In our ongoing effort to provide our readers the finest in easily digestible yet ultimatley informative and entertaining reading, I felt it was time to delve into one of the deepest pockets of punk rock minutiae, one of the richest veins of golden age lore, one of the most profligate wells in music geek fandom, the Angry Samoans Files. Collected over years and years of investigative work into a band that could rightfully take the throne of Greatest Punk Outfit of All Time (and even if they didn't take the throne, they'd be there to throw tomatos at and write songs about whoever did), we shall begin puking forth these files as a running feature. Each installment will tackle a different Samoans-related topic, continuing until we run out of material (which will happen sometime in 2012). If you have any ideas you'd like to see discussed in future installments, by all means let us know.
In this episode, we drop in on Professor Gregg Turner for a little Q&A...
TB: First of all, what do you do in terms of where you work, what you teach, and what kinds of research you do? Feel free to be as specific as you want. i.e.: x'(t) = Ax(t) + Bu(t) and u(t) = Kx[t-r]
Gregg: I'm an assistant professor of mathematics at New Mexico Highlands University, a small liberal arts college in Las Vegas, NM (that's about 60 miles from Santa Fe). I teach the full spectrum of upper and lower division mathematics courses to our majors and as service courses to other departments. My teaching load is 12 credits per semester (roughly 4 classes) which more than sucks - that does not support release time for extensive research. When I completed my PhD in 1991, the job market in mathematics had tanked (suddenly). The net consequence was that you'd have routinely like 300-500 applications per advertised tenure track position. Friends of mine at UCLA, to brighten an otherwise grim prognosis, would wallpaper their offices with the myriad of rejection notices they'd receive! I finished my degree at Claremont Graduate University and my dissertation mentor was a rather well known biomathematician at Pomona College (Dr. Kenneth Cooke - his mentor at Stanford was a really famous guy, Richard Bellman! So I guess despite winding up in the sticks with a less than prestigious Ivy League employer, I can at least enjoy this genealogical pedigree!!). Despite this, there were only post doc opportunities -- so I completed a 2-year post doc at UCLA (where I went undergraduate). Now it was 1993 and the job market wasn't substantially better than ‘91 (despite all the assurances that all the old guys employed circa Sputnik in the 60's would have to be retiring soon -- problem was, their positions were not being readily replaced). So the only tenure track position I was offered was at this small private liberal arts art-college in Santa Fe, NM, the College of Santa Fe. It wasn't Princeton (ha ha) not even say uhm University of Texas at El Paso. But it was an offer and there were still a lot of talented fresh PhD’s with no offer at all. So I grabbed it - and that was my ticket out of Los Angeles and into New Mexico. Anyway -- I apologize for rambling here, just thought I'd set the context for my entry into the professional big leagues. By the way, this move essentially ceased any continuance with the Angry Samoans, but that for all intents and purposes had dead-ended in ‘91. I was loaded down with teaching, sometimes 15 credits per semester, otherwise 12. The pay wasn't red hot either, so it necessitated teaching summer school. This played itself out over a 5-year period. The major downside, due to the heavy teaching, advising responsibilities and committee work, etc, was virtually a dead end as far as time to produce quality research papers. I didn't realize at the time that if tenure wasn't a sure thing at the College of SF, then shopping the next position would become doubly difficult (then don't provide these insights into academic politics in grad school). But after 5 years, the College decided it could not recruit sufficient numbers of math majors to justify tenuring my position (convenient for them to fall back on this). Eventually I moved down the road and wound up at NM Highlands U. My research area of interest is in applied linear algebra, matrix theory, finite dimensional control theory, and delay differential equations. My 3 (count em!) published results have involved stabilization of hybrid control systems in the presence of feedback delays. This involves producing algebraic criteria to stabilize an essentially continuous system with a discrete control vector involving a delay.
where [.] denotes the greatest integer function. In the above, A, B, K are time-invariant and my main result characterized the spectrum of K in terms of A to assure stabilizability.
TB: In your opinion, what has been the most important advancement in mathematics since differential and integral calculus was discovered independently by Isaac Newton and Gottfried Leibniz in the 17th century?
Gregg: This is probably not shared by most mathematicians, but personally I feel the work of Georg Cantor in the late 1800's - his creation of hierarchical infinities, or more precisely, the notion of equivalently sized infinite sets, was a watershed. This allowed a rigor into analysis that previously did not exist.
TB: The concept of zero (a number that symbolizes the absence of a number) did not come about until its discovery in India in the 7th century, and was not introduced into Western mathematics until the 9th century. What took so long? Was it the lack of need for such a number in primitive mathematics, or do you think it was more of an intuitive leap?
Gregg: There are many ways of looking at this, I think. The structure of the way we understand the real numbers, for example, depends on an additive identity. So, from the modern constructive viewpoint, 0 functions as a necessary logical piece of the puzzle for additive groups, rings and so forth. One could make the case that 1 is equally significant (multiplicative identity for mult. groups, rings etc). And then there's the whole numerological schtick - but let's resist the temptation to indulge that (otherwise we'd be knocking on 666's door next!). Phi, the empty set, can be analogized to 0 as a quantitative measure. Why the delay with baptizing 0 into the fold of other numeric symbols? Probably (I'm guessing) because there was greater utility for symbols describing tangible quantities. I mean, when did -1 come into vogue? e, pi or i? These were all context derived per need. E.g., how to solve x2 + 1 = 0? So you need to invent an i and then the Pandora’s box that opens. Pythagorean cultists did not believe in irrational numbers - i.e. all quantities were to be expressed as the ratio of integers. But that didn't fly with the length of the hypotenuse of an isosceles right triangle with sides = 1. There's a very elementary proof demonstrating that the square root of 2 cannot be rational.
Suppose √ 2 is rational. Then it can be expressed as a ratio of 2 integers a and b. I.e. √ 2 = a / b and without any loss of generality we may assume that a / b is an irreducible fraction (cos if not, then reduce it until it is). Square each side, get 2 = a2 / b2 or equivalently that 2b2 = a2. This implies, then, that a2 must be an even integer. But this would imply that a must be an even integer. So we can then write a = 2k where k is some integer. Hence we now have 2b2 = (2k)2 = 4k2 or that b2 = 2k2. But this means that b2 must even and consequently that b must be even. Finally we arrive at a contradiction, a logical incongruity. Since from the initial premise that √ 2 can be expressed as a / b with a / b irreducible, it follows that a and b are both even! Therefore √ 2 cannot be expressed as the ratio of two integers, and it is therefore irrational. Well that was big deal for the Pythagoreans. Supposedly, the story goes, that the fellow who demonstrated this reality to the higher up Pythies was forced to walk the plank (if you were a Pythagorean, one of the major taboos was to [accidentally or intentionally] walk over someone else’s used toenail clippings [I'm not making this up]).
TB: Although calculus can be a powerful tool, it does have its limitations; for example, it could never predict a sharp drop in a population curve. Could there be something out there that we haven't discovered? Something that could serve as a better tool for predicting our fate?
Gregg: Any scientific model is a randomization of the real world it attempts to describe. Equations, formulas, symbols are just window dressing of short hand for symbolic description of these randomized estimates. Something different? Mandelbrot in describing fractal geometries as models of natural physical processes? If nothing else, this spawned the tidal wave (to some degree - it was there before Julia sets became fashionable wallpaper) of chaos and complexity to model discontinuous and nonlinear systems. Some of these new models remain suspect in the eyes of peers - until the theory delivers on its promise to explain phenomena that otherwise had remained elusive or impenetrable. There was this guy, Rene Thom, who was the figurehead of what was called Catastrophe Theory. This was very hot a few decades ago. He constructed this simple machine -- two nails on a short board of wood, connected by a twisted rubber band. If you move the rubber band in a continuous fashion, at one point it snaps back - voila! The discontinuity (catastrophe!).....Soon everyone who had any sense of scientific fashion had one of these contraptions on their desk in their ivory tower academe offices. But my understanding is that the hype never delivered the goods (results), tho some of the mathematics created was quite elegant. Now no one has any Zeeman catastrophe machines in plain view.
