**HẾT thời hạn thắc mắc.**

Cách cộng, trừ:

- Vắng 1 buổi trừ 2 điểm chuyên cần.
- Điểm kiểm tra lấy cho hai cột: thứ hai và thứ ba.
- Một dấu cộng vào cột kiểm tra, cộng full cột 2 thì cộng sang cột 3.
- Một dấu trừ vào cột kiểm tra.
- Các lớp trưởng đều được cộng tương ứng với 1 hoặc 2 dấu cộng.

Nếu ai có thắc mắc gì thì hãy gửi về mail: hpdung83@gmail.com, thời hạn: Hết 24h, ngày 29/05/2016. Sau thời hạn đó, thầy sẽ gửi bảng điểm và mọi thắc mắc không còn hiệu lực nữa. File bảng điểm có 4 tab tương ứng với các lớp.

Đây là link file bảng điểm thành phần: Toan cao cap2-Nhom-1-2-3-6

Chúc các em thi tốt.

D.

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References: Paper of Krasinski, *On the \L ojasiewicz exponent at infinity of polynomial mappings*, Acta. Math. Vietnam. Vol. 32, No. 2-3, 2007.

**Definition:**

Let be a polynomial mapping. The *\L ojasiewicz exponent of at infinity* is defined as the best exponent for which the following inequality holds

for some constant and sufficiently large .

Note that .

**Definition**

Let be a polynomial mapping and be unbounded set. *The \L ojasiewicz exponent of at infinity on * is defined as the best exponent for which the following inequality holds

for some constant and sufficiently large in . This exponent is denoted by :

For example

- We have .
- And where .

\L ojasiewicz exponent has relationship with properness of mappings

Theorem 1if and only if is a proper mapping.

Remark 1Recall the properness of map. is called proper mapping if sastisfies the following property: is a compact set if is a compact set. This fact is equivalent to .

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Remark: has some interesting properties: where .

- .

Indeed, we can prove it by induction.

, Suppose that , we have . Moreover, suppose that , we have . - .

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Câu 3. Cho , là không gian vec tơ con của sinh bởi hai vec tơ và .

- Tìm điều kiện để .
- Tìm .
- Tìm .

Hầu hết các em chỉ làm được ý 3 và làm sai hai ý còn lại. Ý 1 nhiều em đưa ra được hệ phương trình nhưng lại lúng túng.

1. có thể giải đơn giản như sau: nếu và chỉ nếu tồn tại sao cho . Điều này tương đương với hệ sau có nghiệm :

Đến đây các em có thể dùng hạng ma trận để đưa ra liên hệ giữa hoặc có thể xử lý trực tiếp hệ:

Từ hai pt đầu có , pt thứ 3 có do đó: , mặt khác nên và kéo theo , đó chính là điều kiện cần tìm.

2. Dựa vào ý 1, dự đoán rồi chứng minh . Tại sao dự đoán như này? Vì quan sát cơ sở của hai không gian con , hợp của hai cơ sở này có đến 3 vec tơ độc lập tuyến tính.

Có thể phân tích .

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Hence, .

We can apply above formula: .

https://artofproblemsolving.com/community/c7t443f7h1390328_convergence_of_sine_series

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Source: http://www.cut-the-knot.org/proofs/SylvesterGallai.shtml

The Sylvester Problem has been posed by James Joseph Sylvester in 1893 in Educational Times:

*Let n given points have the property that the line joining any two of them passes through a third point of the set. Must the n points all lie on one line?*

T. Gallai’s proof has been outlined by P. Erdös in his submission of the problem to The American Mathematical Monthly in 1943.

Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be *connecting*. Among the connecting lines, those that pass through exactly two points of Π are called *ordinary*.

