Author Stephen Lynch provides an introduction to the theory of dynamical systems. With the aid of Mathematica, this book’s hands-on approach first deals with continuous systems using ordinary differential equations, while the second part is devoted to the study of discrete dynamical systems. Lynch takes the reader from basic theory to recently published research material. Emphasized throughout are numerous applications to biology, chemical kinetics, economics, electronics, epidemiology, nonlinear optics, mechanics, population dynamics and neural networks.
Groups and Manifolds is an introduction to the mathematics of symmetry, with a variety of examples for physicists. Authors Pietro Giuseppe Frè and Alexander Fedotov cover both classical symmetry—as seen in crystallography—and the mathematical concepts used in supersymmetric field theories. After a basic introduction to group theory, they discuss Lie algebras and basic notions of differential geometry. Mathematica allows readers to develop group-theoretical constructions.
In this undergraduate textbook, Mark A. Cunningham discusses the nature of the microscopic universe from a modern perspective, based on Einstein’s notions of relativity and Noether’s proof of the emergence of conservation laws from symmetries of the equations of motion. These ideas drove the development of the Standard Model of particle physics and subsequent attempts to define a unified (string) theory. The second half of the book explores various aspects of many-body physics, ranging from chemical systems to plasmas to black holes. Cunningham makes extensive use of Mathematica to enable students to explore the meanings of different equations in a graphical manner. Students will gain an appreciation of the current state of physical theory in preparation for more detailed, advanced study as upperclassmen.
Author Gautam Dasgupta presents a highly original treatment of the fundamentals of FEM, developed using computer algebra, based on undergraduate-level engineering mathematics and the mechanics of solids. The book is divided into two distinct parts of nine chapters and seven appendices. The appendices give a short introduction to Mathematica, followed by truss analysis using symbolic codes that could be potentially used in all FEM problems to assemble element matrices and solve for all unknowns. All Mathematica codes for theoretical formulations and graphics are included, with extensive numerical examples.
Originally written in Greek, Euclidean Economics is now updated and translated to English. Author Sophocles Michaelides contends that economics can be studied and utilized on the basis of a minimum of fundamental hypotheses and the general laws of mathematics. He uses the Wolfram Language exclusively in the book’s calculations. Part two of the book includes all the notebooks in printed form.
Author Mikio Tohyama is a engineer who plays the piano every day, so he is attuned to sound on both scientific and experiential levels. He uses the Wolfram Language extensively to generate figures illustrating the nature of sound, focusing on the characteristics of sound waves in the context of time structures. This time-domain approach provides an informative and intuitively understandable description of various acoustic topics, such as sound waves traveling in an acoustic tube or in other media where spectral or modal analysis can be intensively performed.
Authors Cliff Hastings, Kelvin Mischo and Michael Morrison have updated the book hailed as “a much-needed compliment to existing support material for Mathematica… that puts to rest, once and for all, the claim that Mathematica is difficult and not accessible to the school and college levels of our computational universe” (Fred Szabo at Concordia University in Montreal, author of Actuaries’ Survival Guide). And now this second edition of the definitive guide to the Wolfram Language is available in Japanese.
Stephen Wolfram’s An Elementary Introduction to the Wolfram Language, now in its second edition, remains the premium gateway for anyone interested in learning programming and computational thinking through the Wolfram Language.
While available in print from all major booksellers, the complete text is also available free online, and forms the basis of a newly launched fully interactive course through Wolfram U. Wolfram U expands educational opportunities with free courses and video classes on everything from data science and statistics to machine learning and image processing. Teachers seeking to incorporate coding in the classroom should also consider Stephen Wolfram’s blog post, “Machine Learning for Middle Schoolers,” which includes an overview of the book.
An Elementary Introduction to the Wolfram Language is available in English, Chinese and Korean, and is coming soon in Spanish and Russian.
]]>Whew! So much has happened in a year. Consider this number: we added 230 new functions to the Wolfram Language in 2017! The Wolfram Blog traces the path of our company’s technological advancement, so let’s take a look back at 2017 for the blog’s year in review.
