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Folders and files, repository files navigation, solutions to exercises & problems in real analysis, collaborations wanted.

You may read the next few sections first for a crude understanding of this project. Here are some brief descriptions:

  • Overleaf based
  • Work any time

Collaborators are desired to meet these requirements:

  • Good undergraduate discipline of Mathematics, especially a comprehensive practice in elementary analysis and linear algebra.
  • (If your mother language is not English) Comfort with English writing.
  • Eagerness to share your ideas.
  • No fear of making mistakes.
Simple is better than complex. Flat is better than nested. -- Excerpted from the Zen of Python, not exactly

Go check it yourself if there are doubts. Please contact me via email at [email protected] or [email protected] if you have any idea.

What is it?

As its name suggests, this cute little book serves as a collection of my solutions to exercises and problems in Real Analysis: Measure Theory, Integration and Hilbert Spaces (Stein & Shakarchi, 2005) . It is still in progress and the current work covered the first four chapters:

  • Ch1: Measure Theory
  • Ch2: Integration Theory
  • Ch3: Differentiation and Integration
  • Ch4: Hilbert Spaces: An Introduction

My plan is to cover

  • Ch6: Abstract measure and Integration Thoery
  • Ch7: Hausdroff Measure and Fractals

How to use it?

Unless otherwise noted, the solutions in this book are the fruits of independent effort. If you are a student enrolled in a class whose textbook is exactly this one, please do not plagiarize any content from this document as your homework. If you are an amateur of solving puzzles and read this book purely for fun, you are suggested to seek its help after giving the conundrum a third shot at least.

Organization of the book

Each chapter is divided into three sections, namely exercises, problems and supplementaries (optional). Presented in supplementaries are auxiliary materials that can better understanding of other content in the textbook.

My schedule is usually crammed, unfortunately, so I make my best effort not to dull my minds with typing answers to those easy and warm-up exercises. For the rest of them, I try my best to answer. If the problem you are looking for is missing, three possible cases can be identified:

  • I decided not to include it, because either the hint in the book reveals virtually all the stories, or it is simple and is not in connection with further problems.
  • I have not worked it out either.
  • I may type a solution soon.

Though I think I am into preciseness and rigor when it comes to Mathematics, I do not mean that every single proof presented in this little book is correct or the most elegant possible. In fact, it is just highly likely to be correct but is unlikely to be the most enlightening.

Error Report

If you find errors, please please do report it. A BIG THANK to U in advance! 🥰

You may report a grammar mistake, a typo, a misformat or the incorrectness of a proof ( Sincerely hope this is not the case 🥺 ). You can send email either to [email protected] or [email protected] and I will reply as soon as possible. Please do not send me messages over Github since I am likely to miss it. Thank you for your understanding!

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Gemini 1.5: Our next-generation model, now available for Private Preview in Google AI Studio

February 15, 2024

real analysis problem and solution pdf

Last week, we released Gemini 1.0 Ultra in Gemini Advanced. You can try it out now by signing up for a Gemini Advanced subscription . The 1.0 Ultra model, accessible via the Gemini API, has seen a lot of interest and continues to roll out to select developers and partners in Google AI Studio .

Today, we’re also excited to introduce our next-generation Gemini 1.5 model , which uses a new Mixture-of-Experts (MoE) approach to improve efficiency. It routes your request to a group of smaller "expert” neural networks so responses are faster and higher quality.

Developers can sign up for our Private Preview of Gemini 1.5 Pro , our mid-sized multimodal model optimized for scaling across a wide-range of tasks. The model features a new, experimental 1 million token context window, and will be available to try out in  Google AI Studio . Google AI Studio is the fastest way to build with Gemini models and enables developers to easily integrate the Gemini API in their applications. It’s available in 38 languages across 180+ countries and territories .

1,000,000 tokens: Unlocking new use cases for developers

Before today, the largest context window in the world for a publicly available large language model was 200,000 tokens. We’ve been able to significantly increase this — running up to 1 million tokens consistently, achieving the longest context window of any large-scale foundation model. Gemini 1.5 Pro will come with a 128,000 token context window by default, but today’s Private Preview will have access to the experimental 1 million token context window.

We’re excited about the new possibilities that larger context windows enable. You can directly upload large PDFs, code repositories, or even lengthy videos as prompts in Google AI Studio. Gemini 1.5 Pro will then reason across modalities and output text.

