Noether symmetries, dynamical constants of motion, and spectrum generating algebras

When discussing consequences of symmetries of dynamical systems based on Noether’s first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time-independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether’s first theorem. Rather a much less restrictive statement applies, namely, that the corresponding Noether charge as an observable over phase space may in fact possess an explicit time dependency, and yet define a constant of the motion by having a commutator with the Hamiltonian which is nonvanishing, thus indeed defining a dynamical conserved quantity. Furthermore, and this certainly within the Hamiltonian formulation, the converse statement is valid as well, namely, that any dynamical constant of motion is necessarily the Noether charge of some symmetry leaving the system’s action invariant up to some total time derivative contribution. This contribution revisits these different points and their consequences, straightaway within the Hamiltonian formulation which is the most appropriate for such issues. Explicit illustrations are also provided through three general but simple enough classes of systems.

Impact of odour with high annoyance potential on human feeling, activity, and mood in odorimetric studies

Introduction. It is known that an extraneous odour can cause an adverse emotional reaction of a person and, as a result, reduce the quality of life, which is most likely the reason for a large number of complaints of air pollution from the population. The converse statement about the influence of a person’s emotional state on his/her perception of a smell can also be true. An assessment of the relationship between odour characteristics and emotional state of a person can be investigated in a laboratory olfactory-odorimetric study with the participation of specially trained investigators. The aim of this study was to analyze in laboratory conditions the impact of odour with high annoyance potential on human feeling, activity, and mood. Materials and methods. Ten 26-71-years healthy investigators of both genders, who have been psychologically tested with a block of standard psychological questionnaires to evaluate emotional stress expression and quality of life, took part in 2 series of odorimetric studies on the ECOMA T08 olfactometer. During this study, they assessed the odour intensity and annoying effect of multicomponent odorant with isopropyl mercaptan, ethyl mercaptan, and 2-butanethiol largest contribution (unpleasant odour). The influence of the odour on investigators’ feeling, activity, and mood before, during, and after odorimetry was determined using a FAM (feeling, activity, and mood) test card. The study results showed the perception of unpleasant smell to be connected with investigators’ age, emotional state at the time of odorimetry, and some indices of the life quality. It is established that the perception of the smell in its tangible concentrations can lead to increased activity and improved mood. At the same time, exposure to the odour in higher concentrations, in most cases, was associated with decreased activity and mood, as well as - for some investigators - with decrease of feelings regardless of age. The given work results comparison with data obtained in the only similar study of odour emissions of chewing gum production carried out with a common methodic approach and on the same equipment, demonstrated a qualitative similarity in the perception of an extraneous odour of different hedonic tone: a connection with age, background emotional state, and quality of life indicators. Moreover, it was revealed odours of different hedonic tones to influenced in the same way on investigators’ activity and mood indices. Conclusion. Although the obtained results do not allow us to unambiguously conclude which odour strength (concentration of odorous substances) is an unacceptable value for all investigators, the similar research implementation is necessary to study the odour impact on the human adaptation systems. In addition, the data obtained allow us to conclude that all complaints of the presence of an extraneous odour of any character and hedonic tone, are most often justified, deserve careful attention and a quick response from the sanitary services.

On semirings all of whose semimodules have injective envelopes

It was shown in [Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997) 121–131.] that each semimodule over an additively regular semiring has an injective envelope. We show the converse statement is true as well; moreover, the result holds even if each finitely generated semimodule has an injective envelope.

Sufficient conditions for the improved regular growth of entire functions in terms of their averaging

Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (an entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that $\log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log |{f(te^{i\varphi})}|}{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.

Quasi-automatic semigroups

Quasi-automatic semigroups are extensions of a Cayley graph of an automatic group. Of course, a quasi-automatic semigroup generalizes an automatic semigroup. We observe that a semigroup [Formula: see text] may be automatic only when [Formula: see text] is finitely generated, while a semigroup may be quasi-automatic but it is not necessary finitely generated. Similar to the usual automatic semigroups, a quasi-automatic semigroup is closed under direct and free products. Furthermore, a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with a zero element adjoined is graph automatic, and also a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with an identity element adjoined is graph automatic. However, the class of quasi-automatic semigroups is a much wider class than the class of automatic semigroups. In this paper, we show that every automatic semigroup is quasi-automatic but the converse statement is not true (see Example 3.6). In addition, we notice that the quasi-automatic semigroups are invariant under the changing of generators, while a semigroup may be automatic with respect to a finite generating set but not the other. Finally, the connection between the quasi-automaticity of two semigroups [Formula: see text] and [Formula: see text], where [Formula: see text] is a subsemigroup with finite Rees index in [Formula: see text] will be investigated and considered.

Embedded Markov chain approximations in Skorokhod topologies

We prove a J1-tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner. To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J1, the linear interpolation embedding forM1, the multistep embedding for J2 and a more general embedding for M2. We show that the convergence of the step function embedding in J1 implies the convergence of the other embeddings in the corresponding topologies. For the converse statement, a J1-tightness condition for embedded time-homogeneous Markov chains is given.Additionally, it is shown that J1 convergence is equivalent to the joint convergence in M1 and J2.

Secant Parallels in Azimuthal Projections

Map projections are commonly approached as mapping onto developable surfaces; cylindrical projections onto the lateral surface of a cylinder, conic projections onto the lateral surface of a cone, and azimuthal projections onto a plane. If an intermediate developable surface intersects the Earth’s sphere or ellipsoid, the projection is referred to as a secant projection. The intersection of a developable surface and the Earth’s sphere or ellipsoid, e.g. secant parallel is considered a standard parallel. In this paper the definitions of secant and standard parallel in azimuthal projections are given. The first conclusion is that the secant and standard parallels are two distinct notions. The second one is that a standard parallel, if such a parallel exists in an azimuthal projection, is a secant parallel, while the converse statement is not true in general. Furthermore, it is shown that there are azimuthal projections with only one secant parallel that is not standard, with only one standard parallel which is also secant one, with two different secant parallels, and with one standard and one secant parallel.

On viscosity and weak solutions for non-homogeneous p-Laplace equations

In this manuscript, we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower-order term depending on x, u and {\nabla u} . More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending results presented in Juutinen, Lindqvist and Manfredi [9], and Julin and Juutinen [6]. Moreover, we provide a converse statement in the full case under extra assumptions on the data.

Residuation in orthomodular lattices

We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.