The year 2017 saw two Wolfram Language releases, a major release of Wolfram SystemModeler, the new Wolfram iOS Player hit the app store, Wolfram|Alpha pumping up its already-unmatched educational value and a host of features and capabilities related to these releases. We’ll start with the Wolfram Language releases.
Stephen Wolfram says it’s “a minor release that’s not minor.” And if you look at the summary of new features, you’ll see why:
Stephen continues, “There’s a lot here. One might think that a .1 release, nearly 29 years after Version 1.0, wouldn’t have much new any more. But that’s not how things work with the Wolfram Language, or with our company. Instead, as we’ve built our technology stack and our procedures, rather than progressively slowing down, we’ve been continually accelerating.”
The launch of Wolfram Language 11.2 continues the tradition of significant releases. Stephen says, “We have a very deliberate strategy for our releases. Integer releases (like 11) concentrate on major complete new frameworks that we’ll be building on far into the future. ‘.1’ releases (like 11.2) are intended as snapshots of the latest output from our R&D pipeline—delivering new capabilities large and small as soon as they’re ready.”
“It’s been one of my goals with the Wolfram Language to build into it as much data as possible—and make all of that data immediately usable and computable.” To this end, Stephen and company have been working on the Wolfram Data Repository, which is now available. Over time, this resource will snowball into a massive trove of computable information. Read more about it in Stephen’s post. But, more importantly, contribute to the Repository with your own data!
Our post about Wolfram|Alpha Pro upgrades was one of the most popular of the year. And all the web traffic around Wolfram|Alpha’s development of step-by-step solutions is not surprising when you consider that this product is the educational tool for anyone studying (or teaching!) mathematics in high school or early college. Read the post to find out why students and forward-thinking teachers recommend Wolfram|Alpha Pro products.
John Fultz, Wolfram’s director of user interface technology, announced the release of a highly anticipated product—Wolfram Player for iOS. “The beta is over, and we are now shipping Wolfram Player in the App Store. Wolfram Player for iOS joins Wolfram CDF Player on Windows, Mac and Linux as a free platform for sharing your notebook content with the world.” Now Wolfram Notebooks are the premium data presentation tool for every major platform.
The Wolfram MathCore and R&D teams announced a major leap for SystemModeler. “As part of the 4.1, 4.2, 4.3 sequence of releases, we completely rebuilt and modernized the core computational kernel of SystemModeler. Now in SystemModeler 5, we’re able to build on this extremely strong framework to add a whole variety of new capabilities.”
Some of the headlines include:
Earlier last year Markus Dahl, applications engineer, announced another advancement within the SystemModeler realm—the integration of OPC Unified Architecture (OPC UA). “Wolfram SystemModeler can be utilized very effectively when combining different Modelica libraries, such as ModelPlug and OPCUA, to either create virtual prototypes of systems or test them in the real world using cheap devices like Arduinos or Raspberry Pis. The tested code for the system can then easily be exported to another system, or used directly in a HIL (hardware-in-the-loop) simulation.”
In 2017 we had some blog posts that made quite a splash by showing off Wolfram technology. From insights into the science behind movies to timely new views on history, the Wolfram Language provided some highlight moments in public conversations this year. Let’s check out a few…
The story of mathematician Katherine Johnson and two of her NASA colleagues, Dorothy Vaughan and Mary Jackson, was in the spotlight at the 2017 Academy Awards, where the film about these women—Hidden Figures—was nominated for three Oscars. Three Wolfram scientists took a look at the math/physics problems the women grappled with, albeit with the luxury of modern computational tools found in the Wolfram Language. Our scientists commented on the crucial nature of Johnson’s work: “Computers were in their early days at this time, so Johnson and her team’s ability to perform complicated navigational orbital mechanics problems without the use of a computer provided an important sanity check against the early computer results.”
