VILNIUS UNIVERSITY

FACULTY OF MATHEMATICS AND INFORMATICS

{STUDY PROGRAMME} STUDY PROGRAMME

{Work type}

Work Title in English

Work Title in Lithuanian

{Name Surname}

Supervisor	:	{pedagogical/scientific title Name Surname}
Scientific advisor	:	{pedagogical/scientific title Name Surname}
Reviewer	:	{pedagogical/scientific title Name Surname}

Acknowledgements

The author is thankful the Information Technology Research Center, Faculty of Mathematics and Informatics, Vilnius University, for providing the High-Performance Computing (HPC) resources for this research.

You can also add here acknowledgements for various other things, such as your supervisor, university, company, etc.

Summary

Summary in English.

Keywords: work related keywords, with a minimum of 3 keywords, but can be more.

Santrauka

Darbo santrauka.

Raktiniai žodžiai: čia surašomi su darbu susiję raktiniai žodžiai, minimalus raktinių žodžių kiekis – 3, tačiau jų gali būti ir daugiau.

List of Figures

1 pav. Pirmas paveikslėlis	9

List of Tables

1 lentelė. Pavyzdinė lentelė	8

Contents

Summary	5

Santrauka	6

List of Figures	7

List of Tables	8

List of symbols	10

Introduction	12

1.	Formatting	13

1.1.	Mathematical text	13

1.2.	Formulating mathematical results	13

1.3.	Tables	15

1.4.	Images, graphs, charts, photos	15

2.	Presentation of software code	17

2.1.	Algorithms	17

2.1.1.	Subsubsection example	17

Results and conclusions	18

Appendix 1. Examples of citations	21

List of symbols

This section is for when symbols are used. For example:

denotes the mean of the random variable .

List of abbreviations

This section is for when abbreviations are used. For example:

u.d.i.r.v.	uniformly distributed independent random variables.

Introduction

For any written work, you must refer to the methodology guidelines of the respective study programme. These contain all the guidelines for citations, structure, length, etc.

Formatting

This section will give you examples of how to format mathematical text, tables and figures, and describe how to correctly formulate the mathematical results of your final thesis.

Mathematical text

Mathematical formulas can be embedded in paragraphs of text. Example: .

However, formulas will look much nicer if they are separated into separate equations, starting the formula code from a new line. Example equation:

Here you can use the Microsoft Word function to enter mathematical equations. Sometimes formulas take several lines:

(1)

Don't forget to put a full stop (.) at the end of the formula if it is the end of a sentence.

If the formula is needed later, it does not need to be rewritten each time. The formula you need can always be quoted. For example, the formula with the number above is quoted as follows: equation (1).

Formulating mathematical results

Example of a definition:

1.2.1 Definition. A number is called a prime number if it is divisible only by 1 and itself. The set of prime numbers is denoted by .

Example of a proposition:

1.2.2 Proposition. The average of the product of two independent random variables is equal to the product of the averages of the two original variables .

provided that the averages of and exist.

Important mathematical statements are called theorems:

1.2.3 Theorem (First theorem on isomorphism). Suppose that is a homomorphism between the groups and . Then the image of the group is isomorphic to the factor group , that is

.

The shorter auxiliary statements are called lemmas. However, the wording of lemmas can also be quite complex:

1.2.4 Lemma (Vector substitution lemma). Suppose that the vectors of the linear space over the body

(2)

are linearly independent, and that each vector of this family is linearly expressed in vectors

.	(3)

Then , and there exists a subfamily of the vector family(3), which we replace by the vectors , we obtain a family of vectors(2) equivalent to the family(2).

Remark for minor observations:

1.2.5 Remark. The condition of the theorem that the interval is compact and the function is continuous on that interval is essential.

Another environment Example is used for short numerical or formula examples:

1.2.6 Example. Systems of equations

Kramer's formula for solutions:

here

Below you can find a proposition and a proof for that proposition.