TB: Mathematics can be seen as a science; a study of the rules that govern the behavior of numbers. Its direct application to physics and other sciences seems to support this claim. However, it can also be viewed as a philosophy, an abstraction of reality. What side of the fence are you on?
Gregg: Both. My work is oriented to applications. However, what turned me on to mathematics in the first place was its reliance on its own abstract structure for justification -- of its own importance. The reasoning is that it never (ever) is clear when a seemingly abstract concept or theorem that for hundreds of years (say) has no utility other than spawning similar abstract results. Then one day, there is a bizarre superimposition of where and how this actually fits in with reality. E.g., the algebraic construct of group theory is one of the major tools for quantum mechanics, as well as elements of encryption theory and so forth. Often, physical and natural reality pose the problem, someone devises a mathematical construct. Mathematical knot theory, e.g., the topology of interconnected shapes, is motivated by the geometry of DNA strands, etc.
TB: Physicist Eugene Wigner once said, "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." It is true that mathematics is a powerful tool not only in physics, but virtually any aspect of life. But what if it wasn't? Is there any area on the cutting edge of math that has absolutely no practical use?
Gregg: All bets are off as far as "no practical use" in perpetuity! (see above). But e.g., in ’93 Andrew Wiles at Princeton proved one of the most famous open problems of all time. This was known as Fermat's Last Theorem. Simple to state (for those who haven't heard?) there are no solutions in integers x, y, z for n > 2 of the equation xn + yn = zn. The proof of this amounted to synthesizing a very sophisticated set of new (last 30 years or so) developments in the theory of elliptic curves. The proof was over 200 pages I believe. Fermat, in editing a book of algebra, would fill in unproved propositions with his own quick proofs - in the margin. This was back in the mid 1600's. However for this particular theorem, he merely noted that he had a truly wonderful proof, but unfortunately its length exceeded the marginal space provided! So it took some 350+ years to fill in the margin, but Wiles' proof was certainly not what Fermat had in mind. In any case, what I'm leading up to, is that the proving of Fermat was covered in the NY Times and papers and periodicals all over the world (to the lay public). But other than tackling a very famous and perplexing riddle it is pretty much agreed that Wile's proof and the solving of this outstanding problem, despite no one doubting it being a monumental achievement, will not probably contribute to a cure for SARS. Or offer a blueprint to the Republicrats on how to dump President Moron.
TB: Euclid took some shit in his day for proving things that were intuitively obvious, i.e. that a straight line is the shortest distance between two points. Is there anything intuitively obvious in modern mathematics that remains to be proven, or is it all just a matter of constructing new knowledge from here on out?
Gregg: My feeling is that it moves in both directions - backwards and forwards. Euclid would not have imagined that sometime in the future there would be the necessity for a NON-Euclidean geometry. There are so many outstanding unsolved problems -- the Riemann Hypothesis e.g. (the new Holy Grail of conjectures yet to be proved. The biography of John Nash [much better than the feeble parable the movie offered] alluded to Nash's schizophrenia precipitated by his inability to prove the Riemann Hypothesis) -- that it would be really reactionary to believe that what's in store, is the next generation (literally) of new ideas, new formalities, new modeling constructs.
TB: It has been proposed that our base-ten counting system (that is, order of magnitude increasing as multiples of the number 10) arises from the simple fact that humans have ten fingers. Trying to count in a number scheme that isn't base-ten proves to be exceptionally confusing (the first twenty numbers of a base-eight system would be 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24), and trying to come up with any sort of algorithm to convert from one counting system to another is even harder. Is there something inherently simple about our base-ten system, or is it just the fact that it's what we're accustomed to? Are there any other cultures or groups you know of that use another base number? Any advantages to such an approach?Define x(n+1) = 3x(n) if x(n) £ 0.5, and x(n+1) = -3x(n) + 3 if x(n) > 0.5
Gregg: I believe the Babylonians used base 16. Anyway, there are actually pretty simple algorithms to just as easily compute in any other base. Certainly base 2 has more utility than 10 (insofar as how binary arithmetic is used in digital computers). There's a simple formulation of what's called the Cantor Set in base 3 (although you don't need to really just use base 3). Here's the Cantor Set Game:
To start the game, you need to define an initial point x(0). If you look at the sequence of points generated by a particular x(0), various things can happen. If the sequence goes to -oo, we say the sequence is an escaping sequence, and that x(0) is an escapee. However, if you select x(0) = 0, then the entire sequence that is generated remains at 0. We call such an initial value x(0) whose sequence eventually winds up at 0, a prisoner. Note, e.g., that x(0) = 1/3 or x(0) = 1/9 are prisoners as well (since several iterates wind up at 0). The interesting question is in the interval [0,1] how to identify prisoners from escapees. How many prisoners are there in [0,1]? What is the ratio of prisoners to escapees in this interval? Given ANY input initial value x(0) an algorithm can be devised to determine whether this is an escapee or a prisoner (aside of course from iterating til you're blue in the face!). The answer involves decimal arithmetic with base 3. And beyond this being a mere mathematical curiosity, the above iterative scheme has companion results in the theory of dynamical feedback systems.
TB: I touched on the importance of mathematics in science earlier, and with theoretical physics in particular. Do you have any particular interest in the concepts that readily go hand-in-hand with the math? Have you ever collaborated with scientists?
Gregg: I've worked on a few projects involving chaos and nonlinear systems (there's a group at Los Alamos National Laboratory, a mathematical division called the Center for Nonlinear Studies), but really circumstances with my present position disallows too many adventures outside of collaboration with colleagues close at hand -- and of course teaching. But that's ok. You are aware that my interests are not constipated w/in technical stuff (despite how dry alla the above may or may not sound) -- I've written a movie treatment that we may get to work on this summer, called "Hell's Whores" (orig titled: "Necromaniacs From Hell") and am working on an anthology of short stories called "The Tapeworm Story and Other Gastrointestinal Nightmares"......and the music stuff once in a while.
TB: The visualization of extra spatial dimensions is something that has intrigued me for quite a while. Here is something that I have been pondering. Suppose there are two 2-dimensional creatures living on a sheet of paper. Arbitrarily, let's just assume that they are both shaped like the letter "F". Now suppose you reached into their two dimensional world and picked one of them up, and flipped him over, so that he appeared as a backwards "F". Of course, he would not be able to comprehend what had just happened, but his friend would be able to tell that something was strange, because no combination of two dimensional movements could cause the creature to appear "backwards". In other words, only a reversal in the 3rd dimension could produce such an effect. My question is this: If a 3-dimensional human was picked up by a 4-dimensional creature, reversed with respect to that 4th dimension, and placed back into his 3-dimensional world, what changes would his friends notice?
Gregg: Well, they haven't "found" Saddam or Osama or the weapons of mass distraction yet, so we have to assume that such technology is not yet available to the warlords in Washington. And I would posit that the result would be different if the 4D critters were Klingon as opposed to, say, polygamous Mormons. But all in all, I guess I'd speculate that the difference the friends would notice is that our 3D abducted into 4D and placed back into 3D human, would suddenly feel compelled to get tickets to John Edward's Crossing Over. Either that, or he/she would start compulsively drinking coffee out of a Klein bottle.
Pretty fucking punk, huh? If you want to read a less intellectual Gregg Turner interview go here. Thanks to whoever it was that sent this to me, and bailing me out since I didn't have anything finished for the next Files yet. Drop me a line at termibore-at-aol-dotcom if you want credit, I'm sorry I lost track of your contact info. Next issue: a more "punk" Samoans Files, which will either be Vom-related or "Angry Samoans FAQ", whichever gets finished first. Both are in the works. And Gregg Turner, drop us a line or answer our e-mails please!