Another Best Picture nominee in 2017 was Arrival, a film for which Stephen and Christoper Wolfram served as scientific advisors. Stephen wrote an often-cited blog post about the experience, Quick, How Might the Alien Spacecraft Work?. On the set, Christopher was tasked with analyzing and writing code for a fictional nonlinear visual language. On January 31, he demonstrated the development process he went through in a livecoding event broadcast on LiveEdu.tv. This livecoding session garnered almost 60,000 views.
Wolfram celebrated the birthday of the late, great Muhammad Ali with a blog post from one of our data scientists, Jofre Espigule-Pons. Using charts and graphs from histograms and network plots, Espigule-Pons examined Ali’s boxing career, his opponent pool and even his poetry. This tribute to the boxing icon was one of the most-loved blog posts of 2017.
For the Fourth of July holiday, Swede White, Wolfram’s media and communications specialist, used a variety of functions in the Wolfram Language to analyze the social networks of the revolutionaries who shaped our nation. (Yes, social networks existed before Facebook was a thing!) The data visualizations are enlightening. It turns out that Paul Revere was the right guy to spread the warning: although he never rode through towns shouting, “The British are coming,” he had the most social connections.
So you say there’s no X in espresso. But are you certain? Vitaliy Kaurov, academic director of the Wolfram Science and Innovation Initiatives, examines the history behind this point of contention. This blog post is truly a shining example of what computational analysis can do for fields such as linguistics and lexicology. And it became a social media hit to boot, especially in certain circles of the Reddit world where pop culture debates can be virtually endless.
Just in time for the holiday board game season, popular Wolfram blogger Jon McLoone, director of technical communication and strategy, breaks down the exact probabilities of winning Risk. There are other Risk win/loss estimators out there, but they are just that—estimations. John uses the Wolfram Language to give exact odds for each battle possibility the game offers. Absolute candy for gamer math enthusiasts!
We had a great year at Wolfram Research, and we wish you a productive and rewarding 2018!
]]>“Stereo Vision” and “Rise Up”
“Clay, I think, pays special attention to expressing a math concept behind the art,” Kaurov says. “It is there, like a hidden gem, but a layman will not recognize it behind the beautiful visual. So Clay’s art is a thing within a thing, and there is more to it than meets the eye. That mystery is intriguing once you know it is there. But it’s not easy to express something as abstract and complex as math in something as compact and striking as a piece of art perceivable in a few moments. This gap is bridged with help from the Wolfram Language, because it’s a very expressive, versatile medium inspiring the creative process.”
Shonkwiler is a mathematics professor at Colorado State University and an avid visual artist, specializing in Wolfram Language–generated GIF animations and static images based on nontrivial math. “I am interested in geometric models of physical systems. Currently I’m mostly focused on geometric approaches to studying random walks with topological constraints, which are used to model polymers,” he says.
In describing how he generates ideas, he says, “There are some exceptions, but there are two main starting points. Often I get it into my head that I should be able to make an animation from some interesting piece of mathematics. For example, in recent months I’ve made animations related to the Hopf fibration.”
“Stay Upright”
✕
DynamicModule[{n = 60, a = \[Pi]/4, viewpoint = {1, 1.5, 2.5}, \[Theta] = 1.19, r = 2.77, plane, cols = RGBColor /@ {"#f43530", "#e0e5da", "#00aabb", "#46454b"}}, plane = NullSpace[{viewpoint}]; Manipulate[ Graphics[{Thickness[.003], Table[{Blend[cols[[;; -2]], r/\[Pi]], InfiniteLine[ RotationMatrix[\[Theta]].plane.# & /@ {{Cot[r] Csc[a], 0, Cot[a]}, {0, Cot[r] Sec[a], -Tan[a]}}]}, {r, \[Pi]/(2 n) + s, \[Pi], 2 \[Pi]/n}]}, Background -> cols[[-1]], PlotRange -> r, ImageSize -> 540], {s, 0., 2 \[Pi]/n}]] |
Like many artists, Shonkwiler draws inspiration from existing art and attempts to recreate it or improve upon it using his own process. He says, “Whether or not I actually succeed in reproducing a piece, I usually get enough of a feel for the concept to then go off in some new direction with it.”
