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What is Class Counter in Maths



In 2009 Dusa Macduff goes to a talk at the Numerical Sciences Exploration Foundation in Berkeley, California. The speaker was Katrin Wehrheim, who was then an associate teacher at the Massachusetts Establishment of Innovation. In his discourse, Wehrheim provoked the symplectic math local area to face blunders in basic strategies grew over 10 years sooner.  reviewed Macduff, who was one of Wehrheim’s doctoral proposal analysts.

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Barnard School mathematician Dusa Macduff battled for a really long time to address what she saw as holes in the underpinnings of symplectic calculation.

The test was private for Macduff. In 1999 he composed an overview article that depended on hazardous major methods by one more sets of mathematicians, Posse Liu and Pack Tian. Presently, after 10 years, Wehrheim was bringing up that Macduff’s paper – in the same way as other early papers in symplectic math – including Fukaya’s – had mistakes, specifically how to move from nearby to worldwide in the worldwide calculation of focuses. Subsequent to paying attention to Wehrheim, Macduff concluded that she would attempt to address any slip-ups.

“I had a terrible heart about what I composed in light of the fact that I knew some way or another it was not completely exact,” she said. “I commit errors, I comprehend individuals commit errors, yet on the off chance that I commit an error, I attempt to address it, and in the event that I can’t, it’s off-base.”

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Macduff and Wehrheim started work on a progression of papers that were brought up and fixed as missteps in Fukaya’s treatment of transversality. In 2012 Macduff and Wehrheim moved toward Fukaya with their interests. Following 16 years in which his work was ignored by the numerical local area, he was happy he was intrigued.

“In 2012 we got K. There was an unmistakable protest from Wehrheim. We were extremely glad to have this since it was the primary significant numerical reaction we got on our work.

To examine the protests, mathematicians framed a Google Gathering in mid 2012 that included Macduff, Wehrheim, Fukaya and Ono along with two of Fukaya’s later partners, Yong-gyun Gracious and Hiroshi Ohta. The conversation by and large follows this structure: Wehrheim and Macduff will bring up issues about Fukaya’s work. Fukaya and his associates would then compose long, itemized replies.

Whether those answers were agreeable relied upon who was understanding them. According to Fukaya’s perspective, his work on Qur’anic structures was finished and right all along. I don’t think there was anything missing,” he said.

Others clash. After the Google Gathering conversation finished, Fukaya and his associates posted a few papers on Quranic structures that together ran for in excess of 400 pages. Hofer feels that the length of Fukaya’s reactions is proof that Macduff and Wehrheim’s incitement was essential.

“By and large, [Fukaya’s approach] worked, however it required more clarification than was initially given,” he said. “I think Fukaya and Ono’s unique paper was a little north of 100 pages, and because of this conversation on the Google bunch they delivered a 270-page composition and a couple hundred pages making sense of the first outcomes. So certain. Obviously required explanation.”

Abujaid concurs that there was a slip-up in Fukaya’s unique work.Simultaneously, he feels that Qur’anic develops are by and large the correct method for managing transversality issues. He views the blunders in the 1996 paper as happening on the grounds that the symplectic math local area was not adequately evolved at an opportunity to audit the new work appropriately.

“The paper ought to have been refereed all the more cautiously. My perspective is that with a decent ref report of a few adjusts that the paper would have been faultless and everything looks great,” said Abujaid.

In August 2012, after a Google Gathering conversation, McDuff and Wehrheim posted an article they had started composition before the conversation definite ways of revising Fukaya’s methodology. He later refined and distributed that paper with two others, and plans to compose a fourth paper regarding the matter. In September 2012, Fukaya and his co-creators posted a portion of their reactions to the issues raised by Macduff and Wehrheim. To Fukaya, Macduff and Wehrheim’s papers didn’t propel this region essentially.

“It is my viewpoint that the papers he composed don’t contain new and significant thoughts. There is absolutely a few contrast from the previous papers of our own and others. Notwithstanding, the thing that matters is just on a little detail,” FukAya said in an email.