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The SFB 1270/2 "Electrically active implants - ELAINE" involves about 70 scientists from the fields of electrical engineering, computer science, mechanical engineering, material sciences, physics, mathematics, biology and medicine.
The following lecture series will also take place in the 2nd funding period:
"Women in Natural Sciences and (Bio-)Engineering" Homepage
The series will begin on April 29, 2022, with lectures to be held 1-2 per month on Fridays at 2 p.m.
With this lecture series, we would like to provide insights into the everyday professional life of successful female scientists and exciting answers to questions about individual decision-making processes for a life path in STEM fields. STEM is an acronym for the fields science, technology, engineering, and mathematics and is a broad term used to group together these academic disciplines.
Of course, the lecture series is not only addressed to girls and young women - the career path and scientific topics will be interesting for all knowledge-thirsty people.
When: during the semester, Friday 2 - 3 p.m.
Where: hybrid (via Zoom), Hörsaal Ex04, Experimentalgebäude, Albert-Einstein-Str. 2 (Südstadt-Campus)


MP4 | Video: h264, 1280×720 | Audio: AAC, 44.1 KHz
Language: English | Size: 691 MB | Duration: 2h 10m
A Short Calculus Course That You Can Actually Finish
What you’ll learn
Understand What a Limit Is Conceptually
How To Write and Interpret Limit Notation
How to Find Limits from a Graph
How to Find Limits Algebraically
How to Find Limits by Factoring
How to Find Limits by Using the Difference of Squares Formula
How to Find Limits by Using the Difference of Cubes Formula
How to Find Limits with Complex Fractions
How to Find Limits by Subtracting Rational Functions
How to Find Limits with Different Variables Other Than x
How to Find Limits Involving Delta x and Delta z
How to Find Limits by Rationalizing the Numerator
How to Find Limits by Rationalizing the Denonimator
Some Algebra Skills are Helpful
This course does not require any trigonometry:)
This course is intended for beginners who want to learn about basic limits in calculus. The only pre-req is a very basic knowledge of algebra. People who already know calculus could also benefit from this course as it will serve as a good review. This course has lots of examples but at the same time it’s short enough that you should actually be able complete the entire course. There is a final exam at the end of the course that you can use to test your knowledge. This course also comes with full video solutions to every single problem on the final exam.
Here are some things I like about this course.
You can actually FINISH this course as it is not extremely long, although it does have tons of examples!
This course is pretty much self contained. You only need some very basic algebra skills.
Anyone with a basic knowledge of algebra can take this course and understand most of it.
This course has a final exam at the end with full video solutions.
Zero knowledge of trigonometry is required for this course.
Since this course has no trigonometry, it is perfect for those who don’t know any trigonometry or anyone taking a business calculus or concepts of calculus course. The problems in this course often appear in those types of courses, and they also appear in regular Calculus 1 courses as well.
I hope you enjoy this course:)
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Anyone Who Wants to Learn Calculus
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Copying from https://bugs.freedesktop.org/show_bug.cgi?id=94551 :
Khaled Hosny 2017-08-03 11:48:13 UTC
(In reply to Akira TAGOH from comment #41)
> (In reply to Behdad Esfahbod from comment #34)
> > (In reply to Akira TAGOH from comment #31)
> > > Okay. updated the orth.
> > Thanks. Got orth for math as well? Don't know what should go there. :)
> Hmm, me neither.. we can get some from
> http://www.unicode.org/Public/math/revision-15/MathClass-15.txt say, I guess
> extra glyphs may be included in even a classification.
Math fonts vary greatly in their coverage and there is no clear cut what a math symbol and what is not. I don’t know if MathClass-XX.txt is used in anything in practice, but I guess we can make an orthography file out of it and test it with the existing math fonts, possibly reducing it until we get them all covering math.
Added und_zmth.orth to git.
$ ./build/fc-list/fc-list :lang=und-zmth
/usr/share/fonts/lm-math/latinmodern-math.otf: Latin Modern Math:style=Regular
/usr/share/fonts/tex-gyre-math/texgyrebonum-math.otf: TeX Gyre Bonum Math:style=Regular
/usr/share/fonts/tex-gyre-math/texgyretermes-math.otf: TeX Gyre Termes Math:style=Regular
/usr/share/fonts/Asana-Math/Asana-Math.otf: Asana Math:style=Regular
/usr/share/fonts/tex-gyre-math/texgyredejavu-math.otf: TeX Gyre DejaVu Math:style=Regular
/usr/share/fonts/tex-gyre-math/texgyreschola-math.otf: TeX Gyre Schola Math:style=Regular
/usr/share/fonts/tex-gyre-math/texgyrepagella-math.otf: TeX Gyre Pagella Math:style=Regular
/usr/share/fonts/xits/xits-math.otf: XITS Math:style=Regular
/usr/share/fonts/stix/STIXMath-Regular.otf: STIX Math:style=Regular
I couldn't test with Cambria Math, Minion Math and Lucida Math. if anyone can do that, that would be appreciated.
Cambria Math, Minion Math, Lucida Math and Libertinus Math (https://github.com/khaledhosny/libertinus) don’t seem to be recognized as math fonts.
(In reply to Khaled Hosny from comment #3)
> Cambria Math, Minion Math, Lucida Math and Libertinus Math
> (https://github.com/khaledhosny/libertinus) don’t seem to be recognized as
> math fonts.
Thank you for testing. adjusted orth for Libertinus Math. can you give me a log for fc-validate -l und-zmth for Cambria Math and Minion Math? so I can improve it much more...
meant fc-validate -l und-zmth -v /path/to/font
(In reply to Akira TAGOH from comment #5)
> meant fc-validate -l und-zmth -v /path/to/font
cambria.ttc:1 Missing 14 glyph(s) to satisfy the coverage for und-zmth language
LucidaBrightMathOT.otf:0 Missing 14 glyph(s) to satisfy the coverage for und-zmth language
Minion Math seems to be split into multiple font files and I’m not sure if my copy is the latest, so I suggest we ignore it for now.
Thank you. updated. hope I'm not missing anything else.
All good now!
Any other math fonts we should support? maybe getting rid of code points to add und-zmth to all of fonts that has "Math" in their family names isn't a realistic and ends up to have it in lang for non-math fonts. it has to be avoided. that said, if there are anything else having enough coverage to support, good to work before closing this.
I think the current orth files covers pretty much all publicly available OpenType math fonts. I’d have been inclined to say any font that has otlayout:math should be considered as math font, but I have seen broken fonts that has otlayout:math but no usable math support.
Okay. let's close this with it so far. we could revisit or do something on new one to polish the orth file as needed. thank you.