Choose any point p1∈Π. If p1 lies on an ordinary line we are done, so we may assume that p1 lies on no ordinary line. Project p1 to infinity and consider the set of connecting lines containing p1. These lines are all parallel to each other, and each contains p1 and at least two other points of Π. Any connecting line not through p1 forms an angle with the parallel lines; let s be a connecting line (not through p1) which forms the smallest such angle:

Then s must be ordinary! For suppose s were to contain three (or more) points of Π, say, p2, p3, p4 named so that p3 is between p2 and p4:

The connecting line through p3 and p1 (being not ordinary) would contain a third point of Π, say p5, and now either the line p2p5 or the line p4p5 would form a smaller angle with the parallel lines than does s.

- P. Borwein, W. O. J. Moser, A survey of Sylvester’s problem and its generalizations,
*Aequationes Mathematicae*40 (1990) 111 – 135 - P. Erdös, R. Steinberg, Problem 4065 [1943, 65],
*The American Mathematical Monthly*, Vol. 51, No. 3 (Mar., 1944), pp. 169-171 - J. J. Sylvester,
*Educational Times*, Mathematical Question 11851, vol. 59 (1893), p. 98

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Nếu ai có thắc mắc gì thì hãy gửi về mail: hpdung83@gmail.com, thời hạn: Hết 24h, ngày 05/01/2016. Sau thời hạn đó, thầy sẽ gửi bảng điểm và mọi thắc mắc không còn hiệu lực nữa. File bảng điểm có 4 tab tương ứng với các lớp.

Đây là link file bảng điểm thành phần: diem-thanh-phan-dai-so-nhom-1-6-9-16

Chúc các em ôn thi tốt.

D.

**Hết thời hạn gửi thắc mắc.**

* Bảng điểm đã update một số trường hợp – 04/1:* diem-thanh-phan-dai-so-nhom-1-6-9-16-updated-4-1

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*Famed mathematician Alan Turing proved in his 1936 paper, “On Computable Numbers,” that a universal algorithmic method of determining truth in math cannot exist.*

Alan Turing was born on June 23, 1912, in London. In his seminal 1936 paper, he proved that there cannot exist any universal algorithmic method of determining truth in mathematics, and that mathematics will always contain undecidable propositions. That paper also introduced the “Turing machine. His papers on the subject are widely acknowledged as the foundation of research in artificial intelligence.

English scientist Alan Turing was born Alan Mathison Turing on June 23, 1912, in Maida Vale, London, England. At a young age, he displayed signs of high intelligence, which some of his teachers recognized, but did not necessarily respect. When Turing attended the well-known independent Sherborne School at the age of 13, he became particularly interested in math and science.

After Sherborne, Turing enrolled at King’s College (University of Cambridge) in Cambridge, England, studying there from 1931 to 1934. As a result of his dissertation, in which he proved the central limit theorem, Turing was elected a fellow at the school upon his graduation.

In 1936, Turing delivered a paper, “On Computable Numbers, with an Application to the Entscheidungsproblem,” in which he presented the notion of a universal machine (later called the “Universal Turing Machine,” and then the “Turing machine”) capable of computing anything that is computable: The central concept of the modern computer was based on Turing’s paper.

Over the next two years, Turing studied mathematics and cryptology at the Institute for Advanced Study in Princeton, New Jersey. After receiving his Ph.D. from Princeton University in 1938, he returned to Cambridge, and then took a part-time position with the Government Code and Cypher School, a British code-breaking organization.

During World War II, Turing was a leading participant in wartime code-breaking, particularly that of German ciphers. He worked at Bletchley Park, the GCCS wartime station, where he made five major advances in the field of cryptanalysis, including specifying the bombe, an electromechanical device used to help decipher German Enigma encrypted signals. Turing’s contributions to the code-breaking process didn’t stop there: He also wrote two papers about mathematical approaches to code-breaking, which became such important assets to the Code and Cypher School (later known as the Government Communications Headquarters) that the GCHQ waited until April 2012 to release them to the National Archives of the United Kingdom.

