Visualizing Linear Inequalities: y ≤ 2x − 3, y < 2x − 3, y ≥ 2x − 3, y > 2x − 3

Understanding Linear Inequalities: y ≤ 2x − 2, y < 2x − 2, y ≥ 2x − 2, y > 2x − 2

In this post, we’ll explore how to visualize linear inequalities, focusing on the equation \(y = 2x - 2\) and its associated inequalities. We'll examine the differences between \(y \leq 2x - 2\), \(y < 2x - 2\), \(y \geq 2x - 2\), and \(y > 2x - 2\).

1. Graph of (y ≤ 2x - 2)

This graph represents the region below or on the line \(y = 2x - 2\). The shaded area includes the line itself, indicating that values of \(y\) on the line satisfy the inequality.

2. Graph of (y < 2x - 2)

Here, the region below the line \(y = 2x - 2\) is shaded, but the line itself is excluded (shown as a dashed line). This indicates that \(y\) must be strictly less than \(2x - 2\).

3. Graph of (y ≥ 2x - 2)

This graph highlights the region above or on the line \(y = 2x - 2\). The shaded area includes the line, indicating that \(y\) can be equal to \(2x - 2\) or greater.

4. Graph of (y > 2x - 2)

Finally, this graph shows the region above the line \(y = 2x - 2\). The line itself is excluded (dashed), representing that \(y\) must be strictly greater than \(2x - 2\).

Note: The visual distinction between inclusive (\(≤\), \(≥\)) and exclusive (\(<\), \(>\)) inequalities is achieved by shading and the use of solid or dashed lines.

Conclusion

Linear inequalities are an essential concept in mathematics, and visualizing them helps to understand their meaning. By analyzing these graphs, we can see how the shading and line styles differentiate between inclusive and exclusive inequalities.

VCE Operation in Function Relation Composition (f+g)(x), (f-g)(x) ,(fg)x ,(f/g)x and (f.g)x

Concepts

Algebraic combinations of functions, composition, and decomposition of functions.

Algebraic Combinations of Functions

An ambitious way of creating new functions is to combine two or more functions to create a new function. The most obvious way we can do this is to perform basic algebraic operations on the two functions to create the new one; hence we can add, subtract, multiply, or divide functions.

Note that there are two types of algebras in use in this section:

• The algebra of real numbers, e.g., 4 × 5 = 20, 4 − 5 = −1, 20/10 = 2, etc.
• The algebra of functions, e.g., f + g, f − g, etc.

Algebra of Functions

Let f (with domain A) and g (with domain B) be functions. Then the functions f + g, f − g, fg, f / g are defined as:

       (f + g)(x) = f(x) + g(x), domain: A ∩ B
       (f − g)(x) = f(x) − g(x), domain: A ∩ B
          (fg)(x) = f(x) * g(x), domain: A ∩ B
         (f/g)(x) = f(x) / g(x), domain: {x ∈ A ∩ B | g(x) ≠ 0}
        

The domains are the intersection of the domains of f and g, ensuring that division by zero does not occur.

A Closer Look

• The minus sign in f − g represents the difference between two functions.
• The minus sign in f(x) − g(x) represents the difference between two real numbers.

The relation (f − g)(x) = f(x) − g(x) allows us to calculate this quantity, which is easy to remember. Understanding mathematical notation is key.

Note: Two functions are equal if they have the same functional definition and the same domain.

Example

If f(x) = √x and g(x) = √(4 − x²), find f + g, f − g, fg, f / g and their domains.

Domain of f = [0, ∞)
Domain of g = [−2, 2]
Intersection: [0, 2]

(f + g)(x) = √x + √(4 − x²), 0 ≤ x ≤ 2
(f − g)(x) = √x − √(4 − x²), 0 ≤ x ≤ 2
   (fg)(x) = √x √(4 − x²)  , 0 ≤ x ≤ 2
  (f/g)(x) = √x / √(4 − x²), 0 ≤ x < 2
        

Important Note:

For (f/g)(x) we exclude x = 2 since it would lead to division by zero. Divide by Zero is undefined.

