LINEARIZATION OF POISSON–LIE STRUCTURES ON THE 2D EUCLIDEAN AND (1 + 1) POINCARÉ GROUPS

The paper deals with linearization problem of Poisson-Lie structures on the \((1+1)\) Poincaré and \(2D\) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.

Simultaneous Identification of Bridge Structural Damage and Moving Loads Using the Explicit Form of Newmark-β Method: Numerical and Experimental Studies

Bridge infrastructures are always subjected to degradation because of aging, their environment, and excess loading. Now it has become a worldwide concern that a large proportion of bridge infrastructures require significant maintenance. This compels the engineering community to develop a robust method for condition assessment of the bridge structures. Here, the simultaneous identification of moving loads and structural damage based on the explicit form of the Newmark-β method is proposed. Although there is an extensive attempt to identify moving loads with known structural parameters, or vice versa, their simultaneous identification considering the road roughness has not been studied enough. Furthermore, most of the existing time domain methods are developed for structures under non-moving loads and are commonly formulated by state-space method, thus suffering from the errors of discretization and sampling ratio. This research is believed to be among the few studies on condition assessment of bridge structures under moving vehicles considering factors such as sensor placement, sampling frequency, damage type, measurement noise, vehicle speed, and road surface roughness with numerical and experimental verifications. Results indicate that the method is able to detect damage with at least three sensors, and is not sensitive to sensors location, vehicle speed and road roughness level. Current limitations of the study as well as prospective research developments are discussed in the conclusion.

Relating Semantics for Epistemic Logic

The aim of this paper is to explore the advantages deriving from the application of relating semantics in epistemic logic. As a first step, I will discuss two versions of relating semantics and how they can be differently exploited for studying modal and epistemic operators. Next, I consider several standard frameworks which are suitable for modelling knowledge and related notions, in both their implicit and their explicit form and present a simple strategy by virtue of which they can be associated with intuitive systems of relating logic. As a final step, I will focus on the logic of knowledge based on justification logic and show how relating semantics helps us to provide an elegant solution to some problems related to the standard interpretation of the explicit epistemic operators.

The q-analogue of the Higgs algebra

In this paper, the q-generalization of the Higgs algebra is considered. The realization of this algebra is shown in an explicit form using a nonlinear transformation of the creation-annihilation operators of the q-harmonic oscillator. This transformation is the performance of two operations, namely, a “correction” using a function of the original Hamiltonian, and raising to the fourth power the creation and annihilation operators of a q-harmonic oscillator. The choice of the “correcting” function is justified by the standard form of commutation relations for the operators of the metaplectic realization Uq(SU(1,1)). Further possible directions of research are briefly discussed to summarize the results obtained. The first direction is quite obvious. It is the consideration of the problem when the dimension of the operator space increases or for any value N. The second direction can be associated with the analysis of the relationship between q-generalizations of the Higgs and Hahn algebras.

Least number of n-periodic points of self-maps of $$PSU(2)\times PSU(2)$$

Let $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

The Canticle of the Creatures as Hypotext behind Dante’s Pater Noster

The article analyses Dante’s explanatory paraphrase and exegesis of the Lord’s Prayer, which opens the eleventh canto (v. 1–24) of Purgatory. The author reminds us that the prayer is the only one fully recited in the entire Comedy and this devotional practice is in line with the Franciscan prescription to recite it in the sixth hour of the Divine Office when Christ died on the cross. The prayer is reported by the poet on the first terrace of Purgatory, where the proud and vainglorious must learn the virtue of humility, and therefore it symbolizes the perfect reciprocity between man and Godhead. Dante collates and amplifies the two complementary Latin versions of the Lord’s Prayer from Matthew 6: 9–13 and Luke 11: 2–4. The two synoptic texts are supplemented by the Gospel of John, from which Dante takes the concept of celestial bread (manna) – the flesh and the blood of Christ – which nourishes, liberates and sanctifies Christians. Apart from the Bible, Dante also draws upon the Augustinian and Tomistic traditions. However, the main hypotext behind the prayer, which is neither cited nor acknowledged in any explicit form in the Comedy, is the Franciscan Laudes creaturarum (“Canticle of the Creatures”), also known as the Canticle of the Brother Sun. Written in vernacular by St. Francis himself, who is also the author of the Expositio in Pater noster, the Canticle was still recited and sung together with the Lord’s Prayer in the Franciscan communities in Dante’s time. Moreover, following the parallel readings popular nowadays in Dante studies, the author argues that Purgatorio 11 may be elucidated in the context of Paradiso 11, which is the Franciscan canto par excellence, and taken together they both offset cantos 10, 11, 12 of Inferno, which are based on the sin of pride (superbia). The denunciation of pride in and around canto 11 of Inferno alludes to humility – the remedy of such pride in Purgatory 11, which in turn prepares the reader for the encounter with St. Francis – the paragon of humility – in Paradiso 11. The author concludes that the Dantean paraphrase of the Lord’s Prayer is no less than an elaborate exegesis and homage to Christ and His teachings, something which is encompassed in a nutshell in the Sermon on the Mount.

Structure of the plane waves for a spin 3/2 particle, massive and massless cases, gauge symmetry

The structure of the plane waves solutions for a relativistic spin 3/2 particle described by 16-component vector-bispinor is studied. In massless case, two representations are used: Rarita – Schwinger basis, and a special second basis in which the wave equation contains the Levi-Civita tensor. In the second representation it becomes evident the existence of gauge solutions in the form of 4-gradient of an arbitrary bispinor. General solution of the massless equation consists of six independent components, it is proved in an explicit form that four of them may be identified with the gauge solutions, and therefore may be removed. This procedure is performed in the Rarita – Schwinger basis as well. For the massive case, in Rarita – Schwinger basis four independent solutions are constructed explicitly.

Series expansion of the Gamma function and its reciprocal

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.

On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).