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catch the egg game lesson
Prior learning/place of lesson in scheme of work
led→unplot, and acceleration
input → accelerationto turn on and off LED lights on the LED screen. We will be learning how to create a catch the egg game app using global variables, forever loop, local variable, input acceleration, math min, math max, math random, math mod, if (conditionals), game library as well as simple commands, such as led plot, led unplot, and pause.
- variables : read more...
- forever : read more...
- unplot : read more...
- plot : read more...
- if : read more...
- acceleration : read more...
- math minimum number : read more...
- math maximum number : read more...
- math random number : read more...
- math modulus : read more...
- show number : read more...
- pause : read more...
- learn how to create a global variable as a place where you can store data so that you can use it later in your code, accessible across functions and in nested code blocks
- learn how to repeat code in the background forever
- learn how to turn off a LED light on the LED screen
- learn how to turn on a LED light on the LED screen
- learn how to learn how to conditionally run code depending on whether a condition is true or not
- learn how to learn how to get the acceleration value (g-force), in one of three specified dimensions
- learn how to return the smaller of two numbers
- learn how to return the larger of two numbers
- learn how to return a random number
- learn how to return the modulus
- learn how to show a number of the BBC micro:bit screen
- learn how to pause your code for the specified number of milliseconds
Progression Pathways / Computational Thinking Framework
- Designs solutions (algorithms) that use repetition and two-way selection, ie if, then and else.(AL)
- Uses logical reasoning to predict outputs, showing an awareness of inputs (AL)
- Recognises that different solutions exist for the same problem (AL) (AB) Understands that iteration is the repetition of a process such as a loop (AL)
- Represents solutions using a structured notation (AL) (AB)
Programming & Development
- Creates programs that implement algorithms to achieve given goals (AL)
- Declares and assigns variables(AB)
- Understands the difference between, and appropriately uses if and if, then and else statements(AL)
- Uses a variable and relational operators within a loop to govern termination (AL) (GE)
- Has practical experience of a high-level textual language, including using standard libraries when programming(AB) (AL)
- Uses a range of operators and expressions e.g. Boolean, and applies them in the context of program control. (AL)
- Selects the appropriate data types(AL) (AB
- time: 20 min.
- Extended Activity: challenges


Oops, sorry, must have crashed – dark, cold, rain, silence around these country parts, lie down, zonk.
How would one arrive at 50? 🙂 Difficult for you to start listing the angles you found, time consuming. I found there was a barrier beyond which I could find no further angles. I guessed 50 but could not be sure.
If EBCD were a cyclic quadrilateral, alpha would be 50 degrees because ECD is 50 degrees. But that can’t be right because alpha+CBE would be 90 degrees; hence EC would be a diameter and CDE would also have to be 90 degrees…
PS. I’ve drawn the figure using AutoCAD and I now know the answer. But I can’t see any way of getting to it by geometry alone.
Exactly. This is what I was hoping someone would explain. One would need a rightangle somewhere perhaps.
First thing to remember is that the drawing above is NOT TO SCALE!
It is possible using geam/trig, but you need to add 2 more lines –
Draw a line from D to line BC, at an angle of 20 deg to line DC, to intersect BC at a point we’ll call G.
Draw in another new line EG.
Angle GDC was drawn as 20 deg. and angle DCG is 80 deg, so angle DGC is also 80 deg, which means that line DC = line DG (as triangle DCG is isosceles)
Similarly, angle DCE is 50 deg, angle CBE is 80 deg, so angle DEC is 50 and line DE = line DC (this triangle is also isosceles).
From this DE = DG, and since angle EDG is 60 deg, triangle EDG is equilateral.
And triangle GBD is isosceles with angle GBD = 40 deg.
And finally, triangle GBE is isosceles with angle 70 deg, so angle alpha is 70-40 = 30 deg.
Simple plodding but my adding up has turned to mush today. Calculate F by 180 – ( 60 + 50) = 70. As the lines cross the side opposite F is the same as F. That lets you calculate the other angles in the middle as (360 – 2F) / 2. = 110 You now have two angles at the right and left.
On the right you have 110 + 30 to give the angle in the corner by alpha 40
On the left you have 110 +20 to give the angle to complete that one as 50
A is 180 – ( 20 + 60 ) – (30 + 50) = 20
So now we have to go to lengths and use sins and cosigns to calculate it out.
So full length of left side = 1 Have no idea but we don’t need size to be right just angles.
Full length of right side = 1 as well as both angles = 80 at bottom
So need to calculate size of top component on left using sins so dist between A and B is 0.53209
Top of left using same calcs = 0.653
So we now calculate the angle in the top of B using cosigns as 110
So alpha = 180 – 110 – 40 = 30
That should be right.
Can’t go to lengths because the drawing is not to scale. It can’t be – just look at the 70 and 110.
Sure you can. Lengths can not be determined but can be used to calculate the angles as we have just demonstrated.
I agree with Lord T that alpha is 30 degrees. I have different working.
I am quite happy to use algebraic values for lengths, and more geometry, and cope with the diagram not being to scale. On this, I don’t really understand James’ objection (if it is an objection).
I’ll type up my working, which might take some time.
First, I calculated and wrote in all the angles that can be simply calculated from sum angles of a triangle and sum angles on one side of a straight line.
Triangles DAC, CDE and ABD are each isosceles – with (in each triangle) the centre vertexes being the unequal ones.
To simplify things, I define the side DC to have unit length (DC=1.0). This differs from Lord T’s definition of DA=AB=1.0.
The following two algebraic equalities are used.
(i) The Law of Sines is that, for any triangle (vertices PQR, angles pqr), the length of each side divided by the sine of the opposite angle. Thus PQ/sin(r) = QR/sin(p) = RP/sin(q).
(ii) The sine of a sum of angles is given by sin(p+q) = sin(p).cos(q) + cos(p).sin(q).
Length EC is 1.28558 which is given from the Law of Sines for triangle CDE: EC/sin(80) = 1.0/sin(50)
Length EB (first equation) is given from the Law of Sines for triangle BDE: EB/sin(20) = 1.0/sin(alpha)
Length EB (second equation) is given from the Law of Sines for triangle BCE: EB/sin(30) = EC/sin(alpha+40)
Solving simultaneously by eliminating EB and putting in the length EC gives: sin(20)/[sin(alpha).sin(30)] = 1.28558/sin(alpha+40).
This ‘simplifies’ to: sin(alpha+40)/sin(alpha) = 1.28558.sin(30)/sin(20).
Using the sine of sum of angles for sin(alpha+40) gives: [sin(alpha).cos(40)+cos(alpha).sin(40)] / sin(alpha) = 1.87939
Cancelling terms and rearranging gives: cotan(alpha) = [1.87939-cos(40)] / sin(40) = 1.73206
Taking the reciprocal gives: tan(alpha) = 0.5775; hence alpha = 29.99984 degrees.
Assuming a ‘simple’ answer and that there is a bit of arithmetic rounding to allow for, we have alpha = 30 degrees.
Hoping I have made no mistakes of copying or algebra.
Which makes Lord T correct and that we must never admit, I might lose money on it. 🙂
Let us give thanks and praise for the mathematically minded. I can make fine coffee. And hum a tune.
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Or download our app "Guided Lessons by Education. Which of the figures is the top view of the solid? Therefore, Figure 1 is the correct answer. Get a free sample copy of our Math Salamanders Dice Games book with each donation!
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But first, we have to verify your age! This triangular prism makes for fun math learning and a great decoration.
Which of the figures is not a possible view of a square pyramid? Word Problems Word Problems. Help Support the Site. Solid shapes worksheets contain differentiating between 2D and 3D shapes; relating solid shapes and real life objects; labeling naming shapes; finding number of faces, xhapes and vertices; movements like roll, slide and stack; solid nets; printable charts and more.
Solid 3D Shapes Worksheets
To get the best deal on Tutoring, call Toll Free. Using these sheets will help your child to: Characteristics of 3d Shapes Each worksheet has 12 problems identifying the attributes and characteristics of 3 dimensional shapes.
Sort 2D and 3D Shapes. A right triangular prism can have no other triangle at its base except a right triangle. Figure 3 is not a possible view of the prism whose base is a right triangle.
Online Practice Online Practice. Kindergarten Section identify 3d shapes and 2d shapes from pictures 1st Grade Section identify different types of 3d shapes e. Choose which type of app you would like to use. I have read and agree to Education. This lesson includes printable activities: Identify the three-dimensional figure that has the right, the top, worksyeet the front views shown.
What would make you love Education.
Super Shapes: 3-D Shape Drawing | Worksheet | diadem-project.info
The view of a solid with respect to a plane will be two syapes. Which of the figures is not a possible view of a prism whose base is a right triangle? Color-1 Color-2 Grab 'em All Level 3: Students will get to practice identifying 3-D shapes such as rectangular prisms, cubes, spheres, and cones. These riddles are all about problem solving with 2d shapes. F Grade 2 2. Standards Mathematics Kindergarten K.
Welcome to shapws Math Salamanders 3d Shapes Worksheets. Geometry practice in kindergarten can help bolster the skills needed for higher-level concepts in the later grades. How to Print support. Color-1 Color-2 Grab 'em All Level 2: Each worksheet has 3 problems identify if a shape is translating slidingrotating turning or reflecting flipping.
Super Shapes: 3-D Shape Drawing
Figure 1 is the front view of a cylinder Figure 2 is the right view of a cylinder. Here you will find our free printable geometry riddles from 1st to 5th grade. Here's how students can access Education. Rectangular Prism Triangular Prism.
Write the name below each shape.
Which of the figures shows the top, the front, and the right views of the solid represented by the base plan?


