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We consider a finite dimensional k-space V with a linear operator T which is nilpotent. We are interested in the subspaces U subsets of V which are T-invariant. To be precise, we look for all such triples (V,U,T) and ask for a classification of the isomorphism classes...

We illustrate an application. In control theory, there is a derivative operator acting on the state space, while the controllable and the non-observable subspaces are both invariant under this operator. I will report about a project of my former student Audrey Moore where she determines the possible systems, up to isomporphy, in case the operator acts nilpotently with nilpotency index at most 4.

As time permits, I will speak about a recent research project with Claus Michael Ringel. Any nilpotent linear operator acting on a finite dimensional space is determined by the number b and the sizes of its Jordan blocks. We investigate properties of the Jordan mean v/b ... and the Jordan level u/b of the systems (V,U,T) in the category S(n).

Keywords: Linear operators, invariant subspaces, isomorphism invariants, controllable and non-observable subspaces, Jordan mean and Jordan level.

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This talk focuses on a survey and some remarks on inverse and ill-posed problems for anomalous diffusion phenomena. We present new results on regularization methods that have been recently established for backward and generalized elliptic equations, along with unresolved questions in this field.

Keywords: Ill-posed problems, backward problem, time-fractional evolution equations, Generalized elliptic equations, regularization methods.

B. Kaltenbacher, W. Rundell. Inverse Problems for Fractional Partial Differential Equations

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Nabla discrete fractional Mittag-Leffler (ML) functions are the key to discrete fractional calculus within nabla analysis, since they extend nabla discrete exponential functions. We first give a quick review of nabla and delta discrete ML functions. Then, we define two new nabla discrete ML functions depending on the Cayley-exponential function on time scales. While, the nabla discrete ML function of (lambda, t) converges for absolute value of lambda less than 1,
both of the defined discrete functions converge for more relaxed lambda.
The nabla discrete Laplace transforms of the newly defined functions are calculated and confirmed as well. Some illustrative graphs for the two extensions are provided.

Keywords: Nabla fractional sum, Nabla Caputo fractional difference, discrete exponential function, discrete ML function, discrete Cayley-exponential function.

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We prove the existence of solutions for a boundary value problem involving the p-Laplacian, where S is a nested fractal set (we especially consider the Sierpinski Gasket as a specific example) on R^{N-1}$ for N> 3,
S_0 is its boundary, a: S -> R are appropriate functions and alpha, beta and p are reals satisfying an adequate hypothesis.

Keywords: Nested fractals, Sierpinski gasket, Critical point,  p-Laplacian, Variational, Nehari Manifold

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Multiplicative calculus offers an alternative framework to classical (additive) calculus, particularly suited for modeling processes characterized by proportional growth or exponential behavior. In this talk, we explore the foundational concepts of multiplicative calculus, including multiplicative derivatives and integrals, and highlight their distinct properties compared to their classical counterparts. We also discuss recent developments in the field, focusing on multiplicative fractional operators and their potential in extending the analytical toolkit for nonlinear and scale-invariant systems. Applications in various disciplines, such as mathematical biology, finance, and physics, will be outlined to demonstrate the utility and relevance of this emerging approach. The presentation aims to provide both a comprehensive overview of the theory and a glimpse into ongoing research directions and open problems in the domain of multiplicative calculus.

Keywords: Multiplicative calculus, framework, proportional growth, multiplicative derivatives and integrals.