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Omran Kouba

Publications and source records attributed to Omran Kouba.

At least 19 recordsLinked to original sources
arXiv2023

Self-Excited Dynamics of Discrete-Time Lur'e Models with Affinely Constrained, Piecewise-C1 Feedback Nonlinearities

Self-excited systems (SES) arise in numerous applications, such as fluid-structure interaction, combustion, and biochemical systems. In support of system identification and digital control of SES, this paper analyzes discrete-time Lur'e models with affinely constrained, piecewise-C1 feedback nonlinearities. The main result provides sufficient conditions under which a discrete-time Lur'e model is self-excited in the sense that its response is 1) bounded for all initial conditions, and 2) nonconvergent for almost all initial conditions.

eess.SY↗
arXiv2022

Euler's Equation via Lagrangian Dynamics with Generalized Coordinates

Euler's equation relates the change in angular momentum of a rigid body to the applied torque. This paper fills a gap in the literature by using Lagrangian dynamics to derive Euler's equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, unit quaternions.

math.DS↗
arXiv2022

New identities obtained from Gegenbauer series expansion

Using the expansion in a Fourier-Gegenbauer series, we prove several identities that extend and generalize known results. In particular, it is proved among other results, that \begin{equation*} \sum_{n=0}^\infty\frac{1}{4^n}\binom{2n}{n}\frac{z-2n}{\binom{z-1/2}{n}}\binom{z}{n}^3 =\frac{\tan(πz)}π \end{equation*} for all complex numbers $z$ such that $\Re(z)>-\frac{1}{2}$ and $z\notin\frac{1}{2}+\mathbb{Z}$.

math.CA↗
arXiv2022

On a combinatorial identity of Chaundy and Bullard

We give two new proofs of the Chaundy-Bullard formula $$ (1-x)^{n+1} \sum_{k=0}^m {n+k\choose k} x^k +x^{m+1}\sum_{k=0}^n {m+k\choose k} (1-x)^k=1 $$ and we prove the "twin formula" $$ \frac{ (1-x)^{(n+1)}}{(n+1)!} \sum_{k=0}^m \frac{n+1}{n+k+1} \frac{ x^{(k)}}{k!} + \frac{ x^{(m+1)}}{(m+1)!} \sum_{k=0}^n \frac{m+1}{m+k+1} \frac{ (1-x)^{(k)}}{k!}=1, $$ where $z^{(n)}$ denotes the rising factorial. Moreover, we present identities involving the incomplete beta function and a certain combinatorial sum.

math.GM↗
arXiv2021

Self-Excited Dynamics of Discrete-Time Lur'e Systems

Self-excited systems arise in numerous applications, such as biochemical systems, fluid-structure interaction, and combustion. This paper analyzes a discrete-time Lur'e system with a piecewise-linear saturation feedback nonlinearity. The main result provides sufficient conditions under which the Lur'e system is self-excited in the sense that its response is bounded and nonconvergent.

eess.SY↗
arXiv2020

A Discrete-Time, Time-Delayed Lur'e Model with Biased Self-Excited Oscillations

Self-excited systems arise in many applications, such as biochemical systems, mechanical systems with fluid-structure interaction, and fuel-driven systems with combustion dynamics. This paper presents a Lur'e model that exhibits biased self-excited oscillations under constant inputs. The model involves asymptotically stable linear dynamics, time delay, a washout filter, and a saturation nonlinearity. For all sufficiently large scalings of the loop transfer function, these components cause divergence under small signal levels and decay under large signal amplitudes, thus producing an oscillatory response. A bias-generation mechanism is used to specify the mean of the oscillation. The main contribution of the paper is a detailed analysis of a discrete-time version of this model.

nlin.AO↗
arXiv2019

Counting Colorful Necklaces and Bracelets in Three Colors

A necklace or bracelet is \textit{colorful} if no pair of adjacent beads are the same color. In addition, two necklaces are \textit{equivalent} if one results from the other by permuting its colors, and two bracelets are \textit{equivalent} if one results from the other by either permuting its colors or reversing the order of the beads; a bracelet is thus a necklace that can be turned over. This note counts the number $K(n)$ of non-equivalent colorful necklaces and the number $K'(n)$ of colorful bracelets formed with $n$-beads in at most three colors. Expressions obtained for $K'(n)$ simplify expressions given by OEIS sequence A114438, while the expressions given for $K(n)$ appear to be new and are not included in OEIS.

math.CO↗
arXiv2016

Elementary evaluation of $\int_0^\infty{sin^p t/t^q} dt$

Let $p$ and $q$ be two positive integers, the goal of this note is to demonstrate, in a very simple and elementary way without using advanced tools, a formula to express the value of the integral $I(p,q)=\int_0^\infty{\sin^p t\over t^q}dt$ when it converges.

math.GM↗
arXiv2016

Lecture Notes, Bernoulli Polynomials and Applications

In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications to these polynomials are presented, including a unified approach to the asymptotic expansion of the error term in many numerical quadrature formulae, and many new and sharp inequalities, that bound some trigonometric sums.

math.CA↗
arXiv2014

A Generalization of Riemann Sums

We generalize the property that Riemann sums of a continuous function corresponding to equidistant subdivision of an interval converge to the integral of that function, and we give some applications of this generalization.

math.CA↗
arXiv2014

A Reverse Hilbert-like Optimal Inequality

We prove an inequality on positive real numbers, that looks like a reverse to the well-known Hilbert inequality, and we use some unusual techniques from Fourier analysis to prove that this inequality is optimal.

math.CA↗
arXiv2013

A Mixed Parseval-Plancherel Formula

In this note, a general formula is proved. It expresses the integral on the line of the product of a function $f$ and a periodic function $g$ in terms of the Fourier transform of $f$ and the Fourier coefficients of $g$. This allows the evaluation of some oscillatory integrals.

math.CA↗
arXiv2012

An Optimal Inequality For The Tangent Function

In this note we deal with some inequalities for the tangent function that are valid for $x$ in $(-π/2,π/2)$. These inequalities are optimal in the sense that the best values of the exponents involved are obtained.

math.CA↗
arXiv2012

An Inequality for Bounded Functions

In this note we prove optimal inequalities for bounded functions in terms of their deviation from their mean. These results extend and generalize some known inequalities due to Thong (2011) and Perfetti (2011)

math.CA↗
arXiv2012

The Sum of Certain Series Related to Harmonic Numbers

In this paper, we consider three families of numerical series with general terms containing the harmonic numbers, and we use simple methods from classical and complex analysis to find explicit formulas for their respective sums.

math.CA↗