International Monthly Seminar on Time Scales Analysis

In this talk, we shall investigate the validity of the Wirtinger inequality within the framework of time scales, a unified approach to continuous and discrete analysis. By constructing explicit counterexamples, we demonstrate that the classical Wirtinger inequality does not hold universally across all time scales. Motivated by this finding, we propose a reformulation of the inequality by modifying its underlying conditions. Additionally, we establish several new improved Wirtinger-like inequalities, extending the theoretical foundation of the inequality on time scales.

1. INTRODUCTION
2. PRELIMINARIES
   3.PRODUCT OF TOEPLITZ MATRIX AND kTH-ORDER SLANT TOEPLITZ MATRIX.

In this work, the uniform stabilization of certain hyperbolic systems with Wentzell boundary conditions is considered, and a uniform energy decay rate for the problem is established, taking into account both internal localized damping and boundary feedback. The exponential stabilization is attained by constructing a new multiplier and using multiplier methods.

In this work, we investigate the existence and decay properties of global solutions for a class of second-order evolution equations incorporating memory effects, a nonlinear delay term, and a time-varying weight function. The model reflects realistic dynamics observed in viscoelastic and thermoelastic systems with hereditary characteristics and delayed feedback. Using appropriate energy methods and the construction of a Lyapunov functional, we establish the global existence of solutions under suitable assumptions on the kernel, delay, and nonlinearity. Furthermore, we derive general decay estimates for the energy, which unify and extend various known exponential and polynomial decay results. These findings contribute to the understanding of long-term behavior in complex dynamical systems with combined memory and delay effects.

In this work, we introduce a perturbed non-convex sweeping process with a class of subsmooth moving sets depending on the time and the state. In the first result we study the existence of solution and we present some topological properties of the attainable set, the perturbation considered here is an upper semi-continuous set-valued mapping with nonempty closed convex values unnecessarily bounded. In the second result we prove the existence to the minimal time problem and we give a description to the attainable set of control systems.

In this conversation, examines the long-term existence of solutions for a system of weakly coupled equations involving fractional evolution and various nonlinearities. The main focus is on analyzing the relationship between the regularity of initial data, memory terms, and the allowable range of exponents in a specific equation. Using L^p–L^q estimates for solutions of associated linear fractional $σ$-evolution equations with vanishing right-hand sides, along with a fixed-point method, the study establishes the existence of small-data solutions with in certain admissible exponent ranges.

We discuss existence results for solutions to some classes of nonlinear impulsive dynamic equations on time scales. By applying modern techniques in fixed point theory, specifically expansive mappings, -set contractions, and cone methods, we derive criteria for the existence of at least one and multiple solutions. Our methodology removes the standard boundedness assumptions often required for nonlinear terms, thereby allowing for more general and flexible terms in the governing equations. Furthermore, we discuss the extension of these frameworks to fractional impulsive systems, emphasizing the integration of fixed point index theory with expansive operators. The theoretical findings are supported by illustrative examples that demonstrate their practical utility and relevance.

This talk provides a definition of an α-integrated semi-group
on a time scale, and we prove some properties associated with this concept. It also examines the relationship between the infinitesimal generator of the α-integrated semi-group on a time scale.

Using tools from generalized quantum calculus, we derive new delay-dependent versions of classical integral inequalities of Chebyshev, Poincaré, Sobolev, Opial, and Ostrowski type. These results extend and refine existing inequalities by incorporating delayed arguments, leading to improved bounds. The proposed inequalities are well suited for applications in the analysis of delayed dynamic equations on time scales and related quantum models.

In extreme value theory (EVT), estimating the tail index of heavy-tailed distributions is crucial for understanding rare and extreme events. Traditional estimators such as the Hill and Maximum Likelihood Estimators (MLE) perform well with large samples but struggle
with small sample sizes due to increased bias and variance. In this paper, we introduce a novel estimation technique the Maximum Lq-Likelihood Estimator (MLqE), which incorporates a distortion parameter q, making it more robust to extreme observations and more accurate in small-sample scenarios. We demonstrate that the MLqE is consistent and asymptotically normal, outperforming the classical MLE in terms of mean squared error in
moderate and small sample sizes. Moreover, we present simulation results that highlight the superior performance of the MLqE, particularly when comparing it to the MLE in tail
index estimation. This method not only offers a significant improvement in the accuracy of heavy-tailed distribution parameter estimation but also provides a versatile tool for various
real-world applications, including finance, hydrology, and risk management.

The present presentation investigates the two-dimensional Euler-Boussinesq system with critical fractional dissipation and a general source term, where we assume that the initial data are of Yudovich type.