Davide Masoero obtained the Master in Theoretical Physics in 2006 at the University of Turin, under the supervision of Roberto Tateo, and the PhD in Mathematical Physics in 2010 at the International School of Advanced Studies (SISSA, Trieste), under the supervision of Boris Dubrovin.
In 2010, he moved to the Group of Mathematical Physics of Lisbon University, where he has been working since then – but for a brief period as a Research Associate at the University of Sydney. He currently holds a FCT Researcher position and he is the Principal Investigator of the FCT research projects “Irregular Connections on Algebraic Curves and Quantum Field Theory” and “A mathematical framework for the ODE/IM correspondence”.
He held visiting positions at the Research Institute for Mathematical Sciences (RIMS, Kyoto), at The University of Sydney, at Centro di Ricerca Matematica Ennio De Giorgi (Pisa), and at The Simons Center for Geometry and Physics (Stony Brook, US).
He is a leading expert in the asymptotic analysis of irregular singularities and he applies this tool to address a great variety of problems in mathematical physics. In particular he has achieved the following results:
1. The ODE/IM correspondence.
He established an ODE/IM correspondence for any affine Kac Moody algebra (collaboration with Andrea Raimondo and Daniele Valeri). See Comm. Math. Phys. 2016, 2017, 2020.
He showed that the monster potentials defined by Bazhanov-Lukyanov-Zamolodchikov are in bijection with the states of the Quantum KdV model (collaboration with Ricardo Conti). See arXiv 2020.
2. Painlevé equations.
He computed the asymptotic distribution of the poles of the tritronnquee solution of the Painlevé first equation and obtained a proof of the bijectivity of the Riemann-Hilbert correspondence for the same equation. See JPhysA 2010, Nonlinearity 2010.
He computed the asymptotic distribution of the roots of the generalized Hermite polynomials, and thus proved Clarkson’s conjecture (collaboration with Pieter Roffelsen). See SIGMA 2018, arXiv 2019.
3. Dispersionless limit of 1 dimensional nonlinear waves.
He obtained a classification of the universality classes of critical behaviour of scalar nonlinear waves (collaboration with Andrea Raimondo). See Lett Math Phys 2012, IMRN 2015, Physica D 2015.
By solving a model of the E. Coli’s metabolism stoichiometry, he obtained an inverse linear law relating the growth rate and the ability of responding to external stimuli of colonies of E. Coli. This finding suggests that suboptimal phenotypes may be actually favoured by varying external conditions (collaboration with Daniele De Martino). See JStat 2016.


Ramnath Kumar and Arun Sai Suggala from Google Research have introduced a variant of the classical stochastic gradient descent (SGD) algorithm called Stochastic Re-weighted Gradient Descent (RGD). RGD is a lightweight algorithm that re-weights data points during each optimization step based on their difficulty. Unlike traditional algorithms like SGD, which give equal importance to all samples, RGD gives more importance to points that the model identifies as more difficult. This re-weighting is done by calculating the difficulty of a point using its loss and reweighting it by the exponential of its loss. RGD can be implemented with just two lines of code and can be combined with popular optimizers like SGD, Adam, and Adagrad.
The authors demonstrate that the RGD reweighting algorithm improves the performance of various learning algorithms across different tasks, including supervised learning, meta-learning, domain adaptation, and class imbalance. They show improvements over state-of-the-art methods on benchmarks like DomainBed, Tabular classification, GLUE, and ImageNet-1K.
The RGD algorithm is inspired by distributionally robust optimization (DRO), which assumes a “worst-case” data distribution shift may occur and aims to make the model robust to these perturbations. The authors develop RGD as a technique for solving the DRO objective, specifically focusing on Kullback-Leibler divergence-based DRO.
The authors acknowledge that RGD may not provide performance improvements in scenarios where training data has a high volume of corruptions. However, they suggest that applying an outlier removal technique to the RGD algorithm could potentially handle such scenarios.
In conclusion, RGD is a promising technique for improving the performance of deep neural networks across various domains. It is simple to implement and can be easily integrated into existing algorithms. The authors express their gratitude to the anonymous reviewers and other members involved in the research.


They say math is the universal language, and without it we would not be the technologically advanced species we are today. Now, Saturn has given us proof that math truly is everywhere in nature.
Thanks to the Cassini spacecraft, we can see with the upmost clarity the beauty of the geometric storm on the pole of Saturn. How the storm forms in a precise hexagon shape is unknown. What we do know is that this storm has been raging for a very long time!
Next time, if anyone asks you how many languages you know, always remember to include the language of math because it’s spoken even in space!
Remember, comment you thoughts below and follow me on twitter for instant updates on blog postings!


Find The Fake Coin
You have twelve coins. You know that one is fake. The only thing that distinguishes the fake from the real ones is that its weight is slightly different. You have a balance scale that only tells you whether or not both sides are even. Find a method to determine which coin is fake and whether it is heavier or lighter than a real one using the scale only three times.
Use only the twelve coins themselves and no others, no other weights, no cutting coins, no pencil marks on the scale... etc.


To define a column whose data type is
DECIMAL we use the following syntax:
column_name DECIMAL(P, D)
In the syntax about:
- P is the
precisionthat represents the number of significant digits. The range of P is 1 to 65.
- D is the
scalethat represents the number of digits after the deciamal point. The range of D is 0 to 30. MySQL requires D is less than or equal to P.
cost DECIMAL(10, 2)
cost column can store 10 digits with 2 decimal places. Therefore, the range of
cost is from 99999999.99 to -99999999.99


Some strategies allow gamblers to win real money online casino for free
Online casinos and their apps work honestly, paying winnings to visitors. Each game provides a percentage of recoil. It can vary depending on the institution, an average of 70-99%. RTP says that the slot returns only 70-99% of the money raised, the rest – the income of the club.
The slot machine first saves money for a prize, and then presents the player with coins. According to statistics, only 2 out of 5 gamblers win. The rest are transferred to another slot machine or end a gaming session, considering it hopeless.
Win real money online casino for free: some hints
Slots have a concept – slots dispersion. The essence of this concept is simple:
- Highly dispersive slots give very rarely, but can give a good money. Here gamblers have to make 300-500 spins (for example, Novomatic slots);
- There are moderately dispersive slots online (NetEnt slots). Here gamblers will need to make about 100 spins to win online casino;
- And there are low dispersion slots. In these games, gamblers have enough 30-50 shots to win. But they, as a rule, give prizes privately, but very small. Almost every provider has such slots.
The greater the maximum gain for an individual machine, the higher its dispersion.
Strategies to win real money online casino
The only way to intervene legally is to develop individual betting system which allows online casino win real money, which would be most subordinate to the laws of the theory of probability and could significantly increase the chances of a return on investment. By betting systems, gambling algorithms prescribe specific amounts of sums that should be put on the outcome of a predetermined event at a given step.
Since this is a system that involves a series of bets and helps to win real money online casino for free, gamblers should take care of a sufficient amount of money in their account before starting to play, with the loss of which they must agree in advance.
Win money online casino: Parlay system
Corresponds to the 2n: 1, 2, 4, 8 progression…
The essence is simple: regardless of the outcome of the event, each next bet should double the previous one, until the expected gain is received, after which the bets start again from one.
It is perhaps the oldest strategy of the roulette game, suggesting to double the bet when losing until the required amount of winnings is reached after which the return to the initial bet occurs.
With such a betting system, a one-time gain compensates for previous losses.
Is based on the progression of the form 1, 2, 3, 5, 8, 13 and so on. In this progression, starting from the third position, each value is the sum of the two previous ones. As a result, each big win online casino actually compensates for the two previous losses.
When losing a bet, the next one is on the list. If you win, the next bet should be two positions lower (for example, if you won the bet 8, then the next one should be the bet 3).


