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In the present paper we deal with non-constant curvature Thurston geometries \cite{M97}, \cite{S}, \cite{Sz22-3},\cite{W06}. We define and determine the generalized trans\-lation-like Apollonius surfaces and thus also bisector surfaces as a special case. Moreover, we give a possible definition of the "surface of a translation-like triangle" in each investigated geometry. In our work we will use the projective model of Thurston geometries described by E. Moln\'ar in \cite{M97}., Comment: arXiv admin note: text overlap with arXiv:1705.04207

In this paper, we present a new record for the densest geodesic congruent ball packing configurations in $\mathbf{H}^2\!\times\!\mathbf{R}$ geometry, generated by screw motion groups. These groups are derived from the direct product of rotational groups on $\mathbf{H}^2$ and some translation components on the real fibre direction $\mathbf{R}$ that can be determined by the corresponding Frobenius congruences. Moreover, we developed a procedure to determine the optimal radius for the densest geodesic ball packing configurations related to the considered screw motion groups. The highest packing density, $\approx0.80529$, is achieved by a multi-transitive case given by rotational parameters $(2,20,4)$. E. Moln\'{a}r demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere $\mathcal{PS}^3(\mathbf{V}^4, \boldsymbol{V}_4, \mathbf{R})$. We use this projective model of $\mathbf{H}^2\!\times\!\mathbf{R}$ to compute and visualize the locally optimal geodesic ball arrangements., Comment: 27 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2311.12260

After having investigated the geodesic triangles and their angle sums in Nil and $Sl\times\mathbb{R}$ geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to $\pi$. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work we will use the projective model of Sol described by E. Moln\'ar in \cite{M97}.

After having investigated several types of geodesic ball packings in $\mathbf{S}^2 \times \mathbf{R}$ space, in this paper we study the locally optimal geodesic of simply and multiply transitive ball packings with equal balls to the space groups generated by rotations in $\mathbf{H}^2 \times \mathbf{R}$ geometry. These groups can be derived by direct product of the isometries on hyperbolic plane $\mathbf{H}^2$ and the real line $\mathbf{R}$. Moreover, we develop a procedure to determine the densities of the above locally densest geodesic ball packing configurations. Additionally, we examine the monotonicity properties of the densities within infinite series of the considered space groups. E. {Moln\'ar} showed, that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere $\mathcal{PS}^3(\mathbf{V}^4,\boldsymbol{V}_4, \mathbf{R})$. In our work, we use this projective model of $\mathbf{H}^2 \times \mathbf{R}$ to visualize the locally optimal ball arrangements., Comment: 5 Figures in EPS format. arXiv admin note: text overlap with arXiv:1206.0566, arXiv:1210.2202

We intend to continue our previous papers (\cite{MSz17} and \cite{MSz18}, as indicated there) on dense ball packing hyperbolic space $\HYP$ by equal balls, but here with centres belonging to different orbits of the fundamental group $Cw(2z, 3 \le z \in \bN$, odd number), of our new series of {\it tube or cobweb manifolds} $Cw = \HYP/\BCw$ with $z$-rotational symmetry. As we know, $\BCw$ is a fixed-point-free isometry group, acting on $\HYP$ discontinuously with appropriate tricky fundamental domain $Cw$, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every $Cw(2z)$ is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group $\BW(u, v, w = u)$ that is a complete Coxeter orthoscheme reflection group, extended by the half-turn $\Bh$ $(0 \leftrightarrow 3, 1 \leftrightarrow 2)$ of the complete orthoscheme $A_0A_1A_2A_3 \sim b_0b_1b_2b_3$ (Fig.~1). The vertices $A_0$ and $A_3$ are outer points of the $\HYP$, as $1/u + 1/v \le 1/2$ is required, $3 \le u = w, v$ for the above orthoscheme parameters. The situation is described first in Figure 1 of the half trunc-orthoscheme $W$ and its usual extended Coxeter diagram, moreover, by the scalar product matrix $(b^{ij}) = (\langle \Bb^i, \Bb^j \rangle)$ in formula (1.1) and its inverse $(A_{jk}) = (\langle \BA_j, \BA_k \rangle)$ in (1.3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme $W$, then its ball packings, densities, then those of the manifolds $Cw(2z)$. As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molekules by equal balls, for general $W(u;v;w=u)$ as well, summarized at the end of our paper., Comment: 24 pages, 7 figures

