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Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety over $k$. We use the $K$-theory of a $C^*$-algebra $A_V$ associated to $V(k)$ to define a height of points of $V(k)$. The corresponding counting function is calculated and we show that it coincides with the known formulas for $n=1$. As an application, it is proved that the set $V(k)$ is finite, whenever the sum of the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the $n$-dimensional Minkowski question-mark function studied by Panti and others., Comment: 12 pages

We use $K$-theory of the $C^*$-algebras to study the Arakelov geometry, i.e. a compactification of the arithmetic schemes $V\to Spec ~\mathbf{Z}$. In particular, it is proved that the Picard group of $V$ is isomorphic to the $K_0$-group of a Cuntz-Pimsner algebra associated to $V$. We apply the result to the finiteness problem for the algebraic varieties over number fields., Comment: 7 pages. arXiv admin note: text overlap with arXiv:2404.12179

We implemented the Bayesian analysis to the polarised neutron reflectivity data. Reflectivity data from a magnetic TbCo thin film structure was studied using the bundle of a Monte-Carlo Markov-chain algorithm, likelihood estimation, and error modeling. By utilizing the Bayesian analysis, we were able to investigate the uniqueness of the solution beyond reconstructing the magnetic and structure parameters. This approach has demonstrated its expedience as several probable reconstructions were found (the multimodality case) concerning the isotopic composition of the surface cover layer. Such multimodal reconstruction emphasizes the importance of rigorous data analysis instead of the direct data fitting approach, especially in the case of poor statistically conditioned data, typical for neutron reflectivity experiments. The analysis details and the discussion on multimodality are in this article.

We recast the local factors of the Hasse-Weil zeta function at infinity in terms of the Cuntz-Pimsner algebras. The nature of such factors is an open problem studied by Deninger and Serre., Comment: 8 pages

Given an undirected graph $G = (V,E)$, the cut polytope $\mathrm{CUT}(G)$ is defined as the convex hull of the incidence vectors of all cuts in $G$. The 1-skeleton of $\mathrm{CUT}(G)$ is a graph whose vertex set is the vertex set of the polytope, and the edge set is the set of geometric edges or one-dimensional faces of the polytope. We study the diameter and the clique number of 1-skeleton of cut polytopes for several classes of graphs. These characteristics are of interest since they estimate the computational complexity of the max-cut problem for certain computational models and classes of algorithms. It is established that while the diameter of the 1-skeleton of a cut polytope does not exceed $|V|-1$ for any connected graph, the clique number varies significantly depending on the class of graphs. For trees, cacti, and almost trees (2), the clique number is linear in the dimension, whereas for complete bipartite and $k$-partite graphs, it is superpolynomial., Comment: 15 pages, 10 figures, 1 table

Using the ideas of Deninger, we prove that the Artin $L$-functions coincide with such of the noncommutative tori. This result can be viewed as the Langlands reciprocity for noncommutative tori., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2309.05779

We have studied the potential barriers for the penetration of atomic beryllium or boron inside the C60 fullerene by performing ab initio density functional theory (DFT) calculations with three variants for the exchange and correlation: B3LYP (hybrid functional), PW91 and PBE. Four principal trajectories to the inner part of C60 for the penetrating atom have been considered: through the center of six-member-carbon ring (hexagon), five-member-carbon ring (pentagon), and also through the center of the double C-C bond (D-bond) and the center of the single C-C bond (S-bond). Averaging over the three DFT variants yields the following barriers for beryllium penetrating inside a deformable fullerene: 3.2 eV (hexagon), 4.8 eV (S-bond), 5.3 eV (D-bond), 5.9~eV (pentagon). These barriers correspond to the slow and adiabatic penetration of Be, in contrast to the fast (non-adiabatic) penetration through the rigid cage of C60 resulting in 5.6 eV (hexagon), 16.3 eV (pentagon), 81.8 eV (S-bond) and 93.4 eV (D-bond). The potential barriers for the boron penetrating inside deformable/rigid C60 are: 3.7/105.4 eV (D-bond), 4.0/86.8 eV (S-bond), 4.7/7.8 eV (hexagon), 6.8/14.0 eV (pentagon). The potential barriers for Be and B escaping from the inner part of C$_{60}$ are higher by the value of 0.84 eV for Be and 0.81 eV for B. The considerable reduction of the potential barriers for the deformable fullerene is ascribed to the formation of the Be-C and B-C bonds. We discuss the difference between Be and B, compare three variants of DFT, and analyze the role of the dispersion interaction., Comment: 9 pages, 5 figures

