At Wolfram Research, we are excited for the April 8 total solar eclipse and plan to observe this extraordinary event in several ways. Read about the science and math of this rare phenomenon in Stephen Wolfram’s new book, *Predicting the Eclipse: A Multimillennium Tale of Computation*, and then find eclipse specifics for your location with the Wolfram precision eclipse website. Now that you know why and where, prepare for your upcoming watch party with these Sun-related recipes using two new functions from the Wolfram Function Repository: `RecipeGraph` and `NutrientComparisonBarChart`.

`RecipeGraph` relies on a large language model (LLM) to help create a graph of the ingredients and instructions for a recipe. The recipe ingredients and instructions form the vertices of the graph. The edges (lines connecting the vertices) represent the flow of the preparation and cooking process. Each ingredient connects to the instruction in which it is used. `NutrientComparisonBarChart` creates a dual bar chart comparing the calories and macronutrients (protein, carbohydrates, fat and fiber) in a list of foods.

Now you’re ready to celebrate. Prepare your favorite recipes, grab your ISO-compliant eclipse glasses and make special memories with family and friends on this historic occasion.

Rise and shine the day of the eclipse with a sunshine smoothie and sunny-side-up eggs:

If you’re limiting carbohydrates or maximizing protein, `NutrientComparisonBarChart` is an efficient way to compare relative carbs and protein per gram of food, so you can make informed nutritional choices for your smoothie. We’re using the `"SolarColors"` chart style in recognition of the eclipse:

Customize your sunny-side-up eggs graph with chart style options:

The kids can help celebrate the eclipse by mixing the ingredients for baked granola with sunflower seeds. They can stir in chocolate chips after baking for an extra treat. Use `NutritionReport` to assess this nutritious after-school snack:

This recipe yields 20 servings of 1/3 cup each, for about 200 calories and 4.5 grams of protein per serving:

Make a tasty eclipse-themed dinner of sunburst salad and baked pasta with sun-dried tomatoes.

Use the LLM to create the salad recipe:

Use `ImageSynthesize` to create an original sun image for the vertices of the baked pasta graph:

Visit the Wolfram Function Repository to learn more about these resource functions:

`RecipeGraph``NutrientComparisonBarChart``NutritionReport``FoodCompassPlot``TotalSolarEclipse2024Explorer`

Visit Wolfram Community or the Wolfram Function Repository to embark on your own computational adventures! |

This year’s Global Astronomy Month is off to an exciting start for North America in anticipation of the total solar eclipse** **on April 8. In light of this momentous event, the following is a list of resources that bring Wolfram Language and astronomy together—including expert video guides, projects and books—for computational astronomers at every level.

Watch astronomical phenomena in action on April 8. This total solar eclipse will be the last one visible from North America until 2044. You can find out exactly when the eclipse will be visible to you using the Wolfram Precision Eclipse Computation website. Simply plug in your location and get your ISO-compliant eclipse glasses ready.

If you’ve ever wondered why black holes don’t collapse in on themselves or about the gravitational limits of a planet, we suggest participating in Stephen Wolfram’s livestreams for the chance to learn about varying topics in the world of science and technology and for a behind-the-scenes look into his life and work. His weekly Science & Technology Q&A for Kids & Others is an open, live Q&A session dedicated to answering your questions.

While the streams are not bound to a single topic, part 140 looks into dark matter, light and other space-related topics. Part 107 features an in-depth conversation about black holes and light, and part 109 looks at questions about gravity and pressure in the vacuum of space. These livestreams will surely intrigue anyone looking to learn more about space!

Do you have more space or other questions for Stephen? You can submit a question to be answered in a future Science & Technology Q&A for Kids & Others or History of Science & Technology Q&A livestream.

The Wolfram Demonstrations Project offers more than 12 thousand interactive Wolfram Language Demonstrations in varying fields, including over two hundred astronomy Demonstrations. Manipulate and learn from unique Demonstrations like the following.

**View of Our Solar System**By: Becky Johnsen

Johnsen’s Demonstration explores the relative distances between the Sun, planets and the dwarf planet Pluto. All bodies are shown larger than scale size but in correct relative proportion except for the Sun and Pluto (for aesthetic reasons).

**How Old Would You Be on Another Planet (or Pluto)?**By: Chris Boucher

The planets in our solar system (and Pluto) rotate on their axes at different rates and take differing amounts of time to complete an orbit of the Sun. Boucher’s Demonstration allows you to calculate how old you would be on different planets (and Pluto).

**Make Your Own Solar System**By: Stephen Wolfram

Wolfram’s Demonstration allows you to create your own 3D solar system by adjusting the size of a central star and the sizes and distances of four planets.

In addition to being an ever-expanding searchable database with knowledge spanning the computation of physics mechanics to providing detailed timelines for historical events, Wolfram|Alpha also gives topical example queries to get your research started in the right direction. Check out the collection of space and astronomy examples to start researching astronomical events and learn to calculate astrophysics problems.

If you’re already an astronomy whiz and are ready to move on to more advanced Wolfram Language computations, you will find these projects offer the inspiration you need to move forward with your exploration.

More advanced Demonstrations are available for those looking to observe and interact with different astronomical concepts.

**Phases of Planets**By: Jeff Bryant

Like the Moon, planets can also have phases. Bryant’s Demonstration offers a view of Mercury, Venus and Earth when viewed from any of these three planets. Planets in inferior orbits undergo complete phase changes like the Moon when viewed from a planet with a superior orbit. Planets in superior orbits only go though minor changes in phase when viewed from a planet with an inferior orbit.

**Solar and Lunar Eclipses**By: Jeff Bryant

A solar eclipse occurs when the Moon’s shadow moves across the face of the Earth. Similarly, a lunar eclipse occurs when the Earth’s shadow moves over the Moon. Bryant’s Demonstration allows you to see a model of solar and lunar eclipses by adjusting the position and distance of the Moon.

**Life Cycle of a Star**By: Allison Jung

Stars evolve from birth to death much as animals or plants do. New stars form in stellar nebulae, made of clouds of plasma, hydrogen and helium. The lifetime of a star varies according to its mass; more massive stars have shorter lifespans than average-sized stars. The dividing line between the two types is around eight times the mass of the Sun. Jung’s Demonstration shows the life cycles of stars by adjusting an average and massive star’s evolutions.

Wolfram Language 13.2 introduced several new astronomy-focused functions, including `AstroPosition` and `AstroGraphics`, for getting started as a computational astronomer. Version 14 overhauled the function `SolarEclipse`, which performs detailed local computations for solar eclipses—just in time for calculating the 2024 North American solar eclipse. The 14.0 feature pages give you a chance to experiment with all of the functions released in this version. You can also use the Astronomical Computation & Data guide for a full list of astronomical functions and available data.