In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space, develop a method to determine its volume and total surface area. We prove that the common part of the above congruent fibre-like cylinders with the base plane are Euclidean circles and determine their radii. Using the former classified infinite or bounded congruent regular prism tilings with generating groups $\mathbf{pq2_1}$ we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in $\mathbf{H}X\mathbf{R}$ space In our work we use the projective model of $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ introduced by E. Moln\'ar in \cite{M97}., Comment: 23 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:1403.3192, arXiv:1304.0546

Graphite has been intensively studied, yet its electron spins dynamics remains an unresolved problem even 70 years after the first experiments. The central quantities, the longitudinal ($T_1$) and transverse ($T_2$) relaxation times were postulated to be equal, mirroring standard metals, but $T_1$ has never been measured for graphite. Here, based on a detailed band structure calculation including spin-orbit coupling, we predict an unexpected behavior of the relaxation times. We find, based on saturation ESR measurements, that $T_1$ is markedly different from $T_2$. Spins injected with perpendicular polarization with respect to the graphene plane have an extraordinarily long lifetime of $100$ ns at room temperature. This is ten times more than in the best graphene samples. The spin diffusion length across graphite planes is thus expected to be ultralong, on the scale of $\sim 70~\mu$m, suggesting that thin films of graphite -- or multilayer AB graphene stacks -- can be excellent platforms for spintronics applications compatible with 2D van der Waals technologies. Finally, we provide a qualitative account of the observed spin relaxation based on the anisotropic spin admixture of the Bloch states in graphite obtained from density functional theory calculations.

After investigating the $3$-dimensional case [35], we continue to address and close the problems of optimal ball and horoball packings in truncated Coxeter orthoschemes with parallel faces that exist in $n$-dimensional hyperbolic space $\overline{\mathbb{H}}^n$ up to $n=6$. In this paper, we determine the optimal ball and horoball packing configurations and their densities for the aforementioned tilings in dimensions $4 \leq n \leq 6$, where the symmetries of the packings are derived from the considered Coxeter orthoscheme groups. Moreover, for each optimal horoball packing, we determine the parameter related to the corresponding Busemann function, which provides an isometry-invariant description of different optimal horoball packing configurations., Comment: This is an original scientific research manuscript. It contains 7 figures, and some tables, in total 35 pages. arXiv admin note: text overlap with arXiv:2205.03945, arXiv:2107.08416

After having investigated the geodesic and translation triangles and their angle sums in $\SOL$ and $\SLR$ geometries we consider the analogous problem in $\NIL$ space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in $\NIL$ geometry and we provide a new approach to prove that it can be larger than or equal to $\pi$. Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of $\NIL$ isoptic surfaces of translation-like segments and as a special case of this we examine the $\NIL$ translation-like Thales sphere, which we call {\it Thaloid}. In our work we will use the projective model of $\NIL$ described by E. Moln\'ar in \cite{M97}., Comment: 22 pages, 6 figures. arXiv admin note: text overlap with arXiv:1710.02394, arXiv:1703.06646

We determine the optimal horoball packing densities for the Koszul-type Coxeter simplex tilings in $\mathbb{H}^3$. We give a family of horoball packings parameterized by the Busemann function and symmetry group that achieve the simplicial packing density upper bound $d_3(\infty) = \left( 2 \sqrt{3} \Lambda\left( \frac{\pi }{3} \right) \right)^{-1} \approx 0.853276$ where $\Lambda$ is the Lobachevsky function., Comment: 21 pages, 3 figures, 8 tables. arXiv admin note: substantial text overlap with arXiv:1907.00595, arXiv:1809.05411, arXiv:1401.6084

Of the Thurston geometries, those with constant curvature geometries (Euclidean $\EUC$, hyperbolic $\HYP$, spherical $\SPH$) have been extensively studied, but the other five geometries, $\HXR$, $\SXR$, $\NIL$, $\SLR$, $\SOL$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions., Comment: Survey, 65 pages, 22 figures