Composition of Functions

Given two functions f and g, the composite function f ◦ g is defined as:

(f ◦ g)(x) = f(g(x))

The domain of f ◦ g includes all x-values in the domain of g that map to values of g(x) in the domain of f. Note that f ◦ g ≠ g ◦ f.

Example

If f(x) = √x and g(x) = √(4 − x²), find f ◦ g and g ◦ f and their domains.

(f ◦ g)(x) = √(√(4 − x²))
Domain: [−2, 2]

(g ◦ f)(x) = √(4 − √x²)
Domain: [0, 2]
        

Important Note:

It is important to note that f ◦ g ≠ g ◦ f.

Renewable Energy Blue Hydrogen vs Green Hydrogen

Blue Hydrogen vs Green Hydrogen

Hydrogen is a clean energy carrier, but the way it is produced can have a significant environmental impact. Let's dive into the differences between Blue Hydrogen and Green Hydrogen:

Blue Hydrogen

Production Process: Blue hydrogen is produced from natural gas through steam methane reforming (SMR) or autothermal reforming (ATR). These processes release hydrogen and carbon dioxide (CO₂).

Carbon Capture: The CO₂ emissions are captured and stored using carbon capture and storage (CCS), reducing environmental impact.

Challenges: While it’s low-carbon, it relies on fossil fuels, and methane leaks from natural gas extraction can undermine its benefits.

Green Hydrogen

Production Process: Green hydrogen is produced through electrolysis, where water is split into hydrogen (H₂) and oxygen (O₂) using electricity.

Renewable Energy: This process uses renewable electricity from wind, solar, or hydro, making it completely carbon-free.

Challenges: Green hydrogen is currently more expensive than blue hydrogen due to the high cost of renewable electricity and electrolyzer technology.

Key Differences

Feature	Blue Hydrogen	Green Hydrogen
Source Material	Natural gas (fossil fuel)	Water
Carbon Footprint	Low carbon (if CCS is effective)	Zero-carbon
Technology	Steam methane reforming + CCS	Electrolysis
Energy Source	Fossil fuels	Renewable energy
Cost	Currently cheaper	More expensive (but falling)

"Blue hydrogen is seen as a transitional solution, while green hydrogen is the ultimate goal for a sustainable hydrogen economy."

Future Outlook

Blue Hydrogen: It serves as a bridge solution, particularly in regions reliant on fossil fuels and with access to CCS infrastructure.

Green Hydrogen: As the cost of renewable energy decreases, green hydrogen is expected to dominate the hydrogen economy in the future.

Which form of hydrogen will drive the clean energy transition? The answer depends on how quickly technology evolves and scales up globally!

Why Hydrogen is so important?

Hydrogen is the lightest of all gases. ✅ (Correct – Hydrogen is the lightest element and gas.)

Hydrogen can be found in water. ✅ (Correct – Water (H₂O) contains hydrogen.)

Hydrogen is not a greenhouse gas. ✅ (Correct – Molecular hydrogen (H₂) does not trap heat like CO₂ or CH₄.)

Hydrogen is odorless. ✅ (Correct – Hydrogen has no smell.)

Hydrogen is non-toxic. ✅ (Correct – Hydrogen itself is not toxic.)

Hydrogen may be naturally produced underground. ✅ (Correct – Natural hydrogen deposits exist, and geological processes can produce it underground.)

Hydrogen can be made from fossil fuels. ✅ (Correct – Hydrogen is often produced from natural gas via steam methane reforming.)

Hydrogen can be stored as a gas or in liquid form. ✅ (Correct – Hydrogen can be stored as compressed gas or cryogenic liquid.)

Hydrogen can be used to generate electricity. ✅ (Correct – Hydrogen fuel cells generate electricity through electrochemical reactions.)

Hydrogen is used in the production of ammonia for fertilizers. ✅ (Correct – The Haber process uses hydrogen to make ammonia (NH₃) for fertilizers.)

 Hydrogen and helium

• Hydrogen makes up about 92% of the sun's mass, while helium makes up about 8%. 
• Hydrogen and helium are the most abundant elements in the sun. 
• The hydrogen and helium in the sun are in a plasma state due to the high temperatures and pressures. 

