Saturday, September 5, 2015

The special linear group is normal in the general linear group

Let $F$ be a field and $n$ a positive integer. Prove that $SL_n(F)$ is normal in $GL_n(F)$ and describe the isomorphism type of $GL_n(F)/SL_n(F)$ .

Let $A \in SL_n(F)$ and $B \in GL_n(F)$ . Now $\mathsf{det}(BAB^{-1}) = \mathsf{det}(B) \mathsf{det}(A) \mathsf{det}(B)^{-1} = \mathsf{det}(A) = 1$ , since multiplication in $F$ is commutative. Thus $BAB^{-1} \in SL_n(F)$ . Hence $SL_n(F)$ is normal in $GL_n(F)$ .
Define a mapping by .
1. (Well-defined) Suppose $A,B \in GL_n(F)$ such that $AB^{-1} \in SL_n(F)$ . Then $\mathsf{det}(AB^{-1}) = 1$ , so that $\varphi(A) = \mathsf{det}(A) = \mathsf{det}(B) = \varphi(B)$ . Thus $\varphi$ is well defined.
2. (Homomorphism) We have $\varphi(\overline{A} \overline{B}) = \varphi(\overline{AB}) = \mathsf{det}(AB)$ $= \mathsf{det}(A) \mathsf{det}(B) = \varphi(\overline{A}) \varphi(\overline{B})$ . Thus $\varphi$ is a homomorphism.
3. (Injective) Suppose $\varphi(\overline{A}) = \varphi(\overline{B})$ . Then $\mathsf{det}(A) = \mathsf{det}(B)$ , and we have $\mathsf{det}(AB^{-1}) = 1$ , so that $AB^{-1} \in SL_n(F)$ . Hence $\overline{A} = \overline{B}$ , and $\varphi$ is injective.
4. (Surjective) For all $q \in F^\times$ , note that the matrix with $q$ in the $(1,1)$ entry, 1 in all other diagonal entries, and 0 in all off diagonal entries has determinant $q$ . Thus $\varphi$ is surjective.
Thus is a group isomorphism, so that

Sunday, February 16, 2014

Computer Science Med.Mathe 2nd year

Master 2^nd year COMPUTER SCIENCE ( Math ED )

Question Patton

Total Mark :20

Unit	No. Of Question	Marks
1	1	*15=5**
2	1	*15=5**
4	1	*15=5**
6	1	*15=5**

MODEL QUESTIONs

Unit One

1. What do you mean by computer? Define its characteristics.

2. Classify the different types of computer.

3. What are input and out put device of personal computer?

4. What is memory ? list out the different types of memory.

5. What is computer network?

6. What do you mean by internet?

7. Define URL.

8. How can charge our education system through internet?

9. Describe the different type of communication media.

10. What do you mean by www ?

11. Define the term internet , Extra-net with example.

12. What are the components of multimedia?

13. What are the application of computer?

Unit Two

1. What is operating system? List out most popular operating system name?

2. What are the main functions of operating system?

3. What are the types of operating system?

4. What are the different between Graphical User Interface(GUI) and Command Line Interface (CLI) operating system?

5. Write short note on Windows Operating System.

6. Describe about DOS.

Unit Four

1. What are differences between data and information?

2. Describe the term Data , Database, Database Management System?

3. What is SQL.

4. Define the term DDC, DML, DCL .

5. Explain about the Relational Database Management System?

6. What does u mean by Normalization? Define the types of Normalization.

7. What is the importance of the Database Management System?

8. Describe about MS Access.

Unit Six

1. What are the different types of programming language?

2. What do you mean by programming language?

3. Define the term event and procedure with suitable example in Visual Basis.

4. What is loop in the programming language?

5. Define the use toolbox, table form in VB?

6. Write syntax to FOR NEXT loop.

7. Define the control structure? Write IF...THEN, BLSE control structure with example.

8. Define the term High Level Programming Language, compiler And Assembler.

9. What is the type of Anay in VB?

Sunday, August 18, 2013

Module : Micro lecture plan no :1

Lecture Note (lesion plan copy )
Pair observation firm three times report firm als
Text book (content ) analysis report
Material – make a uniqueness material and write a note about it.

Micro teaching

Micro lecture note (1-5 ) 10marks

Module :

Micro lecture plan no :1

Level date

Maths.: ………… time:

Unit:………… period:

Topic: no of student:

1. Specific Object :

At the end of the lesson students will be able to;

2. Teaching material : daily use material

3. Teaching activities : (first day ) Teacher will informs ……level…….topic…….etc

Activities

4. Evaluation.

5. Home work

Tuesday, April 9, 2013

Laws of indices

INDICES

Large or small numbers are better expressed in terms of indices. A given number is written as a base raised to the index, that is (base)^index. Bases and indices can be any real number. In the last chapter we have seen what are squares and cubes of numbers; for squares the index is 2, for cubes the index is 3. For square roots the index is (1/2) and for cube roots the index is (1/3). The base has to be a suitable number that together with the indices gives the correct number. If the index is n, then the resultant number is obtained by multiplying the base n times.