For a deeper dive into the astro features, be sure to check out our streams and video walkthroughs with Wolfram’s developers. Join José Martín-García on “More in Astronomy: Eclipses” and Tom Sherlock on “Astrophotography Image Processing Workflows.”

For an even closer look at all the astro features, take a look at our Live with the R&D Team livestream on astro computation, where researchers José Martín-García and Jeff Bryant discuss reference frames, time systems and different application examples like visualizing solar eclipses or computing the position of Jupiter’s barycenter.

For more unique ways to incorporate your astronomy research and Wolfram Language skills, you can visit the Wolfram Function Repository to explore and share your own astronomical functions:

Here are some ready-to-use examples from the Wolfram Language Example Repository to experiment with:

It’s no secret that Wolfram Community is one of the best places to learn about others’ projects and share or find help with your own work. These recent Community posts are a sampling of some of our favorite astronomy projects.

**A System for Modeling Space Debris Collision**** **[Wolfram High School Summer Research Program 2023]By: Shubhan Bhattacharya

The Wolfram High School Summer Research Program is a two-week, high-school program designed to push students in STEM fields through lectures, guided activities and hands-on workshops. Bhattacharya’s final project for the 2023 program featured an evaluation of the Kessler syndrome and tracking satellites to model potential collisions.

**Exploring Solar Imagery**

By: Jeff Bryant

Bryant uses his function `SolarImage` to color and process images of the Sun (originally retrieved from the Helioviewer Project). His results are bright, colorful and dynamic images and videos of the center of our solar system. His post was accompanied by the following post, “Imaging the Sun from SDO Orbital Telescope Extreme Ultraviolet Data” from Vitaliy Kaurov. Be sure to check out Bryant’s other solar-related projects, including “Exploring the Origins of Space Weather.”

**Imaging the Sun from SDO Orbital Telescope Extreme Ultraviolet Data**By: Vitaliy Kaurov

Written to follow Bryant’s post on exploring solar imagery, Kaurov explores the history and tools behind the Solar Dynamics Observatory’s quest in capturing images of the Sun.

*Predicting the Eclipse: A Multimillennium Tale of Computation*

By: Stephen Wolfram

While eclipses were initially perceived as mysterious omens, modern astronomers can predict them to within one second of their appearance. In his book *Predicting the Eclipse: A Multimillennium Tale of Computation*, Stephen Wolfram discusses the history of studying eclipses, using the April 8 North American eclipse as a case study, and the impact of this work on the development of science and technology, from witnessing the stars to soaring among them.

For those looking to go even further with advanced astronomy research, the following publications offer in-depth analyses to push your work to the next level.

**Ancient Plagiarism? An Analysis of Claudius Ptolemy’s Star Catalog**

By: Christopher Wolfram

Wolfram reviews one of history’s oldest star maps and potential star scandals from Claudius Ptolemy. This Alexandrian scientist is the author of one of the most influential scientific works, the *Almagest*. Wolfram reviews the history of the text as well as the notion of plagiarism from an even earlier astronomer and compares the works of the two.

**Testing the Speed of Gravity with Black Hole Ringdown**By: Sergi Sirera Lahoz

Lahoz shares his calculations as he investigates how the speed of gravitational waves can be tested with upcoming black hole ringdown observations. He shares how the different elements of black holes affect calculations and the environment surrounding them.

**Possible Spacetime Discretization in Astrophysical Phenomena**** **[Wolfram Science Winter School 2023]

By: Vittoria Tommasini

The annual Wolfram Science Winter School gives students an opportunity to participate in research projects with Stephen Wolfram and other Wolfram employees, in addition to developing their own research projects with a team of Wolfram mentors.

Tommasini featured a unique look into computational astronomy with her independent project that focused on connecting quantum mechanics on larger-scale objects like black holes. She focused on modeling discretized spacetime geometries for Minkowski and Schwarzschild spacetime graphs.

**Effects of Dimensions D ≠ 3 on Galactic Rotational Velocity Curves** [Wolfram Science Winter School 2023]

By: John Blakely

In another feature from this year’s Winter School, Blakely worked with the Wolfram Physics Project to evaluate the discrete space dimensional effect on galactic rotational velocity curves by creating a model of the flattened curves for observation.

“Dynamical Gravastars”

By: Stephen L. Adler

Adler’s article, published by *Physical Review D*, uses Wolfram Language to look into the structure and behavior of gravastars with the Tolman–Oppenheimer–Volkoff equation. You can find a description of Adler’s work and his notebooks on Wolfram Community.

*Geometric Optics: Theory and Design of Astronomical Optical Systems Using Mathematica*, Second Edition

By: Antonio Romano & Roberto Caveliere

*Geometric Optics: Theory and Design of Astronomical Optical Systems Using Mathematica* from Antonio Romano and Roberto Caveliere combines the computational abilities of Wolfram Language with the optical elements of astronomy.

Wolfram has always been committed to pushing boundaries in pursuit of the idea of computational X, or the coming together of technology and the rest of the world. This idea is carried through the world of Wolfram with the help of the Wolfram developers, who work to make each new version as exciting as possible, and the users, who share their own projects and discoveries through Wolfram Community and their own publication sources.

Looking for more great resources to find your computational X? Check out our collection of courses at Wolfram U and varying events and workshops to learn more about Wolfram Language and its different application areas.

]]>Explore the contents of this article with a **free Wolfram System Modeler trial**. A wind turbine gearbox, susceptible to erratic wind loads, frequently fails well before its intended lifespan. Such failures, occurring globally, not only cause significant downtime but also lead to substantial economic losses. Can simulations help avoid this?

In addition to many other exciting updates to Wolfram System Modeler, we just released the Rotating Machinery library. This library is a powerful tool to simulate an array of critical rotating machinery components, such as bearings, gears, flexible shafts and discs, with high precision. These can be modeled both individually and as part of a larger system capturing dynamic behavior with high fidelity.

In this blog, I will delve into the intricacies of wind turbine design challenges and solutions using the Rotating Machinery library. I will also focus the analysis on maintaining gear contact pressures and foundation forces within design thresholds.

In the first part of our experiment, I will show how to model and simulate the gearbox specifics of the widely used ACCIONA wind turbine, specifically the AW-100/3000 model. Using this, I will verify that the gearbox is within allowable stress limits.