As to the artists who inspire him, Shonkwiler says, “There’s an entire community of geometric GIF artists on social media that I find tremendously inspiring, including Charlie Deck, davidope, Saskia Freeke and especially Dave Whyte. I should also mention David Mrugala, Alberto Vacca Lepri, Justin Van Genderen and Pierre Voisin, who mostly work in still images rather than animations.” If you want to see other “math art” that has inspired Shonkwiler, check out Frank Farris, Kerry Mitchell, Henry Segerman, Craig Kaplan and Felicia Tabing.
Another artistic element in Shonkwiler’s pieces is found in the title he creates for each one. You’ll find clever descriptors, allusions to ancient literature and wordplay with mathematical concepts. He says he usually creates the title after the piece is completely done. “I post my GIFs in a bunch of places online, but Wolfram Community is usually first because I always include a description and the source code in those posts, and I like to be able to point to the source code when I post to other places. So what often happens is I’ll upload a GIF to Wolfram Community, then spend several minutes staring at the post preview, trying to come up with a title.” Although he takes title creation seriously, Shonkwiler says, “Coming up with titles is tremendously frustrating because I’m done with the piece and ready to post it and move on, but I need a title before I can do that.”
“Interlock”
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Stereo[{x1_, y1_, x2_, y2_}] := {x1/(1 - y2), y1/(1 - y2), x2/(1 - y2)}; With[{n = 30, m = 22, viewpoint = 5 {1, 0, 0}, cols = RGBColor /@ {"#2292CA", "#EEEEEE", "#222831"}}, Manipulate[ Graphics3D[{cols[[1]], Table[Tube[ Table[Stereo[ RotationTransform[-s, {{1, 0, 0, 0}, {0, 0, 0, 1}}][ 1/Sqrt[2] {Cos[\[Theta]], Sin[\[Theta]], Cos[\[Theta] + t], Sin[\[Theta] + t]}]], {\[Theta], 0., 2 \[Pi], 2 \[Pi]/n}]], {t, 0., 2 \[Pi], 2 \[Pi]/m}]}, ViewPoint -> viewpoint, Boxed -> False, Background -> cols[[-1]], ImageSize -> 500, PlotRange -> 10, ViewAngle -> \[Pi]/50, Lighting -> {{"Point", cols[[1]], {0, -1, 0}}, {"Point", cols[[2]], {0, 1, 0}}, {"Ambient", RGBColor["#ff463e"], viewpoint}}], {s, 0, \[Pi]}]] |
(This code plots the curves with fewer points so as to increase the responsiveness of the Manipulate.)
Other Wolfram Community members have complimented Shonkwiler on the layers of color he gives his geometric animations. Likewise, his use of shading often enhances the shapes within his art. But interestingly, his work usually begins monochromatically. “Usually I start in black and white when I’m working on the geometric form and trying to make the animation work properly. That stuff is usually pretty nailed down before I start thinking about colors. I’m terrible at looking at a bunch of color swatches and envisioning how they will look in an actual composition, so usually I have to try a lot of different color combinations before I find one I like.”
Shonkwiler says that the Wolfram Language makes testing out color schemes a quick process. “If you look at the code for most of my animations, you’ll find a variable called cols so that I can easily change colors just by changing that one variable.”
“Magic Carpet” and “Square Up”
I asked Shonkwiler if he conceives the visual outcome before he starts his work, or if he plays with the math and code until he finds something he decides to keep. He said it could go either way, or it might be a combination. “‘Magic Carpet’ started as a modification of ‘Square Up,’ which was colored according to the z coordinate from the very earliest versions, so that’s definitely a case where I had something in my head that turned out to require some extra fiddling to implement. But often I’m just playing around until I find something that grabs me in some way, so it’s very much an exploration.”