Laws of indices

We will state a few facts about indices and try to see their validity using all sorts of numbers. The laws are valid for all real numbers, but for the present syllabus, it is sufficient to consider only rational numbers.

1. a^m * aⁿ = a ^(m+n).

2. a ^(-m) = 1 / a^m.

3. a^m / aⁿ = a ^(m-ⁿ⁾, a

4. (a^m)ⁿ = a ^{(m * n)}.

5. (a * b)^m = a^m * b^m.

6. (a / b) ^m = a^m / b^m , b

7. a⁰ = 1.

1. To prove that a^m * aⁿ = a ^(m+n).

Example 1 : Let us consider the base a = 2, let the value of indices m and n be m=4, n=5.

LHS : a^m * aⁿ = 2⁴ * 2⁵ = (2 * 2 * 2 * 2) * (2 * 2 * 2 * 2 * 2) = 16 * 32 = 512.

RHS : a ^(m+n) = 2⁹ = 512.

Since LHS = RHS, hence it is proved that a^m * aⁿ = a ^(m+n).

Example 2 : Let us consider the base a = 10, let the value of indices m and n be m=2, n=3.

LHS : a^m * aⁿ = 10² * 10³ = (10 * 10) * (10 * 10 * 10) = 100 * 1000 = 100000.

RHS : a ^(m+n) = 10⁵ = 100000.

Since LHS = RHS, hence it is proved that a^m * aⁿ = a ^(m+n).

2. To prove that a ^(-m) = 1 / a^m.

Multiply both the LHS and the RHS by a^m

a ^(-m) * a^m = (1 / a^m) * a^m.

From first law of indices proved above, LHS becomes a ^(m-m) = a⁰.

In the RHS, the numerator and the denominator cancel each other out to give a resultant of 1.

Thus a ^(m-m) = 1, which means that a ^(-m) * a^m = 1.

This gives a ^(-m) = 1/ a^m.

3. From the above two proofs, the third law of indices

a^m / aⁿ = a ^(m-n) , a

can be easily deduced.

The condition a

0 is important, because if a = 0, the expression will become infinite.

If m > n, the index of the expression on the RHS, that is (m-n), will be a positive number.

If m < n, the index of the expression on the RHS, the is (m-n), will be a negative number. In this case the second law of indices will have to be applied to obtain the value.

4. To prove that (a^m)ⁿ = a ^{(m * n)}

Example 1 : Let us consider the base a = 7, let the value of indices m and n be m=2, n=5.

LHS : (a^m)ⁿ= (7²)⁵ = ( 7 * 7 )⁵ = 49⁵ = 49 * 49 * 49 * 49 * 49 = 282475249 .

RHS : a ^{(m * n)} = 7¹⁰ = 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 = 282475249 .

Thus, LHS = RHS, hence it is proved that (a^m)ⁿ= a ^{(m * n)} .

Example 2 : Let us consider the base a = 1/2, let the value of indices m and n be m=1, n=2.

LHS : (a^m)ⁿ = ( (1/2)¹ )² = (1/2)² = ‡ * ‡ = º .

RHS : a ^{(m * n)}= (1/2)² = ‡ * ‡ = º .

Thus LHS = RHS, hence it is proved that (a^m)ⁿ= a ^{(m * n)}.

5. To prove that (a * b)^m = a^m * b^m.

Example 1 : Let us consider the base a = 2 and b = 3, let the value of index m be m = 2 .

LHS : (a * b)^m = ( 2 * 3 )² = 6² = 6 * 6 = 36 .

RHS : a^m * b^m = 2² * 3² = 4 * 9 = 36 .

Thus LHS = RHS, hence it is proved that (a * b)^m = a^m * b^m.

Example 2 : Let us consider the base a = 4 and b = 9, let the value of index m be m = ‡ .

LHS : (a * b)^m = ( 4 * 9 )^1/2 = 36^1/2 = 6 .

RHS : a^m * b^m = 4^1/2 * 9^1/2 = 2 * 3 = 6.

Thus LHS = RHS, hence it is proved that (a * b)^m = a^m * b^m.

6. From the 1^st , 2^nd and the 5^th laws of indices the 6^th law of indices can also be proved. Thus,

(a/b)^m= a^m/ b^m , b

The condition b

0 is necessary, otherwise the expression will become infinite with 0 in the denominator.

