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sunaleisocial · 24 days

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A framework for solving parabolic partial differential equations

New Post has been published on https://sunalei.org/news/a-framework-for-solving-parabolic-partial-differential-equations/

A framework for solving parabolic partial differential equations

Computer graphics and geometry processing research provide the tools needed to simulate physical phenomena like fire and flames, aiding the creation of visual effects in video games and movies as well as the fabrication of complex geometric shapes using tools like 3D printing.

Under the hood, mathematical problems called partial differential equations (PDEs) model these natural processes. Among the many PDEs used in physics and computer graphics, a class called second-order parabolic PDEs explain how phenomena can become smooth over time. The most famous example in this class is the heat equation, which predicts how heat diffuses along a surface or in a volume over time.

Researchers in geometry processing have designed numerous algorithms to solve these problems on curved surfaces, but their methods often apply only to linear problems or to a single PDE. A more general approach by researchers from MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) tackles a general class of these potentially nonlinear problems.

In a paper recently published in the Transactions on Graphics journal and presented at the SIGGRAPH conference, they describe an algorithm that solves different nonlinear parabolic PDEs on triangle meshes by splitting them into three simpler equations that can be solved with techniques graphics researchers already have in their software toolkit. This framework can help better analyze shapes and model complex dynamical processes.

“We provide a recipe: If you want to numerically solve a second-order parabolic PDE, you can follow a set of three steps,” says lead author Leticia Mattos Da Silva SM ’23, an MIT PhD student in electrical engineering and computer science (EECS) and CSAIL affiliate. “For each of the steps in this approach, you’re solving a simpler problem using simpler tools from geometry processing, but at the end, you get a solution to the more challenging second-order parabolic PDE.”

To accomplish this, Da Silva and her coauthors used Strang splitting, a technique that allows geometry processing researchers to break the PDE down into problems they know how to solve efficiently.

First, their algorithm advances a solution forward in time by solving the heat equation (also called the “diffusion equation”), which models how heat from a source spreads over a shape. Picture using a blow torch to warm up a metal plate — this equation describes how heat from that spot would diffuse over it.  This step can be completed easily with linear algebra.

Now, imagine that the parabolic PDE has additional nonlinear behaviors that are not described by the spread of heat. This is where the second step of the algorithm comes in: it accounts for the nonlinear piece by solving a Hamilton-Jacobi (HJ) equation, a first-order nonlinear PDE.

While generic HJ equations can be hard to solve, Mattos Da Silva and coauthors prove that their splitting method applied to many important PDEs yields an HJ equation that can be solved via convex optimization algorithms. Convex optimization is a standard tool for which researchers in geometry processing already have efficient and reliable software. In the final step, the algorithm advances a solution forward in time using the heat equation again to advance the more complex second-order parabolic PDE forward in time. 

Among other applications, the framework could help simulate fire and flames more efficiently. “There’s a huge pipeline that creates a video with flames being simulated, but at the heart of it is a PDE solver,” says Mattos Da Silva. For these pipelines, an essential step is solving the G-equation, a nonlinear parabolic PDE that models the front propagation of the flame and can be solved using the researchers’ framework.

The team’s algorithm can also solve the diffusion equation in the logarithmic domain, where it becomes nonlinear. Senior author Justin Solomon, associate professor of EECS and leader of the CSAIL Geometric Data Processing Group, previously developed a state-of-the-art technique for optimal transport that requires taking the logarithm of the result of heat diffusion. Mattos Da Silva’s framework provided more reliable computations by doing diffusion directly in the logarithmic domain. This enabled a more stable way to, for example, find a geometric notion of average among distributions on surface meshes like a model of a koala.

Even though their framework focuses on general, nonlinear problems, it can also be used to solve linear PDE. For instance, the method solves the Fokker-Planck equation, where heat diffuses in a linear way, but there are additional terms that drift in the same direction heat is spreading. In a straightforward application, the approach modeled how swirls would evolve over the surface of a triangulated sphere. The result resembles purple-and-brown latte art.

The researchers note that this project is a starting point for tackling the nonlinearity in other PDEs that appear in graphics and geometry processing head-on. For example, they focused on static surfaces but would like to apply their work to moving ones, too. Moreover, their framework solves problems involving a single parabolic PDE, but the team would also like to tackle problems involving coupled parabolic PDE. These types of problems arise in biology and chemistry, where the equation describing the evolution of each agent in a mixture, for example, is linked to the others’ equations.

Mattos Da Silva and Solomon wrote the paper with Oded Stein, assistant professor at the University of Southern California’s Viterbi School of Engineering. Their work was supported, in part, by an MIT Schwarzman College of Computing Fellowship funded by Google, a MathWorks Fellowship, the Swiss National Science Foundation, the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, the U.S. National Science Foundation, MIT-IBM Watson AI Lab, the Toyota-CSAIL Joint Research Center, Adobe Systems, and Google Research.

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jcmarchi · 24 days

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A framework for solving parabolic partial differential equations

New Post has been published on https://thedigitalinsider.com/a-framework-for-solving-parabolic-partial-differential-equations/

A framework for solving parabolic partial differential equations

To accomplish this, Da Silva and her coauthors used Strang splitting, a technique that allows geometry processing researchers to break the PDE down into problems they know how to solve efficiently.

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harrelltut · 3 years

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rrbalpb · 4 years

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RRB ALP Syllabus

The Questions will be of objective type with multiple choices and are likely to include questions pertaining to:

CBT Stage 1 :Mathematics :

Number system, BODMAS, Decimals, Fractions, LCM, HCF, Ratio and Proportion, Percentages, Mensuration, Time and Work; Time and Distance, Simple and Compound Interest, Profit and Loss, Algebra, Geometry and Trigonometry, Elementary Statistics, Square Root, Age Calculations, Calendar & Clock, Pipes & Cistern etc.

