Deriving Intersection Points of Algebraic Curves

Deriving Intersection Points of Algebraic Curves

by | Mar 15, 2024 | DS Articles

[This article was first published on Saturn Elephant, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

Want to share your content on R-bloggers? click here if you have a blog, or here if you don’t.

I will soon release an update of qspray on CRAN as well
as a new package: resultant. This post shows an
application of these two packages.

Consider the two algebraic curves
(f(x,y)=0) and
(g(x,y)=0) where
[ f(x, y) = y^4 – y^3 + y^2 – 2x^2y + x^4 quad text{ and } quad
g(x, y) = y – 2x^2. ]
We will derive their intersection points. First of all, let’s plot them. 

f <- function(x, y) {
  y^4 - y^3 + y^2 - 2*x^2*y + x^4
}

g <- function(x, y) {
  y - 2*x^2
}

# contour line for f(x,y)=0
x <- seq(-1.2, 1.2, len = 2000)
y <- seq(0, 1, len = 2000)
z <- outer(x, y, f)
crf <- contourLines(x, y, z, levels = 0)
# contour line for g(x,y)=0
x <- seq(-1, 1, len = 2000)
y <- seq(0, 1.5, len = 2000)
z <- outer(x, y, g)
crg <- contourLines(x, y, z, levels = 0)

Theoretically there is only one contour line for both. But for some
technical reasons, crf is split into several pieces: 

length(crf)
## [1] 16

So we need a helper function to construct the dataframe that we will
pass to ggplot2::geom_path: 

intercalateNA <- function(xs) {
  if(length(xs) == 1L) {
    xs[[1L]]
  } else {
    c(xs[[1L]], NA, intercalateNA(xs[-1L]))
  }
}
contourData <- function(cr) {
  xs <- lapply(cr, `[[`, "x")
  ys <- lapply(cr, `[[`, "y")
  data.frame("x" = intercalateNA(xs), "y" = intercalateNA(ys))
}

datf <- contourData(crf)
datg <- contourData(crg)

I also plot the intersection points that we will derive later:

datPoints <- data.frame("x" = c(-0.5, 0, 0.5), "y" = c(0.5, 0, 0.5))

library(ggplot2)
ggplot() +
  geom_path(aes(x, y), data = datf, linewidth = 1, color = "blue") +
  geom_path(aes(x, y), data = datg, linewidth = 1, color = "green") +
  geom_point(aes(x, y), data = datPoints, size = 2)

Now we compute the resultant of the two polynomials with
respect to (x): 

# define the two polynomials
library(qspray)
x <- qlone(1)
y <- qlone(2)
P <- f(x, y)
Q <- g(x, y)
# compute their resultant with respect to x
Rx <- resultant::resultant(P, Q, var = 1) # var=1 <=> var="x"
prettyQspray(Rx, vars = "x")
## [1] "16*x^8 - 32*x^7 + 24*x^6 - 8*x^5 + x^4"

We need the roots of the resultant
(R_x). I use
giacR to get them: 

library(giacR)
giac <- Giac$new()
command <- sprintf("roots(%s)", prettyQspray(Rx, vars = "x"))
giac$execute(command)
## [1] "[[0,4],[1/2,4]]"
giac$close()
## NULL

Thus there are two roots: (0) and
(1/2) (the output of
GIAC also provides their multiplicities). Luckily, they
are rational, so we can use substituteQspray to replace
(y) with each of these roots in
(P) and
(Q). We firstly do the substitution
(y=0): 

Px <- substituteQspray(P, c(NA, "0"))
Qx <- substituteQspray(Q, c(NA, "0"))
prettyQspray(Px, "x")
## [1] "x^4"
prettyQspray(Qx, "x")
## [1] "(-2)*x^2"

Clearly, (0) is the unique common
root of these two polynomials. One can conclude that
((0,0)) is an intersection point.

Now we do the substitution (y=1/2): 

Px <- substituteQspray(P, c(NA, "1/2"))
Qx <- substituteQspray(Q, c(NA, "1/2"))
prettyQspray(Px, "x")
## [1] "x^4 - x^2 + 3/16"
prettyQspray(Qx, "x")
## [1] "1/2 - 2*x^2"

It is easy to see that
(x= pm 1/2) are the roots of the
second polynomial. And one can check that they are also some roots of
the first one. One can conclude that
((-1/2, 1/2)) and
((1/2, 1/2)) are some intersection
points.

