counting to ten and graphing their results during your spring and summer lessons.
Students will enjoy finding, counting, and graphing the frogs, butterflies, and lily pads in this fun and adorable math activity that is perfect for spring and summer.
This activity pack covers the following skills:
• counting to 10
• writing numbers to 10
• graphing numbers to 10
• visual discrimination (finding the various images)
• fine motor skills (coloring and writing)
• analyzing data (how many, how many more, least, most, more, less)
The following pages are included in this pack:
• I Spy page featuring frogs, butterflies, and lily pads
• Graphing page with traceable numbers
• Instruction page with answers
This is a digital product. You will receive a PDF only.
Please note: This Frog Pond count and graph activity is available, along with 31 other themes, in my year-long count and graph bundle! Check it out!


Problem: Two 1.2 kip vertical loads are applied to pin B of the assembly shown. Knowing that a 0.6 inch diameter pin is used at each connection, determine the maximum value of the average normal stress in links AB and BC. Also determine the shearing stresses in the pins and the bearing stress at C.
Solution: The links are two-force members, so a free body diagram of pin B is:
Now I compute the normal stresses...
Next, I find the shearing stresses in the pins...
Note that a factor of two was used in each denominator since both pins are in double shear.
The bearing stress in member BC at C is:
1 This is Problem 1.16
in Beer & Johnston's Mechanics of Materials, 2nd Edition,


HISTORY TELLS HOW Euclid, the Greek mathematician and philosopher, who flourished 300 years B. C. while expounding the problem of subdividing the circle to King Ptolemy, was interrupted by the irate monarch exclaiming: “I am wearied by such dull lessons, and refuse to burden my memory with stupid rules!” Whereupon the great mathematician replied: “Then your majesty will graciously permit me to resign the position of Imperial instructor, for none but a fool knows of a Royal Road to Mathematics.”
“Right you are, Euc!” interjected Beppo, the court jester, as he pushed his way to the blackboard, “and, in accepting the position so gracefully tendered. I will proceed to demonstrate how the great principles of higher mathematics can be taught by simple kindergarten methods which children may understand and remember.”
“Philosophers say:” ‘what is learned with pleasure is never forgotten, but knowledge cannot be beaten into the head with a wormwood club.’ “Pupils should not be made to commit rules to memory; everything should be explained so that they can formulate rules in their own language. A pedagogue who teaches rules would be a good one to train parrots!”
“Dry mathematical problems are more digestible when presented in palatable form, and the mind becomes stored with valuable information when the illustrations are gleaned from the classical tidbits of history.”
“Mathematics, which constitutes the most important branch of learning, forms the groundwork of the arts and sciences, and is so essential to the successful man of affairs, as well as the development of a clear brain, that parents should realize the advantage of encouraging an early love for puzzles, tricks and problems among their children.”
“With the kind permission of your majesty,” continued Beppo. “we will now elucidate the subdivision of the circle by asking Tommy Riddles, the court crier whose learning is limited to the science of simple addition, to show into how many pieces it is possible to divide a German Pancake with seven straight cuts of a knife?”
“Furthermore, to add a point to the moral of the story of the sword of Damocles, which is shown to be suspended over our heads by single thread, we will proceed to impress it indelibly upon the memory by connecting it with a scientific and practical problem; Why is the blade of that scimitar always shown to be curved ?”
“Noting with pleasure the presence of the ‘Pons asinorum.’ the ass's bridge problem, which my distinguished predecessor has made famous as his 47th proposition, which proves that the square described on the long side of a right-angled triangle, termed the hypotenuse, is equal to the sum of the squares of the other two aides, I will ask the author of the 47th proposition to tell how many rails of equal length it would require to enclose a right-angled triangular field if one of the three sides was 47 rails long?”
“The clown's 47th proposition” will doubtless prove that many good mathematicians have much yet to learn regarding the wonderful principles of Pons asinorum which may be said to underlie the foundations of mathematics and geometry.
NOTE.— “Pons asinorum” originally applied to the 5th proposition -- First Book of Euclid -- that “The angles at the base of an isosceles triangle are equal to one another.”
The clown's 47 problem is solved by the rule which produces a right angled triangle from any odd number, viz.: Square it; 47x47=2,209, Divide by 2, and make the hypoth. one greater than the base, Ans.: 47, 1,104 and 1,105. Strange that the fool should have hit upon a dimension to which we can find but one answer in whole numbers. There would? have been ten answers if he had said 48 rails. To find a triangle from any even number, square the given number and divide by 4. The quotient will be 1 more than the base and 1 less than the hypoth., viz.: 8x8=64, divided by 4=18, So the triangle would be 8, 17 and 19.
I really blush to give the fool's answer to "Why was the blade of that scimitar curved?" It was curved so that it would fit the scabbard!