General Intelligence and reasoning :

Analogies, Alphabetical and Number Series, Coding and Decoding, Mathematical operations, Relationships, Syllogism, Jumbling, Venn Diagram, Data Interpretation and Sufficiency, Conclusions and Decision Making, Similarities and Differences, Analytical reasoning, Classification, Directions, Statement – Arguments and Assumptions etc.

General Science:

Physics, Chemistry and Life Sciences of the 10th standard level.

General awareness & current affairs:

Science & Technology, Sports, Culture, Personalities, Economics, Politics.

CBT Stage 2 :

Mathematics :

General Intelligence and reasoning :

Analogies, Alphabetical and Number Series, Coding and Decoding, Mathematical operations, Relationships, Syllogism, Jumbling, Venn Diagram, Data Interpretation and Sufficiency, Conclusions and Decision Making, Similarities and Differences, Analytical reasoning, Clocks, Classification, Directions, Statement – Arguments and Assumptions etc.

Basic Science & Engineering:

Engineering Drawing (Projections, Views, Drawing Instruments, Lines, Geometric figures, Symbolic Representation), Units, Measurements, Mass Weight and Density, Work Power and Energy, Speed and Velocity, Heat and Temperature, Basic Electricity, Levers and Simple Machines, Occupational Safety and Health, Environment Education, IT Literacy etc.

General awareness & current affairs:

Science & Technology, Sports, Culture, Personalities, Economics, Politics.

The part B of the Assistant Loco Pilot CBT 2 syllabus is a bit different from the other sections of the ALP Exam.

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rrbalppreparation-tips · 4 years

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RRB ALP Exam Syllabus

The Questions will be of objective type with multiple choices and are likely to include questions pertaining to:

CBT Stage 1:

Mathematics:

General Intelligence and Reasoning:

General Science:

Physics, Chemistry and Life Sciences of the 10th standard level.

General awareness & Current affairs:

Science & Technology, Sports, Culture, Personalities, Economics, Politics.

CBT Stage 2:

Mathematics:

General Intelligence and Reasoning:

Analogies, Alphabetical and Number Series, Coding and Decoding, Mathematical operations, Relationships, Syllogism, Jumbling, Venn Diagram, Data Interpretation and Sufficiency, Conclusions and Decision Making, Similarities and Differences, Analytical reasoning, Clocks, Classification, Directions, Statement – Arguments and Assumptions etc.

Basic Science & Engineering:

General awareness & Current affairs:

Science & Technology, Sports, Culture, Personalities, Economics, Politics.

The part B of the Assistant Loco Pilot CBT 2 syllabus is a bit different from the other sections of the ALP Exam

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geekmystic · 7 years

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My math teacher is coming out...

I kind of wish we could rename “imaginary numbers”. They are numbers just like 0, 1, pi, sqrt 2, etc. “Complex numbers” better captures the idea as a whole. Imaginary numbers are just an extension of what we use in everyday life. In fact, imaginary numbers actually have practical uses (an example being electrical engineering). Also, using the complex plane (a plane whose axes are labeled R and I for real and imaginary) instead of the xy plane can make geometric concepts easier to understand.

For numbers, we build from the bottom up. We start with Natural Numbers. Caveman Joey can count how many chickens he has with the Naturals. But what if he has 8 chickens but owes his neighbor 10 chickens? How many chickens will he have? With the Naturals, he can’t answer this question as he doesn’t have negative numbers. So he moves up to the Integers. The Integers are the Naturals plus their additive inverses and 0. Now he can deal with issues such a debt and loss.

Now it’s dinner time. Caveman Joey roasted 3 chickens for his family of 5. How much chicken does each family member get? Well, here, the Integers fail to answer the question. We can’t say 0 chickens but we can’t say 1 either. The answer has to be somewhere in between. So maybe what Caveman Joey did was divide each chicken into five parts and gave a part to each family member. So each family member got three “fifths” of a chicken. We just invented fractions or Rational Numbers. I always told my students that when you see the word “rational” in a math setting, think “fractions”.

Now, let’s say Caveman Joey wants to build a chicken coop. His current coop is 1 unit long and 2 units wide. The area is 2 square units. Think of the coop as literally being 2 squares. Caveman Joey is correct to assume that the area covered is equal to the product of the lengths of the sides.

He thinks the space is adequate for his chickens but wants a square coop. After several attempts, he can’t find a fraction that best represents the length of the side he needs for the coop. Let’s say he has infinite patience and tries to measure 14/10 of a unit. The area is 196/100. Not quite 2. He then tries 142/100. The area is now 20164/10000. That is more than 2. He tries several more times to narrow it down but soon gives up. No fraction can represent the length Caveman Joey needs to build a square coop with an area of 2. We have now discovered Irrational Numbers. (And if you wanted, you could go on a side quest and discover limits and Calculus.)

Collectively, we call this set of numbers the Real Numbers. And it’s an unfortunate name because numbers themselves are just abstract concepts. They are not anymore real than the Imaginary Numbers. You have Descartes to thank for that. Like modern 8th/9th grade students, he thumbed his nose at the notion of Imaginary Numbers and gave them the name we have for them today. Damn you, Descartes.

Up until 8th grade or so, this was sufficient for most problems. But there’s a niggling little problem with our set of numbers. In our coop scenario, areas are always positive numbers. So squares were always positive. Now, we know we can multiply positives and negatives and get negative numbers. But squaring a number will always give a positive answer. Even if we’re squaring a negative number. Can we, in some way, square a number and get a negative answer?