Hofer imagines that this understanding short sells the commitments of Macduff and Wehrheim. From his perspective, the pair accomplished something other than fix little specialized subtleties in Fukaya’s work – they tackled significant level issues with Fukaya’s methodology.

“They comprehended the various pieces well overall and how they cooperated, so you couldn’t simply say: ‘Here, assuming it’s risky, I fixed it locally,'” he said. “You should know where pretty much another issue would emerge. He grasped it at an exceptionally undeniable level.”

How mathematicians assess the significance of mistakes in Fukaya’s 1996 paper and the commitments Wehrheim and Macduff made to address them mirrors a polarity in the manner we ponder the act of science.

“Arithmetic has two ideas,” Abuzaid said. “Arithmetic is this way: the money of math is thought. Also, science is the cash of arithmetic is verification. It is hard for me to say which side individuals stand on. My own perspective is: overwhelmingly significant in science is thought, And there’s proof to ensure that contemplations don’t wander off.”

Fukaya is a geometer with a sense to think in overgeneralized terms. Conversely, Wehrheim is prepared in examination, a field known for its thorough regard for specialized subtleties. In a profile for the MIT site Ladies in Math, she expressed that in math, “we don’t compose great papers any longer,” and looked at mathematicians who don’t tell the subtleties of their work to climbers who arrive at the top. Mountains without leaving snares on the way. “Somebody with little preparation would have not a chance of following it without tracking down a way for themselves,” she said.

These contrasting assumptions for what considers an adequate measure of detail in a proof caused a lot of pressure in the symplectic calculation local area around the protests of Macduff and Wehrheim. Abuzaid contends that it is essential to be shrewd while bringing up botches in crafted by another mathematician, and Wehrheim might not have been strategic enough in this. “Assuming that you present it as: ‘All that has preceded us is off-base and presently we will offer the right response,’ that is probably going to set off some kind of issue of need claims,” he said. .

Wehrheim declined different solicitations to be evaluated for this article, saying she “needs to stay away from additional politicization of the subject.” In any case, Macduff feels that he and Wehrheim had no real option except to be enabled to bring up the defects in Fukaya’s work: it was the best way to definitely stand out.

“It resembles being an informant,” she said. out with you. what’s more, not from individuals who can be off-base.”

Notwithstanding who gets the credit for fixing the issues with Fukaya’s paper, they have been fixed. In the beyond couple of years, on account of arithmetic, the debate encompassing his work has been settled.

These issues were felt and ultimately fixed,” Eliashberg said. “Perhaps it superfluously makes a ton of energy in certain gatherings, however generally speaking I think everything was dealt with and things will go on.”


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Unveiling the Power of Binbex Exchange: A Comprehensive Overview




Binbex Exchange has emerged as a noteworthy player in this dynamic field, offering a range of features and services designed to meet the diverse needs of the crypto community. This comprehensive overview aims to delve into the key aspects of Binbex Exchange, exploring its history, features, security measures, trading options, and the unique offerings that set it apart in the competitive crypto exchange market.

 1: The Genesis of Binbex Exchange

1.1 Founding Story:
To understand Binbex’s journey, it’s essential to explore its founding story. The exchange was established in [year] by a team of seasoned professionals in the financial and technology sectors. The founders envisioned creating a platform that addresses the challenges faced by cryptocurrency traders and investors, focusing on reliability, security, and user experience.

1.2 Mission and Vision:
Binbex Exchange’s mission and vision statements are fundamental to its identity. This section will provide insights into the company’s overarching goals, emphasizing its commitment to fostering innovation, financial inclusion, and user empowerment in the cryptocurrency space.

2: Features and Services

2.1 User Interface and Experience:
One of Binbex’s standout features is its user-friendly interface. This chapter will explore the platform’s design, navigation, and overall user experience, shedding light on how Binbex ensures accessibility for both beginners and experienced traders.

2.2 Asset Coverage:
Binbex supports a diverse range of cryptocurrencies. This section will provide an exhaustive list of supported assets and delve into the exchange’s approach to adding new tokens and coins to its platform.

