Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach to characterizing Lie polynomials on the generators [Formula: see text] of an algebra with a defining relation in the form of a twisted commutation relation [Formula: see text]. Here, the twisting map [Formula: see text] is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses [Formula: see text]-deformed Heisenberg algebras, rotation algebras, and some types of [Formula: see text]-oscillator algebras, the deformation parameters of which, are not roots of unity. Thus, we have a general solution for the Lie polynomial characterization problem for these algebras.

An elementary approach to subdiffusion

We consider a basic one-dimensional model which allows to obtain a diversity of diffusive regimes whose speed depends on the moments of a per-site trapping time. This models a discrete subordinated random walk, closely related to the continuous time random walks widely studied in the literature. The model we consider lends itself to a detailed elementary treatment, based on the study of recurrence relation for the time-[Formula: see text] dispersion of the process, making it possible to study deviations from normality due to finite time effects.

Generalization of Heron’s and Brahmagupta’s equalities to any cyclic polygon

It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having $$n>4$$ n > 4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

Necessary Optimality Conditions for a Class of Control Problems with State Constraint

An elementary approach to a class of optimal control problems with pathwise state constraint is proposed. Based on spike variations of control, it yields simple proofs and constructive necessary conditions, including some new characterizations of optimal control. Two examples are discussed.

DOPPELGANGERS AND MATRIXES: A NEW METHOD OF COMPARATIVE ANALYSIS

The paper presents the results of the research of doppelganger topos by means of quantitative analysis, namely, by elementary approach provided by graph theory. The idea to employ the approach belongs to M. Zhuravlev. Seven canonic texts chosen by I. Golovacheva and M. Zhuravlev (E. T. A. Hoffmann’s “Die Doppeltgänger”, E. A. Poe’s “William Wilson”, Fyodor Dostoevsky’s novella “The Double”, Henry James’s “The Jolly Corner”, Joseph Conrad’s “The Secret Sharer” and Vladimir Nabokov’s Despair) are analyzed. Our point of departure was the archetypal attributes of the double highlighted in a seminal work by Otto Rank of 1914. Still, Rankean list of attributes had to be supplemented by a list of other, specifically ‘poetic’, attributes, by a set of diverse invariant motifs found in the above works of fiction, specified by I. Golovacheva and P. de Mauny. Incidence matrixes built for each of the four specific groups of attributes of the double allowed us to view the doppelganger poetics as a whole and each of the chosen seven texts in particular. The quantitative method led to several new qualitative conclusions. For example, it demonstrated that Dostoevsky’s “The Double” and Nabokov’s Despair occupy opposite places in the set of doppelganger texts: Dostoevsky’s novella and Nabokov’s short novel differ much in their comprehensiveness and typicality of doppelganger imagery. Such a result allowed Golovacheva, Zhuravlev and de Mauny to reconsider the notion of the ‘typical’ in literary theory. The incidence matrix method also revealed the similarity of “William Wilson” and Despair. The objectivist comparative analysis and the subsequent close rereading of these two fictions prove, Golovacheva argues, that Poe’s tale is one of the pre-texts of Despair, such intertextuality previously neglected in Nabokov studies.

New Proof That the Sum of the Reciprocals of Primes Diverges

In this paper, we give a new proof of the divergence of the sum of the reciprocals of primes using the number of distinct prime divisors of positive integer n, and the placement of lattice points on a hyperbola given by n=pr with prime number p. We also offer both a new expression of the average sum of the number of distinct prime divisors, and a new proof of its divergence, which is very intriguing by its elementary approach.

Exterior powers of the standard E6-module: An elementary approach

For the standard [Formula: see text]-dimensional representation [Formula: see text] of the exceptional group [Formula: see text] of type [Formula: see text] we prove that [Formula: see text] is a Donkin pair if and only if the characteristic of a ground field is greater than [Formula: see text]. We also develop an elementary approach to describe submodule structure of any exterior power of [Formula: see text].