By this time, we have moved beyond caveman days. The math we built up in the previous scenarios allowed civilization to flourish. With our families fed and housed and loved, we can now sit back and enjoy math for math’s sake. So we sit down with this idea of what quantities we can square to get a negative number. For simplicity’s sake, let’s say we just want the square root of -1.

Well, many people realized that no such real number existed. So, after centuries of using real numbers, mankind added another group of numbers to our set. With Integers, we added the minus symbol. With fractions, we added the bar or division symbol. With Irrational Numbers, we added radicals. What symbol can we add to the Real Numbers that allows us to have negative squares?

This is oversimplified, of course. Real Numbers can also be divided into Algebraic (rational numbers and roots) and Transcendental Numbers. Transcendental Numbers can not be expressed with radicals and rational numbers. So I actually need an extra step in my chicken coop scenario. That of the circular coop. Even with square roots, cube roots, nth roots, we cannot construct a coop of radius 1 and area of pi. The thing is, there’s no symbol to introduce to entirely classify these numbers. They are entirely unpredictable. And, if you study further, you realize they actually outnumber the Algebraic Numbers which is just mind blowing.

Back to our question. The symbol we decided on was i. The definition of i is that i^2 = -1. Now that we have this symbol, we can find square roots of any negative real number. The question is, how do we represent this quantity visually? It’s not a real number so we can’t put it on the number line. What was decided was that, instead of a line, we would visualize these numbers on a plane. The Real Number line is still there. But now, at 0, we have a vertical number line. It looks like the Real Number line except i is attached to every number.

This is the complex plane and it’s how we visualize complex numbers. Complex numbers are the set of Real and Imaginary numbers. Going back through our analogy, we start with the Naturals. We throw in their additive inverses and 0 to get the Integers. We throw in fractions to the get the Rationals. Then, we throw in the Irrationals to get the Reals. The last step is to throw in the Imaginary Numbers to get the Complex Numbers.

You probably learned in school that Complex Numbers look like a+bi. You can treat this as coordinates on the complex plane. 2+3i is at the coordinate (2, 3) on the complex plane. You can also transfer polar coordinates to this plane. Polar coordinates are not in (x, y) form but in (r, a) form or radius and angle. And it actually becomes really poignant to use radius and angle in the complex plane. Through some theories in Calculus, you can express complex numbers as re^(ia). r and a again are the radius and angle (in radians, not degrees) and e is Euler’s number.

And if you play around with this system long enough, you realize there is a completeness to this set of numbers. You can’t add, multiply, nth root, exponentiate, take a logarithm, etc that results in something outside this set of numbers. The technical term is “algebraically closed”. It means that any polynomial equation you can think of has all its solutions in the set of Complex numbers.

I just love this story because it also encodes the history of the human race. From our beginnings, struggling to quantify what we have and don’t have. To a couple thousand years ago when we realized that our chicken coops couldn’t always be built with rational numbers. To a couple hundred years ago, when we could finally let our imagination think up numbers that square to a negative number. And then finding out that there are a multitude of uses for them, especially to engineer things far more advanced than chicken coops.

#math #math is life #math is beautiful #long reads

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ericfruits · 6 years

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Origami spreads its wings

CONVERSATIONS AT THE International Meeting on Origami in Science, Mathematics and Education often pause for a hand to dart into a pocket, emerge with a square of plain paper and fluently fold it up to make a point—both geometrically and rhetorically. The tables at the meeting, which was held in Oxford last September, are strewn with paper constructions that must have taken weeks to make and which participants are nevertheless welcome to handle. But the meeting is not just, or even mostly, about folded paper. It is about the folds themselves: how to design them; how to think about them; how to use them. It is about making creases in everything from steel to sheets of carbon mere atoms thick. It is about differential geometry and elastic moduli. It is about adult nappies and satellite antennae. The more ways are found to fold things up, it seems, the more wide open the field becomes.

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People have folded things up since there were things for them to fold. In Europe, for many centuries, the things folded were mostly cloth; in Japan and China, paper. The uniting of these traditions within a single systematised craft is largely a 20th-century phenomenon, as is its description as “origami” (ori, the Japanese for fold, kami for paper); in Japan the practice was previously known as tatogami.

Friedrich Froebel, the man who invented the kindergarten, was the first person to suggest the systematic use of folding as a way of teaching geometry. Educators steeped in “Froebelian” methods—whose number, intriguingly, included the mother of Frank Lloyd Wright, a revered architect—spread folding for learning and fun as far as kindergarten itself. Some mathematicians took note. In “Geometric Exercises in Paper Folding”, published in 1893, T. Sundara Row took on with folded paper various problems geometers had tackled with compass and straight-edge since the days of Euclid: finding exactly half of the angle between two creases, say, or constructing a figure of three or nine or 15 perfectly equal sides. It could be applied to algebra, too. Mark a spot on the midline of a piece of paper, and fold the paper in as many ways as you can that touch its bottom edge to that spot; the folds will inscribe a parabola, as described by quadratic functions such as y=x {+2}.

Good clean educational fun. But nothing could be folded that could not be equally well done with Euclid’s tools—until, in 1936, Margherita Beloch, an Italian mathematician, developed a form of folding that produced the curves of cubic functions (y=x {+3}). Mastery of such functions lets you “double the cube”—calculate from the length of the side of one cube what the length of the sides of a cube with twice its volume would be. It is a problem that cannot be solved with just edge and compass, and had stumped the ancients. Folding was more than just drawing lines without a pencil.

