In Hilbert spaces, we are interested in proving the existence of bounded variation continuous solutions to a first-order evolution problem involving time and state dependent maximal monotone operators with perturbations.
This new result is established, under a compactness assumption on the domain of the operators. As an example, we provide the corresponding existence result for the perturbed sweeping process.

Alexander Grothendieck suggested creating a new branch of topology, called by him ``topologie modérée''. In the paper ``On Grothendieck's tame topology" by N. A'Campo, L. Ji and A. Papadopoulos (Handbook of Teichmüller Theory, Volume VI. IRMA Lectures in Mathematics and Theoretical Physics Vol. 27 (2016), pp. 521-533) the authors conclude that no such tame topology has been developed on the purely topological level. We see our theory of sets with distinguished families of subsets, which we call smopologies, as realising Grothendieck's idea and the demands of the mentioned paper. Dropping the requirement of stability under infinite unions makes getting several equivalences of categories of spaces with categories of lattices possible. We show several variants of Stone Duality and Esakia Duality for categories of small spaces or locally small spaces and some subclasses of strictly continuous (or bouned continuous) mappings. Such equivalences are better than the spectral reflector functor for usual topological spaces. In particular, spectralifications of Kolmogorov locally small spaces can be obtained by Stone Duality. Small spaces or locally small spaces seem to be generalised topological spaces. However, it is better to look at them as topological spaces with additional structure. The language of smopologies and bounded continuous mappings simplifies the language of certain Grothendieck sites and permits us to glue together infinite families of definable sets in structures with topologies, which was important in the case of developing o-minimal homotopy theory.

The desymmetrized PSL(2, Z) group and its 'square-box' one-cusp congruence subgroups
The desymmetrized PSL(2, Z) group is considered.
The Fourier coefficients of the non-holomorphic one-cusp Eisenstein series of the PSL(2, Z) group are summed ; for this purpose, the
well-posedness (i.e. the meromorphic continuability of the pertinent objects) is controlled. As a further result, a new
dependence on the Euler's gamma constant is also found.
The structures related to the the desymmetrized PSL(2, Z) congruence subgroups are investigated. As a first instance,
the 'square-box' one-cusp congruence subgroup is constructed.
As a second instance, the commutator subgroups of the 'square-box' one-cusp congruence subgroups of the desymmetrized
PSL(2, Z) group, endowed with two (hyperbolic) reflections, are exaxmined. The corresponding leaky tori are defined.
The possible relations with the modular Monster group are envisaged.

A novel family of distributions called the odd beta prime-G is introduced and studied. Some statistical properties of the proposed odd beta prime-G family are studied. The expressions for the moments, incomplete moments, moment generating function, order statistics, and quantile function are derived. The estimates of the parameters of the new family are obtained using the method of maximum likelihood. Finally, three real data sets are used to illustrate the usefulness and efficiency of the new family.

This paper deals with fractional order three-phase-lag (TPL) thermo-elasticity in a micropolar thermoelastic half space medium with voids. The subsequent non-dimensional coupled equations are solved by using the normal mode analysis and eigenvalue approach methods. By doing numerical computations of the physical fields for a substance that resembles a magnesium crystal in the presence of an electromagnetic field, the issue is proven to exist. The effect of the fractional order, the phase lags on the components of temperature, displacement, the stress, and changes in volume fraction field have been depicted graphically. Additionally, a graphic comparison of several types of models employing phase delays and the influence of the magnetic field is displayed.

Problems are at the heart of mathematics and statistics. The posing of mathematical and statistical problems is more important than the solving of problems. Today, mathematics teachers are realizing that there are many benefits of problem-driven teaching, but they also face a number of challenges, such as a lack of confidence, not knowing how to design high-quality problems, and how to deal with the challenges that emerge in problem-driven mathematics and statistics classes. To answer these questions, we propose a model for problem-based instructional design which concludes with three stages: The stage of preparation of the problems (generating new problems based on textbooks; generating new problems based on mathematical, scientific, and life situations, imagining solutions of the prepared problems); The stage of implementation of teaching (teachers teach based on prepared problems and pose new problems in real-time, and students solve them or pose new problems), evaluation and look-back stage (evaluating the quality of teaching and the quality of problems, improving the instruction). Accordingly, we select the topic of 'Estimating the whole from a sample' in the 'General Senior Secondary Textbook, Mathematics, Compulsory Book 2 (People's Education Press, 2019)” to design a problem-driven instruction and as an application of problem posing. In the instruction, we will pose several interesting problems related to the topic. These problems can successfully stimulate students' interest in learning statistics. Through these problems, the maths teacher can guide the students to think actively, and pose and solve the problem themselves, thus understanding the related concepts and methods in statistics. In addition, the design model applied here is a ready-to-use tool for teachers in designing specifically problem-driven instruction and also can improve teachers' problem-posing skills.

In this paper, we consider simple opportunity-based block replacement problems, where the inter-arrival times of replacement opportunities arrvie randomly. We give two replacement models under replacement first and replacement last disciplines in the sense of Zhao and Nakagawa (2012). Then, we derivethe optimal opportunity-based block replacemnt policies under replacement first and replacement last disciples. Furthermore, numerical examples are presented to compare these repacement polices.

TThe strong approximation of a function is a useful tool to analyze the convergence of its Fourier series. It is based on the summability techniques. However, unlike matrix summability methods, it uses non-linear methods to derive an auxiliary sequence using approximation errors
generated by the series under analysis. In this paper, we give some direct results on the strong means of Fourier series of functions in generalized Holder and Zygmund spaces. To elaborate its use, we deduce some corollaries and a discussion follows the results.

The problem of counting independent sets of a graph G, denoted by i(G), is not only mathematically relevant and interesting, but also has many applications in physics, mathematics, and theoretical computer science. Regarding hard counting problems, the computation of i(G) for a graph G has been a key for determining the frontier between efficient counting and intractable counting procedures. In this article, a novel algorithm for counting independent sets on grid-like structures is presented.

We propose a novel algorithm for the computation of i(Gm,n) for a grid graph with m rows and n columns, based on the ‘Branch and Bound’ design technique. The splitting rule in our proposal is based on the well-known vertex reduction rule. The vertex in any subgraph from Gm,n to be selected for the vertex reduction rule must have degree 4. Our proposal consists of decomposing a grid graph until obtaining subgraphs without vertices of degree 4, which are called ‘basic subgrids’. We show how to compute efficiently the number of independent sets for those basic subgrids graphs, and we show which are the new graph topologies expressed by such basic subgrids.

The resulting time-complexity of our proposal for computing the number of independent sets for grid graphs is dramatically inferior to the time-complexity that the classic transfer matrix method requires for computing the same value.

Mathematical models of computer systems and telecommunication networks are based on distributions with heavy tails. This paper falls into category exploring the classical models with Gnedenko-Weibull, Burr, Benktander, log-normal, log-gamma distributions. The moment`s asymptotics for remaining time have been derived especially for this heavy-tailed distributions. An asymptotic expansions are also obtained for the tails moments of regularly varying functions.