Humans of Data: 003
“I find it relaxing to work with data. I’m a mathematician by training and much more into applied mathematics, so I find recursive formulas very relaxing and linear algebra is like a fun puzzle, like a crossword.
I like problem solving. ‘Big data’ is an excellent field for problem solving. I like finding elegant solutions to complex problems. I approach problem solving slightly off-kilter from others – I would often get weird grades in school, but it also means that if people give me problems they’re struggling with, I could look at it and come up with something different from them.
This is my first data science meeting. I’m enjoying the opportunity and being around mathematicians and database people and folks who get excited by data. And I’m pleased that there are other women I can talk to.”


Machine Learning Practitioners have different personalities. While some of them are “I am an expert in X and X can train on any type of data”, where X = some algorithm, some others are “Right tool for the right job people”. A lot of them also subscribe to “Jack of all trades. Master of one” strategy, where they have one area of deep expertise and know slightly about different fields of Machine Learning. That said, no one can deny the fact that as practicing Data Scientists, we will have to know basics of some common machine learning algorithms, which would help us engage with a new-domain problem we come across. This is a whirlwind tour of common machine learning algorithms and quick resources about them which can help you get started on them.
1. Principal Component Analysis(PCA)/SVD
PCA is an unsupervised method to understand global properties of a dataset consisting of vectors. Covariance Matrix of data points is analyzed here to understand what dimensions(mostly)/ data points (sometimes) are more important (ie have high variance amongst themselves, but low covariance with others). One way to think of top PCs of a matrix is to think of its eigenvectors with highest eigenvalues. SVD is essentially a way to calculate ordered components too, but you don’t need to get the covariance matrix of points to get it.
This Algorithm helps one fight curse of dimensionality by getting datapoints with reduced dimensions.
2a. Least Squares and Polynomial Fitting
Remember your Numerical Analysis code in college, where you used to fit lines and curves to points to get an equation. You can use them to fit curves in Machine Learning for very small datasets with low dimensions. (For large data or datasets with many dimensions, you might just end up terribly overfitting, so don’t bother). OLS has a closed form solution, so you don’t need to use complex optimization techniques.
As is obvious, use this algorithm to fit simple curves / regression
2b. Constrained Linear Regression
Least Squares can get confused with outliers, spurious fields and noise in data. We thus need constraints to decrease the variance of the line we fit on a dataset. The right method to do it is to fit a linear regression model which will ensure that the weights do not misbehave. Models can have L1 norm (LASSO) or L2 (Ridge Regression) or both (elastic regression). Mean Squared Loss is optimized.
Use these algorithms to fit regression lines with constraints, avoiding overfitting and masking noise dimensions from model.
3. K means Clustering
Everyone’s favorite unsupervised clustering algorithm. Given a set of data points in form of vectors, we can make clusters of points based on distances between them. It’s an Expectation Maximization algorithm that iteratively moves the centers of clusters and then clubs points with each cluster centers. The input the algorithm has taken is the number of clusters which are to be generated and the number of iterations in which it will try to converge clusters.
As is obvious from the name, you can use this algorithm to create K clusters in dataset
4. Logistic Regression
Logistic Regression is constrained Linear Regression with a nonlinearity (sigmoid function is used mostly or you can use tanh too) application after weights are applied, hence restricting the outputs close to +/- classes (which is 1 and 0 in case of sigmoid). Cross-Entropy Loss functions are optimized using Gradient Descent. A note to beginners: Logistic Regression is used for classification, not regression. You can also think of Logistic regression as a one layered Neural Network. Logistic Regression is trained using optimization methods like Gradient Descent or L-BFGS. NLP people will often use it with the name of Maximum Entropy Classifier.
This is what a Sigmoid looks like:
Use LR to train simple, but very robust classifiers.
5. SVM (Support Vector Machines)
SVMs are linear models like Linear/ Logistic Regression, the difference is that they have different margin-based loss function (The derivation of Support Vectors is one of the most beautiful mathematical results I have seen along with eigenvalue calculation). You can optimize the loss function using optimization methods like L-BFGS or even SGD.
Another innovation in SVMs is the usage of kernels on data to feature engineer. If you have good domain insight, you can replace the good-old RBF kernel with smarter ones and profit.
One unique thing that SVMs can do is learn one class classifiers.
SVMs can used to Train a classifier (even regressors)
Note: SGD based training of both Logistic Regression and SVMs are found in SKLearn’s
sklearn.linear_model.SGDClassifier, which I often use as it lets me check both LR and SVM with a common interface. You can also train it on >RAM sized datasets using mini batches.
6. Feedforward Neural Networks
These are basically multilayered Logistic Regression classifiers. Many layers of weights separated by non-linearities (sigmoid, tanh, relu + softmax and the cool new selu). Another popular name for them is Multi-Layered Perceptrons. FFNNs can be used for classification and unsupervised feature learning as autoencoders.
FFNNs can be used to train a classifier or extract features as autoencoders
Comparision of self-normalizing MLPs with regular MLPs
7. Convolutional Neural Networks (Convnets)
Almost any state of the art Vision based Machine Learning result in the world today has been achieved using Convolutional Neural Networks. They can be used for Image classification, Object Detection or even segmentation of images. Invented by Yann Lecun in late 80s-early 90s, Convnets feature convolutional layers which act as hierarchical feature extractors. You can use them in text too (and even graphs).
Use convnets for state of the art image and text classification, object detection, image segmentation.
8. Recurrent Neural Networks (RNNs):
RNNs model sequences by applying the same set of weights recursively on the aggregator state at a time t and input at a time t (Given a sequence has inputs at times 0..t..T, and have a hidden state at each time t which is output from t-1 step of RNN). Pure RNNs are rarely used now but its counterparts like LSTMs and GRUs are state of the art in most sequence modeling tasks.
RNN (If here is a densely connected unit and a nonlinearity, nowadays f is generally LSTMs or GRUs ). LSTM unit which is used instead of a plain dense layer in a pure RNN.
Use RNNs for any sequence modelling task specially text classification, machine translation, language modelling
Models and examples built with TensorFlow (Many cool NLP research papers from Google are here)
RNNs in Tensorflow, a Practical Guide and Undocumented Features
9. Conditional Random Fields (CRFs)
CRFs are probably the most frequently used models from the family of Probabilitic Graphical Models (PGMs). They are used for sequence modeling like RNNs and can be used in combination with RNNs too. Before Neural Machine Translation systems came in CRFs were the state of the art and in many sequence tagging tasks with small datasets, they will still learn better than RNNs which require a larger amount of data to generalize. They can also be used in other structured prediction tasks like Image Segmentation etc. CRF models each element of the sequence (say a sentence) such that neighbors affect a label of a component in a sequence instead of all labels being independent of each other.
Use CRFs to tag sequences (in Text, Image, Time Series, DNA etc.)
10. Decision Trees
Let’s say I am given an Excel sheet with data about various fruits and I have to tell which look like Apples. What I will do is ask a question “Which fruits are red and round ?” and divide all fruits which answer yes and no to the question. Now, All Red and Round fruits might not be apples and all apples won’t be red and round. So I will ask a question “Which fruits have red or yellow color hints on them? ” on red and round fruits and will ask “Which fruits are green and round ?” on not red and round fruits. Based on these questions I can tell with considerable accuracy which are apples. This cascade of questions is what a decision tree is. However, this is a decision tree based on my intuition. Intuition cannot work on high dimensional and complex data. We have to come up with the cascade of questions automatically by looking at tagged data. That is what Machine Learning based decision trees do. Earlier versions like CART trees were once used for simple data, but with bigger and larger dataset, the bias-variance tradeoff needs to solved with better algorithms. The two common decision trees algorithms used nowadays are Random Forests (which build different classifiers on a random subset of attributes and combine them for output) and Boosting Trees (which train a cascade of trees one on top of others, correcting the mistakes of ones below them).
Decision Trees can be used to classify datapoints (and even regression)
TD Algorithms (Good To Have)
If you are still wondering how can any of the above methods solve tasks like defeating Go world champion like DeepMind did, they cannot. All the 10 type of algorithms we talked about before this was Pattern Recognition, not strategy learners. To learn strategy to solve a multi-step problem like winning a game of chess or playing Atari console, we need to let an agent-free in the world and learn from the rewards/penalties it faces. This type of Machine Learning is called Reinforcement Learning. A lot (not all) of recent successes in the field is a result of combining perception abilities of a Convnet or LSTM to a set of algorithms called Temporal Difference Learning. These include Q-Learning, SARSA and some other variants. These algorithms are a smart play on Bellman’s equations to get a loss function that can be trained with rewards an agent gets from the environment.
These algorithms are used to automatically play games mostly :D, also other applications in language generation and object detection.
These are the 10 machine learning algorithms which you can learn to become a data scientist.