And we’re done.

To leave a comment for the author, please follow the link and comment on their blog: Saturn Elephant.

R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you’re looking to post or find an R/data-science job.

Want to share your content on R-bloggers? click here if you have a blog, or here if you don’t.

Continue reading: An application of the resultant

Long-Term Implications and Future Developments

The author has announced updates for the qspray package and introduced a new package named resultant, both for the R programming language. This update and new package will provide a beneficial tool for the R community, streamlining the process of finding intersection points between two distinct algebraic curves. This functionality will, in the long term, simplify and speed up various mathematical and coding tasks.

As a future development, this technology could expand to more complex equations or higher-dimensional forms. Further development of code and packages specialised in algebraic computations is also an area to anticipate.

Actionable Advice

For developers and users who frequently work with mathematical computations and R language for data analysis or modelling tasks, getting familiar with these packages could improve efficiency and productivity. Both ‘qspray’ and ‘resultant’ packages can streamline the procedure of finding intersection points, which are often used in solving complex mathematical equations or optimization problems.

Learning how to use qspray and resultant

  • Using qspray and resultant packages as demonstrated in the article could become a regular part of your R programming toolkit. Take the time to understand the example given, and apply it to your scenarios.
  • Continuously monitor the updates and improvements of these two packages. The developers often introduce significant enhancements in new releases.
  • Reach out to the developer community if you have questions or need advice. The R programming community is typically very supportive and can provide valuable guidance.

Long-term strategy for leveraging resultant and qspray

  • Consider training sessions or workshops to help your team gain a deep understanding and hands-on experience with these packages. This will improve overall team productivity and efficiency when working with R.
  • Keep an eye on similar packages or developments. The field of R packages for mathematical computations is fast evolving, and continual learning is required to keep up to date.

In conclusion, these packages are set to make a meaningful contribution to the R programming environment. Individuals and teams who leverage these packages early could see significant long-term benefits in terms of efficiency and productivity.

Read the original article

“Empower Your AI: Running Large Language Models Locally”

“Empower Your AI: Running Large Language Models Locally”

by | Mar 15, 2024 | DS Articles

Run large language models on your local PC for customized AI capabilities with more control, privacy, and personalization.

Implications and Future Developments of Running Large Language Models Locally

Bringing the utilization of large language models directly to local PCs offers a hefty arena for the advancement of artificial intelligence (AI) technology. This practice not only raises noteworthy benefits for individual users and businesses alike, but also significant challenges and considerations for the future of AI.

Advantages and Long-Term Implications

The shift to localhost usage has a profound impact on data privacy, personalization, and control over AI applications. By running large language models locally, users can ensure data sensitivity as it remains on the device, eliminating risks associated with data transmission to servers.

Secondly, individuals seize greater control over the model customization as per their unique requirements. Local PCs’ running AI language models will host dynamic, personalized, and potentially superior AI functionalities. Innovations in this domain could provoke a transformative leap in the way users traditionally interact with AI.

Future trends in AI might see an exponential growth in AI literacy as users become more acquainted with managing large language models on their PCs. It could also enhance users’ ability to craft AI applications better fit to solve intricate problems in their respective fields.

Challenges and Considerations

While utilizing local PCs’ presents advantages, it also implores significant concerns. One of the most notable is the substantial computational power and storage that large language models require, making it a considerable challenge for average PCs.

The potential incapacity of many devices to manage such power-hungry applications remains a potential roadblock. However, the advancement in hardware technology and efficiency optimization of AI algorithms can mitigate this downside.

Actionable Advice

To harness the potential of running large language models on local PCs, here are some pieces of advice:

  1. Invest in Hardware Upgrades: Purchasing a PC with advanced computational capabilities will be beneficial. It doesn’t necessarily need to be the most expensive model in the market, but one that correctly aligns with your particular AI requirements.
  2. Understand Your Needs: Since AI models can be customized as per individual requirements, it’s crucial to understand the precise functionalities and features you need from your AI applications. This understanding will help set your configurations and streamline your AI usage efficiently.
  3. Learn and Adapt: As the field of AI evolves, it’s essential to be receptive and adaptive to the new developments in executing large language operations locally.
  4. Secure Your Data: Although data is more secure on a local PC compared to the cloud, there’s still a necessity to adopt robust security measures to protect sensitive information.