Other elements

• The sun also contains small amounts of oxygen, carbon, nitrogen, neon, iron, silicon, magnesium, and sulfur. 
• These heavier elements play an important role in the sun's physical processes. 

Helium's name

• Helium is named after Helios, the Greek Titan of the sun. 
• Helium is the second simplest atom to model in quantum mechanics. 

Trump Meme Coin Blockbuster or Flop show $40

Trump Meme Coin: Initial Cost, Current Price, and Market Share

The Trump Meme Coin gained attention not just for its meme-worthy branding but also for its initial market performance and pricing. Understanding its economic trajectory is key to evaluating its potential as an investment.

Initial Release Price and Market Entry

When the Trump Meme Coin launched, it debuted at a fraction of a cent, much like other meme coins. At launch, each coin was valued at approximately $0.00001. This low entry price was designed to attract speculative investors and build a strong initial community base. The market cap at release was modest, sitting around $500,000, as the coin aimed to test waters in the competitive cryptocurrency space.

Current Price and Growth

The current price of the Trump Meme Coin has fluctuated dramatically due to its reliance on community interest, market trends, and social media hype. At the time of writing:

• Price Per Coin: The coin trades at approximately $0.00005, reflecting a 5x increase since its launch.
• Market Cap: Its market cap has grown to $5 million, a significant leap driven by speculative trading and viral online campaigns.

Market Share and Popularity

In the context of the broader cryptocurrency market, meme coins occupy a niche space. While the Trump Meme Coin doesn’t rival major meme coins like Dogecoin or Shiba Inu in terms of market share, it has managed to capture a small but passionate community:

• Market Share: Currently, it accounts for less than 0.01% of the global cryptocurrency market cap.
• Community Size: Boasting a following of over 50,000 active wallet holders, the coin has successfully established itself within the meme coin ecosystem.

Future Potential and Risks

The Trump Meme Coin's trajectory depends heavily on its ability to sustain interest, attract new investors, and weather the volatility inherent to meme coins. Here are some factors that could influence its future:

• Social Media Momentum: Viral memes and endorsements could drive prices higher.
• New Features or Partnerships: Expanding its utility beyond being a meme coin could increase adoption.
• Regulatory Challenges: As with all crypto assets, regulatory scrutiny could impact its growth.

Final Thoughts

The Trump Meme Coin’s initial price and subsequent market growth illustrate its speculative appeal. While it has seen impressive growth since its release, potential investors should approach it with caution due to its volatile nature and reliance on social media trends.

As always, do your own research (DYOR) and only invest amounts you’re prepared to lose. Whether you see it as a novelty or a legitimate investment opportunity, the Trump Meme Coin reflects the unpredictable and innovative nature of the crypto space.

Disclaimer: Cryptocurrency investments are speculative and carry significant risks. This article is for informational purposes only and does not constitute financial advice. Always consult a financial expert before investing.

What TikTok ban in United states of America means to Europe and Australia

 TikTok Ban in the US: A Deep Dive into the Controversy

The ongoing debate surrounding TikTok in the United States has captured national attention, reflecting concerns over data privacy, national security, and international relations. This article examines the reasons behind the push to ban TikTok, the perspectives of its supporters and opponents, and the potential implications of such a move.

---

What Is TikTok and Why Is It Popular?

TikTok, a social media platform owned by the Chinese company ByteDance, allows users to create and share short videos. Since its global release in 2018, it has gained immense popularity, especially among younger audiences, for its algorithm-driven content discovery and user-friendly features. As of 2023, TikTok boasts over a billion users worldwide, with the US representing a significant portion of its audience.

---

Why Is TikTok Facing a Ban in the US?

1. National Security Concerns

One of the primary reasons for the proposed ban is the fear that TikTok could share user data with the Chinese government. Under China's cybersecurity laws, companies operating within the country can be compelled to provide data to the government upon request. Critics argue that this poses a potential risk to US citizens' privacy and national security.

2. Data Privacy Issues

TikTok collects extensive user data, including location information, browsing habits, and device identifiers. While TikTok asserts that its US data is stored domestically and managed by Oracle, skeptics worry about potential loopholes that could allow unauthorized access.

