In addition to its infinite hierarchy of local, commuting symmetries, the Korteweg-de Vries equation (KdV) has scaling and Galilean symmetries, and these can be generalized, by application of the recursion operator, to a second infinite hierarchy, known as the "additional symmetries" of KdV. Unlike the standard symmetries, the additional symmetries are nonlocal, and do not commute, either amongst themselves or with the standard local symmetries. In previous work, we explained how the standard symmetries can be obtained by power series expansion of a single symmetry depending on a parameter, which can be identified as an infinitesimal double Backlund transformation. We called this the Gardner method for symmetries, as it is similar to the original construction of an infinite set of conserved quantities for KdV given in [Miura, Gardner, Kruskal 1968]. We show that a similar generating function can be found for the additional symmetries, and it is also naturally expressed in terms of the functions appearing in Backlund transformations. We determine the Lie algebra of the generating functions of the standard local symmetries and the additional symmetries.

In the early 1990s various groups showed the existence of 4 further nonlocal hierarchies of symmetries generated by application of the inverse recursion operator to the trivial or the scaling symmetry. We discuss the generating function approach to these. Only a single extra generating function needs to be introduced, also corresponding to an infinitesimal double Backlund transformation. The 6 hierarchies are obtained by expansion of 3 generating functions for small and large values of the parameter. There is substantial subtlety in identifying the commutator algebra of the generating functions, related to the general difficulty of correctly defining nonlocal symmetries.

Some properties of the basic invariants of the symmetry groups $G(m,p,n)$, $B_n^m$ were described by O. I. Rudnitskii. Here we continue to study and expand these properties. We study the properties of the basis invariants of the symmetry groups of the complex polytope $\frac{1}{p}$
$\gamma_n^m$ and the generalized $n$-cube $\gamma_n^m$, as well as its subgroups $D_m^n$. We give an explicit construction of all the basis invariants of odd degree of these groups. This invariants of the symmetry groups $G(m,p,n)$, $B_n^m$ are under construction on the basis of Pogorelov's polynomials and it is possible to construct in explicit form all generators of the algebra $I^{B_n^m}$.

Conclusion:

$\Delta R_{mr}^1=J_{m(r-1)}^{*}=-2\frac{(m(p+1))!}{(mp)!}\sum_{i=1}^{n} x_i^{m(r-1)}$

Since $J_{m(r-1)}^{*}=A\sum_{i=1}^{n} x_i^{m(r-1)}$, the form $J_{m(r-1)}^{*}$ is a basic invariant of an odd degree $mr$ of group $B_n^m$.

Thus, it is proved that, on the basis of Pogorelov's polynomials, it is possible to construct in explicit form all generators of the algebra $I^{B_n^m}$.

Reference:

1. O. I. Rudnitskii, Some Properties of Basis Invariants of the Symmetry Groups $G(m,p,n)$, $B_n^m$.
Journal of Mathematical Sciences, VoL 82, No. 2, 1996

2. Anders Bjorner, Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231,© 2005 Springer Science Business Media, Inc

3. Shephard G.C. Unitary groups generated by reflections, Can Journal Math. 1953, vol.5, page 364-383

We consider PT-symmetric quantum mechanical potentials, having an underlying supersymmetry (SUSY). The eigenspectra of both the broken and unbroken phases of PT-symmetry are analytically obtained through SUSY and shape invariance (SI). The SI is characterized by real and imaginary parametric shifting for unbroken and broken phases of PT, respectively. The use of SUSY and PT is shown to yield several complex potential systems, originating from Coulomb, Pöschl-Teller, and other solvable problems. In the PT-symmetric phase, spontaneous breaking of SUSY is observed in some parametric domains, which leads to non-trivial shape invariances to obtain the energy spectra. A conserved non-local correlation in PT-symmetric problems is shown to explain observed transmission and reflection behavior in optical systems, possessing PT-symmetry. Interestingly, the stationary states with real energy emerge in the PT-symmetric phase. The correlation also plays a crucial role in the broken PT case, having the complex conjugate energy eigenvalues and the corresponding states related.

Bell's inequality is investigated in PT-symmetric quantum mechanics, using a recently developed and more straightforward form of the inequality by Maccone [Am. J. Phys. 81, 854 (2013) ], with two PT-symmetric qubits in the unbroken phase. It is shown that the inequality produces a bound that is consistent with the standard quantum mechanics. Therefore, further, it implies that entanglement invariance is not violated in the PT-symmetric formulation of quantum mechanics. The no-signaling principle for a two-qubit system in PT-symmetric quantum theory is preserved. Consequently, it becomes clear that Bell's inequality is a potent tool as the bound obtained is independent of the internal intricacies of the theory except for the assumptions of locality and realism. To enforce our understanding of the broken PT-symmetric case, we study different types of inner product structures in the regimes of frame theory, i.e., by using the concept of bi-orthogonality and recently developed form of the inner product in pseudo-Hermitian systems [J. Math. Phys. 51, 042103 (2010)].

Invariant differential operators play very important role in the description of physical symmetries.
In a recent paper we started the systematic explicit construction of invariant differential operators. We gave an
explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the necessary representations
are induced. Thus we have set the stage for study of different non-compact groups. In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $so^*(10)$. We use the maximal Heisenberg parabolic subalgebra $p = m \oplus a \oplus n$ with $m = su(3,1) \oplus su(2)\cong so^*(6)\oplus so(3)$. We give the main multiplets of indecomposable elementary representations. This includes the explicit parametrization of the invariant differential operators between the elementary
representations. Due to the recently established parabolic relations the multiplet classification results are valid
also for the algebras $so(p,q)$ (with $p+q=10$, $p\geq q\geq 2$) with maximal Heisenberg parabolic
subalgebra: $p' = m' \oplus a' \oplus n'$, $m' = so(p-2,q-2)\oplus sl(2,R)$, $m'^C\cong m^C$.