A few mathematicians took these ideas further (though for the most part they abandoned paper for idealised folds more suited to mental manipulation). They moved from everyday algebra to differential algebra and, in so doing, from flat sheets to curved ones, and even to non-Euclidean geometries like that with which the theory of relativity describes spacetime.

While the maths of folding raised its ambitions, so did the craft side. In 1958 Lillian Oppenheimer founded the Origami Centre, now known as OrigamiUSA. It did much to bring together traditions from East and West and also opened up, on a personal level, the art’s link to science. Acknowledged “masters” of OrigamiUSA have included Michael LaFosse, a marine biologist, and John Montroll, an electrical engineer. It was surely not a coincidence that one of Oppenheimer’s highly mathematical children, Martin Kruskal, spent time studying the folding-up of spacetime inside black holes.

In 1989 a few such enthusiasts convened a meeting to explore what origami could contribute to science and engineering. Robert Lang, a former laser physicist, says that meeting in Ferrara, Italy, “played an outsized role in the triggering of the explosive growth that we’re now in the middle of, because it brought together isolated individuals and fields.”

He should know. Since hanging up his lab coat and taking origami on full time, Dr Lang has had a hand in a mind-bending array of pursuits both academic and artistic, penning 22 origami how-to books along the way. At the sixth sequel to the meeting in Ferrara, the one in Oxford, he is treated like a rock star.

It works on paper

Dr Lang’s greatest hits have been in formalising and building enthusiasm for the mathematics behind origami with systematic, quantitative studies on how to achieve a particular shape starting from a single flat, square, uncut sheet. He developed software that can compute the folds and their order for almost any beast imaginable. The patterns that the program spits out for deer with multiply-branching antlers, or one praying mantis eating another, are staggering to behold, both in their flat and folded forms. The mathematical operations through which the former becomes the latter, one can only imagine, represents a peculiarly elegant trajectory through a vast and bewildering space of possibilities.

The flashiest early example of origami solving a scientific problem was when Koryo Miura and Masamori Sakamaki, astrophysicists at Tokyo University’s space-science department, devised a new approach to the unfolding and refolding of a satellite’s solar panels, first put into practice in 1995. The obvious approach is to fold them as one does a map. But anyone who has tried to return a good-sized map to its folded state knows the damage it can inflict on the paper. The scientists’ insight was not to fold the panel at right angles, which produces rectangles between folds, but at a slightly skewed angle, producing parallelograms. This creates a panel that can be completely unfolded just by tugging two of its opposed corners out, and refolded by pushing them in.

To have a fold named after you is a rarefied honour in the origami world, but “Miura-ori” has since earned that distinction. Simon Guest, who works on structural mechanics at the University of Cambridge, calls it the “crucial link between origami and science”, and vividly recalls the first time he saw it. Dr Lang says the fold connects “hundreds of moving parts that move in different directions in a synchronised way”—which is just what builders of exotic experiments are often aiming for. “There are so many connections that it shouldn’t be possible for it to move,” he says. “That’s really powerful, and those properties come almost naturally from patterns that arise in the world of origami.”

In the wake of the Miura fold, more scientists and engineers took an interest, and more applications began to crop up in the scientific literature. In 2012 America’s National Science Foundation decided that this sporadic enthusiasm could do with some institutional legitimacy, and set up a programme called Origami Design for Integration of Self-assembling Systems for Engineering Innovation, or ODISSEI; it offered grants to scientists interested in trying an origami-based approach to a problem, on the condition that they collaborate with origami artists. It was, in the rather non-paper-friendly words of Larry Howell, a mechanical engineer at Brigham Young University, in Utah, “like throwing gasoline on a match”. The still-spreading flames lit up the Oxford meeting.

Though origami is at the centre of this applications boom, many of the devices displayed and discussed in Oxford represent a kind of goal reversal. For a recreational folder, the purpose is to finish with a given shape, such as a tato, the traditional paper purse that accessorises a kimono; for many applications, it is the unfolded version of an object that is the useful one.

Quite a few such applications are medical; the human body, like outer space, is best entered with small packages that can be spread out once you reach your destination. There are stents for arteries, retinal implants for the eye, forceps that scrunch up to pass through a tiny incision before getting to work within the body. Not all the bodily uses are interior, though. Dr Howell’s group is developing new designs for nappies, folding away the structures within them to better control the wicking of liquid and to fit to a wider range of body shapes.

Some are for lab use. One group has built a flat, origami-inspired contraption which is folded up by the growth and movement of the cells living on it. Another group is showing off sheets of carbon atoms—graphene—that bend into shape when their environment changes—for example, when it becomes more acid. It is at such scales that self-assembling systems—the SS of ODISSEI—come into their own, beyond the reach of fingers or tweezers.

Back in the visible world there is shape-shifting furniture based on a puzzle called a Yoshimoto cube that folds into a wide array of squishy seating options, to the delight of its child users. A three-metre-tall architectural arch made from fibreglass folds flat for transport. A fairing for locomotives is designed to reduce aerodynamic drag but to fold away when the engines are parked, or used in the middle of a train; it could, its makers say, save millions of dollars a year in fuel. “Origami tubes”—imagine a Miura-folded sheet further folded into an extensible prism—are unusually stiff in some directions. Architects and car designers have taken notice. Thanks to Dr Lang and many others, there is a general, mathematical folding theory underlying all these applications.