1.2.7 Proposition. A square matrix is non-singular if and only if .

Proof. If is non-singular, then there exists a matrix such that . By the determinant property of the product of matrices,

Therefore, . Now suppose that . Let be a transposed adjoint matrix. Then:

So, . Dividing both sides of the identity by , gives . Similarly, we can show that . Hence, is the inverse of the matrix , and hence is non-singular.

Note that mathematical environments must be numbered. Like formulas, mathematical definitions, statements and examples can be quoted elsewhere in the text. For example, we can quote Theorem 1.2.3 or Lemma 1.2.4 in the places in the text where we need to refer to them.

Tables

If tables are presented, the references of the tables should be mentioned in the text, for example: table 1 shows some results.

1 table Tables are numbered at the top, headings at the top

Column 1	Column 2	Column 3

Each table must have a title, which, like the table number, must appear on the same line above the table. All tables shall be numbered consecutively (letter numbering is not recommended, e.g. Table 7a).

Images, graphs, charts, photos

If figures are used in the work, they must be mentioned in the text, e.g. in Figure 1 we see an example of a figure presentation.

1 Figure Figure numbers are at the bottom, caption is at the bottom

Below is the text after the image.

Presentation of software code

This section outlines the way in which software code can be presented.

Algorithms

Algorithms are numbered in the same way as figures or tables. They must be mentioned in the text, e.g. Algorithm 1 is used to find the minimum value of the function .

1 algorithm Gradual descent pseudocode
1: # We assume that is defined in the text
2: Input: dataset
3: Input:initializing random values for parameters
4: Input:learning rate, step size
5: Input:number of epochs
6: for do
7:
8: # The derivative of function is calculated automatically by autograd
9: end for

Subsubsection example

No need to use lots of subsubsections.

Results and conclusions

For details of what needs to be written in this section, please refer to the methodology requirements of the respective programme.

References and sources (Example with numerical numbering)

[1]	OpenAI. ChatGPT. 2024 m. URL: https://www.openai.com/ (žiūrėta 2024-06-06).

[2]	OpenAI. ChatGPT. Atsakymas į klausimą apie citavimo pavyzdį. 2024 m. URL:https://chatgpt.com/share/550cde77-019d-414d-b36a-f3ac9b857b9c (žiūrėta 2024-06-06).

[3]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. „Straipsnio pavadinimas“. Iš: Žurnalo pavadinimas IV (2001 m.), puslapiai 8–17.

[4]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. Knygos pavadinimas. 172 psl. Miestas, šalis. Leidykla, 2003 m.

[5]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. Elektroninės publikacijos pavadinimas. http://example.com/kelias/iki/straipsnio.pdf. 45 KB, tikrinta 2015-02-01. 2004 m.

[6]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. Duomenų rinkinio pavadinimas. 2022 m. URL: nuoroda.

[7]	A. Pavardenis, B. Pavardonis, C. Pavardauskas, D. Pavardinskas „ Straipsnio pavadinimas“. Iš: Rinkinio pavadinimas. Miestas, šalis. Leidykla, 2002 m., puslapiai 3–15.

[8]	A. Pavardonis. „Magistrinio darbo pavadinimas“. Magistro darbas. Universiteto pavadinimas, 2005 m.

[9]	Internetinis puslapis. Puslapio pavadinimas. URL: puslapio nuoroda (žiūrėta 2024-04-25).

[10]	A. Surname, „Ttitle fo PhD thesis“. Disertacija. Title of university, 2005 m.

[11]	A. Surname, B. Tsurname, C. Usurname. „Article Title“. Iš: Journal Title IV (2001 m.), puslapiai 3–15.

[12]	A. Surname, B. Tsurname, C. Usurname. Book title. 172 p. City, country. Publisher, 2003 m.

[13]	A. Surname, B. Tsurname, C. Usurname. Online Publication Title. http://example.com/path/to/the/article.pdf. 45 KB, accessed 2015-02-01. 2004 m.