Upload multiple files and ask questions We’ve added the ability for developers to upload multiple files, like PDFs, and ask questions in Google AI Studio. The larger context window allows the model to take in more information — making the output more consistent, relevant and useful. With this 1 million token context window, we’ve been able to load in over 700,000 words of text in one go. Gemini 1.5 Pro can find and reason from particular quotes across the Apollo 11 PDF transcript. 
[Video sped up for demo purposes]
Query an entire code repository The large context window also enables a deep analysis of an entire codebase, helping Gemini models grasp complex relationships, patterns, and understanding of code. A developer could upload a new codebase directly from their computer or via Google Drive, and use the model to onboard quickly and gain an understanding of the code. Gemini 1.5 Pro can help developers boost productivity when learning a new codebase.  
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Mathematics > Analysis of PDEs

Title: dirichlet problems in perforated domains.

Abstract: In this paper we establish $W^{1,p}$ estimates for solutions $u_\varepsilon$ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, $C^1$ domain $\Omega_{\varepsilon, \eta}$ in $\mathbb{R}^d$. The bounding constants depend explicitly on two small parameters $\varepsilon$ and $\eta$, where $\varepsilon$ represents the scale of the minimal distance between holes, and $\eta$ denotes the ratio between the size of the holes and $\varepsilon$. The proof relies on a large-scale $L^p$ estimate for $\nabla u_\varepsilon$, whose proof is divided into two parts. In the first part, we show that as $\varepsilon, \eta $ approach zero, harmonic functions in $\Omega_{\varepsilon, \eta}$ may be approximated by solutions of an intermediate problem for a Schrödinger operator in $\Omega$. In the second part, a real-variable method is employed to establish the large-scale $L^p$ estimate for $\nabla u_\varepsilon$ by using the approximation at scales above $\varepsilon$. The results are sharp except in the case $d\ge 3$ and $p=d$ or $d^\prime$.

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Trump Fraud Trial Penalty Will Exceed $450 Million

The ruling in Donald J. Trump’s civil fraud case could cost him all his available cash. The judge said that the former president’s “complete lack of contrition” bordered on pathological.

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Donald Trump, wearing a blue suit and blue tie, sits at the defendant’s table in a courtroom.

By Jonah E. Bromwich and Ben Protess

A New York judge on Friday handed Donald J. Trump a crushing defeat in his civil fraud case , finding the former president liable for conspiring to manipulate his net worth and ordering him to pay a penalty of nearly $355 million plus interest that could wipe out his entire stockpile of cash .

The decision by Justice Arthur F. Engoron caps a chaotic, yearslong case in which New York’s attorney general put Mr. Trump’s fantastical claims of wealth on trial. With no jury, the power was in Justice Engoron’s hands alone, and he came down hard: The judge delivered a sweeping array of punishments that threatens the former president’s business empire as he simultaneously contends with four criminal prosecutions and seeks to regain the White House.

Justice Engoron barred Mr. Trump for three years from serving in top roles at any New York company, including portions of his own Trump Organization. He also imposed a two-year ban on the former president’s adult sons and ordered that they pay more than $4 million each. One of them, Eric Trump, is the company’s de facto chief executive, and the ruling throws into doubt whether any member of the family can run the business in the near term.

The judge also ordered that they pay substantial interest, pushing the penalty for the former president to $450 million, according to the attorney general, Letitia James.

In his unconventional style, Justice Engoron criticized Mr. Trump and the other defendants for refusing to admit wrongdoing for years. “Their complete lack of contrition and remorse borders on pathological,” he said.

He noted that Mr. Trump had not committed violent crimes and also conceded that “Donald Trump is not Bernard Madoff.” Still, he wrote, “defendants are incapable of admitting the error of their ways.”

real analysis problem and solution pdf

The Civil Fraud Ruling on Donald Trump, Annotated

Former President Donald J. Trump was penalized $355 million plus interest and banned for three years from serving in any top roles at a New York company, including his own, in a ruling on Friday by Justice Arthur F. Engoron.

Mr. Trump will appeal the financial penalty but will have to either come up with the money or secure a bond within 30 days. The ruling will not render him bankrupt, because most of his wealth is in real estate, which altogether is worth far more than the penalty.

Mr. Trump will also ask an appeals court to halt the restrictions on him and his sons from running the company while it considers the case. In a news conference from his Palm Beach, Fla., home, Mar-a-Lago, on Friday evening, he attacked Ms. James and Justice Engoron, calling them both “corrupt.”

Alina Habba, one of Mr. Trump’s lawyers, described the ruling in her own statement as “a manifest injustice — plain and simple.” She added that “given the grave stakes, we trust that the Appellate Division will overturn this egregious verdict.”