“Renewable Resources” and “Inner Light”
Shonkwiler actually has a lot of pieces that are related to each other mathematically. Regarding the two above, “They’re both visualizations of the same Möbius transformation. A Möbius transformation of a sphere is just a map from the sphere to itself that preserves the angles everywhere. They’re important in complex analysis, algebraic geometry, hyperbolic geometry and various other places, which means there are lots of interesting ways to think about them. They come up in my research in the guise of automorphisms of the projective line and as isometries of the hyperbolic plane, so they’re often on my mind.”
“To make ‘Inner Light,’ I took a bunch of concentric circles in the plane and just started scaling the plane by more and more, so that each individual circle is getting bigger and bigger. Then I inverse-stereographically project up to the sphere, where the circles become circles of latitude and I make a tube around each one. ‘Renewable Resource’ is basically the same thing, except I just have individual points on each circle and I’m only showing half of the sphere in the final image rather than the whole sphere.”
When I asked Shonkwiler about his philosophy on the relationship between math and aesthetics, he said, “Part of becoming a mathematician is developing a very particular kind of aesthetic sense that tells you whether an argument or a theory is beautiful or ugly, but this has probably been overemphasized to the point of cliché.”
However, Shonkwiler continued to mull the question. “I do think that when you make a visualization of a piece of interesting mathematics, it is often the case that it is visually compelling on some deep level, even if not exactly beautiful in a traditional sense. That might just be confirmation bias on my part, so there’s definitely an empirical question as to whether that’s really true and, if it is, you could probably have a metaphysical or epistemological debate about why that might be. But in any case, I think it’s an interesting challenge to find those visually compelling fragments of mathematics and then to try to present them in a way that also incorporates some more traditional aesthetic considerations. That’s something I feel like I’ve gotten marginally better at over the years, but I’m definitely still learning.”
Here is Shonkwiler with one of his GIFs at MediaLive x Ello: International GIF Competition in Boulder, Colorado:
Check out Clayton Shonkwiler’s Wolfram Community contributions. To explore his work further, visit his blog and his website. Of course, if you have Wolfram Language–based art, post it on Wolfram Community to strike up a conversation with other art and Wolfram Language enthusiasts. It’s easy and free to sign up for a Community account.
]]>Wolfram Community recently surpassed 15,000 members! And our Community members continue to impress us. Here are some recent highlights from the many outstanding Community posts.
BVH Accelerated 3D Shadow Mapping, Benjamin Goodman
Shade data converted to solar map
In a tour de force of computational narrative and a fusion of various Wolfram Language domains, Benjamin Goodman designs a shadow mapping algorithm. It’s a process of applying shadows to a computer graphic. Goodman optimized shadow mapping via space partitioning and a hierarchy of bounding volumes stored as a graph, forming a bounding volume hierarchy.
Pairs Trading with Copulas, Jonathan Kinlay
Jonathan Kinlay, the head of quantitative trading at Systematic Strategies LLC in New York, shows how copula models can be applied in pairs trading and statistical arbitrage strategies. The idea comes from when copulas began to be widely adopted in financial engineering, risk management and credit derivatives modeling, but it remains relatively underexplored compared to more traditional techniques in this field.
The Global Terrorism Database (GTD), Marco Thiel
Marco Thiel broke a Wolfram Community record in April when he contributed four featured posts in just three days! He utilized data from the Global Terrorism Database (GTD), an open-source database including information on terrorist events around the world, starting from 1970. It includes systematic data on domestic as well as transnational and international terrorist events, amounting to more than 150,000 cases. Marco analyzes weapon types, geo distribution of attacks and casualties, and temporal and demographical behavior.
Flight Data and Trajectories of Aeroplanes, Marco Thiel
Thiel utilizes the large amounts of data becoming ever more available. Often, however, these datasets are very valuable and difficult to access. Thiel shows how to use air traffic data to generate visualizations of three-dimensional flight paths on the globe and access flight positions and altitudes, call signs, types of planes, origins, destinations and much more.