In the second part, I’ll incorporate the mast, shafts and blades into the model to check if the gearbox continues to meet the design criteria. Adding these components will enhance the model’s accuracy, allowing it to better reflect the actual dynamics at play.

Let’s begin with an in-depth model of the ACCIONA AW-100/3000 gearbox. The gearbox has two parts: a planetary gear and a three-stage gearbox. We can see the gearbox marked in the following image:

Using ready-made components from the library, I model the gearbox system, including a planetary gear, a three-stage gearbox, a shaft in between and a corresponding support, as shown in the following. All the specifics of the respective gearbox are defined by parametrizing things like wheel geometries, number of teeth and profile shifting. Now we are ready to simulate and analyze:

In the simulation, I apply a speed profile starting at a standstill and accelerating to the operating speed of 20 rpm, as shown by the blue line in the following figure. We can also see the rotational speed of the sun wheel (i.e. the center wheel) in green and one of the planet wheels in orange:

Other than the expected difference in speeds due to gear ratios, we also see that the planet wheel seems to vibrate a lot compared to the others. This could be caused by backlash between the different wheels; this might be reduced by changing a variable such as the profile shift (i.e. the teeth geometries and gap), but for now, we will focus on analyzing the contact stresses instead.

Let’s start by taking a look at a visualization of the planetary gear and three-stage gearbox. The images are screenshots from the animation in System Modeler. On the left, we see the planetary gear, and on the right the three-stage gearbox. In both cases, I have marked the contact points that I will be studying. I selected these because they had the largest contact pressure when looking at the simulation results:

In the following, I plot the stress acting on the selected tooth contact pairs with the allowable stress limit for AISI 5160 steel, which is one of the steels used in the ACCIONA turbine, 1800 MPa. Notice that the peak contact stresses are greater in the planetary gear but remain below the maximum allowable limit:

But perhaps there is more to the story than this. The following figure shows the contact pressure at the two locations during the same time interval. Observe that the planetary gear (top figure) rotates slower, and, therefore, each contact takes more time. More interestingly, we can also see that there are a lot of vibrations, especially on the planetary gear. Vibrations are often a cause of failure, so this is something that should be investigated further to understand the potential consequences. This investigation would require its own dedicated blog, so I will not include this here. If you are interested in understanding how to do this kind of frequency analysis, you can read this post:

Building on our understanding of gearbox dynamics, let’s move toward a more complete model. This model also includes a flexible rotor with blades, a tower and an additional bearing:

The following image shows a snapshot of the animation that illustrates the combination of the gearbox with a planetary gear and a three-shaft gearbox, providing a more comprehensive understanding of the system. The ACCIONA AW-100/3000 wind turbine model has a 100-meter hub height, which can be modeled by the flexible beam component from the Rotating Machinery library:

In the following animation, we can observe the rotation of each wheel gearbox. It is easy to see that there is a big difference in speed between the blades and the outgoing shaft. In fact, the propeller rotates at 20 rpm, while the outgoing shaft is doing 1560 rpm, corresponding to a total gear ratio of 77.8. The high speed is tuned to maximize the performance of the generator it drives:

In the following plot, we show the rotational velocities of the different components of the planetary gears, just as we did in the first part of the blog. You can observe that there are more vibrations now. These are caused by the dynamics related to the blades, tower and shafts:

The big question is whether these higher vibrations lead to problems with the allowable limit for contact stress. In the following plot, we can see a comparison of the contact pressures from the gearbox-only model (in orange) and the full system model (blue). It is easy to see with the naked eye that we are now much closer to the allowable limit of 1700 MPa. However, the simulations show that we are still within it:

While in this case we still ended up within the limit, the analysis highlights the necessity of paying attention to small details while considering the entire system. This is exactly where the Rotating Machinery library and System Modeler excel.

For more information on modeling wind turbines, check out the Wolfram System Modeler libraries and interact with examples like “High-Fidelity Wind Turbine Mast“

Don’t miss Wolfram U’s course “Testing and Modeling Turbines, Gears and Drivelines with the Rotating Machinery Library” on Wednesday, March 20. |

Happy Leap Day 2024! A leap day is an extra day (February 29) that is added to the Gregorian calendar (the calendar most of us use day to day) in leap years. While leap years most commonly come in four-year intervals, they sometimes come every eight years. This is because a traditional leap day every four years is actually a slight overcompensation in the calendar. Thus, a leap year is skipped every one hundred years when those years are not divisible by 400 (this is actually the entire difference between the Julian and the Gregorian calendars).

Phileas Fogg (*Around the World in Eighty Days*) traveled around the world in fewer than 80 full days from his start in London, but he counted 81 sunrises because he was traveling opposite to the motion of the Sun in the sky. If he had traveled in the same direction, he would have counted 79 sunrises in the same period of time. If the Earth rotated backward, these numbers would be swapped, and Fogg would have needed to travel toward the west to win his bet.

The same phenomenon happens for all of us every year. The Earth travels a full orbit around the Sun in a year and, in the same time, it rotates approximately 366.25 times (this is the equivalent of 80 days for Fogg) with respect to the stars—well, it’s actually with respect to the vernal equinox point, which itself moves too due to precession, but that gets too complicated.

Because the Earth rotates in the same direction, we count one day fewer, and so we get a year that has, on average, 365.25 solar days. If the Earth rotated backward, we would perceive that a year has 367.25 days!

Let us stop for a moment to measure what a day would be in each case. A full orbit around the Sun takes this amount of time:

It corresponds to this “Foggian” number of our solar days:

If we measure rotations with respect to the stars, we count one more, so the day is shorter:

This is the so-called “sidereal day”:

If the Earth rotated backward, the solar day would have this length:

That is, we would have 367.25 days in a year, but each day would be about eight minutes shorter. Or, perhaps, we should say that days would still have 24 hours, but each one would be about 20 seconds shorter:

It’s so easy to get all these precise numbers with Wolfram Language!

*Disclaimer:* Due to how the solar system was created, it is unlikely that the Earth would rotate backward, and if it did, tidal friction with the Moon would have given a very different duration of the day. But let us ignore all that here and assume that rotation with respect to the stars would have the same angular speed.

Now we can address our question: would we remove leap days if the Earth rotated backward? No. We see that the number of days in a year would be 367.25, so the natural thing would be to have normal years of 367 days and then add a leap day every four years (with a Gregorian correction!) The main consequence for our standard calendar is that we would have two more days. Presumably, February would also have 30 days in normal years, and 31 in leap years. Wouldn’t that be nicely symmetric with all other months?

So, how many total leap days have there been (including Julian and Gregorian calendars)? Has that math been done, and if so, was it right?