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Arabic mathematicians merged the concepts of " number " and " magnitude " into a more general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.
In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.
In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Joseph Liouville showed that neither e nor e 2 can be a root of an integer quadratic equation , and then established the existence of transcendental numbers; Georg Cantor extended and greatly simplified this proof.
Lindemann's proof was much simplified by Weierstrass , still further by David Hilbert , and has finally been made elementary by Adolf Hurwitz  and Paul Gordan.
The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly.
The first rigorous definition was published by Georg Cantor in In , he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite.
Contrary to widely held beliefs, his first method was not his famous diagonal argument , which he published in See Cantor's first uncountability proof.
There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts , which are certain subsets of rational numbers.
Another possibility is to start from some rigorous axiomatization of Euclidean geometry Hilbert, Tarski, etc.
All these constructions of the real numbers have been shown to be equivalent, that is the resulting number systems are isomorphic.
Let R denote the set of all real numbers. The last property is what differentiates the reals from the rationals and from other, more exotic ordered fields.
For example, the set of rationals with square less than 2 has a rational upper bound e. These properties imply Archimedean property which is not implied by other definitions of completeness.
That is, the set of integers is not upper-bounded in the reals. The real numbers are uniquely specified by the above properties.
More precisely, given any two Dedekind-complete ordered fields R 1 and R 2 , there exists a unique field isomorphism from R 1 to R 2 , allowing us to think of them as essentially the same mathematical object.
The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like 3; 3.
For details and other constructions of real numbers, see construction of the real numbers. More formally, the real numbers have the two basic properties of being an ordered field , and having the least upper bound property.
The first says that real numbers comprise a field , with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication.
The second says that, if a non-empty set of real numbers has an upper bound , then it has a real least upper bound.
The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound e.
A main reason for using real numbers is that the reals contain all limits. More precisely, a sequence of real numbers has a limit, which is a real number, if and only if its elements eventually come and remain arbitrarily close to each other.
This is formally defined in the following, and means that the reals are complete in the sense of metric spaces or uniform spaces , which is a different sense than the Dedekind completeness of the order in the previous section.
This definition, originally provided by Cauchy , formalizes the fact that the x n eventually come and remain arbitrarily close to each other.
Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.
The set of rational numbers is not complete. For example, the sequence 1; 1. The completeness property of the reals is the basis on which calculus , and, more generally mathematical analysis are built.
In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.
For example, the standard series of the exponential function. The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
First, an order can be lattice-complete. Additionally, an order can be Dedekind-complete , as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant.
This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field the rationals and then forms the Dedekind-completion of it in a standard way.
These two notions of completeness ignore the field structure. However, an ordered group in this case, the additive group of the field defines a uniform structure, and uniform structures have a notion of completeness topology ; the description in the previous section Completeness is a special case.
We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces , since the definition of metric space relies on already having a characterization of the real numbers.
It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field , and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field".
Every uniformly complete Archimedean field must also be Dedekind-complete and vice versa , justifying using "the" in the phrase "the complete Archimedean field".
This sense of completeness is most closely related to the construction of the reals from Cauchy sequences the construction carried out in full in this article , since it starts with an Archimedean field the rationals and forms the uniform completion of it in a standard way.
But the original use of the phrase "complete Archimedean field" was by David Hilbert , who meant still something else by it.
He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R.
Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers , since that construction starts with a proper class that contains every ordered field the surreals and then selects from it the largest Archimedean subfield.
The reals are uncountable ; that is: there are strictly more real numbers than natural numbers , even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets i.
Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. As a topological space, the real numbers are separable.
This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
By virtue of being a totally ordered set, they also carry an order topology ; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls.
The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation.
The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.
Every nonnegative real number has a square root in R , although no negative number does. This shows that the order on R is determined by its algebraic structure.
Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a real closed field.
Proving this is the first half of one proof of the fundamental theorem of algebra. The reals carry a canonical measure , the Lebesgue measure , which is the Haar measure on their structure as a topological group normalized such that the unit interval [0;1] has measure 1.
There exist sets of real numbers that are not Lebesgue measurable, e. Vitali sets. The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement.
It is not possible to characterize the reals with first-order logic alone: the Löwenheim—Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves.
The set of hyperreal numbers satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R.
This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model which may be easier than proving it in R , we know that the same statement must also be true of R.
The field R of real numbers is an extension field of the field Q of rational numbers, and R can therefore be seen as a vector space over Q.
Zermelo—Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others.
However, this existence theorem is purely theoretical, as such a base has never been explicitly described.
The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on R with the property that every non-empty subset of R has a least element in this ordering.
Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. In mathematical expressions, unknown or unspecified real numbers are usually represented by lowercase italic letters u through z.
The set R gives rise to other sets such as the set of imaginary numbers and the set of complex numbers. The idea of a real number and what makes it "real" is primarily of interest to theoreticians.
Abstract mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.
The set of reals is "dense" in the same sense as the set of irrationals. Both sets are nondenumerable. There are more real numbers than is possible to list, even by implication.
The set R is sometimes called the continuum because it is intuitive to think of the elements of R as corresponding one-to-one with the points on a geometric line.
This notion, first proposed by Georg Cantor who also noted the difference between the cardinalities sizes of the sets of rational and irrational numbers, is called the Continuum Hypothesis.
This hypothesis can be either affirmed or denied without causing contradictions in theoretical mathematics.
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The Biot-Savart law
According to Eq. (316), we can obtain an expression for the electric field generated by stationary charges by taking minus the gradient of Eq. (335). This yields
which we recognize as Coulomb's law written for a continuous charge distribution. According to Eq. (318), we can obtain an equivalent expression for the magnetic field generated by steady currents by taking the curl of Eq. (334). This gives
where use has been made of the vector identity . Equation (337) is known as the Biot-Savart law after the French physicists Jean Baptiste Biot and Felix Savart: it completely specifies the magnetic field generated by a steady (but otherwise quite general) distributed current.
Let us reduce our distributed current to an idealized zero thickness wire. We can do this by writing
where is the vector current (i.e., its direction and magnitude specify the direction and magnitude of the current) and is an element of length along the wire. Equations (337) and (338) can be combined to give
which is the form in which the Biot-Savart law is most usually written. This law is to magnetostatics (i.e., the study of magnetic fields generated by steady currents) what Coulomb's law is to electrostatics (i.e., the study of electric fields generated by stationary charges). Furthermore, it can be experimentally verified given a set of currents, a compass, a test wire, and a great deal of skill and patience. This justifies our earlier assumption that the field equations (277) and (278) are valid for general current distributions (recall that we derived them by studying the fields generated by infinite, straight wires). Note that both Coulomb's law and the Biot-Savart law are gauge independent: i.e., they do not depend on the particular choice of gauge.
Consider an infinite straight wire, directed along the -axis, and carrying a current (see Fig. 33). Let us reconstruct the magnetic field generated by the wire at point using the Biot-Savart law. Suppose that the perpendicular distance to the wire is . It is easily seen that
Thus, according to Eq. (339), we have
which gives the familiar result
So, we have come full circle in our investigation of magnetic fields. Note that the simple result (345) can only be obtained from the Biot-Savart law after some non-trivial algebra. Examination of more complicated current distributions using this law invariably leads to lengthy, involved, and extremely unpleasant calculations.