Analysing “All” of the World’s News—Database of Everything, Marco Thiel
In another clever data collection/analysis project, Thiel works with “the largest, most comprehensive, and highest resolution open database of human society ever created,” according to the description provided by GDELT (Global Database of Events, Language, and Tone). Since 2015, this organization has acquired about three-quarters of a trillion emotional snapshots and more than 1.5 billion location references. Thiel performs some basic analysis and builds supporting visualizations.
How-to-Guide: External GPU on OSX—How to Use CUDA on Your Mac, Marco Thiel
Thiel discusses the neural network and machine learning framework that has become one of the key features of the latest releases of the Wolfram Language. Training neural networks can be very time-consuming, and the Wolfram Language offers an incredibly easy way to use a GPU to train networks and also do numerous other interesting computations. This post explains how to use powerful external GPU units for Wolfram Language computing on your Mac.
Creative Routines Charts, Patrick Scheibe
People are often interested in how creative or successful individuals manage their time, and when in their daily schedules they do what they are famous for. Patrick Scheibe describes how to build and personalize “creative routines” visualizations.
QR Code in Shopping Cart Handle, Patrick Scheibe
Scheibe also brought to Wolfram Community his famous article “QR Code in Shopping Cart Handle.” It explains the image processing algorithm for reading QR code labels when they are deformed by attachment to physical objects such as shopping carts and product packages.
Calculating NMR-Spectra with Wolfram Language, Hans Dolhaine
Hans Dolhaine, a chemist from Germany, writes a detailed walk-through calculating nuclear magnetic resonance spectra with the Wolfram Language. This is a useful educational tool for graduate physics and chemistry classes. Please feel free to share it in your interactions with students and educators.
Computational Introduction to Logarithms, Bill Gosper
Another excellent resource for educators is this elementary introduction to logarithms by means of computational exploration with the Wolfram Language. The Community contributor is renowned mathematician and programmer Bill Gosper. His article is highly instructive and accessible to a younger generation, and it contains beautiful animated illustrations that serve as outstanding educational material.
Using Recursion and FindInstance to Solve Sudoku and The Puzzled Ant and Particle Filter, Ali Hashmi
Finally, Ali Hashmi uses the recursion technique coupled with heuristics to solve a sudoku puzzle and also explains the connection between the puzzled ant problem and particle filters in computer vision.
If you haven’t yet signed up to be a member of Wolfram Community, don’t hesitate! You can join in on these discussions, post your own work in groups of your interest, and browse the complete list of Staff Picks.
We’re fascinated by artificial intelligence and machine learning, and Achim Zielesny’s second edition of From Curve Fitting to Machine Learning: An Illustrative Guide to Scientific Data Analysis and Computational Intelligence provides a great introduction to the increasingly necessary field of computational intelligence. This is an interactive and illustrative guide with all concepts and ideas outlined in a clear-cut manner, with graphically depicted plausibility arguments and a little elementary mathematics. Exploring topics such as two-dimensional curve fitting, multidimensional clustering and machine learning with neural networks or support vector machines, the subject-specific demonstrations are complemented with specific sections that address more fundamental questions like the relation between machine learning and human intelligence. Zielesny makes extensive use of Computational Intelligence Packages (CIP), a high-level function library developed with Mathematica’s programming language on top of Mathematica’s algorithms. Readers with programming skills may easily port or customize the provided code, so this book is particularly valuable to computer science students and scientific practitioners in industry and academia.
The Art of Programming in the Mathematica Software, third edition
Another gem for programmers and scientists who need to fine-tune and otherwise customize their Wolfram Language applications is the third edition of The Art of Programming in the Mathematica Software, by Victor Aladjev, Valery Boiko and Michael Shishakov. This text concentrates on procedural and functional programming. Experienced Wolfram Language programmers know the value of creating user tools. They can extend the most frequently used standard tools of the system and/or eliminate its shortcomings, complement new features, and much more. Scientists and data analysts can then conduct even the most sophisticated work efficiently using the Wolfram Language. Likewise, professional programmers can use these techniques to develop more valuable products for their clients/employers. Included is the MathToolBox package with more than 930 tools; their freeware license is attached to the book.