The physical year (i.e. an orbit of the Earth around the Sun) is called a “tropical year,” known to very good precision:

Put another way:

The difference with 365 days and 6 hours is only a bit more than 11 minutes.

This is the number of days (i.e. turns of the Earth with regards to the Sun) between January 1 of year 1 (in the Julian calendar) and January 1 of year 2025 (in the current Gregorian calendar), including one but not the other, so this is 2,024 full calendar years:

The difference with 2,024 tropical years is only 2.8 days:

This is a very good approximation in more than 700,000 days. But where did those 2.8 days come from?

Imagine all years had 365 days. Then 2,024 years would be:

And there would be a difference of more than a full year with respect to the physical counting of years!

The Julian calendar was introduced in 45 BCE to add one day every four years (extending by six hours the average length of a year). Then 2,024 Julian years would be this number of days:

That’s now too much by 15.8 days:

By the end of year 1581, exactly 395 leap days had been added since year 1, which was about 12 days too many:

The Gregorian reform of the calendar removed 10 days in 1582 (the day following October 4 was October 15). The new calendar also changed the rule of how leap days are added, to avoid accumulating 11 minutes of error every year (or, equivalently, one day every 128 years). Years that are a multiple of 100 but not of 400 are not leap years. This has happened so far for years 1700, 1800 and 1900. Therefore, the Gregorian calendar has corrected 13 days of the 15.8 days of error. The difference is the 2.8 days we saw before, most of it from the removal of 10 instead of 12 days. The other 0.8 is essentially because we are close to correcting another leap day in year 2100.

The important comparison is this: In 400 years of the Gregorian calendar, there are 97 leap days added. Therefore, the average year is:

So there is a difference of only 27 seconds per year, to be compared with the more than 11 minutes of error in the Julian calendar:

It will take more than 3,200 years to accumulate a day of error in the Gregorian calendar, while it takes only 128 years to have a day of error in the Julian calendar:

In short, yes, the math has been done… and it wasn’t *exactly* right—but with more precise computation, we’re getting closer by the second!

(The Newtonian calendar is slightly more precise, but that’s a rabbit hole for another day.)

]]>Learning quantum theory requires dedication and a willingness to challenge classical assumptions. Quantum interference, particularly for massive particles, is a pivotal example in this journey. The Schrödinger equation, inspired by de Broglie’s hypothesis, revolutionized our understanding by revealing the wavelike nature of even massive particles. This phenomenon not only deepens our grasp of nature but also fuels innovations in quantum applications, from quantum sensing to quantum computing. Yet many students don’t have the opportunity to run experiments that require sophisticated hardware. Not anymore!

We’re proud to announce a strategic partnership in quantum education today by joining forces with Infleqtion, a global leader in quantum information—heralding a new era in quantum education and research that can address both theoretical and experimental aspects. Together, we’re committed to the design and development of educational materials, combining our computational prowess with Infleqtion’s quantum matter service Oqtant.

This collaboration aims to bring classrooms closer to “quantum everywhere,” making advanced learning tools for quantum systems more accessible. We believe that with this collaboration, a unique educational experience is now possible. There is but one more ingredient that is crucial to providing quantum education: academic partners.

We are calling upon academic institutions and educators to join this exciting initiative. Joint partnership offers access to Infleqtion’s Oqtant platform and Wolfram Language, providing an unprecedented opportunity to explore quantum mechanics through hands-on experience and interactive learning. Academic partners will receive Wolfram Language licenses and limited sponsored access to Oqtant to foster research and development in quantum education.

Interacting with phenomena is a key component of science education. We envision educational materials where students use Infleqtion’s Oqtant quantum matter service to run experiments with real quantum hardware and analyze the results and theoretical models in an interactive Wolfram Notebook.

We invite academic institutions, researchers and educators passionate about quantum education to partner with us in this venture. Together, we can shape the future of quantum education, making it more interactive and accessible for students around the world.

For academic inquiries and more information on how to get involved, email quantum@wolfram.com.

Let’s try to get a taste of how modeling tools provided in Wolfram Language can help with understanding and modeling the types of experiments students can run using the Oqtant API from Infleqtion. Many students will be familiar with the Schrödinger equation from introductory courses on quantum mechanics. Bose–Einstein condensates (BECs), the system accessible through the Oqtant API from Infleqtion, can be modeled by a nonlinear version of the Schrödinger equation. The source of nonlinearities in the Gross–Pitaevskii equation arises from the interaction term representing the mean-field effects of BECs, and they are not a fundamental correction to the Schrödinger equation.

Unlike the linear version of the Schrödinger equation, numerical techniques are immediately needed to solve the resulting equations. By studying systems like those accessible through Oqtant, students gain practical skills working with real experimental systems, theoretical modeling and numerical simulations.

Utilizing BECs in quantum education is invaluable for several reasons. Firstly, BECs provide a tangible platform for exploring fundamental quantum principles, allowing students to observe and manipulate quantum phenomena firsthand. This hands-on experience fosters a deeper understanding of concepts such as superfluidity, coherence and quantum entanglement, which can be challenging to grasp solely through theoretical study. Additionally, BEC experiments often involve interdisciplinary techniques, exposing students to a range of scientific methodologies and encouraging collaboration across scientific disciplines. Furthermore, by engaging with BECs, students gain practical skills in experimental design, data analysis and problem solving, preparing them for future careers in quantum research and technology development.

Let us start with a simple case of a nonlinear, time-dependent Schrödinger partial differential equation (PDE) operator in 1D:

Define the boundary and initial conditions:

Compute the solution of the Schrödinger time-dependent equation:

Plot the absolute value of the solution:

Including a harmonic potential term makes this 1D example qualitatively similar to a 3D BEC. Strictly speaking, BECs cannot exist in one-dimensional systems; nonetheless, the 1D equation can serve as a valuable pedagogical model:

Find the solution:

Plot the absolute value of the solution:

When comparing to the previous case without a harmonic trap, it’s evident that the trap effectively confines particles around its well, demonstrating its clear influence on the system.

Let’s examine a more realistic scenario, where the system exhibits axisymmetric region symmetry, represented by a truncated cylindrical coordinate system that eliminates the angular variable while retaining radial and axial coordinates:

Set the boundary condition and the initial state:

Compute the solution:

You can see how an initially localized wavefunction spreads out over time when not confined by a trapping potential:

Let’s change the initial state to a superposition of two Gaussians in order to observe interference patterns that emerge from the overlap of two wave packets:

Find the solution, given the previous initial condition:

Even though hardware for creating a BEC is a real 3D system, visualizing the process along just the “main” axis helps make the connection with 1D problems clear for students. As time passes, you can observe interference patterns that emerge:

Since the modeling of the system is done in 3D, you can also look at 2D slices through the “middle” of the system:

Alternatively, you can show the full 3D picture:

It’s important to note that a full 3D image is quite difficult to achieve in an experimental setting with systems like these. Typically, one would have sensors that look at the 3D phenomena from a particular direction and can give an “integrated picture” of the density.