In this paper, we describe and visualize the densest ball and horoball packing configurations belonging to the simply truncated $3$-dimensional hyperbolic Coxeter orthoschemes with parallel faces. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter groups. We use for the visualization the Beltrami-Cayley-Klein ball model of $3$-dimensional hyperbolic space $\boldsymbol{H}^3$ and the pictures were made by the Python software., Comment: 21 pages, 20 figures. arXiv admin note: substantial text overlap with arXiv:2107.08416

In this paper we deal with $\NIL$ geometry, which is one of the homogeneous Thurston 3-geometries. We define the "surface of a geodesic triangle" using generalized Apollonius surfaces. Moreover, we show that the "lines" on the surface of a geodesic triangle can be defined by the famous Menelaus' condition and prove that Ceva's theorem for geodesic triangles is true in $\NIL$ space. In our work we will use the projective model of $\NIL$ geometry described by E. Moln\'ar in \cite{M97}., Comment: 19 pages, 3 figures. arXiv admin note: text overlap with arXiv:2012.06155

High concentration of negatively charged nitrogen-vacancy ($\text{NV}^{-}$) centers was created in diamond single crystals containing approximately 100 ppm nitrogen using electron and neutron irradiation and subsequent thermal annealing in a stepwise manner. Continuous wave electron paramagnetic resonance (EPR) was used to determine the transformation efficiency from isolated N atoms to $\text{NV}^{-}$ centers in each production step and its highest value was as high as 17.5 %. Charged vacancies are formed after electron irradiation as shown by EPR spectra, but the thermal annealing restores the sample quality as the defect signal diminishes. We find that about 25 % of the vacancies form NVs during the annealing process. The large $\text{NV}^{-}$ concentration allows to observe orientation dependent spin-relaxation times and also the determination of the hyperfine and quadrupole coupling constants with high precision using electron spin echo (ESE) and electron-nuclear double resonance (ENDOR). We also observed the EPR signal associated with the so-called W16 centers, whose spectroscopic properties might imply a nitrogen dimer-vacancy center for its origin.

In this paper we consider the ball and horoball packings belonging to $3$-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space $\mathbb{H}^3$. The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices. We prove that the densest packing of the above cases is realized by horoballs related to $\{\infty;3;6;\infty \}$ and $\{\infty;6;3;\infty \}$ tilings with density $\approx 0.8413392$., Comment: 27 pages, 8 figures

After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, we consider the corresponding covering problems. In \cite{MSSz} the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of truncated tetrahedra, and determined the optimal congruent hyperball packing and covering configurations belonging to some of these classes. In this paper we compliment these results with the investigation of the non-congruent covering cases, and the remainig congruent cases. We prove, that between congruent and non-congruent hyperball coverings the thinnest belongs to the $\{7,3,7\}$ Coxeter tiling with density $\approx 1.26829$. This covering density is smaller than the conjectured lower bound density of L.~Fejes~T\'oth for coverings with balls and horoballs. We also study the local packing arrangements related to $\{u,3,7\}$ $(6< u < 7, ~ u\in \mathbb{R})$ doubly truncated orthoschemes and the corresponding hyperball coverings. We prove, that these coverings are achieved their minimum density at parameter $u\approx 6.45953$ with covering density $\approx 1.26454$ which is smaller then the above record-small density, but this hyperball arrangement related to this locally optimal covering can not be extended to the entire $\mathbb{H}^3$. Moreover we see, that in the hyperbolic plane $\mathbb{H}^2$ the universal lower bound of the congruent circle, horocycle, hypercycle covering density $\frac{\sqrt{12}}{\pi}$ can be approximated arbitrarily well also with non-congruent hypercycle coverings generated by doubly truncated Coxeter orthoschemes., Comment: 23 pages, 6 figures

In the present paper we study $\SXR$ and $\HXR$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the "surface of a geodesic triangle". Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\SXR$ and $\HXR$ tetrahedron. Moreover, we generalize the famous Menelaus' and Ceva's theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\SXR$ and $\HXR$ geometries described by E. Moln\'ar in \cite{M97}., Comment: 23 pages, 10 figures

Chemical modification, such as intercalation or doping of novel materials is of great importance for exploratory material science and applications in various fields of physics and chemistry. In the present work, we report the systematic intercalation of chemically exfoliated few-layer graphene with potassium while monitoring the sample resistance using microwave conductivity. We find that the conductivity of the samples increases by about an order of magnitude upon potassium exposure. The increased of number of charge carriers deduced from the ESR intensity also reflects this increment. The doped phases exhibit two asymmetric Dysonian lines in ESR, a usual sign of the presence of mobile charge carriers. The width of the broader component increases with the doping steps, however, the narrow components seem to have a constant line width.

