One unsolved mathematical problem remains the perfect cuboid problem. A perfect cuboid is a rectangular parallelepiped whose edges, face diagonals and space diagonal are all expressed as integers. No such cuboid has yet been discovered and its existence has also not been proven. This paper shows a proof of the non-existence of a perfect cuboid.

We introduce Super Quantum Mechanics (SQM) as a theory that considers states in Hilbert space subject to multiple quadratic constraints. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization. In its simplest form, SQM considers states in the form of unitary operators, where the quadratic constraints are conditions of unitarity. In this case, the stationary SQM problem is a quantum inverse problem with multiple applications in machine learning and artificial intelligence. The SQM stationary problem is equivalent to a new algebraic problem that we address in this paper. The SQM non-stationary problem considers the evolution of a quantum system, distinct from the explicit time dependence of the Hamiltonian, $H(t)$. Several options for the SQM dynamic equation are considered, and quantum circuits of 2D type are introduced, which transform one quantum system into another. Although no known physical process currently describes such dynamics, this approach naturally bridges direct and inverse quantum mechanics problems, allowing for the development of a new type of computer algorithm. Beyond computer modeling, the developed theory could be directly applied if or when a physical process capable of solving an inverse quantum problem in a single measurement act (analogous to wavefunction measurement in traditional quantum mechanics) is discovered in the future.

We consider non-stationary oscillations of an infinite string with time-varying tension. The string lies on the Winkler foundation with a point inhomogeneity (a concentrated spring of negative stiffness). In such a system with constant parameters (the string tension), under certain conditions a trapped mode of oscillation exists and is unique. Therefore, applying a non-stationary external excitation to this system can lead to the emergence of the string oscillations localized near the inhomogeneity. We provide an analytical description of non-stationary localized oscillations of the string with slowly time-varying tension using the asymptotic procedure based on successive application of two asymptotic methods, namely the method of stationary phase and the method of multiple scales. The obtained analytical results were verified by independent numerical calculations based on the finite difference method. The applicability of the analytical formulas was demonstrated for various types of external excitation and laws governing the varying tension. In particular, we have shown that in the case when the trapped mode frequency approaches zero, localized low-frequency oscillations with increasing amplitude precede the localized string buckling. The dependence of the amplitude of such oscillations on its frequency is more complicated in comparison with the case of a one degree of freedom system with time-varying stiffness.

The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $\rho^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} \omega^{(l)} F\left(\varrho^{(l)},\sum_s B_s \rho^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,\sigma)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a mixed unitary quantum channel $A^{OUT}=\sum_s |w_s|^2 \mathcal{U}_s A^{IN} \mathcal{U}_s^{\dagger}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states $\phi_l=\mathcal{U} \psi_l$ to a quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} \rho^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{\rho^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a quantum channel, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more.

The galaxies with photometric redshifts observed in a close angular proximity might be either projection coincidences, strongly lensed images of the same galaxy, or separate galaxies that are in a stage of merging. We search for the groups of galaxies in the Hubble Ultra Deep Field (HUDF09) in $z\sim7$ and $z\sim8$ drop-out samples. We find no close pairs among 50 galaxies in the $z\sim7$ sample, while in the $z\sim8$ sample we find that 6 out of 22 galaxies have a companion within $\sim1'$ (3 pairs). Adopting a numerical simulation and performing forward modeling we show that even though mergers are unlikely to have such a high fraction, the projection coincidences and the strong lensing are even less likely mechanisms to account for all of three pairs. Alternatively, there is a possibility of the contamination in the drop-out catalog from lower redshifts, which potentially can account for all of the groups. Finally, we make projection on the sensitivity to mergers of the James Webb Space Telescope, and discuss the possible applications of the high-redshift merging galaxies for decreasing cosmic variance effect on the luminosity function and for improving the accuracy of photometric redshifts in general.

Bosonic cascades formed by lattices of equidistant energy levels sustaining radiative transitions between nearest layers are promising for the generation of coherent terahertz radiation. We show how, also for the light emitted by the condensates in the visible range, they introduce new regimes of emission. Namely, the quantum statistics of bosonic cascades exhibit super-bunching plateaus. This demonstrates further potentialities of bosonic cascade lasers for the engineering of quantum properties of light useful for imaging applications.

The robust valorization of carbon dioxide (CO2) stays at the center of sustainable development. Since CO2 represents a low-energy compound, its transformation into commercially coveted products is cumbersome. In the present work, we report a revolutionary method to obtain dimethyl carbonate (DMC) out of methanol (CH3OH) and CO2 catalyzed by sodium chloride (NaCl) and similar inorganic salts. The computational exploration revealed a mechanism of favorable catalysis, which was subsequently confirmed experimentally. Unlike all competitive syntheses of DMC, the new one does not produce water and, therefore, the hydrolysis of a carbonate does not occur. No dehydrating agents are necessary. The employed catalyst is cheap and permanently exists in the same phase with the reactants and products. The action of NaCl was compared to those of other alkali metal salts, LiI, LiCl, and KI, and competitive performances were recorded. The experimentally obtained result outperforms most competing technologies according to the DMC yield, 19% with molecular sieves and 17% without molecular sieves. All existing competitors are excelled by the simplicity and cleanness of the synthesis. The reported advance substantially simplifies the synthesis of linear organic carbonates and robustly valorizes CO2. Keywords: Dimethyl carbonate; carbon dioxide utilization; sodium chloride; methanol.