We study a relation between the Drinfeld modules and the even dimensional noncommutative tori. A non-abelian class field theory is developed based on this relation. Explicit generators of the Galois extensions are constructed., Comment: 13 pages, 1 figure. arXiv admin note: text overlap with arXiv:2306.06416

The Drinfeld module is a tool of the explicit class field theory for the function fields. We first observe a similarity of such modules with the noncommutative tori, and then use it to develop an explicit class field theory for the number fields. The case of the imaginary quadratic number fields is treated in detail., Comment: 12 pages, an update

Physics of high-$T_c$ superconducting cuprates is obscured by the effect of strong electronic correlations. One way to overcome the problem is to seek for an exact solution at least within the small cluster and expand it to the whole crystal. Such an approach is in the heart of the cluster perturbation theory (CPT). Here we develop CPT for the dynamic spin and charge susceptibilities (spin-CPT and charge-CPT), within which the correlation effects are explicitly taken into account by the exact diagonalization. We apply spin-CPT and charge-CPT to the effective two-band Hubbard model for the cuprates obtained from the three-band Emery model and calculate one- and two-particle correlation functions, namely, spectral function and spin and charge susceptibilities. Doping dependence of the spin susceptibility was studied within spin-CPT and CPT-RPA that is the CPT generalization of the random phase approximation (RPA). Both methods produce the low energy response at four incommensurate wave vectors in qualitative agreement to the results of the inelastic neutron scattering on overdoped cuprates., Comment: 14 pages, 8 figures

We study the analogy between number fields and function fields in one variable over finite fields. The main result is an isomorphism between the Hilbert class fields of class number one and a family of the function fields $\mathbf{F}_q(C)$ over a desingularized algebraic curve $C$. Our proof is based on the K-theory of the Serre $C^*$-algebras and birational geometry of the curve $C$. We apply the isomorphism to construct explicit generators of the Hilbert class fields coming from the torsion submodules of the Drinfeld module., Comment: 9 pages, 1 figure

We consider the classical minimum and maximum cut problems: find a partition of vertices of a graph into two disjoint subsets that minimize or maximize the sum of the weights of edges with endpoints in different subsets. It is known that if the edge weights are non-negative, then the min-cut problem is polynomially solvable, while the max-cut problem is NP-hard. We construct a partition of the positive orthant into convex cones corresponding to the characteristic cut vectors, similar to a normal fan of a cut polyhedron. A graph of a cone partition is a graph whose vertices are cones, and two cones are adjacent if and only if they have a common facet. We define adjacency criteria in the graphs of cone partitions for the min-cut and max-cut problems. Based on them, we show that for both problems the vertex degrees are exponential, and the graph diameter equals 2. These results contrast with the clique numbers of graphs of cone partitions, which are linear for the minimum cut problem and exponential for the maximum cut problem., Comment: 15 pages, 12 figures, 1 table

The effects associated with exciton Bose condensate formation in strongly correlated spin crossover systems are considered within the effective Hamiltonian obtained from the two-orbital Hubbard-Kanamori model. The collective excitations spectrum at various points of the temperature-crystal field phase diagram is calculated. The role of the electron-phonon interaction is discussed. The exciton and magnetic order photoenhancement (induction) in strongly correlated spin crossover systems a new mechanism based on the cooperative effect of electron-phonon and interatomic exchange interactions and the appearance of a massive collective phase mode is demonstrated.

In three out of five cases considered in our work, DFT calculations presented by Xu and Hou in their Comment give the same ground state confirmations. On the other hand, depending on the choice of the exchange-correlation functional, the geometry optimization within DFT results in different ground state confirmations for B@C60 and B60, Table I of the Comment. Therefore, the energy balance between nearest confirmations in these molecular complexes is subtle, and various methods can give different ground state structures. Consequently, the results of our method - the Hartree-Fock (HF) approach with the second order M{\o}ller-Plesset perturbation theory (MP2) - should be compared with the DFT results on equal ground, we cannot agree that the DFT method used in the Comment is superior to HF-MP2. In the Reply, we also present additional HF calculations with the 6-31G* basis set (used in the Comment for the geometry optimization) to show that the polarization functions do not change the ground state confirmations obtained by us earlier at the HF/6-31G level., Comment: Reply to the Comment of J. Xu and G.-L. Hou, Comment on "Multiple locations of boron atoms in the exohedral and endohedral C60 fullerene",to be published in PRA (November, 2022). The Reply has not been allowed to be published by PRA. (Our initial work: A. V. Bibikov, et al. Phys. Rev. A 105, 022813 (2022).)