Today's great challenges of energy and informational technologies are addressed with a singular compound, the Li and Na doped few layer graphene. All what is impossible for graphite (homogeneous and high level Na doping), and unstable for single layer graphene, works very well for this structure. The transformation of the Raman G line to a Fano lineshape and the emergence of strong, metallic-like electron spin resonance (ESR) modes, attest the high level of graphene doping in liquid ammonia for both kinds of alkali atoms. The spin-relaxation time in our materials, deduced from the ESR line-width, is 6-8 ns, which is comparable to the longest values found in spin-transport experiments on ultrahigh mobility graphene flakes. This could qualify our material as promising candidate in spintronics devices. On the other hand, the successful sodium doping, this being a highly abundant metal, could be an encouraging alternative to lithium batteries., Comment: 10 pages, 5 figures+ Supplementary Materials

In our Novi Sad conference paper (1999) we described Dehn type surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold $\mathcal{S}$, leading to compact fundamental domain $\mathcal{S}(k)$, $k = 2, 3, \dots$ with singularity geodesics of rotation order $k$, but as later turned out with cone angle $2(k-1)/k$. We computed also the volume of $S(k)$, tending to zero if $k$ goes to infinity. That time we naively thought that we obtained orbifolds with the above surprising property. As the reviewer of Math. Rev., Kevin P. Scannell (MR1770996 (2001g:57030)) rightly remarked, "this is in conflict with the well-known theorem of D. A. Kazhdan and G. A. Margulis (1968) and with the work of Thurston, describing the geometric convergence of orbifolds under large Dehn fillings". In this paper we refresh our previous publication. Correctly, we obtained cone manifolds (for $k > 2$), as A. D. Mednykh and V. S. Petrov (2006) kindly pointed out. We complete our discussion and derive the above cone manifold series (Gies.1 and Gies.2) in two geometrically equivalent form, by the half turn symmetry of any ideal simplex. Moreover we obtain a second orbifold series (Gies.3 and 4), tending to the regular ideal simplex as the original Gieseking manifold., Comment: 19 pages, 9 figures

Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Moln\'{a}r et al. in $2006$. As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L.~Fejes~T\'oth for balls and horoballs in $1964$.

After having investigated the packings derived by horo- and hyperballs related to simple frustum Coxeter orthoscheme tilings we consider the corresponding covering problems (briefly hyp-hor coverings) in $n$-dimensional hyperbolic spaces $\mathbb{H}^n$ ($n=2,3$). We construct in the $2-$ and $3-$dimensional hyperbolic spaces hyp-hor coverings that are generated by simply truncated Coxeter orthocheme tilings and we determine their thinnest covering configurations and their densities. We prove that in the hyperbolic plane ($n=2$) the density of the above thinnest hyp-hor covering arbitrarily approximate the universal lower bound of the hypercycle or horocycle covering density $\frac{\sqrt{12}}{\pi}$ and in $\mathbb{H}^3$ the optimal configuration belongs to the $\{7,3,6\}$ Coxeter tiling with density $\approx 1.27297$ that is less than the previously known famous horosphere covering density $1.280$ due to L.~Fejes T\'oth and K.~B\"or\"oczky. Moreover, we study the hyp-hor coverings in truncated orthosche\-mes $\{p,3,6\}$ $(6< p < 7, ~ p\in \mathbb{R})$ whose density function attains its minimum at parameter $p\approx 6.45962$ with density $\approx 1.26885$. That means that this locally optimal hyp-hor configuration provide smaller covering density than the former determined $\approx 1.27297$ but this hyp-hor packing configuration can not be extended to the entirety of hyperbolic space $\mathbb{H}^3$., Comment: 23 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:1505.03338

Alkali intercalated graphite compounds represent a compelling modification of carbon with significant application potential and various fundamentally important phases. We report on the intercalation of graphite with alkali atoms (Li and K) using liquid ammonia solution as mediating agent. Alkali atoms dissolve well in liquid ammonia which simplifies and speeds up the intercalation process, and it also avoids the high temperature formation of alkali carbides. Optical microscopy, Raman, and electron spin resonance spectroscopy attest that the prepared samples are highly and homogeneously intercalated to a level approaching Stage-I intercalation compounds. The method and the synthesis route may serve as a starting point for the various forms of alkali atom intercalated carbon compounds (including carbon nanotubes and graphene), which could be exploited in energy storage and further chemical modifications.