But there might be little Mr. Trump can do to thwart one of the judge’s most consequential punishments: extending for three years the appointment of an independent monitor who is the court’s eyes and ears at the Trump Organization. Justice Engoron also strengthened the monitor’s authority to watch for fraud and second-guess transactions that look suspicious.

Mr. Trump’s lawyers have railed against the monitor, Barbara Jones, saying that her work had already cost the business more than $2.5 million; the decision to extend her oversight of the privately held company could enrage the Trumps, who see her presence as an irritant and an insult.

Ms. James had sought an even harsher penalty, asking for Mr. Trump to be permanently barred from New York’s business world. In the 2022 lawsuit that precipitated the trial, she accused Mr. Trump of inflating his net worth to obtain favorable treatment from banks and other lenders, attacking the foundation of his public persona as a billionaire businessman.

Even though the lenders made money from Mr. Trump, they were the purported victims in the case, with Ms. James arguing that without his fraud, they could have made even more.

The financial penalty reflects those lost profits, with nearly half of the $355 million — $168 million — representing the interest that Mr. Trump saved, and the remaining sum representing his profit on the recent sale of two properties, money that the judge has now clawed back from Mr. Trump and corporate entities he owns.

Before the trial began, Justice Engoron ruled that the former president had used his annual financial statements to defraud the lenders, siding with the attorney general on her case’s central claim. The judge’s Friday ruling ratified almost all of the other accusations Ms. James had leveled against Mr. Trump, finding that the former president had conspired with his top executives to violate several state laws.

The judge’s decision for now grants Ms. James, a Democrat, a career-defining victory. She campaigned for office promising to bring Mr. Trump to justice, and sat calmly in the courtroom as the former president attacked her, calling her a corrupt politician motivated solely by self-interest.

“This long running fraud was intentional, egregious, illegal,” Ms. James said during a Friday evening news conference, adding that “there cannot be different rules for different people in this country, and former presidents are no exception.”

New York Attorney General Speaks on Trump Fraud Case Decision

“today we are holding donald trump accountable,” said the attorney general, letitia james, after a new york judge found donald j. trump’s claims of wealth fraudulent..

No matter how rich, powerful or politically connected you are, everyone must play by the same rules. We have a responsibility to protect the integrity of the marketplace. And for years, Donald Trump engaged in deceptive business practices and tremendous fraud. Donald Trump falsely, knowingly, inflated his net worth by billions of dollars to unjustly enrich himself, his family, and to cheat the system. After 11 weeks of trial, we showed the staggering extent of his fraud and exactly how Donald Trump and the other defendants deceived banks, insurance companies and other financial institutions for their own personal gain. We prove just how much Donald Trump, his family and his company unjustly benefited from his fraud. White-collar financial fraud is not a victimless crime. When the powerful break the law and take more than their fair share, there are fewer resources available for working people, small businesses and families. Today we are holding Donald Trump accountable.

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Her win is Mr. Trump’s second major courtroom loss in two months, following a January jury verdict in a defamation case brought by E. Jean Carroll, a writer whom he was found liable of sexually abusing. The jury penalized him $83.3 million.

Friday’s ruling comes as Manhattan prosecutors are set to try Mr. Trump on criminal charges late next month . He is also contending with 57 more felony counts across three other criminal cases.

But none of his legal troubles seem to have anguished Mr. Trump quite like the fraud case. During the trial, he protested its premise, pleading, “This has been a persecution of somebody that’s done a good job in New York.”

Mr. Trump’s lawyers argued that the fraud did not have a victim in the traditional sense , daring the attorney general to find someone who was harmed. And in a statement on Friday, a Trump Organization spokeswoman noted that the company had “never missed any loan payment or been in default on any loan” and that the lenders “performed extensive due diligence prior to entering into these transactions.”

At trial, Mr. Trump’s lawyers called as witnesses the president’s former bankers, who testified that they had been delighted to have Mr. Trump as a client.

Eric Trump and his brother Donald Trump Jr. also testified, but their efforts to distance themselves from their father’s financial statements fell flat with the judge. Justice Engoron’s decision to bar them from running any New York business for two years — and Mr. Trump for three — will likely strike a nerve with the Trump family.

Before the trial, the fallout from the case seemed to threaten the Trump Organization’s very existence. When Justice Engoron first ruled that Mr. Trump had committed fraud, he ordered the dissolution of much of the former president’s New York empire.