Turing moved to London in the mid-1940s, and began working for the National Physical Laboratory. Among his most notable contributions while working at the facility, Turing led the design work for the Automatic Computing Engine and ultimately created a groundbreaking blueprint for store-program computers. Though a complete version of the ACE was never built, its concept has been used as a model by tech corporations worldwide for several years, influencing the design of the English Electric DEUCE and the American Bendix G-15—credited by many in the tech industry as the world’s first personal computer—among other computer models.

Turing went on to hold high-ranking positions in the mathematics department and later the computing laboratory at the University of Manchester in the late 1940s. He first addressed the issue of artificial intelligence in his 1950 paper, “Computing machinery and intelligence,” and proposed an experiment known as the “Turing Test”—an effort to create an intelligence design standard for the tech industry. Over the past several decades, the test has significantly influenced debates over artificial intelligence.

Homosexuality was illegal in the United Kingdom in the early 1950s, so when Turing admitted to police—who he called to his house after a break-in—in January, 1952, that he had had a sexual relationship with the perpetrator, 19-year-old Arnold Murray, he was charged with gross indecency. Following his arrest, Turing was forced to choose between temporary probation on the condition that he receive hormonal treatment for libido reduction, or imprisonment. He chose the former, and soon underwent chemical castration through injections of a synthetic estrogen hormone for a year, which eventually rendered him impotent.

As a result of his conviction, Turing’s security clearance was removed and he was barred from continuing his work with cryptography at the GCCS, which had become the GCHQ in 1946.

Turing died on June 7, 1954. Following a postmortem exam, it was determined that the cause of death was cyanide poisoning. The remains of an apple were found next to the body, though no apple parts were found in his stomach. The autopsy reported that “four ounces of fluid which smelled strongly of bitter almonds, as does a solution of cyanide” was found in the stomach. Trace smell of bitter almonds was also reported in vital organs. The autopsy concluded that the cause of death was asphyxia due to cyanide poisoning and ruled a suicide.

In a June 2012 BBC article, philosophy professor and Turing expert Jack Copeland argued that Turing’s death may have been an accident: The apple was never tested for cyanide, nothing in the accounts of Turing’s last days suggested he was suicidal and Turing had cyanide in his house for chemical experiments he conducted in his spare room.

Image: Bombe rebuild.

Shortly after World War II, Alan Turing was awarded an Order of the British Empire for his work. For what would have been his 86th birthday, Turing biographer Andrew Hodges unveiled an official English Heritage blue plaque at his childhood home. In June 2007, a life-size statue of Turing was unveiled at Bletchley Park, in Buckinghamshire, England. A bronze statue of Turing was unveiled at the University of Surrey on October 28, 2004, to mark the 50th anniversary of his death. Additionally, the *Princeton University Alumni Weekly* named Turing the second most significant alumnus in the history of the school – James Madison held the number 1 position.

Turing was honored in a number of other ways, particularly in the city of Manchester, where he worked toward the end of his life. In 1999, *Time*magazine named him one of its “100 Most Important People of the 20th century,” saying, “The fact remains that everyone who taps at a keyboard, opening a spreadsheet or a word-processing program, is working on an incarnation of a Turing machine.” Turing was also ranked 21st on the BBC nationwide poll of the “100 Greatest Britons” in 2002. By and large, Turing has been recognized for his impact on computer science, with many crediting him as the “founder” of the field.

Following a petition started by John Graham-Cumming, then-Prime Minister Gordon Brown released a statement on September 10, 2009 on behalf of the British government, posthumously apologized to Turing for prosecuting him as a homosexual.

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The subspace described in above fact is called the span of {or the subspace generated by the elements of ).

We have the following theorem in the book ‘Linear algebra’ of Friedberg et al.

Theorem 1Let be a linearly independent subset of a vector space , and let be an element of that is not in . Then is linearly dependent if and only if .

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Thầy up bảng điểm thành phần của nhóm Toán cao cấp hè, học vào sáng 5,6,7. Các em xem, nếu ai có thắc mắc phải hỏi sớm nhé. File điểm :Diem-thanh-phan-TCC2-he-2016-1

Liên lạc qua mail: hpdung83@gmail.com

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