Koszul type Coxeter simplex tilings exist in hyperbolic $n$-space $\mathbb{H}^n$ up to $ n = 9$, and their horoball packings achieve the highest known regular ball packing densities for $n = 3, 4, 5$. In this paper we determine the optimal horoball packing densities of Koszul simplex tilings in dimensions $6 \leq n \leq 9$, which give new lower bounds for optimal packing density in each dimension. The symmetries of the packings are given by Coxeter simplex groups, and a parameter related to the Busemann function gives an isometry invariant description of different optimal horoball packing configurations., Comment: 16 Pages, 6 Tables, 1 Figure. arXiv admin note: substantial text overlap with arXiv:1809.05411, arXiv:1401.6084

In the present paper we study $S^2 \times R$ and $H^2 \times R$ geometries, which are homogeneous Thurston 3-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and prove, that in $S^2 \times R$ space it can be larger or equal than $\pi$ and in $H^2 \times R$ space the angle sums can be less or equal than $\pi$. In our work we will use the projective model of $S^2 \times R$ and $H^2 \times R$ geometries described by E. Moln\'ar in \cite{M97}., Comment: 22 pages and 10 figures. arXiv admin note: substantial text overlap with arXiv:1703.06646, arXiv:1611.05613

In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. In this paper we prove, using the above results and the results of papers \cite{M94} and \cite{Sz14}, that the density upper bound of the saturated congruent hyperball (hypersphere) packings related to the corresponding truncated tetrahedron cells is realized in a regular truncated tetrahedra with density $\approx 0.86338$. Furthermore, we prove that the density of locally optimal congruent hyperball arrangement in regular truncated tetrahedron is not a monotonically increasing function of the height (radius) of corresponding optimal hyperball, contrary to the ball (sphere) and horoball (horosphere) packings., Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1709.04369, arXiv:1811.03462, arXiv:1803.04948, arXiv:1405.0248

In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. Therefore, in this paper we examine the doubly truncated Coxeter orthoscheme tilings and the corresponding congruent and non-congruent hyperball packings. We proved that related to the mentioned Coxeter tilings the density of the densest congruent hyperball packing is $\approx 0.81335$ that is -- by our conjecture -- the upper bound density of the relating non-congruent hyperball packings too., Comment: 24 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1803.04948

We study quasiparticle scattering effects on the dynamics of a homogeneous Bose-Einstein condensate of ultracold atoms coupled to a single mode of an optical cavity. The relevant excitations, which are polariton-like mixed excitations of photonic and atomic density-wave modes, are identified. All the first-order correlation functions are presented by means of the Keldysh Green's function technique. Beyond confirming the existence of the resonant enhancement of Beliaev damping, we find a very structured spectrum of fluctuations. There is a spectral hole burning at half of the recoil frequency reflecting the singularity of the Beliaev scattering process. The effects of the photon-loss dissipation channel and that of the Beliaev damping due to atom-atom collisions can be well separated. We show that the Beliaev process does not influence the properties of the self-organization criticality.

We describe the optimal horoball packings of asymptotic Koszul type Coxeter simplex tilings of $5$-dimensional hyperbolic space where the symmetries of the packings are generated by Coxeter groups. We find that the optimal horoball packing density of $\delta_{opt}=0.59421\dots$ is realized in an entire commensurability class of arithmetic Coxeter tilings. Eleven optimal arrangements are achieved by placing horoballs at the asymptotic vertices of the corresponding Coxeter simplices that give the fundamental domains. When multiple horoball types are allowed, in the case of the arithmetic Coxeter groups, the relative packing densities of the optimal horoball types are rational submultiples of $\delta_{opt}$, corresponding to the Dirichlet-Voronoi cell densities of the packing. The packings given in this paper are so far the densest known in hyperbolic $5$-space., Comment: 28 pages, 5 figures, 5 tables. J. Geom Dedicata (2019). arXiv admin note: substantial text overlap with arXiv:1401.6084