This graph page provides several ways to visualize the distribution of performance measurements across time intervals.
MyKeynote offers several different distribution graph styles, which you can select in the Distribution Graph Type list on the chart configuration page. This page also provides a control for the interval, that is, the number of buckets or time slices that will be shown in the graph.
Actions for this graph
The Actions menu at the top right of the Distribution graph provides access to several tools to help make use of the graph and its underlying data.
See Chart Actions for information about graph page controls that you can use to save graphs, download data, and refresh the graph.
Distribution graph types
These graph types are available:
This is the default graph style for distribution graphs. Each vertical bar represents one time bucket, with the height of the bar indicating the count of data points that fell within that time bucket. Each time bucket represents a second.
Distribution Graph - Histogram
Frequency polygon graph
Similar to a histogram, this graph type contains one point representing the number of data points for each time bucket.
This is a cumulative frequency graph. Each dot on the graph represents the number of measurements that were at or below the corresponding time segment.
This graph type provides a quick way to visualize the number of measurements that lie within the lowest, second, third, and fourth quartiles of measured data points. It also shows the lowest and highest data points.
The box plot is a graphical display of a five number summary, including: the smallest observation, lower quartile (Q1), median, upper quartile (Q3), and largest observation.
This process is followed to draw a box plot:
- Draw a box from the 25th to the 75th percentile.
- Split the box with a line at the median.
- Draw a thin lines (whisker) from the 75th percentile up to the maximum value.
- Draw another thin line from the 25th percentile down to the minimum value.
The length of the box in a box plot, that is, the distance between the 25th and 75th percentiles, is known as the interquartile range. You can use this box length to detect outliers. If any whisker is more than 1.5 times as long as the length of the box, then we have evidence of outliers. A common variation on the box plot is to draw the whisker to the value which is just shy of 1.5 box lengths away, and highlight each individual data point more than 1.5 box lengths away.
When setting up a distribution graph on the Graph Settings page you have the option to add a legend to the graph. When Study is set to Yes, a legend appears next to the graph with distribution details, and the distribution graph is overlaid with color bands indicating the lower, upper and middle quartiles.
Distribution - Histogram graph with study legend
Distribution - Frequency Polygon graph with study legend
Distribution - Ogive graph with study legend
Distribution - Box Plot graph with study legend


We hope you have had a lovely weekend with plenty of rest and relaxation. Thank you for all the work you did on tens and ones last week. It was great to hear that many of you became more confident at completing the work independently as the week went on.
This week maths is addition.
Write some number words randomly to (10 green/blue group and 20 red/yellow group) in your child's remote learning book e.g.
Ask your child to read the word and then write the correct numeral next to it, like this
If your child is in green group and can easily do to 10 then let them have a go at reading numbers to 20.
Show your child an addition sentence with a missing number in the first part e.g.
5 + =8
Talk about the fact that they can see the answer already. How could they find the missing number? What do they need to do? What could they use to help them?
Use a number line and show them that they can start with their finger on number 5 and then they have got to count how many jumps it takes to get to 8.
Some children find this activity quite tricky. If they are really struggling use objects to demonstrate and just keep practising a little every day with the number line. You don't have to print the addition cards off below you can just use them for ideas and copy them into the remote learning book.
I have added a number line to print off below or you could just as easily write numbers in a line to 20 for your child to use. If you have a 100 square you could also use that.


Residence Edge and True Odds in Slot machine games and Bingo
The main difference between a casino activity and a video poker equipment is that video poker devices require more skill than most casino games. Among this is Texas Holdem, a casino game of luck, where the outcome is pre-determined. Players select a hand and the machine rolls the quantities for the chosen palm and tells the player whether the hand gets the “low” or “high” card. If it has the high card, the ball player will win; if it has the low card, the player loses the game. There is no other way to gain in Holdem.
The results of a random event has a range, that may be studied using mathematics and statistics. Standard deviation is one particular study, which uses probability concept and various assumptions concerning the distribution of results. Among the assumptions, called the binomial distribution, predicts that the likelihood of an event occurring would depend on the number of outcomes that occur. For example, in a standard deviation analysis, the standard deviation is used to estimate the number of possible results. This can be weighed against the lottery wheel, which runs on the finite amount of draws to estimate the likelihood of getting a specific number of “balls”. Each draw takes place in a discrete interval and is in addition to the others.
Another popular type of casino game is the slot machine game game. Slot machines are played on revolving pubs, and each time the ball lands on a particular line, the amount paid out is correspondingly increased. An average slot machine game gives players a chance to win huge amounts of money, but they aren’t random, since there is a larger chance of hitting the winning line. Actually, it is considered a “lier” type of gambling because it would depend on probability.
One other popular type of casino game may be the card game. Most casinos feature a variant of the game, as much players find this more interesting compared to the more traditional slots. Additionally, there are some variation of the card game in non-casinos, such as bingo. Again, because of its dependability on chance, card games are less predictable than slots. However, many players find card games more exciting than the other styles.
An important characteristic of casino games is the house edge. The house edge is the difference between the expected payoff and the amount actually received by the gambling house. In most slot machines, the home edge is about twenty %, while it may be only ten per cent in bingo. The house edge can also increase when game variations, such as card shuffling, increase the number of hands performed.
In comparison to gambling, casino games require extra calculation skills. While skill in gambling may boost with experience, it isn’t necessarily true for casino games. While luck may affect the results of individual spins, it is impossible to depend totally on luck. The home advantage means that there exists a greater risk of losing profits when one partakes in internet casino games than gambling.
Although the house edge does affect the sum of money that may be won or lost, it is not the principal reason a casino user chooses a 카지노 specific machine. Like any other form of gambling, real money is involved. The primary motivation for playing may be the thrill of winning which is easier attained in slots. Alternatively, it is much harder to get at bingo, the typical card game. If the gambler prefers bingo or slot machines depends on their preferences for leisure and the casino’s offerings.
When all factors are considered, it is easy to note that slots win far more than bingo. Slots offer a chance for the player to obtain additional than one chance to win and if the amount wagered is small, as it is in most casino settings, much more likely than not, the home edge will negate the gains of the individual spins. In this scenario, casinos with larger house benefits keep their slots open more, thereby providing more options to win. If the amount wagered is significant, on the other hand, slot players may prefer to play bingo. Regardless of which casino activity is chosen, the real odds of success will be the very same.