In conclusion, the shift towards running large language models on local PCs poses both exciting opportunities and challenges. While we anticipate the scalability in terms of privacy and customization, preparing for the required computational needs and advancing our AI literacy will be key to adapting to this impending AI age.

Read the original article

by | Mar 15, 2024 | DS Articles

Artificial Intelligence is one of the biggest technological waves that has hit the world of technology. With the growing demand for artificial intelligence, several individuals are considering artificial intelligence career option.

Long-term Implications of Artificial Intelligence

Artificial Intelligence (AI) has emerged as one of the most significant technological revolutions in recent times. Its growing demand has not only transformed various industrial sectors but also greatly influenced career trends among individuals aspiring to build paths in technology. Given its tremendous potential, the future of AI holds a plethora of opportunities and challenges.

The Sphere of Influence for AI

AI’s influence is transcending boundaries, impacting several sectors. From healthcare to transportation, education to entertainment, AI is set to transform conventional practices, propel innovations, and redefine standards of operation. As these sectors increasingly integrate AI into their operations, this will engender a surge in demand for AI specialists to drive these innovations.

AI’s Stature in Job Market

As a clear indicator of its burgeoning demand, AI has rapidly grown into a substantial career option. Frost & Sullivan’s report predicts that AI will possibly create around trillion of economic value across various industries globally by the year 2035, indicating the vast expanse of opportunities that it holds for professionals.

Possible Future Developments in AI

The future landscape of AI appears extremely promising with several advancements lined up in the pipeline. AI could potentially transform our lives in ways we can hardly imagine today, thereby unlocking new frontiers for human endeavor.

Advancements in AI Technologies

  • AI-Powered Automation: Future AI developments point toward more intricate automation systems which could perform tasks more efficiently and independently, thereby reducing human intervention.
  • AI in Healthcare Diagnostics: AI’s critical role in enhancing healthcare diagnostics is increasingly being recognized. Future advancements may include AI systems capable of diagnosing diseases with a degree of accuracy surpassing human expertise.
  • AI for Customized Experience: AI is expected to personalize the user experience in diverse settings, such as online retail, entertainment and education, lending a tailored approach to customer engagement.

Actionable Advice

Given the influence and growth of AI, individuals considering AI as a career option should take note of the following points:

  1. Acquire Core Skills: Building solid foundations in Machine Learning, Neural Networks, Robotics, among others, is highly recommended.
  2. Continuous Learning: AI is constantly evolving. It is crucial to stay updated with latest trends and advancements.
  3. Practical Exposure: Getting hands-on experience is equally important. Try projects that can provide practical experience on AI tools and applications.
  4. Networking: Connect with professionals in the field. The more successful people you know in AI, the more opportunities you will have.

AI’s impact in the technological sphere is beyond our imagination. Its influence will continue to surge, showcasing an evident shift in the way we will envisage and actualize technological developments, thereby presenting staggering opportunities for those keen on a career in the AI domain.

Read the original article

Lissajous Animations for Pi Day

Lissajous Animations for Pi Day

by | Mar 14, 2024 | DS Articles

[This article was first published on R on kieranhealy.org, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

Want to share your content on R-bloggers? click here if you have a blog, or here if you don’t.

Some Lissajous animations for Pi Day. Made with R, ggplot, and gganimate.