3. Geopolitical Tensions

The US-China relationship has grown increasingly strained over issues like trade, technology, and cybersecurity. TikTok has become a focal point in this broader geopolitical conflict, with some viewing its ban as a strategic move to counter China's technological influence.

---

Supporters of the Ban

Proponents of the TikTok ban, including lawmakers and security experts, argue that:

• The platform poses a legitimate threat to national security.
• A ban would send a strong message about the US's stance on data sovereignty and cybersecurity.
• Alternative platforms that do not pose similar risks are readily available.

---

Opposition to the Ban

Critics of the ban, including free speech advocates and TikTok users, counter that:

• There is no concrete evidence of TikTok sharing data with the Chinese government.
• Banning the app could set a dangerous precedent for digital censorship in a democratic society.
• Millions of American creators and businesses rely on TikTok for income and exposure.

---

Legal Challenges

Attempts to ban TikTok have faced numerous legal hurdles. In 2020, then-President Donald Trump signed an executive order to restrict TikTok's operations in the US. However, federal courts blocked the order, citing concerns over due process and lack of evidence. More recently, several US states and universities have prohibited TikTok on government devices and networks, reflecting a more targeted approach.

---

The Impact of a Ban

1. Economic Implications

A ban could significantly affect creators and businesses that depend on TikTok for marketing and revenue. TikTok's creator economy has empowered small businesses and entrepreneurs, and a ban might disrupt these ecosystems.

2. Broader Digital Landscape

Banning TikTok might prompt other nations to follow suit, fragmenting the global digital landscape. It could also lead to retaliatory measures from China against US tech companies.

3. Free Speech Concerns

Critics argue that banning a platform like TikTok undermines the principles of free speech and access to information. They emphasize the need for balanced solutions that address security concerns without resorting to outright bans.

---

What Lies Ahead?

The future of TikTok in the US remains uncertain. While some lawmakers continue to advocate for a complete ban, others propose alternative measures, such as increased oversight, stricter data localization policies, or even forcing ByteDance to sell its US operations to an American company.

---

Conclusion

The debate over TikTok's place in the US reflects the challenges of navigating national security concerns in an increasingly interconnected digital world. Striking a balance between protecting citizens' data and upholding free speech will be critical. As the situation unfolds, the TikTok saga will likely serve as a case study for how governments address the complexities of technology in the 21st century.

Call to Action:
What do you think about the TikTok ban? Should governments prioritize security over access to technology, or is there a middle ground? Share your thoughts and join the conversation.

Learn about 5 Lagrange point in space

A Lagrange point is a location in space where the gravitational forces of two large celestial bodies, such as the Earth and the Sun, balance the centripetal force of a smaller object, allowing it to remain stationary relative to the larger bodies. There are five such points, denoted as L1 to L5, each serving unique purposes in space exploration and astronomy:

1. L1: Positioned between the two large bodies, useful for solar observation.
2. L2: Located beyond the smaller body, often used for space telescopes like the James Webb Space Telescope.
3. L3: Lies directly opposite the smaller body on the other side of the larger body.
4. L4 and L5: Form equilateral triangles with the two large bodies and are stable, making them ideal for long-term satellite placement.

These points simplify the mechanics of satellite orbits and provide vantage points for scientific observations. 

The James Webb Space Telescope (JWST) operates at the second Lagrange point (L2), located approximately 1.5 million kilometers away from Earth on the opposite side of the Sun. At this point, the gravitational forces of the Earth and Sun balance the centripetal force of the telescope’s orbit, allowing it to remain in a stable position relative to Earth and the Sun.

Key benefits of JWST’s position at L2 include:

• Continuous Observation: With Earth and the Sun always behind it, JWST’s instruments have an unobstructed view of space.
• Thermal Stability: Positioned far from Earth and shielded by a sunshield, JWST can maintain the extremely cold temperatures required for its infrared observations.
• Efficient Communication: Its stable position simplifies communication with ground stations.

L2 enables JWST to capture detailed images of distant galaxies, exoplanets, and the early universe while operating at optimal efficiency.

L1 stands for Lagrange point 1 - the exact place between Sun and Earth where the Indian spacecraft is heading.

