announcement

Using Artificial Neural Networks in Solving Differential Equations

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The theme of our career event this month is “Generative Artificial Intelligence and Our Profession” and within the scope of the theme, a seminar titled “Using Artificial Neural Networks in Solving Differential Equations” will be given by Dr. Cem ÇELİK in our department. The seminar is open to everyone who is interested.

Speaker: Dr. Cem ÇELİK (DEU, Department of Mathematics)
Title: Using Artificial Neural Networks in Solving Differential Equations
Date and Time: 19.03.2025 (Wednesday),  12:00
Place: B256 (DEU Faculty of Science Department of Mathematics Block B 2nd floor)

Abstract: The intersection of scientific computing and machine learning has given rise to a paradigm known as Scientific Machine Learning (SciML). This talk will explore the principles of SciML, with a particular focus on Physics-Informed Neural Networks (PINNs), a novel approach that integrates the principles of physics with machine learning techniques to solve complex scientific and engineering problems. By embedding physical laws directly into the neural network architecture, PINNs enable the modeling of systems governed by partial differential equations (PDEs) while leveraging the strengths of data-driven methods. This presentation will cover the foundational concepts of PINNs through practical examples, and participants will gain insights into the potential of PINNs to revolutionize problem-solving.

Actuarial Career for Mathematics Students

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The theme of our career event this month is “Actuarial” and a talk will be held with our 2023 graduate Mustafa EROL on the subject of “Actuarial Career for Mathematics Students”. The event is open to all mathematics students and anyone interested.

Speaker: Mustafa Erol (AXA Insurance Actuarial Assistant Specialist)
Subject: Actuarial Career for Mathematics Students
Date and Time: 13.01.2025, 12:00
Place: B256 (Faculty of Science, Department of Mathematics)
Moderator: Asst.Prof.Dr. Celal Cem SARIOĞLU

Fundamental Theorem of Symmetric Polynomials, Newton’s Identities and Discriminants

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Mustafa Eren Taşlı, Dokuz Eylül University. Date: 26th of December, 2024, Wednesday, Time: 15.00 – 16.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will define symmetric polynomials and the elementary symmetric polynomials in n indeterminates over a field F. The elementary symmetric polynomials in the indeterminates x_1, x_2, \ldots, x_n are as follows:

    \begin{align*} \sigma_1=&x_1+x_2+\ldots+x_n \\ \sigma_2=&\sum_{1 \le i < j \le n} x_ix_j \\ \vdots\\ \sigma_n =&x_1 x_2 \ldots x_n \end{align*}

The Fundamental Theorem of Symmetric Polynomials states that any symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials, that is:

Theorem. Let f(x_1, x_2, \ldots, x_n) be a symmetric polynomial in the n indeterminates x_1, x_2, \ldots, x_n over a field F. Then, there exists a polynomial g(y_1, y_2, \ldots, y_n) in the n indeterminates y_1, y_2, \ldots, y_n such that

    \[f(x_1, x_2, \ldots, x_n) = g(\sigma_1, \sigma_2, \ldots, \sigma_n),\]

where \sigma_1, \sigma_2, \ldots, \sigma_n are the above elementary symmetric polynomials of the n indeterminates x_1, x_2, \ldots, x_n. Moreover, the polynomial g(y_1, y_2, \ldots, y_n) is uniquely determined.

We will prove this theorem using the graded lexicographic order for multivariable polynomials.

Using the recurrence relation from the Newton Identities, we will learn how to express the sum of powers of the indeterminates, that is, the polyomials

    \[s_k = x_1^k + x_2^k + \ldots + x_n^k\]

for a positive integer k, as polynomials in terms of the elementary symmetric polynomials. We will reinforce this understanding with examples.

The discriminant in the indeterminates x_1, x_2, \ldots, x_n over the field F is given by:

    \[\Delta = \prod_{1 \leq i < j \leq n} (x_i - x_j)^2 \in F[x_1, \dots, x_n].\]

The discriminant is a symmetric polynomial, and we will express it in terms of the elementary symmetric polynomials using determinants.

This seminar, as part of my graduation project titled Symmetric Polynomials, Newton’s Identities, Discriminants, and Resultants, serves as an introduction to a method for calculating the discriminant (\Delta) of an n-th degree polynomial without finding its roots.