“Rigid origami” needs new maths; it also offers new abilities

As a result the mathematics of origami has moved beyond early efforts to show how much higher maths could be recapitulated in folds (answer: a surprising amount). The folders are now providing the mathematicians with interesting new challenges, which can elicit intriguing mathematical proofs. For example, Erik Demaine, a computer scientist at MIT, has proved that any straight-sided figure—an octagon, a cityscape silhouette or a blocky Bart Simpson—can be extracted with exactly one straight cut if you fold the paper up the right way first (you can make a just-one-cut Christmas tree, and your own Miura fold, at economist.com/origami). This is just the kind of thing Dr Lang relishes: “gaining an understanding of a phenomenon that we see in the world of folding but don’t yet have a mathematical description for”.

The need for such approaches becomes acute when you move to materials other than paper—materials which cannot be treated by assuming that they are infinitely thin and stress-free. Bend a sheet of steel and it will not lie flat. It may also be under considerable strain at and far from the fold. Such “rigid origami” needs new maths; it also offers new abilities. The non-local strains in non-paper materials can be used to generate forces which will make things fold, or unfold, seemingly spontaneously. To make the most of such wonders, though, you need a much richer theory. Dr Demaine, whose particular interest is in curved folds, another frontier with even more demanding analytical requirements, says that work is under way toward a unified theory of rigid origami as good as that now available for paper.

Such a theory, he cautions, may not exist. But he and his colleagues will have a lot of fun looking for it.

This article appeared in the Christmas Specials section of the print edition under the headline "The function of folds"

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peeterjoot · 9 months

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V0.3.5 of Geometric Algebra for Electrical Engineers (and temp hardcover price drop)

Yes, I just published an update last week, but here’s another one. Temporary price drop on hardcover. It’s been 4 years since I printed a copy of the book for myself to mark up and edit. In particular, having added some vector calculus identities and their geometric algebra equivalents to chapter II, it messes up the flow a bit, and I’d like a paper copy to review to help figure out how to…

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#curl as wedge product #Geometric Algebra for Electrical Engineers #hardcover #price drop #spherical coordinates #vector calculus identities

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degreeacademic · 4 years

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The Ten Secrets That You Shouldn’t Know About Engineering Mathematics | engineering mathematics

Engineering mathematics, also known as scientific mathematics, is a branch of traditional applied mathematics mainly dealing with scientific methods and mathematical concepts generally used in industrial and engineering field. It consists of various sub-branches such as computer algebra, linear algebra, discrete mathematics, etc. The main function of engineering mathematics is to provide an exact solution to certain mathematical problems by developing and using numerical techniques.

In engineering mathematics, numerical solutions of mathematical equations are developed using algebraic and geometric procedures. Usually, the equations are related to physical or logical problems and can be solved through the use of appropriate mathematical tools. Most of the problems that can be solved in engineering mathematics are related to scientific, physical and/or mechanical phenomena. Engineers who are working on scientific or technological projects often depend heavily on engineering mathematics.

Engineering mathematics are a great source of information and help. For example, in computer engineering, engineers need to study the mathematical principles and concepts in order to design computer systems. In chemical engineering, engineers need to study the mathematical principles and concepts in order to design chemical processing systems. In mechanical engineering, engineers have to study the mathematical principles and concepts in order to design motor vehicles. In aerospace industry, the mathematical principles and concepts are necessary to design aircraft engines.

Engineering mathematics helps the engineers to develop a better method of solving the problem. Moreover, it helps the engineers to increase the efficiency of their work because it is very hard to solve any given problem if it is not properly understood. Therefore, the mathematical principles and concepts in engineering mathematics have many applications in various fields.

Some of the most important mathematical concepts and principles in engineering mathematics include: calculus, mechanics, statistical mechanics, dynamics, numerical analysis, etc. The application of these concepts in engineering mathematics mainly depends on the application of the calculus, geometric and algebraic procedures. However, different types of engineering mathematics have different application and different type of problems they solve. Different types of applications can be found in mechanics, statistics, electrical, optical, etc. In the process of solving the mathematical problems, one needs to apply different techniques.

There are a number of software programs available in the market which can help the engineers in mastering different engineering mathematics concepts. These programs help to apply several techniques in solving the problems. For example, there are software programs available for mechanical engineering of turbines, there are software programs for aerospace, there are software programs for civil construction, etc. Engineering Mathematics and other software applications are necessary for the success in all sectors of the engineering industry.

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mathematicianadda · 5 years

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Returning to CSE maths 4 years after High School

I know this subreddit says that questions about "learning maths" should be on r/learnmath but I feel that my question is a little more focused (not just about...wanting book or resources) and could be answered here. Nevertheless, I will be posting this on there too.

This is a long one... bear with me. If you will.

I am going into the second year of a Computer Science degree and we have a course called "Engineering Mathematics" (ME3) in the next semester.

I graduated high school a WHILE ago and honestly need a little brushing up before I start learning ME3. But I don't have the time to go through all the maths topics we had then in all the 4 years. I was wondering if someone could help me decide what I should revisit and revise before going on to ME3.

Course Content of ME3

------------------------------------

1 - Linear Differential Equations (LDE)

\- LDE of nth order with constant coefficients \- Method of variation of parameters \- Cauchy's & Legendre's LDE \- Simultaneous & Symmetric Simultaneous DE \- Modelling of Electric Circuits

2 - Transforms

\- Fouriers Transform \- Complex exponential form Fourier series \- Fourier Integral Theorem \- Fourier Sine & Cosine Integrals \- Fourier Sine & Cosine transforms & their inverses \- Z Transform (ZT) \- Standard Properties \- ZT of standard sequences & their inverse

3 - Statistics

\- Measures of Central tendency \- Standard deviation, \- Coefficient of variation, \- Moments, Skewness and Kurtosis \- Curve fitting: fitting of straight line \- Parabola and Related curves \- Correlation and Regression \- Reliability of Regression Estimates.