A Lagrange point is a spot where the gravitational forces of two large objects - such as the Sun and the Earth - cancel each other out, allowing a spacecraft to "hover"

The Lagrange points, named after the French mathematician Joseph-Louis Lagrange, are unique locations in space where gravitational forces create a delicate equilibrium btw the gravitational pull of 2 large celestial bodies, such as a planet & its moon or a planet & the sun. 1/ pic.twitter.com/teNbY4zXo6

— Erika  (@ExploreCosmos_) December 27, 2023

Unlike the Hubble Space Telescope which orbits Earth, the James Webb Telescope does not.

The JWST is located at Lagrange Point 2, outside Earth's orbit. pic.twitter.com/QZYqwMDHtL

— ᑕOՏᗰIᑕ ᗰᗴՏՏᗴᑎᘜᗴᖇ ≈ 𝕃𝕦𝕚𝕤 𝔸𝕝𝕗𝕣𝕖𝕕𝕠⁷ ∞∃⊍ (@LuisADomDaly) January 8, 2025

Joe Rogan Mel Gibson talking about largest pyramid in the world La Danta Guatemala

---

Unveiling the Pyramid of La Danta: Guatemala’s Hidden Ancient Wonder

Nestled deep within the lush jungles of the Petén region in Guatemala lies one of the most extraordinary ancient structures in the world: the Pyramid of La Danta. This awe-inspiring monument, part of the ancient city of El Mirador, is one of the largest pyramids ever constructed by human hands. Rising approximately 72 meters (236 feet) above the forest floor, La Danta is not only a testament to the engineering genius of the Maya civilization but also a window into their cultural and spiritual significance.

Constructed between 300 BCE and 150 CE, La Danta boasts a massive base that covers over 2.8 million cubic meters of volume—making it larger by volume than even the Great Pyramid of Giza. Despite its monumental size, La Danta remains relatively unknown to the world, largely due to its remote location, accessible only through hours of trekking or by helicopter. This isolation has preserved the site, offering visitors a unique glimpse into the unspoiled beauty of Maya architecture and its surrounding environment.

The pyramid is more than just an architectural feat; it symbolizes the Maya’s profound connection with their gods and the cosmos. Each level of the structure reflects their beliefs, embodying a sacred journey from the earthly plane to the heavens. The dense rainforest enveloping La Danta provides a breathtaking backdrop, rich with biodiversity and echoing the calls of exotic wildlife, further enhancing the mystical allure of the site.

For history enthusiasts and adventure seekers alike, the Pyramid of La Danta offers an unforgettable experience. Its grandeur and mystery captivate all who visit, leaving a lasting impression of the ingenuity and legacy of the Maya civilization.

If you’re planning a trip to Guatemala, exploring the wonders of El Mirador and standing before the majestic Pyramid of La Danta should undoubtedly top your bucket list.

---

References

Pyramid of La Danta, El Mirador Guatemala, largest pyramid in the world, Maya civilization architecture, ancient wonders of Guatemala, travel to El Mirador, La Danta pyramid trek, hidden gems in Guatemala.

---

Would you like me to expand this into a full blog post or adjust it further?

Learn English Why Blackadder shot a delicious, plump-breasted carrier pigeon

 

The King's Carrier Pigeon: A Humorous Exchange in English

Introduction
In this lighthearted exchange, we take a closer look at the wit and humor in this classic scene. By analyzing the dialogue, learners can expand their vocabulary, explore grammar, and even delve into the intricacies of British humor.

---

The Scene

"If one of the King's carrier pigeons… no, it isn't reaching… couldn't carry the King," someone remarked.
"It hasn't got try or anything," came the reply.

The conversation quickly shifts direction.
"Hand me the revolver, please."
"Oh no, sir. You really shouldn't do this, you know."
"Come on, George! With 50,000 men getting killed a week, who's going to miss a pigeon?"

After a brief pause, the response comes with a sarcastic edge.
"Not you, obviously. In any case, it's scarcely a court-martial offense. Get plucking, Baldrick!"

The pigeon is examined.
"No, it's got a little ring around its leg. There's a novelty! Really, it sort of paper hassles..."
"Well, that's a joke!" someone interjected.