[14]	A. Surname, B. Tsurname, C. Usurname, D. Vsurname. „Article title“. Iš: Conference book title. City, country. Publisher, 2002 m., puslapiai 3–15.

References and sources (Example with alphabetical numbering)

[Ope24a]	OpenAI. ChatGPT. 2024 m. URL: https://www.openai.com/ (žiūrėta 2024-06-06).

[Ope24b]	OpenAI. ChatGPT. Atsakymas į klausimą apie citavimo pavyzdį. 2024 m. URL:https://chatgpt.com/share/550cde77-019d-414d-b36a-f3ac9b857b9c (žiūrėta 2024-06-06).

[Pav05]	A. Pavardonis. „Magistrinio darbo pavadinimas“. Magistro darbas Universiteto pavadinimas, 2005 m.

[PPP+02]	A. Pavardenis, B. Pavardonis, C. Pavardauskas, D. Pavardinskas. „Straipsnio pavadinimas“. Iš: Rinkinio pavadinimas. Miestas, šalis: Leidykla, 2002 m., puslapiai 3–15.

[PPP01]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. „Straipsnio pavadinimas“. Iš: Žurnalo pavadinimas IV (2001 m.), puslapiai 8–17.

[PPP03]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. Knygos pavadinimas. 172 psl. Miestas, šalis: Leidykla, 2003 m.

[PPP04]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. Elektroninės publikacijos pavadinimas. http://example.com/kelias/iki/straipsnio.pdf. 45 KB, tikrinta 2015-02-01. 2004 m.

[PPP22]	A. Pavardenis, B. Pavardonis, C. Pavardauskas. Duomenų rinkinio pavadinimas. 2022 m. URL: nuoroda.

[pus]	Internetinis puslapis. Puslapio pavadinimas. URL: puslapio nuoroda (žiūrėta 2024-04-25).

[STU+02]	A. Surname, B. Tsurname, C. Usurname, D. Vsurname. „Article title“. Iš: Conference book title. City, country: Publisher, 2002 m., puslapiai 3–15.

[STU01]	A. Surname, B. Tsurname, C. Usurname. „Article Title“. Iš: Journal Title IV (2001 m.), puslapiai 3–15.

[STU03]	A. Surname, B. Tsurname, C. Usurname. Book title. 172 p. City, country: Publisher, 2003 m.

[STU04]	A. Surname, B. Tsurname, C. Usurname. Online Publication Title. http://example.com/path/to/the/article.pdf. 45 KB, accessed 2015-02-01. 2004 m.

[Sur05]	A. Surname, „Ttitle fo PhD thesis“. Disertacija. Title of university, 2005 m.

Appendix 1. Examples of citations

There are 2 sections on „References and sources“, showing different citation styles - numerical and alphabetical - from which you have to choose the one you need (according to the methodology requirements of the respective programme). When citing, the appropriate source designation is added to the text.

The „References and sources“ sections provide examples of some of the most commonly cited types of sources:

web pages [9] or [pus],

datasets [3] or [PPP22],

articles [6, 11] or [PPP01, STU01],

conference papers [7, 14] or [PPP+02, STU+02],

books [5, 12] or [PPP03, STU03],

theses [8, 10] or [Pav05, Sur05],

electronic publications [4, 13] or [PPP04, STU04].

Examples are also given for ChatGPT citations, both in general [1] or [Ope24a] and for a specific conversation [2] or [Ope24b].

Write your study programme (example – Software engineering, Financial and Actuarial Mathematics, etc.)

Bachelor's thesis or Master's thesis

There may be more than one author, in which case each author starts on a new line

If present, otherwise delete

If present, otherwise delete

If you have used IT resources (CPU-h, GPU-h, other IT resources) provided by MIF for your thesis research, please leave the acknowledgement; if you have not, you can delete it.

Added through „Tables of figures“

Pridedamas per „Tables of figures“

Leave if necessary

Leave if necessary

The latest methodology requirements for all programmes can be found here, in the respective programme: https://mif.vu.lt/lt3/en/studies/bachelor-studies