According to Ruckle's 1973 paper 'FK Spaces in which the sequence of coordinate vectors is bounded' in the Canadian Journal of Mathematics (Vol. 25, pp. 973-978), a modulus function f is defined as a function from [0, ∞) to [0, ∞) that satisfies the following conditions:

1. f(x)=0 if and only if x=0,

2. f(x + y) ≤ f(x) + f(x) for all x,y ≥ 0,

3. f is increasing,

4. f is continuous from the right at zero.

The goal of this talk is to define and study a new sequence spaces that are defined using the modulus function and infinite matrix A. Further, some inclusion theorems are proved.

Keywords: σ−convergence, Infinite matrices, inclusion theorems, Banach limits

See full Abstract here
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.

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.

We investigate the geometric properties of functions through their Caputo fractional derivative. We show that concavity and monotonicity properties of functions can be explored via the Caputo derivative, provided that the fractional derivative of the function is of one sign for some values of alpha. Analogues result for the global extrema of a function is obtained. However, to release the condition on (K. Diethelm, 2016) that the fractional derivative of the function is of one sign for all values of alpha in certain domain, higher order fractional inequalities is required. We also present some recent results concerning the fractional derivatives of function at their extreme points. The applicability of the presented results will be discussed, where it is noticed that for some types of functions dealing with fractional derivative is more practical than the integer derivative

Keywords: Fractional Calculus, Caputo derivative, monotone function, function concavity, extreme points, fractional differential equations.

The primary goal of this goal is to propose some new forms of fractional derivatives and integrals. These new forms are developed by using at the same time the proportional and the weighed derivative of a function with respect to another function

Keywords: Riemann-Liouville fractional operators, proportional fractional operators, Caputo Fractional derivatives, weighted fractional operators.

In this study, we consider the problem of finding an initial condition for a class of linear biparabolic equations in the abstract setting. To overcome the difficulties of instability caused by the ill-posed character of the problem in question, we propose a regularization strategy based on the spectral Krylov projection method. We prove the convergence of this approximation, and explicitly establish error estimation in the discrete case. The stopping criterion is based on a posteriori strategy by using a geometric measurement based on the notion of the principal angle between two subspaces. Finally, we ran numerical tests to justify the efficiency and accuracy of the numerical results obtained compared with the theoretical results.

Keywords: Ill-posed problems, Biparabolic problem, regularization, Krylov subspaces, Matrix functions approximation.

In this talk, we give an overview of some classes of ill-posed problems in PDEs and their regularization methods. We show that we can extend these analytical tools for new non-classical problems, and we also give new numerical strategies to approach these problems, which are characterized by an unstable and difficult nature.

Keywords: PDEs, Ill-posed problems, regularization methods.

MSC[2020]: 35R25, 35R30, 65F22, 65F45, 65F60.