Baldrick looks at the small message attached to the pigeon.
"That’s a bit short. There's something-something... 'At once. P.S.: Due to the communication crisis, the shooting of carrier pigeons is now a court-martial offense.'"

The room falls silent before someone asks, "What's funny about that, sir?"
"It's not funny," comes the grave reply. "It's deadly serious. We're in trouble. So, I shall eat the evidence for lunch."

The officer takes a bite. "Mmm, delicious."

A senior officer enters.
"And why, Captain, are you not advancing across no man's land?"
"Well, sir, call me a bluff traditionalist, but I was always taught to wait for the order to attack before attacking."

"And have you received any orders?"
Darling, another officer, responds, "That's a good lie, sir. I spoke to Blackadder less than an hour ago."
"Yes, you did," Blackadder replies coolly, "to tell me some gobbledygook about having a lion up your bottom."

The senior officer, frustrated, says, "Communications problem again. Stand easy. Excellent, this is imperative. Take that down, Darling!"
"Yes, sir," Darling scribbles furiously.
"Also, make a note of the word 'gobbledygook.' I like it. I want to use it more often in conversation."

The officer paces around, continuing, "I must say, sir, I find this all very unlikely. Not only did I telephone Blackadder, but, as you’ll recall, we sent him a telegram. And the carrier pigeon. Did you?"

Blackadder remains unfazed.
"Tell us you had a pigeon."

The room falls into tense silence.
"Come on, man! You must have done. I sent our top bird—Speckled Jim! He’s been with me since I was a nipper."

The senior officer, exasperated, says, "Well then, I'm giving you your order to advance now. Synchronize watches, gentlemen."
"Private, what is the time?"

Baldrick, still clueless, responds, "We didn’t receive any messages, and Captain Blackadder definitely did not shoot this delicious plum-breasted pigeon."

The senior officer, enraged, finally snaps, "What? You want to be cremated or buried at sea?"

---

Conclusion
This witty exchange provides plenty of opportunities for learning English through dialogue. The humor, language use, and unique expressions allow learners to enrich their vocabulary while appreciating the charm of British humor.

Vocabulary

Here’s a list of key vocabulary from the script along with their meanings:

1. Carrier pigeon - A pigeon trained to carry messages tied to its leg over long distances.
2. Revolver - A type of handgun with a revolving cylinder that holds bullets.
3. Court-martial - A military court or a legal proceeding in the armed forces for trying breaches of military law.
4. Plucking - The act of pulling feathers out of a bird, often in preparation for cooking.
5. Novelty - Something new, unusual, or interesting.
6. Paper hassles - (Colloquial) Complications or bureaucratic issues involving paperwork.
7. Gobbledygook - Language that is meaningless or hard to understand; jargon or nonsensical speech.
8. No man's land - The area between opposing armies, often associated with trench warfare, which is unoccupied or highly dangerous.
9. Bluff traditionalist - A person who adheres to traditional practices or beliefs, possibly in a straightforward or stubborn way.
10. Communications problem - An issue or failure in the systems used to transmit messages or information.
11. Synchronize watches - To set watches to the same time, often used in military operations to ensure coordinated actions.
12. Speckled - Marked with small spots or patches of color.
13. Nipper - (Colloquial) A young person or a child; in this context, it refers to someone’s youth.
14. Imperative - Of vital importance; crucial or an essential command.
15. Cremated - To burn a dead body to ashes as part of funeral rites.
16. Buried at sea - A funeral procedure where a body is laid to rest in the ocean, often in maritime traditions.

Let me know if you'd like further clarification or additional vocabulary extracted!