And the really not very efficient code that made them:

r
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76

library(tidyverse)
library(gganimate)
library(transformr)

df_base <- tibble(
  id = seq(1, 1000, 1),
  t_vals = seq(0, 2 * pi, length.out = 1000))


circles <- function(t) {
  x01 <- cos(t * 1)
  y01 <- sin(t * 1)

  x02 <- cos(t * 2)
  y02 <- sin(t * 2)

  x03 <- cos(t * 3)
  y03 <- sin(t * 3)

  x04 <- cos(t * 4)
  y04 <- sin(t * 4)

  x05 <- cos(t * 5)
  y05 <- sin(t * 5)

  x06 <- cos(t * 6)
  y06 <- sin(t * 6)

  x07 <- cos(t * 7)
  y07 <- sin(t * 7)

  x08 <- cos(t * 8)
  y08 <- sin(t * 8)

  x09 <- cos(t * 9)
  y09 <- sin(t * 9)

  x10 <- cos(t * 10)
  y10 <- sin(t * 10)


  tibble(
    tick = seq_along(t),
    x01, x02, x03, x04, x05, x06, x07, x08, x09, x10,
    y01, y02, y03, y04, y05, y06, y07, y08, y09, y10
    )
}

df_out <- circles(t = df_base$t_vals)

df <- bind_cols(df_base, df_out) |>
  select(id, tick, everything()) |>
  pivot_longer(x01:x10, names_to = "x_group", values_to = "x") |>
  pivot_longer(y01:y10, names_to = "y_group", values_to = "y") |>
  mutate(x_group = str_remove(x_group, "x"),
         y_group = str_remove(y_group, "y")) |>
  unite("group_id", x_group, y_group, remove = FALSE)

out <- df |>
  ggplot(aes(x = x, y = y, color = group_id, group = group_id)) +
  geom_point(size = 3) +
  geom_path() +
  facet_grid(x_group ~ y_group) +
  coord_equal() +
  guides(color = "none") +
  theme_void() +
  transition_reveal(tick) +
  ease_aes("linear")



animate(out, duration = 30, fps = 24, height = 1080, width = 1080,
        renderer = ffmpeg_renderer())

anim_save(filename = "lissajous-fixed-lg-2.webm",
          height = 1080, width = 1080)
To leave a comment for the author, please follow the link and comment on their blog: R on kieranhealy.org.

R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you’re looking to post or find an R/data-science job.

Want to share your content on R-bloggers? click here if you have a blog, or here if you don’t.

Continue reading: Pi Day Circles

Implications and Future Developments Based On Lissajous Animations Made with R

The key point in the text is the usage of R statistical programming language, combined with libraries ggplot and gganimate, to create Lissajous animations. The importance of this development lies in the widening potential applications of these animations and the ongoing innovation within the field.

Long-Term Implications

Lissajous curves, visual representations of complex harmonic motion, have a wide array of applications – from physics and astronomy to music and digital signal processing. The increasing integration of the R language and its graphic capabilities signifies a huge shift in how these complex mathematical concepts can be visually represented, manipulated, and understood. This could herald a new era in the analysis and visualisation of complex data, particularly as it becomes more integral to numerous fields – from science to finance.

Future Developments

This trend of combining data visualisation and computation in one efficient language such as R is likely to persist, if not accelerate, in the future. With the rising interest and application of mathematical modeling in various sectors – including healthcare, energy, and the environment – the demand for accessible, user-friendly, and efficiently coded visualization tools can only grow.

Actionable Advice

  • Invest in R and its libraries: Staying on top of developments in this field will require a proficient understanding of R and its various libraries, such as ggplot and gganimate.
  • Prioritize Efficiency: The code used in the referenced text is self-admittedly not very efficient. Consequently, a focus should be placed on finding ways to improve its efficiency.
  • Keep an eye on developing trends: The merging of computation and visualisation holds promising potential for the future. By staying informed about advancements in the R community and the wider field, you can ensure you’re prepared to leverage new tools and methods as they emerge.

“The intersection between computation and visualization are where the most exciting developments are happening. Understanding and leveraging these tools today will give you a significant advantage tomorrow.”

Read the original article

“Mastering AI Application Development with Python and Tools”

“Mastering AI Application Development with Python and Tools”

by | Mar 14, 2024 | DS Articles

Explore the fundamental steps for creating a successful AI Application with Python and other tools.

Understanding the Long-Term Implications of AI Application Development with Python

In the field of Artificial Intelligence, creating effective and efficient applications requires a deep understanding of certain programming languages, like Python, along with various other tools. The potential implications and future developments of this field are expansive and filled with opportunities. Here, we delve into those aspects and provide actionable advice for those looking to optimize their AI application development process with Python.

Long-Term Implications

The potential long-term implications of creating successful AI applications are profound. As technology continues to evolve, so do the opportunities for implementing AI in various industries, from healthcare to agriculture, to finance, and beyond.