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Cardano’s Formula and Casus Irreducibilis

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Çağdaş Çiğdemoğlu, Dokuz Eylül University. Date: 18th of December, 2024, Wednesday, Time: 15.00 – 16.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will start with a general monic cubic equation in the form

    \[ x^3 + bx^2 + cx + d = 0, \]

and transform it into the following form

    \[ y^3 + py + q = 0, \]

using a substitution. Then, we will construct the Cardano Formulas to find the roots of the equation. The roots are expressed as follows:

    \[ y_1 = \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}}, \]

    \[ y_2 = \omega \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega^2 \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}}, \]

    \[ y_3 = \omega^2 \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}}, \]

where \omega = e^{i \frac{2\pi}{3}} = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2} is a primitive cube root of unity and where the above cube roots are one of the three complex cube roots whose product is -\frac{p}{3} and these are fixed in the above formulas.

We will learn about the discriminant \Delta of the above monic cubic polynomial, understand its significance, and examine how the roots change depending on the value of the discriminant.

For the monic cubic polynomial in the form y^3 + py + q, the discriminant is expressed by:

    \[ \Delta = -27q^2 - 4p^3 = (y_1-y_2)^2(y_1-y_3)^2(y_2-y_3)^2. \]

For the general monic cubic polynomial x^3 + bx^2 + cx + d, whose roots are x_1, x_2, x_3, the discriminant is expressed by:

    \[ \Delta = b^2c^2 + 18bcd - 4c^3 - 4b^3d - 27d^2 = (x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2. \]

We will also discuss Casus Irreducibilis, which occurs when the coefficients b, c, d (or p, q) are in a subfield of F of \mathbb{R}, the cubic polynomial is irreducible over F (equivalently, the cubic equation has no roots in F) and when the discriminant is positive. In this case, the cubic equation has three distinct real roots that cannot be expressed using real radicals.

This seminar serves as an introduction to my project, which focuses on understanding the proof of Casus Irreducibilis using Galois Theory.

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Cryptography and Mathematics

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This month’s career event theme is “Cryptography and Mathematics”. We are pleased to host a talk on this subject with Beste AKDOĞAN KÖSEMEN, a 2016 graduate of our department and currently pursuing her Ph.D. in Cryptography at METU Institute of Applied Mathematics. The event is open to all mathematics department students and anyone interested. (Non-mathematics department participants are kindly requested to contact the moderator for attendance.)

Speaker: Res. Asst. Beste AKDOĞAN KÖSEMEN (Ankara Yıldırım Beyazıt University, Mathematics Department & METU Institute of Applied Mathematics)
Topic: Mathematics and Cryptography
Date & Time: December 23, 2024, 12:00 PM
Venue: online.deu.edu.tr
Channel: DEUMatematikKARİYER
Moderator: Asst.Prof.Dr. Celal Cem SARIOĞLU

 

From Mathematics to Codes: Shaping the Future with Artificial Intelligence

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The theme of our career event this month, organized by DEU Faculty of Science, Department of Mathematics in collaboration with DEU Career Planning Center, is “Career Tips from Young Graduates”. Everyone interested in the talk titled “From Mathematics to Codes: Shaping the Future with Artificial Intelligence”, which we will hold with our 2024 graduate of the Department of Mathematics “Şule YALIM”, who works as an expert software developer at Morfoz AI, is invited. (Participants other than students of the Department of Mathematics are kindly requested to contact the moderator for event participation.)

Speaker: Şule YALIM (DEU Mathematics 2024 Graduate / Morfoz AI, Expert Software Developer)
Moderator: Asst. Prof. Dr.  Celal Cem SARIOĞLU
Date and Time: 30.11.2024, 19:00
Place: online.deu.edu.tr
Chanel: DEUMatematikKARİYER

How to Write a TUBITAK 2209 A/B Project? What Should be Considered?

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The theme of our career event this month is “+1 Step in Career: I am Writing a TÜBİTAK 2209 Project”. In our event, information will be given about TÜBİTAK 2209 University Student Research Projects and a conversation will be held on the points to be considered when writing a project. The event is open to all mathematics department students and those who are interested.

Speaker: Dr. Öğr. Üyesi Celal Cem SARIOĞLU
Date and Time: 25.10.2024, 12:15
Place: B256 (DEU Faculty of Science, Department of Mathematics)

 

2024 – 2025 Foreign Language Proficiency Exams

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Foreign language exemption exams will be held by the Directorate of the School of Foreign Languages ​​on September 3-4, 2024, for students who are newly registered to undergraduate programs whose language of instruction is completely or partially in a foreign language in the 2024-2025 academic year (including international students), students who failed the preparatory education the year before, and students who will optionally study in a foreign language. Students who will receive compulsory and optional preparatory education are required to take this exam.