7. The 7^th law of indices a⁰= 1 is already proved above in the 2^nd law of indices.

Summary

The indices are a very easy and convenient method for expressing very large or very small numbers. In case of small numbers, the index is negative and in case of large numbers the index is positive. All the laws of indices are useful in many calculations. These laws are valid for all real numbers, whether integers or fractions.

Monday, April 8, 2013

HISTORY OF MATHEMATICS EDUCATION OF NEPAL

1 Introduction

Education as a system can be called the brain of any society and it is the backbone of any system. Mathematics is a vast adventure in ideas, an exact science and truly saying the mirror of civilization. According to Perry, mathematical education began because it was useful, it continues because of the usefulness of its results. Nowadays, even the social sciences are becoming increasingly mathematical. Most mathematical creations are the result of intuition. The direction of modem mathematics has been greatly influenced by the developments in other disciplines.
The mathematical sciences have changed significantly during the past few decades. The most obvious change is the enormous growth of mathematics. Even the latest scientific and technological developments have extended each branch of mathematics and have proved mathematics as a powerful tool for any scientific achievements. The history of teaching mathematics is as old as the human civilization. Mathematics shows much more durability in its attention to concepts and theories than do other sciences. These days history of mathematics is a powerful tool for a disseminating an understanding of mathematics. We look at history as a way of motivating the learner to see the significance of the area being studied. We consider to history as a route to help the learner understand the path of development to a mathematical concept or process. With the history of mathematics, students will come to know that mathematical science is a work of all civilizations, and teachers will find more confidence in teaching. However, the goals of mathematics education differ according to the country's socio-economic condition and the innovation of science and technology in the society and the existing educational status of a country. Nevertheless, mathematics is taught in all levels of education in every country in the world. The history of mathematics reflects some of the noblest thoughts of countless generations. Nepalese mathematical system is highly influenced by the development of world's mathematical system.
The chapter presents the main trends in the development of mathematics throughout the ages and of the social and cultural settings in which it took place.

2 Historical Background (Pre-1951)

Vedic Era (3000 BC): It is believed that the education in Nepal was started from "Gurukul" where religious teachers, Guru and priests used to teach various methods, techniques and procedures of education. The purposes and contents, the types of education that a learner should receive were decided by the teachers according to the nature, interest and needs of the learners. The different forms of education called Gurukul Shiksha, Rishikul Shiksha, Devkul Shiksha, Rajkul Shiksha and Pitrikul Shiksha were into existence. For example, Brahmans were allowed to study Ved, Vedang, Darshan, Nitishastra, Joytish Shastra, Pitrikul Shiksha and Grammar etc. The main job of educated Brahman was to do Puja Path at Jajman's house and temples to protect and to preserve the religion.

The education given to Chhetri was the Sastra Vidya and the methods and techniques of handling weapons that were necessary for security of the nation. Princes or Kings used to receive Sastra Vidya and Rajkul Shiksha needed to govern the nation. The agriculture skills and business types of education were given to Vaishyas and service oriented part of education was separated for Sudras.

"Ved" were studied and recited in Gurukul and called "Vedic Period". Among them, "Rig Ved" was concerned with mathematics. m Vedic education system mathematics was not studied separately but studied in conjunction with other subjects. At that time astrology was also studied and more emphasis was given in astrology and geometry

.
In ancient period, it is believed that Janakpur was also known as centre for education. The place of education was Rishi-Ashram or Guru Ashram located at Jungle or lonely place, or religious temples "Gumba" and "Gurukul". The education was given to the learner for the protection and preservation of religion and learner has to stay at "Ashram", "Temples", "Gumba" and "Gurukul" and to follow strict discipline of the education institutions. The medium of instruction has been believed to have Sanskrit language. The expense of education was received through donation from people and income from Guthi established by Kings or people for that purpose.
Buddhist Era: The education centers of Buddhist education were Bihars, Gumbas and Buddhist temples, where Buddha Darshan and Buddha Upadesh were studied and recited. The learner has to stay at "Gumba", "Bihars" and to follow strict discipline of the Gumba and Bihars. The medium of instruction was Pali language. Methods of instruction were discussion, question-answer, and religious lectures by Buddhist monk. The purpose of education to produce Buddhist monk for the protection and preservation of Buddhist religion.
Lichchavi Era (143 - 1243 AD): Lichhivi period was concentrated in the development of cultures and arts. For people had belief that education is the religion and religion is the education. The people were attracted towards religion and spend more time in worship of god. They were not attracted in Grihastha Ashram and the population was decreased fairly. To increase population and to attract the people toward Grihastha Ashram, the kings of that period made different sexual Asans of woods and stones in the temples to give the impression to the people that the production of children is also one part of life. We can see such Asans in various temples of Kathmandu, and other parts of Nepal. The mathematics was used to collect tax from the people. The barter system and money were used in business. The simple arithmetic was used. Sumati Tantra and Sumati Siddhanta were found useful for astrologers