2.3 Trading Pairs and Liquidity:
The availability of trading pairs and liquidity is crucial for traders looking to execute orders seamlessly. This chapter will analyze Binbex’s trading pairs, liquidity measures, and the strategies employed to enhance the overall trading experience.

2.4 Advanced Trading Features:
For professional traders, advanced features are paramount. Binbex offers a suite of advanced trading tools, such as margin trading, futures contracts, and more. This section will provide an in-depth exploration of these features, highlighting their benefits and risks.

 3: Security Measures

3.1 Cold and Hot Wallets:
Security is a top priority for any cryptocurrency exchange. Binbex employs a combination of cold and hot wallets to safeguard user funds. This chapter will delve into the security measures in place, exploring how Binbex ensures the integrity of its users’ assets.

3.2 Two-Factor Authentication (2FA) and Biometrics:
To fortify user accounts, Binbex incorporates two-factor authentication and biometric verification. This section will examine these security layers and their effectiveness in preventing unauthorized access.

3.3 Regulatory Compliance:
Compliance with global regulations is essential for the longevity of any cryptocurrency exchange. Binbex’s adherence to regulatory standards will be discussed, offering insights into how the platform navigates the complex regulatory landscape.

4: Trading Experience and Tools

4.1 Order Types:
Binbex supports various order types to cater to the diverse trading strategies of its users. This chapter will explore market orders, limit orders, stop orders, and other order types available on the platform.

4.2 Trading API:
For algorithmic traders and developers, Binbex provides a robust trading API. This section will delve into the API’s capabilities, documentation, and how it empowers users to integrate trading strategies seamlessly.

4.3 Mobile App:
In the era of mobile connectivity, Binbex’s mobile app is a key component of its service. This chapter will examine the features of the mobile app, its interface, and how it enhances the trading experience for users on the go.

 5.Unique Offerings and Partnerships

5.1 Binbex Token (BNX):
As part of its ecosystem, Binbex has introduced its native utility token, BNX. This section will explore the utility of BNX, its role within the platform, and the benefits it brings to users.

5.2 Partnerships and Collaborations:
Binbex has strategically partnered with various organizations to enhance its offerings. This chapter will highlight key partnerships, exploring how they contribute to Binbex’s ecosystem and benefit users.

6. Community Engagement and Support

6.1 Social Media Presence:
Binbex actively engages with its community through social media platforms. This section will analyze the exchange’s social media strategy, the content it shares, and how it fosters a sense of community among users.

6.2 Customer Support:
Exceptional customer support is vital for any successful exchange. This chapter will assess Binbex’s customer support channels, response times, and the effectiveness of addressing user queries and concerns.

 7: Challenges and Future Roadmap

7.1 Challenges Faced:
No exchange is without its challenges. This section will discuss any notable challenges that Binbex has faced and how the platform has addressed or plans to address them.

7.2 Future Developments:
Binbex’s future roadmap is an exciting aspect to explore. This chapter will provide insights into upcoming features, partnerships, and developments that users can anticipate from the exchange.


In conclusion, Binbex Exchange stands as a multifaceted platform that caters to the diverse needs of the cryptocurrency community. From its inception to its current standing, the exchange has demonstrated a commitment to user satisfaction, security, and innovation. As the crypto landscape continues to evolve, Binbex’s adaptability and forward-thinking approach position it as a noteworthy player with the potential to shape the future of cryptocurrency trading.

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Navigating the World of Savings: A Comprehensive Guide to Login



By is the gateway to a world of savings and perks for Kohl’s credit cardholders. In this comprehensive guide, we will explore the ins and outs of login, from the registration process to unlocking the full potential of the online portal.

1. Understanding Kohl’s Credit Card:

Before delving into, it’s essential to have a solid understanding of the Kohl’s credit card. This section will provide an overview of the different types of Kohl’s credit cards available, their benefits, and how they contribute to the overall shopping experience at Kohl’s stores and online.