4 - Probability and Probability Distributions

\- Probability, Theorems on Probability \- Bayes Theorem, \- Random variables, \- Mathematical Expectation \- Probability density function \- Probability distributions: Binomial, Poisson, Normal and Hypergometric \- Test of Hypothesis: Chi-Square test, t-distribution

5 - Vector Calculus

\- Vector differentiation \- Gradient, Divergence and Curl \- Directional derivative \- Solenoid and Irrigational fields \- Vector identities. Line, Surface and Volume integrals \- Green‘s Lemma, Gauss‘s Divergence theorem and Stoke‘s theorem

6 - Complex Variables

\- Functions of Complex variables \- Analytic functions \- Cauchy-Riemann equations \- Conformal mapping \- Bilinear transformation \- Cauchy‘s integral theorem & Cauchy‘s integral formula, \- Laurent‘s series, and Residue theorem

-------------------------------------------------------------------------------------------

Overview of roughly what all we had in the years of High School, a little out of order because I am summarizing all 4 years of Math books.

Real Numbers

\- Laws of Exponents for Real Numbers \- Euclid’s Division Lemma \- Fundamental Theorem of Arithmetic

Polynomials

\- Polynomials in One Variable \- Zeroes of a Polynomia, Remaider Theorem, Factorization of Polynomials \- Relationship between Zeroes and Coefficients of a Polynomial \- Division Algorithm for Polynomials

Pair of Linear Equations in Two variables

\- Linear Equations \- Solution of a Linear Equation \- Pair of Linear Equations in Two Variables \- Graphical Method of Solution of a Pair of Linear Equations \- Substitution Method, Elimination Method & Cross-Multiplication Method

Principles of Mathematical Induction

Complex Numbers

\- Modulus and the Conjugate \- Argand Plane and Polar Representation

Quadratic Equations

\- Factorisation & Completing the Square, Roots of Equations.

Sets----------

\- Sets: Empty, Finite, Infinite, Equal, Subsets, Power Set, Universal Set. \- Venn Diagrams \- Union, Intersection & Complement of a Set

Permutations and Combinations

Binomial Theorem

\- Binomial Theorem for Positive Integral Indices \- General and Middle Terms

Sequences and Series

\- Sequences & Series \- Arithmetic Progressions \- nth Term of an AP, Sum of n terms of an AP \- Geometric Progression \- Relationship Between Arithematic Mean and Geometric Mean

Matrices

\- Types & Operations \- Transpose \- Symmetric and Skew Symmetric Matrices \- Transformation \- Invertible Matrices

Determinants

\- Properties of Determinants \- Area of a Triangle \- Minors and Cofactors \- Adjoint and Inverse of a Matrix \- Applications of Determinants and Matrices

Relations and Functions

\- Cartesian Product of Sets \- Relations & Functions \- Composition of Functions and Invertible Function \- Binary Operations

Limits and Derivatives

\- Limits, Derivatives \- Limits of Trigonometric Functions \- Applications: Rate of Change of Quantities, Increasing and Decreasing Functions

Tangents and Normals, Approximations & Maxima and Minima

Continuity and Differentiability

\- Exponential and Logarithmic Functions \- Logarithmic Differentiation \- Derivatives of Functions in Parametric Forms \- Second Order Derivative \- Mean Value Theorem

Integrals

\- Inverse Process of Differentiation \- Methods of Integration \- Integration by Partial Fractions & by Parts \- Definite Integral \- Fundamental Theorem of Calculus \- Definite Integrals by Substitution \- Properties of Definite Integrals \- Applications: Area under Simple Curves, Area between Two Curves

Differential Equations

\- Basic Concepts \- General and Particular Solutions of Differential Equation \- Differential Equation whose General Solution is given \- Methods of Solving First order, First Degree Differential Equations

Vector Algebra

\- Types of Vectors \- Addition of Vectors, Multiplication of a Vector by a Scalar \- Product of Two Vectors

Linear Programming

Statistics

\- Graphical Representation \- Distribution. Mean, Mode & Median \- Measures of Dispersion, Range, Mean Deviation \- Variance and Standard Deviation

Probability

\- Random Experiments, Events, Axiomatic Approach to Probability \- Conditional Probability , Multiplication Theorem, Independent Events \- Bayes' Theorem \- Random Variables and their Probability Distributions \- Bernoulli Trials and Binomial Distribution

------------------------------------------------------------------------------

Euclids's Geometry

Properties of Lines, Angles, Circles, Triangles, Quadrilaterals, Parallelograms (Too easy to worry about)

Some chapters about Areas & Volumes of Quadrilaterals, Circles, Cylinders, Cuboids & Spheres (Again.. too easy)

Heron's Formula

\- Area of a Triangle – by Heron’s Formula \- Application of Heron’s Formula

Trigonometry

\- Trigonometric Ratios, Identities \- Applications : Heights and Distances

Trigonometric Functions

\- Sum and Difference of Two Angles \- Trigonometric Equations \- Inverse Trigonometric Functions & their Properties

Circles

\- Tangent to a Circle

Straight Lines

\- Slope of a Line \- Forms of Equations of a Line \- Distance of a Point From a Line

Conic Sections

\- Cone, Circle, \- Equations: Parabola, Ellipse & Hyperbola \- Eccentricity, Latus rectum

Three Dimensional Geometry

\- Coordinate Axes and Coordinate Planes in 3D Space \- Coordinates of a Point in Space \- Distance between Two Points \- Section Formula \- Direction Cosines and Direction Ratios of a Line \- Equation of a Line in Space, Angle between Two Lines, Shortest Distance between Two Lines \- Plane \- Coplanarity of Two Lines \- Angle between Two Planes \- Distance of a Point from a Plane \- Angle between a Line and a Plane

------------------------------------------------------------------------------------------------------------------------------------------------

Some of the topics are obvious. Like the entire Calculus section from "Relations & Function" to "Integrals" & Vector Algebra.