Of course, this is only the beginning of the interesting modeling one can do. By combining data from numerical simulations and the Oqtant service, educators can give students hands-on experience with skills that will prepare them for the quantum-literate workforce of the future. Not only will students gain strong computational skills using Mathematica, they will also learn details of quantum experiments and using the theory to handle real experimental data.

For more information on how to partner with us and take part in this quantum leap in education, please contact us at quantum@wolfram.com. |

Practical quantum computers have not entered the mainstream, but that has not stopped researchers and developers from innovating. Simulating quantum results on classical hardware and getting meaningful results from noisy quantum hardware are two important areas with lots of recent innovations.

The Wolfram Quantum Framework is a toolkit for Wolfram Language that offers quantum simulations. The Framework brings quantum experimentation to anyone and opens the door for more research and development of quantum algorithms. This can also be used to connect with external cloud services such as Amazon Braket or IBM Quantum for an even closer look at what running on quantum hardware could be.

Because of the presence of environmental noise, quantum processing units (QPUs) are susceptible to “misfires,” potentially leading to erroneous outcomes when executing a quantum algorithm on a QPU. To address this challenge, Wolfram has partnered with Q-CTRL, making it easier than ever to get useful results out of quantum hardware.

Q-CTRL’s Fire Opal offers a support system that automatically includes error-suppression techniques as the quantum circuit is translated to hardware-specific instructions, and also runs on a QPU. This greatly increases the probability of obtaining an accurate result from the intended algorithm and makes it possible to successfully run more complex algorithms. The following graph shows the improvement in QPU results using Fire Opal when compared with exact predictions of quantum theory for this phase estimation algorithm:

The preceding graph shows an example of what the QPU results look like both with and without Fire Opal in comparison with the exact predictions by the Wolfram Quantum Framework.

The Wolfram Quantum Framework allows you to easily specify quantum circuits and analyze their behavior as predicted by quantum theory. The framework is developed as a paclet in the Wolfram Paclet Repository and can be installed by running the following code in a notebook:

After installing and loading the paclet, create a circuit to implement the standard phase estimation algorithm for a given operator—in this case, a phase operator with the angle 2π/7:

You can show the diagram of this circuit by simply asking for the `"Diagram"` property of the circuit operator:

Next, compute the measurement probabilities for each outcome as predicted by quantum theory. In this case, the results have also been numericized for a speedier result. If you want the exact results, don’t use `N` on the circuit; simply use `qc[]`. The result is returned as a quantum measurement object:

From the measurement object, you can plot the probability of each outcome:

Connections to external services (such as quantum hardware access) can be utilized by establishing a service connection. If you want to use an IBM backend, create an IBM Quantum account.

To create a new connection to IBMQ, you can use the following code:

The first time you attempt to connect to a service that requires an API token, you will see the following window appear:

If you choose to save the connection, you can simply use `ServiceConnect``["IBMQ"]` the next time that you want to connect on the same system.

After the connection is established, you can ask for the queue of all available backends:

With the service connection established, running a circuit can be as simple as the following code structure:

Note that you must replace *backend* with an actual QPU name, such as `"ibm_kyoto"`.

However, this simplest method requires waiting for the job to complete and receiving a response before proceeding with any further computations. This will mean your kernel is busy waiting for a response from the hardware provider.

If you want to do other local computations while waiting for the submitted job to complete, one method is to use a separate kernel to send the request and handle the response. The following code does this and assigns the result to the variable `qpu` in the current kernel session:

Note that the wait time for QPU providers to provide responses can vary quite a lot, and is dependent on the queue for the specific backend. Once the job is done, you can also check on results using the service connection and the appropriate job ID:

If you already know the job is done and want to store the results in the variable `qpu`, you can use the appropriate job ID and perform some post-processing to get the data into shape:

With results in hand, you can visualize them as a bar chart:

So far, you have just seen how to use a service connection to run your circuit on quantum computing hardware. Using Fire Opal as part of the process is as simple as adding another option. Set the `"FireOpal"` parameter to `True` when providing a method option to your quantum circuit operator. Running the code with a valid circuit operator and specified backend will prompt redirection to the Fire Opal webpage, where login is required:

One can first validate the compatibility of circuits for Fire Opal. Note that by default Fire Opal’s “validate” feature is not active. For example, the following circuit returns an error:

As before, you may want to utilize a separate kernel or other method to avoid keeping your system busy waiting for the response from the QPU provider. Remember to fill in the parameter for the desired backend before running the code:

If the job is already done, you can retrieve past results in the same way as before, this time storing them in the variable `fireOpal`:

As before, you can visualize the results:

If you want to immediately analyze the results of an experiment without submitting your own jobs to QPUs, you can use the results here from one experiment:

The exact results are computed using the Wolfram Quantum Framework and are the probabilities one would expect from the mathematics of quantum theory. They provide a nice baseline for comparing what comes back from the inherently noisy QPUs of today. One set of measurements for the phase estimation circuit in the first section was done without using Fire Opal, and another set was done using Fire Opal.

Combine the results and visualize them in a bar chart for easy comparison:

Fire Opal’s improvement of the experimental results is visually evident as the expected peak at a particular measurement outcome. Notice how the theoretical result for this circuit is sharply peaked at one particular measurement value, while the hardware without Fire Opal provides a much greater deviation from the expected results.

To compare the results of the phase estimation quantitatively, one can choose from a variety of statistical tests.

For example, one can compute how “different” one distribution is from another using the Kullback–Leibler divergence. Smaller numbers of this quantity indicate the two distributions are more similar. Compare the results both with and without Fire Opal to the exact reference distribution:

Another test to consider is the Kolmogorov–Smirnov test. The convention for this function is that larger numbers indicate it is more likely the results are from the exact reference distribution:

Finally, let’s compute the Hellinger fidelity as a measure of the distance between distributions. Smaller numbers of this quantity indicate the distributions are more similar:

With three different quantitative evaluations to support our work, Fire Opal has proved itself to be a valuable resource to improve the accuracy of the quantum circuit extension when running on a QPU.