Introduction to Mathematica with Applications
For a more basic introduction to Mathematica, readers may turn to Marian Mureşan’s Introduction to Mathematica with Applications. First exploring the numerous features within Mathematica, the book continues with more complex material. Chapters include topics such as sorting algorithms, functions—both planar and solid—with many interesting examples and ordinary differential equations. Mureşan explores the advantages of using the Wolfram Language when dealing with the number pi and describes the power of Mathematica when working with optimal control problems. The target audience for this text includes researchers, professors and students—really anyone who needs a state-of-the art computational tool.
Geographical Models with Mathematica
The Wolfram Language’s powerful combination of extensive map data and computational agility is on display in André Dauphiné’s Geographical Models with Mathematica. This book gives a comprehensive overview of the types of models necessary for the development of new geographical knowledge, including stochastic models, models for data analysis, geostatistics, networks, dynamic systems, cellular automata and multi-agent systems, all discussed in their theoretical context. Dauphiné then provides over 65 programs that formalize these models, written in the Wolfram Language. He also includes case studies to help the reader apply these programs in their own work.
Our tour of new Wolfram Language books moves from terra firma to the stars in Geometric Optics: Theory and Design of Astronomical Optical Systems Using Mathematica. This book by Antonio Romano and Roberto Caveliere provides readers with the mathematical background needed to design many of the optical combinations that are used in astronomical telescopes and cameras. The results presented in the work were obtained through a different approach to third-order aberration theory as well as the extensive use of Mathematica. Replete with workout examples and exercises, Geometric Optics is an excellent reference for advanced graduate students, researchers and practitioners in applied mathematics, engineering, astronomy and astronomical optics. The work may be used as a supplementary textbook for graduate-level courses in astronomical optics, optical design, optical engineering, programming with Mathematica or geometric optics.
Don’t forget to check out Stephen Wolfram’s An Elementary Introduction to the Wolfram Language, now in its second edition. It is available in print, as an ebook and free on the web—as well as in Wolfram Programming Lab in the Wolfram Open Cloud. There’s also now a free online hands-on course based on the book. Read Stephen Wolfram’s recent blog post about machine learning for middle schoolers to learn more about the new edition. |
Differential Equations with Mathematica, Fourth Edition
The fourth edition of Differential Equations with Mathematica is a supplementing reference that uses the fundamental concepts of Mathematica to solve (analytically, numerically and/or graphically) differential equations of interest to students, instructors and scientists. Authors Martha L. Abell and James P. Braselton include instruction on basic methods and algorithms. They cover the Mathematica functions relevant to differential equations and dependant concepts from calculus and linear algebra. This book contains many helpful illustrations that make use of Mathematica’s visualization capabilities.
Solution Techniques for Elementary Partial Differential Equations, Third Edition
Christian Constanda teaches students to solve partial differential equations through concise, easily understood explanations and worked examples that allow students to see the techniques in action. The third edition includes new sections on series expansions of more general functions, other problems of general second-order linear equations, vibrating string with other types of boundary conditions and equilibrium temperature in an infinite strip. It also includes new and improved exercises with a brief Mathematica program for nearly all of the worked examples, teaching students how to verify their results with a computer.
Differential Equations & Linear Algebra, Fourth Edition
Authors C. Henry Edwards, David E. Penney and David Calvis provide updated and improved figures, examples, problems and applications. With real-world applications and a blend of algebraic and geometric approaches, Differential Equations & Linear Algebra introduces students to mathematical modeling of real-world phenomena and offers an array of problem sets. Alongside this fourth edition, an expanded applications website is now available that includes programming tools from Mathematica and Wolfram|Alpha.