We study the electron-loss-to-continuum (ELC) cusp experimentally and theoretically by comparing the ionization of U$^{89+}$ projectiles in collisions with N$_2$ and Xe targets, at a beam energy of 75.91 MeV/u. The coincidence measurement between the singly ionized projectile and the energy of the emitted electron is used to compare the shape of the ELC cusp at weak and strong perturbations. A significant energy shift for the centroid of the electron cusp is observed for the heavy target of Xe as compared to the light target of N$_2$. Our results provide a stringent test for fully relativistic calculations of double-differential cross sections performed in the first-order approximation and in the continuum-distorted-wave approach.

We calculate the radiative corrections of order O(alpha E_e/m_N) as next-to-leading order corrections in the large nucleon mass expansion to Sirlin's radiative corrections of order O(alpha/pi) to the neutron lifetime. The calculation is carried out within a quantum field theoretic model of strong low-energy pion--nucleon interactions described by the linear sigma-model (LsM) with chiral SU(2)xSU(2) symmetry and electroweak hadron-hadron, hadron-lepton and lepton-lepton interactions for the electron-lepton family with SU(2)_L x U(1)_Y symmetry of the Standard Electroweak Model (SEM). Such a quantum field theoretic model is some kind a hadronized version of the Standard Model (SM). From a gauge invariant set of the Feynman diagrams with one-photon exchanges we reproduce Sirlin's radiative corrections of order O(alpha/pi), calculated to leading order in the large nucleon mass expansion, and calculate next-to-leading corrections of order O(alpha E_e/m_N). This confirms Sirlin's confidence level of the radiative corrections O(alpha E_e/m_N). The contributions of the LsM are taken in the limit of the infinite mass of the scalar isoscalar sigma-meson. In such a limit the LsM reproduces the results of the current algebra (Weinberg, Phys. Rev. Lett. {\bf 18}, 188 (1967)) in the form of effective chiral Lagrangians of pion-nucleon interactions with non--linear realization of chiral SU(2)xSU(2) symmetry. In such a limit the L$\sigma$M is also equivalent to Gasser-Leutwyler's chiral quantum field theory or chiral perturbation theory (ChPT) with chiral SU(2)xSU(2)symmetry and the exponential parametrization of a pion-field (Ecker, Prog. Part. Nucl. Phys. {\bf 35}, 1 (1995)).

Theoretical calculations of the Lamb shift provide the basis required for the determination of the Rydberg constant from spectroscopic measurements in hydrogen. The recent high-precision determination of the proton charge radius drastically reduced the uncertainty in the hydrogen Lamb shift originating from the proton size. As a result, the dominant theoretical uncertainty now comes from the two- and three-loop QED effects, which calls for further advances in their calculations. We review the present status of theoretical calculations of the Lamb shift in hydrogen and light hydrogen-like ions with the nuclear charge number up to $Z = 5$. Theoretical errors due to various effects are critically examined and estimated.

A lot of seismic phenomena with small magnitude has been taking place in the Ladoga lake region. One of them is the earthquake which was recorded on the 31$^\text{th}$ of July, 2010 followed by an earthquake swarm near Nikonovsky cape. There is only one station in the region that is why it is difficult to process the data. In this work we propose a new method for obtaining main event focal mechanism and constructing the synthetic seismograms for this event. The data of swarm events is used to test the method of constructing synthetic seismograms and the data of close events to determine the parameters of the media. Since the amplitude of the wave P is close to zero, the location of the nodal plane can be submitted. That is the reason why we can find the focal mechanism for the earthquake using data of the single seismostation. Resulting fault plane has the same inclination as known tectonics line which the earthquake source is placed on. Our proposed method is based on Green function method with using several kinds of sources and temporal $\delta$-shaped sequences.

We review modern achievements and problems in physics of the van der Waals and Casimir forces which arise due to zero-point and thermal fluctuations of the electromagnetic field between closely spaced material surfaces. This subject attracted great experimental and theoretical attention during the last few years because the fluctuation-induced forces find a lot of applications in both fundamental physics and nanotechnology. After a short introduction to the subject, we describe main experimental and theoretical results obtained in the field during the last fifteen years. In the following presentation, we discuss some of the recent results by the authors and their collaborators which are of high promise for future developments. Specifically, we consider new features of the Casimir force acting between a gold sphere and an indium tin oxide plate, present the experimental and theoretical results on measuring the Casimir interaction between two gold surfaces by means of dynamic atomic force microscope, and outline first measurements of the Casimir interaction between magnetic surfaces and related theory. Special attention is devoted to the Casimir effect for graphene, which is the prospective material for microelectromechanical devices of next generations.