If You Read Nothing Else Today, Read This Report on What Is Mode in Math
Self-awareness is an alternate approach to spell out consciousness. The large part of the proof you must believe is there. paramountessay The major advantage is the fact that it can be looked at repeatedly until the subject is mastered.
It is possible to even press two numbers in sequence. Function notation is an efficient and efficient approach to display functions of all kinds in mathematics. At the same time, it is a simple concept that is easy to use.
New Questions About What Is Mode in Math
It’s written with the conventional y-variable with respect to the x-variable, that is the input variable, or independent variable. This easy example shows just one approach to modeling multivariate data, through using covariance structures. Such a random variable is called discrete.
To locate the median, your numbers need to be listed in numerical order from smallest to largest, which means you may need to rewrite your list before you may discover the median. The median is another manner of specifying the average and works well if we’ve got outlying values. It is also the number that is halfway into the set.
The little sample gives a fair estimate of the whole population. The median is another manner of specifying the average and works https://writingcenter.wustl.edu/ well if we’ve got outlying values. It is important because it describes the behavior of the entire set of numbers.
All the blend mode math happens in the little range between 0 and 1. There’s even an integrated stopwatch and timer function that could be convenient. Before I list the 27 blend modes and the way in which they work, you must fully grasp the way the math in Photoshop works.
Top What Is Mode in Math Choices
There was a stunning means to aid with that issue, however. For instance, a man or woman who sells ice creams might want to understand which flavour has become the most popular. The mode for the quantity of money students spent is 4.
What Is Mode in Math Ideas
By using clearly defined discrete trials, it is simpler to identify what may not be working. The calendar activities that should be done for each day is going to be addressed. Use the negative binomial distribution when you’re interested in the quantity of trials essential to create the event a specified quantity of times.
The majority of the ideas may easily be utilized with different kinds of candy pieces, too. The only means to combat fraction-mayhem is to offer students with a LOT of chances to experience fractions with tangible objects. Look at a few different examples.
What Is Mode in Math – Dead or Alive?
There’s a very clear beginning and end to every trial with prompts and antecedents kept simple and at a suitable level. You will use them many times in the remainder of your math classes and for the remainder of your life! Use the negative binomial distribution when you’re interested in the quantity of trials essential to create the event a specified quantity of times.
The Essentials of What Is Mode in Math That You Can Benefit From Starting Right Away
The history of discrete mathematics has involved numerous challenging problems that have focused attention within regions of the area. The solution to a subtraction problem is known as the how to write a dissertation difference. Whether there are outliers, a median may give a more accurate picture of the real temperament of the scenario.
As a workaround compiling texvc will need to be done offline. LaTeX users are already acquainted with this process, and the syntax is comparable. Before I list the 27 blend modes and the way in which they work, you must fully grasp the way the math in Photoshop works.
While it’s possible to define lines individually, these won’t be aligned. These diagrams show how to get the median from a given set of information. The 2 middle numbers only will need to get averaged while the data set has an even number of information points in it.
While both, in addition to other statistical values, ought to be calculated when describing data, if only one can be used, the median can supply a better estimate of a normal value in a specific data set whenever there are really huge variations between values. To begin with, it isn’t in any respect clear how to figure out the mean value. The middle value in the set of information values is known as the median.
For samples, if it’s known they are drawn from a symmetric unimodal distribution, the sample mean can act as an estimate of the people mode. In this instance, you want to compute the median. It is also the number that is halfway into the set.
It’s an extremely concrete notion, Bialek explained. As previously mentioned, the articles will be peer-reviewed in the standard way. Add your answer only in the event that you feel you’ve got something non trivial to add.
Below you’ll find a detailed step-by-step guide, but I’d like to provide you with a quick overview how it works. Second, the DTT method of one-to-one teaching allows for the programme to be totally individualised for the requirements of each kid. Much like all elements of life, of which the overall public to understand a little’, and people who study astrology is usually divided into two groups.
With properly adjusted VESA code what you’re attempting to do should get the job done. Suppose you’re going to use a lot of Christoffel symbols, but the very first superscript is usually likely to be i. It may be difficult to see in the little graph, but there’s a bar for each batch size.
You should not presume that your mean is going to be one of your initial numbers. It is more commonly known as the average. Finding the mean, also called averaging numbers, is a very helpful point to understand how to do, particularly whenever you desire a precise estimate or an extremely accurate generalization.
You may use the data within this worksheet if you’d love to attempt it. In a big set of data similar to this, instead of analyzing each time recorded, it’s simpler to analyze the full set of information. A set of data values collected by means of a survey or experiment is called a sample.


What are IEP and 504 accommodations for dyscalculia?
The following are some common dyscalculia accommodations that can be included in a child’s IEP or 504 plan for dyscalculia to help them succeed in school:
- To reduce anxiety, don’t call on the student to answer math questions in class unexpectedly.
- Allow the student to take pictures of note materials so they don’t have to copy them down.
- Give the student extra time to complete in-class and homework assignments.
- Allow the student to take math tests in a quiet room.
- Provide the student with lesson outlines and previews before class starts.
- Provide assistive technologies like calculators, graph paper, graphing tools, math notation tools, and graphic organizers for math.
- Allow the student to write answers directly on the test paper instead of having to fill out a scantron.
For more information, see our article Dyscalculia 101.


- Testing normality and linear form: LR, LM and W tests (criteria of likelihood ratio, Wald and Lagrange multiplier)
- Tests for selecting between linear and logarithmic forms (Box-Cox, Bera-McAleer, McKinnon-White- Davidson)
- Autoregressive Conditional Heteroskedasticity model (testing for ARCH, GARCH model)
- Testing the stability of coefficients (criterion Hansen, CUSUM, CUSUMSQ)
- Regression on dummy dependent variables (the LPM, Logit and Probit models)
- Autoregressive and distributed-lag models: Models with infinite number of lags (Koyck, Solow). Models with finite number of lags: The Almon approach to distributed lag models (the Almon or polynomial distributed lag). Empirical distributed lag models (adaptive expectations model, the stock adjustment, or partial adjustment model, combination of adaptive expectations and partial adjustment models)
- Simultaneous-Equation models: The nature of the models, the identification problem (just or exact identified or over-identified equations).
- Simultaneous equation methods (recursive models and OLS, Indirect least squares, two-stage least squares)
- Causality in Economics: the Granger test
- Times series econometrics: Main components of time series. Unit roots, spurious regression, testing for stationarity (Dickey-Fuller, Augmented Dickey-Fuller, Phillips-Peron tests), testing for cointegration (Engle-Granger, cointegrating Regression Durbin Watson). Error correction model and cointegration.