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Question: If your attic is going to have 50 bags of insulation blown into it, does it make much of a difference if it enters flat or lumpy?
Let’s look at an example. If the insulation goes in completely flat, let’s state we have a nice consistent R-value of 30 throughout the attic.
There are all kinds of methods it can enter lumpy, however let’s presume that 50% of the attic has lumps of R-50 insulation, and the other 50% is valleys with only R-10. We wish to discover the average R-value of this lumpy setup. It’s the same quantity of insulation, and the average of R-50 and R-10 is R-30. But, how does it really perform?
Let’s do the math!
Now, we can’t simply average the R-values. If we did that here, we’d get R-30, and we’d be wrong. The heat will take the path of least resistance, and the less resistance you give it, the more heat will stream. If you’ve studied physics, engineering, or structure science, you’ve probably seen the formula for heat flow by conduction:
Heat circulation formula building science
If we computed the amount of heat streaming through the lumps and valleys in this attic, we’d see that the R-10 half (red bar listed above) would allow method more heat to flow than the R-50 half (yellow bar listed above). In fact, the R-10 valleys, having one-third of the R-value, would permit 3 times more heat to flow than if it had R-30. The R-50 lumps would allow only 3/5 the heat flow as R-30.
Also, the total quantity of heat circulation is greater for the lumpy insulation than for the flat insulation As you can see below, the red and yellow parts together amount to a lot more heat flow than the orange (in fact 1.8 times as much).
Heat circulation through flat and lumpy insulation.
Due to the fact that the lumpy insulation allows more heat circulation, it’s got to have a worse average R-value. Clearly, the average R-value is going to be more influenced by the underinsulated part than by the over insulated part.
To find the average R-value the correct way, we first need to transform R-values to U-values. R represents resistance, as in resistance to heat circulation, so the greater the number the better when we’re talking about building materials. U-value, the heat transfer coefficient is the reciprocal of the R-value, so the lower the number, the better.
The equation for average U-value is:.
Average U-value formula, to find the average R-value of insulation.
Taking the attic as 1000 square feet overall and putting in 1/R for U, we get:.
average u-value for R-50 & R-10.
Taking the reciprocal of 0.06, we get the average R-value of about 17, which is much lower than the R-30 we had actually get by improperly balancing the R-values.
As I stated above, heat takes the path of least resistance, so the amount of additional heat going through the R-10 half far exceeds the additional heat flow that’s stopped on the R-50 side. Instead of getting an R-30 average, the lumpy attic has an R-17 average.
In practical terms, this means that if you see an attic with lumpy insulation, act with a rake and smooth it out. In the example I simply exercised, you ‘d almost double the R-value without including any extra insulation!
Another common example of the flat-or-lumpy conundrum is an attic that’s completely insulated except for one little location, say the pull-down attic stairs. We can go through the exact same steps as above and show that an attic that has a uniform R-30 over 99% of the area and 1% at R-1 (the pull-down stairs) will have an average R-value of 23.
That’s right, those pull-down stairs can reduce your overall R-value by 25%. One little uninsulated location reduces the R-value dramatically.


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OCTOPUSES HAVE ZERO BONES: A COUNTING BOOK (HB)
Do you know how many bones an octopus has? How many breaths you take a day? How many times a black bear's heart beats each hour? How many leaves are on an oak tree? What happens when you keep placing zeroes at the end of numbers? Starting with zero instead of one, this whimsical, fact-filled counting book will delight readers of all ages, and encourage them to see numbers hiding in the world around them. Its layered content makes it suitable for the very young, but also offers presents more challenging notions.