But legal experts had questioned the judge’s ability to do that , and in his ruling on Friday, Justice Engoron pulled back. Instead, the judge said any “restructuring and potential dissolution” would be up to Ms. Jones, the independent monitor.

The judge also granted Ms. Jones new authority as part of an “enhanced monitorship,” and asked her to recommend an independent compliance director who will oversee the company’s financial reporting from within its ranks.

The monitorship and other penalties, including a three-year ban on Mr. Trump and his company seeking loans from banks registered in New York, could hamstring the company as it seeks to compete in the state’s crowded real estate market.

However, nothing will hurt quite as much as the financial penalty. If upheld on appeal, it could erase the cushion of liquidity — cash, stocks and bonds — that Mr. Trump built in his post-presidential life.

Mr. Trump claimed under oath last year that he was sitting on more than $400 million in cash, but between Justice Engoron’s $355 million punishment, the interest Mr. Trump owes and the $83.3 million payout to Ms. Carroll, that might all be gone. If so, Mr. Trump might have to sell one of his properties or another asset to cover the payouts.

The symbolism of the punishments cannot be overlooked, either. Mr. Trump is synonymous with the company he ran for decades, and by severing him from its operations, the judge has written an embarrassing epilogue to the former president’s story of his career as a New York mogul.

For now, Mr. Trump has spun his legal misfortunes into what he sees as political gold. He has used the cases to falsely portray himself as a victim of a Democratic cabal led by President Biden, and he has campaigned at every courthouse he has visited.

In Justice Engoron’s courtroom, Mr. Trump delivered a rally-made rant from the witness stand, marking the climax of a monthslong proceeding that was alternately stultifying and scintillating. The former president attacked one of Ms. James’s lawyers, saying: “You and about every other Democrat, district attorney, A.G. and U.S. attorney were coming after me from 15 different sides. All Democrats, all Trump haters.”

He did not spare Ms. James herself, or the judge, calling the attorney general a “political hack” and Justice Engoron an “extremely hostile judge.”

Mr. Trump later delivered his own closing statement, calling Ms. James’s fraud accusation a “fraud on me” and saying that the attorney general was the one who “should pay me.”

He generated drama even when not in the spotlight, rolling his eyes at the defense table and muttering to his lawyers. He was particularly enraged by the testimony of his former fixer, Michael D. Cohen , who linked Mr. Trump directly to the fraud scheme.

Mr. Trump’s lawyers succeeded in rattling Mr. Cohen, and asked, based on apparent contradictions in his testimony, that Justice Engoron throw out the case. When the judge declined, Mr. Trump abruptly stood up and stormed out of the courtroom.

The judge largely tolerated Mr. Trump’s behavior, but early on, he barred the former president from attacking his staff members, most prominently his law clerk, who sat near the judge throughout the trial so they could confer. Mr. Trump twice violated that order, prompting $15,000 in fines from the judge.

Courtroom theatrics notwithstanding, the evidence presented was often tedious, consisting of years-old emails and spreadsheets. Through that documentary evidence, Ms. James’s lawyers showed that Mr. Trump’s company had ignored appraisals and manipulated numbers to inflate the value of properties such as golf clubs and office buildings, sometimes to absurd heights.

The most blatant exaggeration was the listed size of Mr. Trump’s triplex apartment in Trump Tower on Fifth Avenue. For years, the former president had valued it as if it were 30,000 square feet, when it was actually 10,996.

In his ruling, Justice Engoron blasted Mr. Trump and the other defendants, saying that misstating the apartment’s size was the only error to which they would admit.

Justice Engoron wrote that he was not looking to “judge morality” — only to find facts and apply the law.

“The court intends to protect the integrity of the financial marketplace and, thus, the public as a whole,” he wrote.

Justice Engoron added that Mr. Trump’s refusal to admit error left him with no choice but to conclude that the former president would continue to commit fraud unless he was stopped.

William K. Rashbaum , Claire Fahy and Maggie Haberman contributed reporting.

Jonah E. Bromwich covers criminal justice in New York, with a focus on the Manhattan district attorney's office, state criminal courts in Manhattan and New York City's jails. More about Jonah E. Bromwich

Ben Protess is an investigative reporter at The Times, writing about public corruption. He has been covering the various criminal investigations into former President Trump and his allies. More about Ben Protess


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    SOLUTIONS. Practice midterm 1 (PDF) Practice midterm 1 solutions (PDF) Practice midterm 2 (PDF) Practice midterm 2 solutions (PDF) Practice final exam (PDF) Practice final exam solutions (PDF) This section includes practice midterms, and a practice final exam for this course, with solutions.

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