It is shown that, in the cyclic tower of Hanoi with $4$ pegs and $N$ disks, there exists an initial configuration of disks and a valid sequence of moves, after which the configuration has made a half-turn, where the number of disk moves grows in $O(Z^N)$, where $Z\simeq 1.69562$ is the unique real root of the equation $Z^3=Z^2+2$. This result is consistent with some numerical experiments and conjectures made by Paul Zimmermann in 2017. Then, the optimality of the number of disk moves obtained by this process is discussed.

The Leray-alpha model to regularize the forced three-dimensional periodic Boussinesq system is shown to have a global in time unique solution and a universal attractor which is connected. Energy method, compactness argument and absorbing balls are used.

We revisit and generalize the geometric procedure of regularizing a sequence of real numbers with respect to a so-called regularizing function. This approach is due to S. Mandelbrojt and becomes useful and necessary when working with corresponding classes of ultradifferentiable functions defined via weight sequences and analogous weighted spaces. In this note we also study non-standard situations for the construction yielding the (log-)convex minorant of a sequence and allow a ``blow-up'' for the regularizing function.

We establish several consequences of Burnside's vanishing property \cite{b-galois}. One well-known implication of this property is Harada's identity, which concerns the product of all conjugacy class sums of a finite group and is derived from the vanishing property of characters. Expanding on this idea, we present a similar formula applicable to any weakly-integral fusion category. As an application we show that any modular integral tensor category of dimension $p^2q^2r^2d$ with $p\prec q\prec r$ prime numbers and $d$ a square-free integer is weakly-group theoretical.

It is proved that every pair of sufficiently large even integers can be represented as not only a pair of one prime, one square of prime, one cube of prime, one biquadrate of prime and $34$ powers of $2$, but also a pair of two squares of primes, two cubes of primes, two biquadrates of primes and $36$ powers of $2$.

We prove that the loop Heisenberg-Virasoro Lie algebra possesses non-trivial $\frac{1}{2}$-derivations, but it does not admit any non-trivial transposed Poisson algebra structures. Furthermore, we demonstrate that this algebra has a non-trivial Hom-Lie algebra structure.

We study $\sigma$-spanned lattices, that is, complete lattices where the complete join or meet of any set is the join or meet of a countable subset. We derive that for a $\sigma$-finite measure space, the set of equi\-valence classes of measurable functions (with values in a closed subset of $\mathbb{R}\cup\{-\infty,+\infty\}$) modulo equality almost everywhere (a.e.) is an infinitely distributive $\sigma$-spanned lattice, and the same holds for equivalence classes of measurable sets; this slightly improves a result from functional analysis. Finally, we give an interpretation of the essential supremum and infimum in terms of adjunctions. Our results can provide a basis for a lattice-theoretical form of functional analysis and harmonic analysis, in particular in mathematical morphology.

By means of matrix decompositions, we examine three classes of Hankel matrices, whose entries are reciprocal factorials, factorial quotients and binomial coefficients. Their determinants are evaluated in closed form and inverses are explicitly determined.