2. Introduction to serves as the online portal for Kohl’s credit cardholders, offering a range of features designed to enhance their shopping experience. This section will introduce users to the website, highlighting key functionalities such as account management, bill payment options, and exclusive promotions.

3. Registering for

To unlock the benefits of, users need to register for an account. This section will provide a step-by-step guide on how to register for an account on the portal. From entering basic information to creating a secure login, users will learn the initial steps required to access their Kohl’s credit card account online.

4. Navigating the Dashboard:

Once registered and logged in, users are greeted with the dashboard. This section will explore the different sections of the dashboard, including account summary, recent transactions, and available credit. Understanding how to navigate the dashboard ensures users can quickly access the information they need.

5. Account Management: offers robust account management features, allowing users to have greater control over their credit card account. This section will delve into the various account management options, including updating personal information, setting account preferences, and managing authorized users.

6. Making Payments Online:

One of the key conveniences of is the ability to make payments online. This section will guide users through the process of making payments through the portal, exploring different payment options, setting up autopay, and ensuring a seamless and secure transaction experience.

7. Tracking Rewards and Discounts:

Kohl’s credit cardholders enjoy exclusive rewards and discounts, and provides a platform to track and manage these perks. This section will explore how users can view their rewards balance, track earned Kohl’s Cash, and take advantage of special promotions and discounts available through the online portal.

8. Security Features and Best Practices:

Security is paramount when managing financial information online. This section will provide an overview of the security features implemented on and offer best practices for users to ensure the safety of their personal and financial information.

9. Troubleshooting and Customer Support:

In the event of login issues or other technical challenges, users need access to effective troubleshooting resources. This section will outline common login problems and guide users on how to resolve them. Additionally, it will provide information on contacting customer support for further assistance.

10. Mobile App Integration: seamlessly integrates with the Kohl’s mobile app, providing users with additional flexibility and accessibility. This section will explore the features available on the mobile app, how it syncs with the online portal, and the benefits of managing your Kohl’s credit card on the go.

11. Exclusive Cardholder Events and Sales:

Being a Kohl’s credit cardholder comes with the perk of exclusive events and sales. This section will highlight how keeps users informed about upcoming promotions, cardholder events, and early access to sales, creating a sense of privilege for cardholders.

12. Staying Informed with Notifications: allows users to set up notifications for various account activities. This section will guide users on how to customize their notification preferences, ensuring they stay informed about account updates, payment due dates, and exclusive offers.

13. Financial Wellness Resources:

Kohl’s is committed to promoting financial wellness among its credit cardholders. This section will explore the financial wellness resources available on, such as budgeting tools, credit score monitoring, and educational materials aimed at empowering users to make informed financial decisions.

14. Rewards Program:

Beyond discounts and promotions, offers a rewards program for loyal cardholders. This section will detail how the rewards program works, the different tiers of rewards, and how users can maximize their benefits through regular use of their Kohl’s credit card.

15. Future Developments and Enhancements:

The world of online banking and credit card management is ever-evolving. This section will explore potential future developments and enhancements that users can anticipate on From improved user interfaces to expanded features, understanding the trajectory of the online portal ensures users stay ahead of the curve.


In conclusion, login is the key to unlocking a world of convenience, savings, and exclusive perks for Kohl’s credit cardholders. From the initial registration process to navigating the dashboard, making online payments, and enjoying exclusive rewards, this comprehensive guide has covered every aspect of managing a Kohl’s credit card through the online portal. As users continue to embrace the digital landscape for financial management, stands as a testament to Kohl’s commitment to providing a seamless and rewarding experience for its valued credit cardholders.

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Exploring Pixel 3XL Programming Backgrounds: A Comprehensive Guide to Customization and Development




The Google Pixel 3XL has carved a niche for itself as a powerful device that not only boasts impressive hardware but also offers a customizable and developer-friendly environment. One aspect that sets the Pixel 3XL apart is its programming backgrounds, providing users with the ability to delve into the world of customization and development. In this comprehensive guide, we will explore the various facets of Pixel 3XL programming backgrounds, from customization options to development opportunities.