A new generalization of the log-logistic distribution with increasing, decreasing, unimodal and bathtub hazard rates was proposed and studied, extending the log-logistic distribution by adding an extra shape (or skewness) parameter to the existing distribution, leading to greater flexibility in analysing and modeling various data types. Some of its mathematical and statistical properties were derived and model parameters estimated using the classical method especially the maximum likelihood approach and the Bayesian approach. The new hazard rate can be “increasing”, “decreasing”, “unimodal”, and “bathtub” shapes. The flexibility and usefulness of the proposed distribution was applied to three different real-life data sets with symmetric and asymmetric shapes and as well as different failure rate shapes and compared to other competitive parametric survival models. Finally, the proposed distribution is applied to regression survival analysis and verified that it is closed under both proportional hazard and accelerated failure time models that is a great contribution to the field of survival and reliability analysis and other disciplines in the areas of economics and demographic studies.

We study a special class of dynamical systems of Boltzmann-Bogolubov and Boltzmann-
Vlasov type on in…nite dimensional functional manifolds modeling kinetic processes in many-
particle media. Based on algebraic properties of the canonical quantum symmetry current
algebra and its functional representations we proposed a new approach to invariant reducing
the Bogolubov hierarchy on a suitably chosen correlation function constraint and deducing
the related modi…ed Boltzmann-Bogolubov kinetic equations on a …nite set of multiparticle
distribution functions. It is well known that the classical Bogolubov-Boltzmann kinetic equations under the
condition of manyparticle correlations at weak short range interaction
potentials describe long waves in a dense gas medium. The same equation, called the Vlasov
one, as it was shown by N. Bogolubov, describes also exact microscopic solutions of the
in…nite Bogolubov chain or the manyparticle distribution functions, which was widely
studied making use of both classical approaches and making use
of the generating Bogolubov functional method and the related quantum current algebra
representations. In this case the Bogolubov equation for distribution functions in some domain. Remark here that the basic kinetic equation is reversible under the time re‡ection t-->-t, thus it is obvious that it can not describe
thermodynamically stable limiting states of the particle system in contrast to the classical
Bogolubov-Boltzmann kinetic equations, being a priori time nonre-
versible owing to the choice of special boundary conditions. This
means that in spite of the Hamiltonicity of the Bogolubov chain for the distribution func-
tions, the Bogolubov-Boltzmann equation a priori is not reversible. The classical Poisson bracket expression allows a slightly diffeerent
Lie-algebraic interpretation, based on considering the functional space D( M_f_1 ) as a Pois-
sonian manifold, related with the canonical symplectic structure on the di¤eomorphism
group Diff() of the domain   R^3, …first described still in 1887 by Sophus Lie.
These aspects and its di¤erent consequences are analyzed in detail in our report.

We present a symmetry of Brans-Dicke gravity in (electro-)vacuo or with conformally invariant matter and use it as a solution-generating technique within this theory. This technique uses known solutions of the scalar-tensor field equations as seeds and generates new solutions of the field equations. The symmetry of the Brans-Dicke action and field equations consists of a conformal transformation of the metric (with an appropriate power of the scalar field as conformal factor), plus a non-linear rescaling of the Brans-Dicke scalar field. Using this novel general technique, we generate a new 3-parameter family of spherical, time-dependent, spacetimes conformal to a Campanelli-Lousto geometry, plus a family of cylindrically symmetric geometries. Even in spherical symmetry, time-dependent analytic solutions of scalar-tensor gravity are rather rare and the new family found adds to the meagre catalogue.

This talk is based on the article V. Faraoni, D.K. Ciftci & S.D. Belknap-Keet 2018, “Symmetry of Brans-Dicke gravity as a novel solution-generating technique”, Phys. Rev. D 97, 064004 (arXiv:1712.02205)]

It goes without saying that symmetries are one of the cornerstones of natural sciences, in particular it has proved itself a key concept in quantum field theory (QFT). Within the frame of QFT it may happen that a symmetry present classically is broken once we take all quantum effects into account. Such symmetries are known as anomalous and lead, among other things, to macroscopic transport phenomena. We will focus here on the Chiral Magnetic Effect (CME), a current generated in a chirally imbalanced medium in the presence of a magnetic field stemming from the chiral anomaly, reviewing its key features. The CME is expected to be present in the quark-gluon plasma (QGP) and is at current search to two particle accelerators: LHC and RHIC. We will briefly discuss the state-of-the-art of this search. Finally, we construct a holographic model that correctly reproduces the CME and match our parameters to QCD so that we can draw lessons of possible relevance to the realization of the CME in the QGP generated in heavy ion collisions.

Based on the principle of reparametrization invariance, the general structure of physically relevant classical matter systems is illuminated within the Lagrangian framework. In a straightforward way, the matter Lagrangian contains background interaction fields, such as a 1-form field analogous to the electromagnetic vector potential and symmetric tensor for gravity. The geometric justification of the interaction field Lagrangians for the electromagnetic and gravitational interactions are emphasized. The generalization to E-dimensional extended objects (p-branes) embedded in a bulk space M is also discussed within the light of some familiar examples. The concept of fictitious accelerations due to un-proper time parametrization is introduced, and its implications are discussed. The framework naturally suggests new classical interaction fields beyond electromagnetism and gravity. The simplest model with such fields is analyzed and its relevance to dark matter and dark energy phenomena on large/cosmological scales is inferred. Unusual pathological behavior in the Newtonian limit is suggested to be a precursor of quantum effects and of inflation-like processes at microscopic scales.