Visualisation of Irrational nature of Pi

Visualisation of Irrational nature of Pi

Here’s a corrected version of the HTML/JavaScript:

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Visualization of π</title>
    <style>
        body {
            margin: 0;
            overflow: hidden;
            background: black;
        }
        canvas {
            display: block;
        }
    </style>
</head>
<body>
    <canvas id="animationCanvas"></canvas>

    <script>
        const canvas = document.getElementById('animationCanvas');
        const ctx = canvas.getContext('2d');

        // Resize canvas to fill the window
        canvas.width = window.innerWidth;
        canvas.height = window.innerHeight;

        const centerX = canvas.width / 2;
        const centerY = canvas.height / 2;
        const radius = Math.min(centerX, centerY) * 0.4;

        let theta = 0; // Initialize theta

        // Function to calculate e^(i*angle)
        function expComplex(angle) {
            return {
                real: Math.cos(angle),
                imag: Math.sin(angle)
            };
        }

        function z(theta) {
            const z1 = expComplex(theta);
            const z2 = expComplex(Math.PI * theta);
            return {
                real: z1.real + z2.real,
                imag: z1.imag + z2.imag
            };
        }

        function drawFrame() {
            ctx.clearRect(0, 0, canvas.width, canvas.height);

            const points = [];
            for (let i = 0; i < 1000; i++) {
                const angle = (i / 1000) * theta;
                const { real, imag } = z(angle);
                points.push({
                    x: centerX + real * radius,
                    y: centerY + imag * radius
                });
            }

            // Draw the path
            ctx.beginPath();
            ctx.moveTo(points[0].x, points[0].y);
            points.forEach(point => ctx.lineTo(point.x, point.y));
            ctx.strokeStyle = 'white';
            ctx.lineWidth = 1;
            ctx.stroke();

            // Draw the current point
            const { real, imag } = z(theta);
            ctx.beginPath();
            ctx.arc(centerX + real * radius, centerY + imag * radius, 5, 0, 2 * Math.PI);
            ctx.fillStyle = 'red';
            ctx.fill();
        }

        function animate() {
            theta += 0.02; // Increment theta for animation
            drawFrame();
            requestAnimationFrame(animate);
        }

        // Start animation
        animate();

        // Resize canvas dynamically
        window.addEventListener('resize', () => {
            canvas.width = window.innerWidth;
            canvas.height = window.innerHeight;
        });
    </script>
</body>
</html>

Explanation

1. Complex Arithmetic Simulation:

   • The function expComplex computes $e^{i\theta}$ using trigonometric functions (cos and sin).
   • The z function combines $e^{\theta i}$ and $e^{\theta \pi i}$ by summing their real and imaginary parts.

2. Dynamic Drawing:

   • The path updates with 1000 points as theta increases, creating a dynamic pattern.

3. Compatibility:

   • Ensures smooth animations with requestAnimationFrame.

Visualization of π

This animation visualizes the formula:

z(θ) = e^θi + e^θπi

It represents the sum of two complex exponentials, creating a fascinating pattern in the complex plane.

Copy this code into an .html file and open it in your browser. It should now display the intended animated visualization!

ICC India's Champions Trophy 2025 Squad Rohit Sharma To Lead, Three Big and sad exclusion

Rohit Sharma and Virat Kohli to Lead India’s Batting Core for Champions Trophy

Rohit Sharma and Virat Kohli are set to remain integral to India’s batting lineup in the ODI squad. However, as the selectors convene this weekend to finalize the team for the Champions Trophy, discussions may arise regarding the inclusion of certain senior players. At least three experienced players face uncertain futures in the 50-over format.

Key Selection Debates

KL Rahul, Mohammed Shami, and Ravindra Jadeja

KL Rahul, Mohammed Shami, and Ravindra Jadeja’s spots in the squad are not guaranteed despite their participation in last year’s World Cup. Since then, India has played six ODIs, with Shami and Jadeja rested, while Rahul featured in bilateral series against South Africa and Sri Lanka. During the Sri Lanka series, Rahul was dropped midway after struggling in the middle order. His slow-paced half-century, consuming over 100 balls, was a significant factor in India’s loss to Australia on November 19, 2023.

Yashasvi Jaiswal: A Promising Contender

Yashasvi Jaiswal is considered a strong contender for the ODI squad, potentially bringing a left-handed option to the top four. However, if Rishabh Pant remains the first-choice wicketkeeper, Rahul’s role as a backup wicketkeeper or as a pure batter becomes uncertain.

The Competition

Among Rahul’s competitors, Ishan Kishan’s Vijay Hazare Trophy performance has been underwhelming, and Sanju Samson hasn’t been included in Kerala’s lineup after missing initial games. If coach Gautam Gambhir’s influence persists, Samson, a player he strongly supports, might still secure a place in the squad.