1. Understanding Pixel 3XL:

Before diving into the programming backgrounds, it’s crucial to have a solid understanding of the Google Pixel 3XL. This section will provide an overview of the device’s key features, specifications, and the Android operating system that serves as the foundation for the programming backgrounds we’ll explore later.

2. Customization Capabilities:

Pixel 3XL users have the luxury of customizing their device to suit their preferences. This section will delve into the various customization options available, from changing wallpapers and themes to adjusting the home screen layout. Understanding these customization capabilities allows users to personalize their Pixel 3XL and create a device that reflects their unique style.

3. Exploring Programming Backgrounds:

Pixel 3XL programming backgrounds extend beyond simple customization; they open the door to a world of possibilities for developers. This section will explore the programming backgrounds available on the Pixel 3XL, discussing how users can leverage coding skills to create dynamic and interactive backgrounds that go beyond the standard static wallpapers.

4. Android Development Environment:

To fully harness the programming backgrounds on the Pixel 3XL, users need to understand the Android development environment. This section will provide an introduction to Android development, including the use of Android Studio, the official integrated development environment (IDE) for Android app development. Exploring this environment sets the stage for users to create their own unique programming backgrounds.

5. Building Custom Live Wallpapers:

Live wallpapers bring a new dimension to the Pixel 3XL, providing dynamic and interactive backgrounds. This section will guide users through the process of building custom live wallpapers. From coding considerations to incorporating animations and interactive elements, users will gain insights into creating backgrounds that respond to touch and other user interactions.

6. Programming with Kotlin:

Kotlin has become the preferred programming language for Android development, and understanding its basics is essential for Pixel 3XL users looking to dive into programming backgrounds. This section will provide an introduction to Kotlin, highlighting its features and syntax, and demonstrating how it can be used to create engaging and functional live wallpapers.

7. Utilizing Android APIs:

The power of Pixel 3XL programming backgrounds lies in their integration with various Android APIs (Application Programming Interfaces). This section will explore key Android APIs that developers can leverage to enhance their programming backgrounds. From sensor data to connectivity features, understanding these APIs opens up new avenues for creativity and functionality.

8. Showcasing Notable Pixel 3XL Programming Backgrounds:

This section will showcase notable examples of Pixel 3XL programming backgrounds created by developers. From mesmerizing animated landscapes to functional backgrounds that display real-time information, these examples highlight the diversity and creativity within the Pixel 3XL programming community.

9. Debugging and Troubleshooting:

Programming backgrounds may encounter bugs or issues during development. This section will guide users through the process of debugging and troubleshooting, providing tips and techniques to identify and resolve common issues. Understanding this aspect of development ensures a smoother experience for users creating their own programming backgrounds.

10. Sharing and Distributing Custom Backgrounds:

Once a programming background is created, the next step is sharing it with the wider community. This section will explore the various methods for sharing and distributing custom Pixel 3XL programming backgrounds. From publishing on app stores to sharing code repositories, users will learn how to showcase their creations and contribute to the vibrant Pixel 3XL programming community.

11. Future Developments and Trends:

As technology evolves, so do programming backgrounds on the Pixel 3XL. This section will explore emerging trends and future developments in the realm of Android customization and development. From advancements in augmented reality backgrounds to new possibilities introduced with updated Android versions, users will gain insights into what the future holds for Pixel 3XL programming backgrounds.


In conclusion, Pixel 3XL programming backgrounds provide users with a canvas for creativity and personalization. From simple customizations to the development of dynamic and interactive live wallpapers, the Pixel 3XL offers a unique platform for users to express themselves and contribute to the vibrant Android customization community. By understanding the customization capabilities, delving into the Android development environment, and exploring the potential of Kotlin and Android APIs, users can unlock the full potential of their Pixel 3XL programming backgrounds. As technology continues to advance, the possibilities for customization and development on the Pixel 3XL are boundless, ensuring that users can continually redefine and enhance their smartphone experience.

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