As quantum hardware advances, the possibilities for quantum computing become more exciting. Using the Wolfram Quantum framework for algorithm design, cloud service providers for quantum hardware access and Fire Opal for increased hardware performance, it has never been easier to innovate.

For more information on how to partner with us and make your own quantum noise, please contact us at quantum@wolfram.com. |

Hypergeometric series appeared in the mid-seventeenth century; since then, they have played an important role in the development of mathematical and physical theories. Most of the elementary and special functions are members of the large hypergeometric class.

Hypergeometric functions have been a part of Wolfram Language since Version 1.0. The following plot shows the implementation timeline of different hypergeometric functions during the evolution of our system:

The Gauss hypergeometric _{2}*F*_{1}, Kummer hypergeometric _{1}*F*_{1} and confluent hypergeometric _{0}*F*_{1} functions were implemented in Wolfram Language Version 1.0, and in Versions 3.0, 4.0 and 7.0, powerful updates were made that implemented four very general functions: the generalized hypergeometric _{p}*F*_{q} function, the “monster” superfunction `MeijerG`, the `AppellF1` function and the so-called *q*-hypergeometric function, implemented as `QHypergeometricPFQ`. All these general functions significantly increased the integration, summation and other symbolic manipulation capabilities of Wolfram Language.

During the last three years, we have made a strong effort to implement the remaining computable hypergeometric functions. Three Appell functions (`AppellF2`, `AppellF3` and `AppellF4`) were implemented in Version 13.3; further generalization of `MeijerG`—the `FoxH` function—was implemented a little earlier, in Version 12.3; and, finally, for Version 14.0, we’re presenting the doubly infinite hypergeometric function of one variable—the so-called bilateral hypergeometric function—as `BilateralHypergeometricPFQ`.

The term “hypergeometric series” appears to have first been used by John Wallis in his 1655 book *Arithmetica Infinitorum*, and then these hypergeometric series were treated by Leonhard Euler.

Starting from the works of Carl Gauss and continuing with Ernst Kummer, Bernhard Riemann, Paul Appell and other great scholars, these functions were systematically studied, along with the differential equations they satisfy and their vast applications in different engineering, physical and other applications.

A hypergeometric series is a power series , where the ratio of successive coefficients is a rational function of *n* (, where *A*(*n*) and *B*(*n*) are polynomials in *n*).

Let’s take a look at the Taylor series of the exponential function:

Calculate the ratio of successive coefficients (this can be done via `DiscreteRatio`):

This ratio is obviously a rational function of *n*, and for this case *A*(*n*) = 1, *B*(*n*) = *n* + 1, hence the Taylor series of `Exp` is hypergeometric.

In fact, various well-known series are hypergeometric, so having a comprehensive theory of such series-based functions is interesting as well as very useful in different areas of science. So let’s switch to the class of hypergeometric functions and start with the leading one—the generalized hypergeometric function _{p}*F*_{q}—and then move on to the well-known Kummer _{1}*F*_{1} and Gauss _{2}*F*_{1} hypergeometric functions that frequently arise in different physical and mathematical applications.

The main function of the hypergeometric class is the generalized hypergeometric function _{p}*F*_{q}, which is defined by the following series:

where (*a _{i}*)

The ratio of successive terms of _{p}*F*_{q} is obviously rational:

The generalized hypergeometric function _{p}*F*_{q} is implemented in Wolfram Language as `HypergeometricPFQ``[a;b;z]`. Here, the number of parameters in the *a* and *b* lists is not fixed; they might even be empty lists.

The *q*-analog of _{p}*F*_{q} is the basic hypergeometric function _{r}*Φ*_{s}, which has the series expansion

where (*a*;*q*)_{n} is the *q*-Pochhammer symbol. The basic hypergeometric function _{r}*Φ*_{s}, implemented in Wolfram Language as `QHypergeometricPFQ`, becomes the generalized hypergeometric function _{p}*F*_{q} in the limit *q* → 1.

_{p}*F*_{q} plays an important role in the theory of differential equations. A large set of ordinary differential equations (ODEs) can be solved in terms of _{p}*F*_{q} functions (we refer to such equations as hypergeometric ODEs). Following, we present such an ODE that is solved in terms of _{p}*F*_{q} functions:

_{p}*F*_{q} has a well-developed theory and various fundamental applications in science (one might take a look at the Applications section of the `HypergeometricPFQ` reference page).

Another remarkable application example is the trinomial equation *x ^{n}* –

The trinomial equation has *n* roots. Let’s generate one of them for, say, *n* = 5 and *t* = 2:

Now we generate a table of five solutions and check that they really solve the trinomial equation:

_{p}*F*_{q} is extensively used for integration and summation as well as for symbolic expression simplification. For example, here is a seemingly simple integration example:

And here is an example of an infinite sum:

Other hypergeometric functions can be written in terms of `HypergeometricPFQ`:

The following table shows some special cases of _{p}*F*_{q}:

Although _{p}*F*_{q} is a very general and important function, its special cases are even more popular. They significantly affected mathematical and physical theories of the nineteenth and twentieth centuries. Two of the most famous special cases are the Gauss hypergeometric function _{2}*F*_{1} and the Kummer confluent hypergeometric function _{1}*F*_{1}.

The well-known _{2}*F*_{1} function is defined by the following series:

It is a solution of the Gauss differential equation, which is a singular second-order linear ODE:

`ComplexPlot3D` demonstrates the pole of _{2}*F*_{1} at the singular point 1:

Why is this function of fundamental importance? Because every second-order linear ODE with three regular singular points can be transformed to it, hence the Gauss differential equation is the “basic” ODE with three singular points.

Second-order linear ODEs with a low number of singularities (the majority of ODEs that describe some physical phenomenon) can often be treated as special or limiting cases of the Gauss hypergeometric equation. This means that the powerful _{2}*F*_{1} incorporates most of the known special functions as special cases, including the famous Bessel functions, Legendre polynomials and others.

More information about the second-order linear ODEs, their solutions and their singularities is available in the author’s earlier blog post, titled “From Sine to Heun,” as well as a comprehensive tutorial on Wolfram Language’s `DSolve` function.

Aside from its mathematical importance, the Gauss hypergeometric function has various applications in physics, statistics and other areas of science. The twentieth-century quantum mechanical potentials can typically be solved in terms of hypergeometric functions.

Some of the applications are presented on the reference page of `Hypergeometric2F1`.

The confluent hypergeometric function _{1}*F*_{1} is defined by the following series:

It is a solution of the Kummer confluent differential equation *x**y*"(*x*) + (*b* – *x*)*y*'(*x*) – *a**y*(*x*) = 0. This differential equation can be obtained from the Gauss differential equation for _{2}*F*_{1} via the complex procedure of merging two regular singularities (coalescence).