We analyze the correlation coefficient T(E_e), which was introduced by Ebel and Feldman (Nucl. Phys. 4, 213 (1957)). The correlation coefficient T(E_e) is induced by the correlations of the neutron spin with the antineutrino 3-momentum and the electron spin with the electron 3-momentum. Such a correlation structure is invariant under discrete P, C and T symmetries. The correlation coefficient T(E_e), calculated to leading order in the large nucleon mass m_N expansion, is equal to T(E_e) = - 2 g_A(1 + g_A)/(1 + 3 g^2_A) = - B_0, i.e. of order |T(E_e)| ~ 1, where $g_A$ is the axial coupling constant. Within the Standard Model (SM) we describe the correlation coefficient $T(E_e)$ at the level of 10^{-3} by taking into the radiative corrections of order O(\alpha/\pi) or the outer model-independent radiative corrections, where \alpha is the fine-structure constant, and the corrections of order O(E_e/m_N), caused by weak magnetism and proton recoil. We calculate also the contributions of interactions beyond the SM, including the contributions of the second class currents.

We study the spatial performance of the entangling gate proposed by H. Levine et al. (Phys. Rev. Lett. 123, 170503 (2019)). This gate is based on a Rydberg blockade technique and consists of just two global laser pulses which drive nearby atoms. We analyze the multilevel Zeeman structure of interacting $^{87}$Rb Rydberg atoms and model two experimentally available excitation schemes using specific driving beams geometry and polarization. In particular, we estimate the blockade shift dependence on inter-atomic distance and angle with respect to the quantization axis. In addition, we show that using Rydberg $d$-states, in contrast to $s$-states, leads to a pronounced angular dependence of the blockade shift and gate fidelity.

The Gamma Factory initiative proposes to develop novel research tools at CERN by producing, accelerating and storing highly relativistic, partially stripped ion beams in the SPS and LHC storage rings. By exciting the electronic degrees of freedom of the stored ions with lasers, high-energy narrow-band photon beams will be produced by properly collimating the secondary radiation that is peaked in the direction of ions' propagation. Their intensities, up to $10^{17}$ photons per second, will be several orders of magnitude higher than those of the presently operating light sources in the particularly interesting $\gamma$--ray energy domain reaching up to 400 MeV. This article reviews opportunities that may be afforded by utilizing the primary beams for spectroscopy of partially stripped ions circulating in the storage ring, as well as the atomic-physics opportunities afforded by the use of the secondary high-energy photon beams. The Gamma Factory will enable ground breaking experiments in spectroscopy and novel ways of testing fundamental symmetries of nature.

We develop a microscopic calculation scheme for the excitation spectrum of a single-electron atom localized near a dielectric nanostructure. The atom originally has an arbitrary degenerate structure of its Zeeman sublevels on its closed optical transition and we follow how the excitation spectrum would be modified by its radiative coupling with a mesoscopicaly small dielectric sample of arbitrary shape. The dielectric medium is modeled by a dense ensemble of $V$-type atoms having the same dielectric permittivity near the transition frequency of the reference atom. Our numerical simulations predict strong coupling for some specific configurations and then suggest promising options for quantum interface and quantum information processing at the level of single photons and atoms. In particular, the strong resonance interaction between atom(s) and light, propagating through a photonic crystal waveguide, justifies as realistic the scenario of a signal light coupling with a small atomic array consisting of a few atoms. As a potential implication, the directional one-dimensional resonance scattering, expected in such systems, could provide a quantum bus by entangling distant atoms integrated into a quantum register.

The Rayleigh scattering of x-rays by many-electron highly charged ions is studied theoretically. The many-electron perturbation theory, based on a rigorous quantum electrodynamics approach, is developed and implemented for the case of the elastic scattering of (high-energetic) photons by helium-like ion. Using this elaborate approach, we here investigate the many-electron effects beyond the independent-particle approximation (IPA) as conventionally employed for describing the Rayleigh scattering. The total and angle-differential cross sections are evaluated for the x-ray scattering by helium-like Ni$^{26+}$, Xe$^{52+}$, and Au$^{77+}$ ions in their ground state. The obtained results show that, for high-energetic photons, the effects beyond the IPA do not exceed 2% for the scattering by a closed $K$-shell.

We present calculations of the self-energy correction to the $E1$ transition amplitudes in hydrogen-like ions, performed to all orders in the nuclear binding strength parameter. Our results for the $1s$-$2p_{1/2}$ transition for the hydrogen isoelectronic sequence show that the perturbed-orbital part of the self-energy correction provides the dominant contribution, accounting for approximately 99\% of the total correction for this transition. Detailed calculations were performed for $ns$-$n'p$ and $np$-$n'd$ transitions in H-like caesium. We conclude that the perturbed-orbital part remains dominant also for other $ns$-$n'p$ transitions, whereas for the $np$-$n'd$ matrix elements this dominance no longer holds. Consequently, the self-energy corrections for the $np$-$n'd$ one-electron matrix elements cannot be well reproduced by means of effective QED operators constructed for energy levels.