An efficient route to synthesize macroscopic amounts of graphene is highly desired and a bulk characterization of such samples, in terms of the number of layers, is equally important. We present a Raman spectroscopy-based method to determine the distribution of the number of graphene layers in chemically exfoliated graphene. We utilize a controlled vapor-phase potassium intercalation technique and identify a lightly doped stage, where the Raman modes of undoped and doped few-layer graphene flakes coexist. The spectra can be unambiguously distinguished from alkali doped graphite, and a modeling with the distribution of the layers yields an upper limit of flake thickness of five layers with a significant single-layer graphene content. Complementary statistical AFM measurements on individual few-layer graphene flakes find a consistent distribution of the layer numbers.

We investigate the quantum measurement noise effects on the dynamics of an atomic Bose lattice gas inside an optical resonator. We describe the dynamics by means of a hybrid model consisting of a Bose--Hubbard Hamiltonian for the atoms and a Heisenberg--Langevin equation for the lossy cavity field mode. We assume that the atoms are prepared initially in the ground state of the lattice Hamiltonian and then start to interact with the cavity mode. We show that the cavity field fluctuations originating from the dissipative outcoupling of photons from the resonator lead to vastly different effects in the different possible ground state phases, i.e., the superfluid, the supersolid, the Mott- and the charge-density-wave phases. In the former two phases with the presence of a superfluid wavefunction, the quantum measurement noise appears as a driving term leading to excess noise depletion of the ground state. The time scale for the system to leave the ground scale is determined analytically. For the latter two incompressible phases, the quantum noise results in the fluctuation of the chemical potential. We derive an analytical expression for the corresponding broadening of the quasiparticle resonances.

In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular octahedron and cube tilings. These are derived from the Coxeter simplex tilings $\{p,3,4\}$ $(7\le p \in \mathbb{N})$ and $\{p,4,3\}$ $(5\le p \in \mathbb{N})$ in $3$-dimensional hyperbolic space $\mathbb{H}^3$. We determine the densest hyperball packing arrangement and its density with congruent and non-congruent hyperballs related to the above tilings in $\mathbb{H}^3$. We prove that the locally densest congruent or non-congruent hyperball configuration belongs to the regular truncated cube with density $\approx 0.86145$. This is larger than the B\"or\"oczky-Florian density upper bound for balls and horoballs. Our locally optimal non-congruent hyperball packing configuration cannot be extended to the entire hyperbolic space $\mathbb{H}^3$, but we determine the extendable densest non-congruent hyperball packing arrangement related to a regular cube tiling with density $\approx 0.84931$., Comment: 33 pages, 12 figures. arXiv admin note: text overlap with arXiv:1510.03208, arXiv:1505.03338

Previous discoveries of the first author (1984-88) on so-called hyperbolic football manifolds and our recent works (2016-17) on locally extremal ball packing and covering hyperbolic space $\HYP$ with congruent balls had led us to the idea that our "experience space in small size" could be of hyperbolic structure. In this paper we construct an infinite series of oriented hyperbolic space forms so-called cobweb (or tube) manifolds $Cw(2z, 2z, 2z)=Cw(2z)$, $3\le z$ odd, which can describe nanotubes, very probably., Comment: 22 pages, 8 figures

In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. We develop a method to determine the centre and the radius of the circumscribed translation sphere of a given {\it translation tetrahedron}. A further aim of this paper is to study lattice-like coverings with congruent translation balls in Nil space. We introduce the notion of the density of the considered coverings and give upper estimate to it using the radius amd the volume of the circumscribed translation sphere of a given {\it translation tetrahedron}. The found minimal upper bound density of the translation ball coverings $\Delta \approx 1.42783$. In our work we will use for computations and visualizations the projective model of Nil described by E. Moln\'ar in \cite{M97}., Comment: 25 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1705.04207, arXiv:1105.1986

We study the spin-1 bilinear-biquadratic model on the complete graph of N sites, i.e., when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of su(3) and su(2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.