The radial wavefunction for the continuous spectrum for the hydrogen atom is written in terms of the _{1}*F*_{1} function:

Here is a plot of the solution:

Plotting the solution in 3D gives more insight about the behavior of the radial wavefunction for the hydrogen atom:

Finally, here is a differential equation that can be solved in terms of _{1}*F*_{1}:

So far, we’ve talked about hypergeometric functions of one variable. _{p}*F*_{q} is a very general function with an unlimited number of parameters, but it has only one argument. What if we turn to hypergeometric functions of two or more arguments? Does that make sense?

The answer is yes. Further extensions to two or more variables are possible and yes, they open some new possibilities.

The first class is the Appell hypergeometric functions of two variables, named after French mathematician Paul Émile Appell.

Appell was a remarkable French mathematician who contributed to various fields of mathematics (projective geometry, algebraic functions, differential equations, complex analysis, etc.). Appell polynomials and Appell’s equations of motion in mechanics are named after him. Appell hypergeometric functions were introduced by him in 1880, and in 1926 he authored a treatise on these functions with another famous French mathematician, Joseph Kampé de Fériet:

There are four Appell functions. These functions have the following double series definitions around the origin (presented here with their convergence regions):

Appell functions reduce to `Hypergeometric2F1` when *x* = 0 or *y* = 0.

As noted earlier, `AppellF1` was introduced in Wolfram Language 4.0 back in 1999, while we’ve implemented the `AppellF2`, `AppellF3` and `AppellF4` functions only in 2023 in Wolfram Language 13.3.

Here are plots of a family of `AppellF2` functions:

The series expansions of Appell functions can be written in `Hypergeometric2F1` functions:

As with `HypergeometricPFQ`, we use `AppellF1` for integration:

And here is another general example of a whole class of integrands:

All four Appell functions solve the corresponding Horn PDEs with polynomial coefficients (we might think about these PDEs as a generalization of Gauss hypergeometric ODEs). This is the PDE that `AppellF3` solves:

And as for `HypergeometricPFQ`, many elementary and special functions are to be considered as special cases of the Appell functions:

The Appell functions are the first four functions in the set of 34 Horn hypergeometric functions of two variables.

The Appell functions are special cases of the Kampé de Fériet function, which is the general hypergeometric function of two variables. The Kampé de Fériet function can be used to represent the derivatives of _{p}*F*_{q} with respect to parameters and multiple integrals of the Meijer G-function.

Further hypergeometric generalizations to *n* dimensions include the Lauricella functions, which are very general and very complex. For *n* = 2, they reduce to the Appell *F*_{1}–*F*_{4} functions, while for *n* = 1 we get the _{2}*F*_{1} Gauss hypergeometric function.

Another generalization of the hypergeometric _{p}*F*_{q} function is the doubly infinite hypergeometric function (the bilateral hypergeometric function). It is written as

with a very similar definition to _{p}*F*_{q} except that for the bilateral series, the sum is computed from negative infinity to infinity. This function is available in Wolfram Language 14.0 as `BilateralHypergeomtricPFQ`.

There are two completely different subcases of the bilateral hypergeometric function: the “good” case when

*p* = *q* (i.e. _{2}*H*_{2}) and the “bad” case when *p* ≠ *q*.

For the first case, we can think about the bilateral function as a sum of two ordinary generalized hypergeometric functions. For example:

In the following, we calculate the value of _{2}*H*_{2} (1/2, 3/4; 1/4, 1/3; 5.4) and plot this function:

And for this “good” case of `BilateralHypergeometricPFQ`, simplifications are possible:

For the second case, where *p* ≠ *q*, the bilateral hypergeometric series is divergent. Usually for the calculation of such sums, various regularization methods are used. The blog post “The ABCD of Divergent Series” gives comprehensive information about this topic.

For calculation of the bilateral hypergeometric function, we use the Borel regularization technique:

Following is the series expansion for `BilateralHypergemoetricPFQ` at the origin:

The bilateral hypergeometric series has its unique and important role: it can be used for summing doubly infinite series:

So to sum a doubly infinite series, we internally first sum it to `BilateralHypergeometricPFQ` (as in the previous example) and then, where possible, simplify it—as in the following example:

The use of `BilateralHypergeometricPFQ` gives a huge speedup in the summation of doubly infinite hypergeometric series. As an example, the summation of the previous series in Wolfram Language 13.3 (without using `BilateralHypergeometricPFQ`) took more than 46 seconds, but now we’re able to reduce the calculation time by a factor of 1,000!

Hypergeometric functions have been at the core of Wolfram Language since the first version was launched more than 35 years ago. We constantly improve them, along with implementing new ones.

Version 14.0 contains the whole set of hypergeometric functions of one variable; the four Appell functions; the bilateral hypergeometric function and related ones (the monster superfunctions `MeijerG`/`FoxH` and others); and the *q*-analog of _{p}*F*_{q}—the basic hypergeometric function _{r}*Φ*_{s}.

It seems that we now have an almost complete “hypergeometric” infrastructure needed by researchers. This infrastructure includes powerful symbolic and numeric computational abilities as well as documentation that is being updated in almost every new version of Wolfram Language.

To close this blog post, we would like to thank all the Wolfram Research developers that contributed to this huge project.

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In days past, life sciences was reserved for those who had access to the proper equipment to observe and experiment with the organisms of the physical world. For today’s scientist, exploration doesn’t end with access to physical encounters. Whether you’re classifying an animal for the first time or using a protein visualizer to develop medication, Wolfram Language holds the tools and power to support your computational life science endeavors. The following is a collection of biology resources, projects and functions in Wolfram Language for any skill level.

The fields of life science cover a lot of ground—understandably so, given how expansive life itself is. Begin your computational biology journey with basic educational tools and virtual experiments.

Stephen Wolfram’s Science & Technology Q&A for Kids & Others is a weekly stream where Wolfram answers questions in hopes of breaking down the complexities of science and technology in an approachable way for those unfamiliar.

Each stream is an impromptu discussion that is not bound to a particular topic, but often finds common themes as the discussions unfold. Episode 120 features an in-depth look into DNA, genetic engineering and neurology. In episode 121, Wolfram looks into synthetic biology and the future of biotechnology. Have a question? You can submit your own questions to be answered in a future stream.

Wolfram|Alpha’s searchable database gives budding computational scientists the tools to find reliable information and calculations to support just about any field of work—including biology and life sciences. Example queries for biology are available to instantly learn about anatomy, taxonomy and genomics.