“Creating successful AI applications has the potential to revolutionize industries, drive innovation, and create efficiencies that can lead to significant cost savings and improved customer service.”

However, there is potential for challenges as well, particularly regarding ethics and data privacy. Striking a balance between leveraging AI’s capabilities and maintaining ethical practices and privacy safeguards will be crucial.

Future Developments

As technology advances, future developments in AI application creation will likely include an increased use of machine learning, deep learning, and other subfields of artificial intelligence. The increasing proficiency of these technologies will enable more sophisticated, adaptable, and intelligent applications.

Furthermore, with Python’s reputation for simplicity and readability, this programming language will remain integral to the development of AI applications. Given this, understanding Python will continue to be an essential skill for successful AI developers.

Actionable Advice

  1. Stay Current: Technology, and especially AI, evolves rapidly. It’s vital to stay updated with the latest developments and tools to create effective AI applications.
  2. Master Python: If you’re serious about AI application development, mastering Python is necessary. This language stands out for its simplicity and readability, making it ideal for AI development.
  3. Consider Ethics & Privacy: In the age of data-driven solutions, considering the ethical implications and privacy concerns of your AI applications is of utmost importance. Strive to create applications that are user-friendly, transparent, and respect user privacy.
  4. Embrace Collaboration: The complex nature of AI application development means collaboration is key. Online communities, open-source projects, and collaboration with professionals in the field can help you navigate through challenges and enhance your AI creations.

The future of AI application development promises a world filled with expansive opportunities and potential challenges. With Python at the core, a diligent focus on ethics and privacy, and an intentional commitment to staying current, you can navigate this space with confidence and success.

Read the original article

by | Mar 14, 2024 | DS Articles

Discover how API management tools drive digital transformation, enhancing security, innovation, and seamless integration for success.

The Role of API Management Tools in Digital Transformation

In today’s digital world, API (Application Programming Interface) management tools play a crucial role in enhancing security, driving innovation, and enabling seamless integration. Their potential to drive digital transformation cannot be overstated. But what are the long-term implications of these tools? And how could they shape the future?

Long-term Implications

Improved Business Efficiency

API management tools improve business efficiency by automating processes, reducing manual errors, and speeding up service delivery. With APIs connecting and integrating various digital assets, businesses are more capable of optimizing their operations.

Enhanced Security

APIs provide a secure way of transmitting data between different software applications. Strong API management tools thus present potential long-term benefits in guarding businesses against cybersecurity threats, safeguarding both enterprises and consumer information.

Foster Innovation

The use of APIs allows developers to create new software applications more quickly and efficiently. Over time, this fuels creativity and innovation within a company, helping to keep it competitive within its industry.

Possible Future Developments

API management tools are constantly evolving, opening up new possibilities for the future. Advanced API design, applications in emerging technologies, and business-specific APIs could all be part of this evolution.

Advanced API design

Future API tools might incorporate advanced design elements, making it easier for developers to work with complex data structures and software architectures. This could drive a faster and more efficient application development process.

Emerging Technologies

APIs could play an important part in the rise of emerging technologies such as AI and machine learning, IoT, and blockchain. By making it easier to integrate these technologies into existing systems, APIs could bring about a new era of digital transformation.

Business-specific APIs

In the future, there could be an increase in APIs that are specifically designed to meet the needs of individual businesses. These could be tailored to address unique challenges and opportunities within specific sectors, providing a level of customization that is currently rare.

Actionable Advice

  1. Invest in API Management: Given the multiple benefits of API management tools, businesses should consider investing in high-quality options that meet their specific needs.
  2. Consider API Security: As APIs become more central to business operations, their security should be a key concern. Businesses should ensure that their API management tools have robust security features.
  3. Stay Updated: The world of APIs is always evolving, so it’s essential to stay updated with the latest tools and trends in the field.
  4. Plan for Customization: With the possibility of more tailored APIs in the future, businesses should be prepared to take advantage of these personalized solutions when they become available.

“The potential of API management tools to drive digital transformation across industries is vast. It’s crucial for businesses to stay ahead of the curve and leverage these tools for growth.”

Read the original article