And Stats and Probability.

But what about Binomial Theorem, Sequences & Series, Matrices & Determinants. And Complex Numbers.

Polynomials, Quadratics is fairly easy.

And what about he Geometry-ish section. Especially the entire Conic Sections and 3D Geometry. I am completely blanked on that. I can't remember it at all.

I can remember a fair amount of Trig and Straight Lines (Slope & distance etc). Not sure if that is needed. Trig Functions is probably important. (sine, cosine etc)

Thank very very much for even taking the time to read.

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tutordoctoruae · 3 years

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rrbalpb · 4 years

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RRB ALP SYLLABUS

The Questions will be of objective type with multiple choices and are likely to include questions pertaining to:

CBT Stage 1 :

Mathematics :

General Intelligence and reasoning :

General Science:

Physics, Chemistry and Life Sciences of the 10th standard level.

General awareness & current affairs:

Science & Technology, Sports, Culture, Personalities, Economics, Politics.

CBTStage 2 :

Mathematics :

General Intelligence and reasoning :

Analogies, Alphabetical and Number Series, Coding and Decoding, Mathematical operations, Relationships, Syllogism, Jumbling, Venn Diagram, Data Interpretation and Sufficiency, Conclusions and Decision Making, Similarities and Differences, Analytical reasoning, Clocks, Classification, Directions, Statement – Arguments and Assumptions etc.

General awareness & current affairs:

Science & Technology, Sports, Culture, Personalities, Economics, Politics.

The part B of the Assistant Loco Pilot CBT 2 syllabus is a bit different from the other sections of the ALP Exam.

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airolcus · 6 years

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Check Part A syllabus for the RRB ALP Second CBT exam

The results for the first stage RRB ALP CBT have already been declared by the board. The exams were help in the months of August and September. According to an official more than 5 lakh students have successfully qualified the first stage of assistant loco pilots and technicians exam. The advertisement was published for 64,371 posts for recruitment of assistant loco pilots and technicians. Candidates who had appeared for the exam are required to visit the official website of the regional RRB to check their individual score. The second stage for the exam is scheduled from 12 to 14 December, 2018. The second stage of the CBT will be conducted in two parts – Part A and Part B. Candidates below can check the syllabus for the part A exam. Mathematics: Number system, BODMAS, Decimals, Fractions, LCM, HCF, Ratio and Proportion, Percentages, Mensuration, Time and Work; Time and Distance, Simple and Compound Interest, Profit and Loss, Algebra, Geometry and Trigonometry, Elementary Statistics, Square Root, Age Calculations, Calendar & Clock, Pipes & Cistern etc. General Intelligence and Reasoning: Analogies, Alphabetical and Number Series, Coding and Decoding, Mathematical operations, Relationships, Syllogism, Jumbling, Venn Diagram, Data Interpretation and Sufficiency, Conclusions and decision making, Similarities and differences, Analytical reasoning, Classification, Directions, Statement - Arguments and Assumptions etc. Basic Science and Engineering: Engineering Drawing (Projections, Views, Drawing Instruments, Lines, Geometric figures, Symbolic Representation), Units, Measurements, Mass Weight and Density, Work Power and Energy, Speed and Velocity, Heat and Temperature, Basic Electricity, Levers and Simple Machines, Occupational Safety and Health, Environment Education, IT Literacy etc. General Awareness: Current Affairs in Science & Technology, Sports, Culture, Personalities, Economics, Politics and any other subjects of importance. Time limit for Paper A will be 90 minutes and candidates will have to attempt 100 questions. Marks scored by the candidates in Paper A will be used for shortlisting the candidates alone given that they clear the minimum cut-off in Paper B as well.

Source: https://www.brainbuxa.com/education-news/check-part-a-syllabus-for-the-rrb-alp-second-cbt-exam-8842

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meanbeau · 6 years

Text

Check Part A syllabus for the RRB ALP Second CBT exam

The results for the first stage RRB ALP CBT have already been declared by the board. The exams were help in the months of August and September. According to an official more than 5 lakh students have successfully qualified the first stage of assistant loco pilots and technicians exam. The advertisement was published for 64,371 posts for recruitment of assistant loco pilots and technicians. Candidates who had appeared for the exam are required to visit the official website of the regional RRB to check their individual score. The second stage for the exam is scheduled from 12 to 14 December, 2018. The second stage of the CBT will be conducted in two parts -- Part A and Part B. Candidates below can check the syllabus for the part A exam. Mathematics: Number system, BODMAS, Decimals, Fractions, LCM, HCF, Ratio and Proportion, Percentages, Mensuration, Time and Work; Time and Distance, Simple and Compound Interest, Profit and Loss, Algebra, Geometry and Trigonometry, Elementary Statistics, Square Root, Age Calculations, Calendar & Clock, Pipes & Cistern etc. General Intelligence and Reasoning: Analogies, Alphabetical and Number Series, Coding and Decoding, Mathematical operations, Relationships, Syllogism, Jumbling, Venn Diagram, Data Interpretation and Sufficiency, Conclusions and decision making, Similarities and differences, Analytical reasoning, Classification, Directions, Statement - Arguments and Assumptions etc. Basic Science and Engineering: Engineering Drawing (Projections, Views, Drawing Instruments, Lines, Geometric figures, Symbolic Representation), Units, Measurements, Mass Weight and Density, Work Power and Energy, Speed and Velocity, Heat and Temperature, Basic Electricity, Levers and Simple Machines, Occupational Safety and Health, Environment Education, IT Literacy etc. General Awareness: Current Affairs in Science & Technology, Sports, Culture, Personalities, Economics, Politics and any other subjects of importance. Time limit for Paper A will be 90 minutes and candidates will have to attempt 100 questions. Marks scored by the candidates in Paper A will be used for shortlisting the candidates alone given that they clear the minimum cut-off in Paper B as well.