Exploring Calculus: Labs and Projects with Mathematica
Authors Crista Arangala and Karen A. Yokley created a hands-on lab manual that can be used in class every day to guide the exploration of the theory and applications of differential and integral calculus. Each lab consists of an explanation of material with integrated exercises. The exercise sections integrate problems, technology, Mathematica R visualization and the Computable Document Format (CDF) to help students discover the theory and applications of differential and integral calculus in a meaningful and memorable way.
Calculus and Differential Equations with Mathematica
In this book, Pramote Dechaumphai offers a clear and easy-to-understand presentation of how to use Mathematica to solve calculus and differential equation problems. It contains essential topics that are taught in calculus and differential equation courses, including differentiation, integration, ordinary differential equations and Laplace and Fourier transforms, as well as special functions normally encountered in solving science and engineering problems. Numerical methods are employed when the exact solutions are not available. Additionally, the finite element method in Mathematica is used to analyze partial differential equations for problems with complex geometry. These partial differential equations could be in elliptic, parabolic and hyperbolic forms. Many examples are presented with detailed derivation for their solutions before using Mathematica to confirm the results.
Geometry, Language and Strategy Vol. 2: The Dynamics of Decision Processes
The first volume, Geometry, Language and Strategy, extended the concepts of game theory, replacing static equilibrium with a deterministic dynamic theory. It also opened up many applications that were only briefly touched on. To study the consequences of the deterministic approach and the extent of these applications in contrast to standard Bayesian approaches requires an engineering foundation and discipline, which this volume supplies. It provides a richer list of applications, such as the prisoner’s dilemma, expanding the relevance of volume 1 to more general time-dependent and transient behaviors.
Mathematica for Mathematics, Physics and Engineers
Mehrzad Ghorbani expands on an earlier work, Applied Mathematical Softwares: Mathematica, developed over the course of more than 10 years of teaching mathematics software and Mathematica code in Iranian universities. This new title includes more elegant and basic mathematical problems from a range of specializations including calculus, number theory, numerical analysis, vector and matrix algebra, complex variables, graph theory, engineering mathematics and mathematical physics. Although applicable to undergraduate and graduate studies in math and science, Ghorbani’s book is additionally relevant to those who use Mathematica in computational scientific branches that need symbolic or numerical code.
]]>If aliens actually visited Earth, world leaders would bring in a scientist to develop a process for understanding their language. So when director Denis Villeneuve began working on the science fiction movie Arrival, he and his team turned to real-life computer scientists Stephen and Christopher Wolfram to bring authentic science to the big screen. Christopher specifically was tasked with analyzing and writing code for a fictional nonlinear visual language. On January 31, he demonstrated the development process he went through in a livecoding event you can watch on YouTube.
Scientists and general viewers alike were interested in the story of the Wolframs’ behind-the-scenes contributions to the movie, from Space.com to OuterPlaces.com and others. SlashFilm.com went further, pointing readers to the Science vs. Cinema Arrival episode featuring interviews with the Wolframs, other scientists, Jeremy Renner, Amy Adams and Villeneuve. Wired magazine also interviewed Christopher Wolfram on the subject of the Wolfram Language code he created to lend validity to the computer screens shown in the film. Watch Christopher Wolfram walk you through his development process.
Wolfram Research has a track record of contributing to film and TV. From the puzzles in the television show NUMB3RS to the wormhole experience in Interstellar, Wolfram technology and expertise have enriched some beloved popular art and entertainment. With Arrival, however, Stephen and Christopher consulted more extensively on what Stephen calls “the science texture” of the film.
Science and technology shape our world now more than ever. Science fiction movies are finding a wider audience, and we find these stories are crafted into films by some of the most skilled filmmakers around. If filmmakers such as Villeneuve continue to recognize the importance of getting the science right, science fiction will continue to live up to Arthur C. Clarke’s claim that “science fiction is escape into reality…. [It] concern[s] itself with real issues: the origin of man; our future.”
For more information on the Wolframs’ involvement in Arrival, read Stephen Wolfram’s blog post, “Quick, How Might the Alien Spacecraft Work?”
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