.
Malla Era (1243-1741): In Malla period, the arts and architects at palaces, buildings and temples show that Malla were familiar in higher form of geometry. At that time, "hat" was used for measuring length, "mana", "pathi" for amount and "dhak" for weight. Malla were experts in making of soil pots. For future prediction and to make calendars easily, astrologers of Mall era also took the help to Sumati Tantra and Sumati Siddhanta. Many great astrologers made suitable calendars based on Surya Sidhhanta.
Shah era (1742-1846): This period is also called negligence period in the history of education of Nepal. Education did not flourished properly because the state was engaged in battles and wars for the unification of Nepal. However, "Gurukul" or teachers hired at home for education, or Banaras (Kashi) were the main places of education at that period.

Rana era (1846 - 1951): The modem education system in Nepal started in 1854 AD. Returning from Britain, Prime Minister Jung Bahadur Rana opened the Durbar School at the Gol Baithak, Thapathali and mathematics was taught at that school in English medium. This was the first formal school in the history of education in Nepal. Children from higher class Rana families were allowed to study in this school. Similar School was also opened at Hanuman Dhoka Durbar for the members of royal family. Durbar school was affiliated to Calcutta University. Arithmetic, algebra was taught by the Indian and European / British teachers. British type of education was given in this school. The medium of instruction was English. After the death of Jung Bahadur Rana, Durbar School was shifted to Rani Pokhari and opened for the children of other Rana families.

During the Prime-Ministerial time of Dev Shumsher, Durbar School was opened for common people. Many Bhasha Pathshalas were opened throughout the country. Education was made free and stationers were distributed freely.

2.1 Mathematics Education after Democracy (1951 - )

With the advent of democracy in Nepal in 1951, the political situation of the country influenced the education system. Various commissions, education boards, advisory committee such as National Education Planning Commission, 1954, All Round National Education, Committee, 1961, National Education System Plan, 1971, National Education Commission, 1992, Higher Level National Education Commissions, constituted in different period have given mathematics a significant place at all levels of school education.

National Education System Plan (NESP) was introduced in 1971 in Nepal. It was a revolutionary step made by those people who had some ideas about European and American system of education. Annual examination system was replaced by the semester system. New topics in mathematics were introduced and new books were prescribed. NESP was the milestone and pioneering work hi the history of education of Nepal. Mathematics was accepted an essential requirement for literacy. The organization of primary education was made from grade 1 to 3, lower secondary from grade 4 to 7 and secondary from grade 8 to 10. In 1981, the structure of education again changed into 5+3+2 type of school education, hi the process of the implementation of recommendations of NEC, 1992, the school education was structured 5+3+2+2 type of education e.g. primary level education was made from grade 1 to 5, secondary education has three tiers: lower secondary (classes 6-8), secondary (classes 9-10), higher secondary (classes 11-12). The mathematics carrying at least 100 marks has been is made compulsory from class one to class ten and optional in classes 11 and 12

2.2 Mathematics Education at University Level

Higher education of modem mathematics in Nepal started from intermediate level at Trichandra College in 1918 (Arts) and in 1926 (Science). Mathematics classes in B. A. and B.Sc. were started in 1932 and 1942 respectively at the same college. The mathematics curriculum at Bachelor level at that time included topics from Algebra, Trigonometry, Analytical Geometry and Calculus. Classical English textbooks on these subjects were taught for many years. However, master level classes in mathematics were started in 1959 with the establishment of the Mathematics Department at Tribhuvan University.

Tribhuvan University introduced three-year bachelor program from the academic session of 1996 and two-year master program from the academic session of 1999. In this way, our country has embarked on a 10+2+3+2 type of education. A drastic change in mathematics curriculum has taken place in different levels from intermediate to post-graduate. Many new topics in different subjects are included in Bachelor and Master Levels that can corroborated the advanced syllabi of the universities of SAARC countries.
Since establishment of FOE, it has been producing trained teachers for primary, lower secondary, secondary and tertiary levels. Beyond this, FOE also prepares curriculum specialists, evaluation experts, supervisor and administrators for various governmental and non-governmental educational organizations. Faculty of Education has been launching PCL in education, three years B. Ed. one year B. Ed., M. Ed. and Ph. D. programme with specialization in Math Education.