The Wolfram Demonstrations Project offers more than thirteen thousand interactive Wolfram Language Demonstrations in varying fields, including nearly two hundred biology Demonstrations. Set unique conditions and watch experiments unfold from Demonstrations like the following.

**The Cell Cycle**By: Rachel Lian and Stacy Hu

This Demonstration shows a visual model of the phases of mitosis.

**DNA Replication**By: Priyanka Multani

Multani’s Demonstration shows how the DNA helix unwinds and uses the old DNA strand as a template to create two daughter helices.

**3D Skeletal Anatomy of the Arm**By: Stewart Dickson

Dickson’s Demonstration offers an interactive skeletal model of the human arm—complete with rotating views and highlighting of different bones for easy identification.

**Predator-Prey Dynamics with Type-Two Functional Response**By: Wilfried Gabriel

Gabriel’s Demonstration uses simplified Lotka–Volterra equations to demonstrate simple predator-prey cycles. You can adjust the model by altering each part of the equation from predator competition to prey death rates.

When you’re ready to start creating your own computational life science experiments, Wolfram Language’s biology functions give you the power to build an interactive stage for exploration and experimentation. The most recent published entities include:

`"TaxonomicSpecies"`—This feature offers detailed information for the taxa

`AnatomicalStructure`—This feature offers detailed information for more than ninety thousand human anatomical parts.

The Wolfram Function Repository offers an ever-expanding collection of Wolfram Language functions developed by both Wolfram teams and users. With over 2,500 functions available, there are plenty of biology tools to go around for the computational biologist.

`DNAAlignmentPlot`creates a colorful visual for DNA sequences.`TaxonomicNearest`generates taxa`FoodWeb`generates graphs displaying predator-prey relationships for a given animal.`TaxonomyGraph`displays a taxonomy graph for a given species.

Wolfram System Modeler is an interactive modeling lab that gives you the chance to run dynamic simulations for varying environments. The Bio Chem library offers modeling, simulation and visualization of biological and biochemical systems. You can learn about how the Bio Chem library is used for safe drug research and development with FDA-approved models.

Computational biology in Wolfram Language doesn’t stop with informational entities and projects. The following resources show applications for using Wolfram technologies to complete life science experiments and research.

The Wolfram|Alpha biology team walks through its more advanced content and features in livestreams, Wolfram Technology Conference talks and blog posts.

**Video Walkthroughs**

- New Biology Content in the Wolfram Language
- Computational Taxonomy (Biology)
- Representing Biological Sequence Data in the Wolfram Language

**Wolfram Research Blog**

- “Brain, Neurons, Cognition: Computational Neuroscience”
- “Visualizing Anatomy”
- “Dissecting the New Anatomy Content in the Wolfram Language”

The Wolfram Function Repository also offers more advanced functions to keep you progressing with your computational biology work, including utilizing the Global Biodiversity Information Facility’s data.

From Pictures of Animals, Try to Reconstruct the Tree of Life (Wolfram High School Summer Research Program 2022)

By: Maya Viswanathan

The Wolfram High School Summer Research Program is an opportunity for high-school students to participate in their own research projects with mentors from the Wolfram team and Stephen Wolfram.

Viswanathan’s research project used Wolfram Language’s image-processing capabilities to make a taxonomical tree of life. The diagrams are used to organize different organisms into different classifications, including taxonomy and evolution. Viswanathan’s diagrams build trees solely off of how Wolfram Language interpreted images of different organisms, resulting in an impressive and colorful display.

Water and Heat Exchanges in Mammalian Lungs

By: Benoit Haut

Haut’s project uses a mathematical model that evaluates how varying mammalian lungs use water and heat to self-regulate temperature. Haut spares no effort in creating visually stunning models for easy reading.

Computational Anatomy Visualizations, Animations, Web-Deployment

By: Martijn Froeling

Froeling is an assistant professor specializing in quantitative neuromuscular MRI techniques to better understand muscle functions and diseases. He found himself in a project that required many images of anatomical models of lower-extremity muscles. He decided to use Wolfram Language to generate interactive models to use in his project rather than taking the time to search the web for the exact angles needed.

In 2023, Froeling was awarded a Wolfram Innovation Award for his paclet QMRITools. This paclet was developed as a toolkit for experimental design, data analysis and teaching. The paclet has been credited as a tool in over 50 scientific papers and currently offers more than 450 functions. QMRITools has helped to simplify quantitative MRI analysis.

Wolfram technology is currently being used in a variety of advanced research projects that push the current understanding of life sciences further and further. Combining Wolfram and the life sciences at a higher level offers an affordable and quicker way to test hypotheses and conduct analyses.

Mathematica in Cell Biology: Image Segmentation and Analysis of 3D Tumor Spheroids

Sabine Fischer discusses the work of the physical biology group at Goethe University Frankfurt in cell biology—particularly its work in image segmentation and assessing tumor spheroids.

Bioinformatics in the Wolfram Language

John Cassel discusses the Wolfram|Alpha Scientific Group’s work on computational bioinformatics in Wolfram Language and different applications to the life sciences.

*Mathematical Models in the Biosciences 1*

By: Michael Frame

Frame’s *Mathematical Models in the Biosciences 1 *offers a look into using Wolfram Language to aid in the mathematical foundations of biosciences, including chemotherapy, predator-prey relations, nerve impulses and more.

Introducing the Wolfram ProteinVisualization Paclet!

By: Soutick Saha

Saha’s ProteinVisualization paclet is designed to create intricate, colorful, 3D visualizations of biomolecules, including proteins, nucleic acids and their complexes. The paclet also allows for computing elements such as contact maps, graphs and dihedral angles.

The Wolfram Language Paclet Repository offers additional tools to be used within Wolfram Language. Check out the current available biology paclets to bolster your computational biology work, including CompartmentalModeling and StickyDBSCAN. You can help build the Repository by submitting your own paclets.

Wolfram has always been committed to pushing boundaries in pursuit of the idea of computational X, or the coming together of technology and the rest of the world. The Wolfram Language we know and love today was founded on the basis of supporting Stephen Wolfram’s passion for physics. This idea of pushing boundaries in different fields is carried through by the efforts of Wolfram developers, who strive to make exciting breakthroughs with every new version, and the users, who share their own projects and discoveries.

Looking for more great resources to find your computational X? Check out our collection of courses at Wolfram U and varying events and workshops to learn more about Wolfram Language and its different application areas. If you’re currently working on a project, be sure to share it to Wolfram Community, or contact us for the chance to be featured in an upcoming blog post.

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