In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new procedure to get a decomposition of 3-dimensional hyperbolic space $\HYP$ into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices., Comment: 13 pages, 3 figures. arXiv admin note: text overlap with arXiv:1405.0248

In the present paper we study the $\SOL$ geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. Moreover, we develop a method to determine the centre and the radius of the circumscribed translation sphere of a given {\it translation tetrahedron}. In our work we will use for computations and visualizations the projective model of $\SOL$ described by E. Moln\'ar in \cite{M97}., Comment: 17 pages, 8 figures. arXiv admin note: text overlap with arXiv:1703.06646

We investigate magnetic properties of strongly interacting four component spin-3/2 ultracold fermionic atoms in the Mott insulator limit with one particle per site in an optical lattice with honeycomb symmetry. In this limit, atomic tunneling is virtual, and only the atomic spins can exchange. We find a competition between symmetry breaking and liquid like disordered phases. Particularly interesting are valence bond states with bond centered magnetizations, situated between the ferromagnetic and conventional valence bond phases. In the framework of a mean-field theory, we calculate the phase diagram and identify an experimentally relevant parameter region where a homogeneous SU(4) symmetric Affleck-Kennedy-Lieb-Tasaki-like valence bond state is present.

After having investigated the geodesic and translation triangles and their angle sums in $\NIL$ and $\SLR$ geometries we consider the analogous problem in $\SOL$ space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of translation triangles in $\SOL$ geometry and prove that it can be larger or equal than $\pi$. In our work we will use the projective model of $\SOL$ described by E. Moln\'ar in \cite{M97}, Comment: 13 pages, 4 figures

This presentation starts with the regular polygons, of course, then with the Platonic and Archimedean solids. The latter ones are whose symmetry groups are transitive on the vertices, and in addition, whose faces are regular polygons (see only I. Prok's home page [11] for them). Then there come these symmetry groups themselves (starting with the cube and octahedron, of course, then icosahedron and dodecahedron). Then come the space filling properties: Namely the cube is a space filler for the Euclidean space E^3. Then we jump for the other regular solids that cannot fil E^3, but can hyperbolic space H^3, a new space. These can be understood better if we start regular polygons, of course, that cannot fil E^2 in general, but can fil the new plane H2, as hyperbolic or Bolyai-Lobachevsky plane. Now it raises the problem, whether a football polyhedron - with its congruent copies - fil a space. It turns out that E^3 is excluded (it remains an open problem - for you, of course, in other aspects), but H^3 can be filled as a schematic construction show this (Fig. 5), far from elementary. Then we mention some stories on Buckminster Fuller, an architect, who imagined first time fullerens as such crystal structures. Many problems remain open, of course, we are just in the middle of living science., Comment: 20 pages, 8 figures

A compact hyperbolic "cobweb" manifold (hyperbolic space form) of symbol $Cw(6,6,6)$ will be constructed in Fig.1,4,5 as a representant of a presumably infinite series $Cw(2p,2p,2p)$ $(3 \le p \in \bN$ natural numbers). This is a by-product of our investigations \cite{MSz16}. In that work dense ball packings and coverings of hyperbolic space $\HYP$ have been constructed on the base of complete hyperbolic Coxeter orthoschemes $\mathcal{O}=W_{uvw}$ and its extended reflection groups $\bG$ (see diagram in Fig.~3. and picture of fundamental domain in Fig.~2). Now $u=v=w=6 (=2p)$. Thus the maximal ball contained in $Cw(6,6,6)$, moreover its minimal covering bal l (so diameter) can also be determined. The algorithmic procedure provides us with the proof of our statements., Comment: 14 pages, 5 figures

We report a detailed study of the electrical transport properties of single crystals of Pr$_4$Fe$_2$As$_2$Te$_{1-x}$O$_4$, a recently discovered iron-based superconductor. Resistivity, Hall effect and magnetoresistance are measured in a broad temperature range revealing the role of electrons as dominant charge carriers. The significant temperature dependence of the Hall coefficient and the violation of Kohler's law indicate multiband effects in this compound. The upper critical field and the magnetic anisotropy are investigated in fields up to 16 T, applied parallel and perpendicular to the crystallographic c-axis. Hydrostatic pressure up to 2 GPa linearly increases the critical temperature and the resistivity residual ratio. A simple two-band model is used to describe the transport and magnetic properties of Pr$_4$Fe$_2$As$_2$Te$_{1-x}$O$_4$. The model can successfully explain the strongly temperature dependent negative Hall coefficient and the high magnetic anisotropy assuming that the mobility of electrons is higher than that of holes.