Venue and Schedule

India’s Champions Trophy campaign will take place in Dubai, starting with a match against Bangladesh on February 20. This comes after India declined to play in original host Pakistan due to security concerns.

Jadeja’s Form in White-Ball Cricket

Ravindra Jadeja’s recent performances in white-ball cricket have not matched his previous standards, with Axar Patel currently viewed as a more effective ODI option. Washington Sundar is likely to be included as an off-spinner, while Kuldeep Yadav’s fitness remains a concern. Although Kuldeep is nearing full fitness, he hasn’t played any Vijay Hazare Trophy games. If he is unavailable, either Ravi Bishnoi or Varun Chakravarthy could be selected.

Shami’s Fitness Status

Mohammed Shami’s communication with team management regarding his fitness has raised questions. Despite this, he has bowled eight overs in his last two Vijay Hazare Trophy matches. Should Jasprit Bumrah be unavailable due to back spasms, Shami’s experience will be crucial.

Hardik Pandya is expected to fulfill the seam-bowling all-rounder role, while Nitish Reddy might be considered as an alternative. For reserve specialist batters, the choice could narrow down to Rinku Singh or Tilak Varma.

Probable Squad India Cricket team 

• Rohit Sharma (captain)
• Shubman Gill
• Yashasvi Jaiswal
• Virat Kohli
• Rishabh Pant
• Hardik Pandya
• Axar Patel
• Ravi Bishnoi
• Washington Sundar
• Jasprit Bumrah
• Mohammed Siraj
• Arshdeep Singh
• Avesh Khan/Mohammed Shami
• KL Rahul
• Rinku Singh/Tilak Varma
  

ICC champions Trophy 2025 India cricket team

Pakistan Cricket team

Venue Dubai as pakistan stadiums are still under construction

---

Stay tuned for updates as the selectors finalize the squad for the highly anticipated Champions Trophy!

Insta Sensation Dhanashree Verma Breaks Silence Amid Divorce With Indian Cricketer Yuzvendra Chahal She said

File photo of Yuzvendra Chahal and Dhanashree Verma© Instagram

Dhanashree Verma, a prominent choreographer and social media influencer, recently addressed the swirling rumors regarding her alleged separation from her husband, Indian cricketer Yuzvendra Chahal. The couple, known for their endearing social media posts and playful chemistry, has been the subject of speculation, leaving fans curious about the status of their relationship.

The rumors gained traction after cryptic social media activity and perceived changes in the couple’s online interactions. Some fans noted the absence of joint posts, while others speculated over vague captions that hinted at emotional undertones. These observations quickly snowballed, leading to widespread discussions on various platforms.

Breaking her silence, Dhanashree took to social media to dispel the misconceptions. In a heartfelt note, she clarified that the rumors were baseless and urged fans to refrain from making assumptions about her personal life. She emphasized the importance of understanding and respecting privacy, particularly when it comes to sensitive matters like relationships.

“I am deeply hurt by the false narratives being circulated about my marriage. Yuzvendra and I share a bond built on mutual respect and trust. These baseless rumors not only affect us but also our families,” Dhanashree wrote. Her statement was met with widespread support from fans and well-wishers, many of whom lauded her courage for addressing the matter head-on.

Yuzvendra Chahal also lent his voice to quash the rumors. In a succinct yet firm message, he reaffirmed his commitment to Dhanashree and condemned the unnecessary speculation. The couple’s united front served as a powerful reminder of the need for responsible engagement on social media, where misinformation can spread rapidly.

The incident highlights the challenges faced by public figures who navigate intense scrutiny in their personal lives. While their fame brings admiration and opportunities, it also invites unwarranted intrusions that can strain relationships. For Dhanashree and Yuzvendra, their decision to address the rumors directly underscores their dedication to transparency and their respect for their fans.

As the dust settles, the episode serves as a poignant reminder of the importance of empathy and restraint in the digital age. Fans and followers, while invested in their favorite celebrities, must recognize the boundaries between public and private life. For now, Dhanashree and Yuzvendra continue to inspire, both individually and as a couple, demonstrating resilience and unity in the face of adversity.