Source: https://www.brainbuxa.com/education-news/check-part-a-syllabus-for-the-rrb-alp-second-cbt-exam-8842

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thrivous · 7 years

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Let’s start with simple things: the rest of this issue of Pulse news is challenging.

I don’t know much about nootropics but I do know, and love, broccoli. Therefore I was intrigued to see that the American Association for the Advancement of Science (AAAS) recommends broccoli as a secret weapon against diabetes. Concentrated broccoli sprout extract may help type 2 diabetes patients manage their blood sugar, according to a new study published in Science Translational Medicine.

Those who are into meditation will certainly be pleased to hear that, in a new study published in Frontiers in Immunology, which reviews over a decade of studies analyzing how the behavior of our genes is affected by different Mind-Body Interventions (MBIs) including mindfulness and yoga, researchers at the University of Coventry and Radboud have found that MBIs can ‘reverse’ the molecular reactions in our DNA which cause ill-health and depression.

The following news might require meditation of the intellectual sort. Especially the last item: close your eyes, and try to imagine how groups of neurons in your three-dimensional brain could be operating, in some sense, as computing circuits in an 11-dimensional space (string theory anyone?). If you are into science fiction, you could find related food for thought in Greg Egan’s “Diaspora” (look for the magic carpets).

Watching how DNA operates in real-time. Using sophisticated imaging technology, researchers at UC Davis watched DNA from bacteria as it replicated and measured how fast enzyme machinery worked on the different strands. The study, published in Cell, shows how scientists have been able to watch individual steps in the replication of a single DNA molecule, and reports unexpected findings. It seems likely that the ability to watch DNA as it operates will permit a better understanding of how life works and open the way to futuristic medical technologies.

Ultrasound-driven nanoparticles boost the efficacy of cancer drugs. Scientists at the Wyss Institute for Biologically Inspired Engineering at Harvard have used ultrasound-sensitive nanoparticles to deliver toxic doses of chemotherapy drugs to tumors, while minimizing toxicity to nearby healthy tissues. The new drug delivery platform uses ultrasound waves to trigger the dispersal of chemotherapy-containing nanoparticles precisely at tumor sites. The research results, published in Biomaterials, show that concentrating nanoparticles at the tumor site resulted in a significantly greater reduction in tumor volume compared to tumors treated with a 20-fold higher dose of the free drug.

Toward bioengineered human liver tissue. Using bioenginered human liver tissue, researchers led by a Cincinnati Children's Hospital Medical Center team have uncovered networks of genetic-molecular crosstalk that control the liver’s developmental processes. The study, published in Nature, illuminates previously inaccessible aspects of human liver development, opening the possibility to generate healthy and usable human liver tissue from human pluripotent stem cells.

Organ Chips with sensors for studying human organs. Scientists at the Wyss Institute for Biologically Inspired Engineering at Harvard have developed “Organ Chips” - synthetic testbeds to study the physiology of human organs and tissues - with embedded electrodes that enable accurate and continuous monitoring and evaluation of the electrical activity of living cells. The research results, published in Lab Chip (1, 2), show how electrically active Organ Chips help to open a window into how living human cells and tissues function within organs, as demonstrated in a Heart Chip model.

Finding the best patients for new cancer drugs. Researchers at the Walter and Eliza Hall Institute have found a way to identify the right patients for promising new anti-cancer drugs called FGFR (fibroblast growth factor receptor) inhibitors, which are being investigated for treating lung squamous cell carcinoma. In the study, published in Molecular Cancer Therapeutics, the scientist show that the presence of a specific ‘biomarker’ indicates the patients who would best respond to the treatment.

Nerve bridges with Pac Man cells repair damaged nerves. Scientists at Duke University have shown that macrophages - known as the Pac-Man of the immune system - can play an important role in “nerve bridges” to regenerate damaged nerves. The research results, published in PNAS, documents how the scientists used nerve bridges filled with a biological signal able to attract younger, undifferentiated cells destined to become pro-healing macrophages, and achieved enhanced axonal regeneration and muscle reinnervation in lab rats, with results comparable to the best nerve regeneration technique operational today.

Advances in brain-inspired computing. Researchers at Georgia Institute of Technology and University of Notre Dame have created a new computing system inspired by the human brain, where processing is handled collectively with a massively parallel neural oscillatory network, rather than with a central processor. The study, published in Scientific Reports (open access), shows that the new computer system has an impressive performance on “hard” computational problems like mathematical graph coloring.

Evidence for high-dimensional data processing in the brain. Using the sophisticated mathematical techniques of algebraic topology, a team of neuroscientists led by the Blue Brain Project have uncovered “a universe of multi-dimensional geometrical structures and spaces within the networks of the brain.” Here, “dimension” refers to the complexity of the connections between neurons, which can be analyzed with the mathematics of higher-dimensional spaces. “In some networks, we even found structures with up to eleven dimensions,” said Blue Brain project leader Henry Markram. Futurism has a very readable explanation, and Wired has a somewhat critical explanation. The study, published in Frontiers in Computational Neuroscience, could have significant implications for our understanding of the brain, according to the authors.

Originally published at thrivous.com on June 19, 2017 at 09:26PM.

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