Ultrafast quantum dynamics relaxation of a photoexcited state in a strongly correlated spin crossover system LaCoO3 under a sudden perturbation is considered with the density matrix generalized master equation. The magnetization and cobalt-oxygen bond length oscillations were found. The evolution of the electronic band structure during relaxation is calculated in the framework of the LDA+GTB method.

The effects associated with interatomic hoppings of excitons and the excitonic Bose condensate formation in the strongly correlated spin crossover systems are considered in the framework of the effective Hamiltonian for the two-band Kanamori model. The appearance of antiferromagnetic ordering due to the exciton order is found even in the absence of interatomic exchange interaction. The spectrum of excitonic excitations is calculated at various points of the "temperature vs. crystal field" phase diagram. Outside the region of exciton ordering, the spectrum has a gap, which vanishes at the boundary of the exciton condensate phase. The non-uniform spectral weight distribution over the Brillouin zone is found. The role of electron-phonon interaction is discussed as well., Comment: 10 pages, 10 figures

The B20-type cobalt germanide CoGe was investigated by measuring the specific heat, resistivity, and $^{59}$Co nuclear magnetic resonance (NMR). We observed a phase transition at $T_Q=13.7$ K, evidenced by a very narrow peak of the specific heat and sharp changes of the nuclear spin-spin ($T_2^{-1}$) and spin-lattice ($T_1^{-1}$) relaxation rates. The fact that the entropy release is extremely small and the Knight shift is almost independent of temperature down to low temperatures as anticipated in a paramagnetic metal indicates that the $T_Q$ transition is of non-magnetic origin. In addition, we detected a crossover scale $T_0\sim30$ K below which the resistivity and the NMR linewidth increase, and $T_1^{-1}$ is progressively distributed in space, that is, a static and dynamical spatial inhomogeneity develops. While the order parameter for the $T_Q$ transition remains an open question, a group-theoretical analysis suggests that the finite electric quadrupole density arising from the low local site symmetry at cobalt sites could drive the crystal symmetry lowering from the P2$_1$3 symmetry that is commensurate to the R3 symmetry with an incommensurate wavevector, which fairly well accounts for the $T_Q$ transition. The quadrupole-order-driven commensurate-incommensurate phase transition may be another remarkable phenomenon arising from the structural chirality inherent in the noncentrosymmetric B20 family., Comment: 9 pages including appendix; 5 figures; published in Physical Review B

A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. We consider the second Hamiltonian decomposition problem: for a 4-regular multigraph find 2 edge-disjoint Hamiltonian cycles different from the given ones. This problem arises in polyhedral combinatorics as a sufficient condition for non-adjacency in the 1-skeleton of the travelling salesperson polytope. We introduce two integer linear programming models for the problem based on the classical Dantzig-Fulkerson-Johnson and Miller-Tucker-Zemlin formulations for the travelling salesperson problem. To enhance the performance on feasible problems, we supplement the algorithm with a variable neighbourhood descent heuristic w.r.t. two neighbourhood structures, and a chain edge fixing procedure. Based on the computational experiments, the Dantzig-Fulkerson-Johnson formulation showed the best results on directed multigraphs, while on undirected multigraphs, the variable neighbourhood descent heuristic was especially effective.

This work demonstrates the effectiveness of the high-pressure method for the production of graphite and diamond with a high degree of boron doping using adamantanecarborane mixture as a precursor. At 8 GPa and $1700 ^{o}C$, graphite is obtained from adamantane $C_{10}H_{16}$, whereas microcrystals of boron-doped diamond (2{\div}2.5 at.% of boron) are synthesized from a mixture of adamantane and ortho-carborane $C_{2}B_{10}H_{12}$ (atomic ratio B:C = 5:95). This result shows convincingly the catalytical activity of boron in the synthesis of diamond under high pressure. At pressures lower than 7 GPa, only graphite is synthesized from the adamantane and carborane mixture. Graphitization starts at quite low temperatures (below $1400 ^{o}C$) and an increase in temperature simultaneously increases boron content and the quality of the graphite crystal lattice. Thorough study of the material structure allows us to assume that the substitutional boron atoms are distributed periodically and equidistantly from each other in the graphite layers at high boron concentrations (>1 at.%). The theoretical arguments and model ab initio calculations confirm this assumption and explain the experimentally observed boron concentrations.