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2. On the Exceptional Set of Transcendental Entire Functions in Several Variables

Author

Alves, Diego, Lelis, Jean, Marques, Diego, and Trojovský, Pavel

Subjects
Mathematics - Number Theory, Mathematics - Complex Variables
Abstract

In this paper, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountable many transcendental entire functions over $\mathbb{C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions.

Published
2023

3. A Note On Transcendental Analytic Functions With Rational Coefficients Mapping $\mathbb{Q}$ Into Itself

Author

Lelis, Jean, Marques, Diego, Moreira, Carlos Gustavo, and Trojovský, Pavel

Subjects
Mathematics - Number Theory, Mathematics - Complex Variables, Primary 11J82, Secondary 30Dxx
Abstract

In this note, the main focus is on a question about transcendental entire functions mapping $\mathbb{Q}$ into $\mathbb{Q}$ (which is related to a Mahler's problem). In particular, we prove that, for any $t>0$, there is no a transcendental entire function $f\in\mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq\mathbb{Q}$ and whose denominator of $f(p/q)$ is $O(q^{t})$, for all rational numbers $p/q$, with $q$ sufficiently large.

Published
2023

4. Algebraic periodic points of transcendental entire functions

Author

Krumm, David, Marques, Diego, Moreira, Carlos Gustavo, and Trojovský, Pavel

Subjects
Mathematics - Number Theory, Primary 37P10, Secondary 11Jxx
Abstract

We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and in addition having a prescribed number of $k$-periodic algebraic orbits, for all $k\geq 1$. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.

Published
2023

33. Deep learning model for deep fake face recognition and detection

Author

Suganthi ST, Mohamed Uvaze Ahamed Ayoobkhan, Krishna Kumar V, Nebojsa Bacanin, Venkatachalam K, Hubálovský Štěpán, and Trojovský Pavel

Subjects
Deep fake, Fisherface, LBPH, DBN, RBM, Deep learning, Electronic computers. Computer science, QA75.5-76.95
Abstract

Deep Learning is an effective technique and used in various fields of natural language processing, computer vision, image processing and machine vision. Deep fakes uses deep learning technique to synthesis and manipulate image of a person in which human beings cannot distinguish the fake one. By using generative adversarial neural networks (GAN) deep fakes are generated which may threaten the public. Detecting deep fake image content plays a vital role. Many research works have been done in detection of deep fakes in image manipulation. The main issues in the existing techniques are inaccurate, consumption time is high. In this work we implement detecting of deep fake face image analysis using deep learning technique of fisherface using Local Binary Pattern Histogram (FF-LBPH). Fisherface algorithm is used to recognize the face by reduction of the dimension in the face space using LBPH. Then apply DBN with RBM for deep fake detection classifier. The public data sets used in this work are FFHQ, 100K-Faces DFFD, CASIA-WebFace.

Published
2022
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44. On Diophantine equations involving Lucas sequences

Author

Trojovský Pavel

Subjects
p-adic order, fibonacci number, lucas sequence, order of appearance, tridiagonal matrix, 11b37, 11b39, Mathematics, QA1-939
Abstract

In this paper, we shall study the Diophantine equation un = R(m)P(m)Q(m), where un is a Lucas sequence and R, P and Q are polynomials (under weak assumptions).

Published
2019
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