Synthesis of Pyrazolo[3,4‐b]pyridin‐6‐ones

The procedures for the synthesis of substituted pyrazolo[3,4‐b]pyridine‐6‐ones and some of their properties are reviewed.     

Benzothiazinones are suicide inhibitors of mycobacterial decaprenylphosphoryl-β-D-ribofuranose 2'-oxidase DprE1

Benzothiazinones (BTZs) are antituberculosis drug candidates with nanomolar bactericidal activity against tubercle bacilli. Here we demonstrate that BTZs are suicide substrates of the FAD-dependent decaprenylphosphoryl-β-D-ribofuranose 2'-oxidase DprE1, an enzyme involved in cell-wall biogenesis. BTZs are reduced by DprE1 to an electrophile, which then reacts in a near-quantitative manner with an active-site cysteine of DprE1, thus providing a rationale for the extraordinary potency of BTZs. Mutant DprE1 enzymes from BTZ-resistant strains reduce BTZs to inert metabolites while avoiding covalent inactivation. Our results explain the basis for drug sensitivity and resistance to an exceptionally potent class of antituberculosis agents.     

Fractional quantum Hall States of Dirac electrons in graphene

We have investigated the fractional quantum Hall states of Dirac electrons in a graphene layer in different Landau levels. The relativistic nature of the energy dispersion relation of electrons in graphene significantly modifies the interelectron interactions. This results in a specific dependence of the ground state energy and the energy gaps for electrons on the Landau-level index. For the valley-polarized states, i.e., at nu=1/m, m being an odd integer, the energy gaps have the largest values in the n=1 Landau level. For the valley-unpolarized states, e.g., for the 2/3 state, the energy gaps are suppressed for n=1 as compared to those at n=0. For both n=1 and n=0, the ground state of the 2/3 system is fully valley-unpolarized.     

The bondage number of graphs on topological surfaces and Teschner’s conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, and improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of graphs. Also, we present stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of graph genera. This settles Teschner’s Conjecture in affirmative for almost all graphs. As an auxiliary result, we show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus.     

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)?B(t)y(t) = Δ(t) f(t) on an interval ${\mathcal{I}=[a,b) }$ with the regular endpoint a. It is assumed that the deficiency indices n _±(T _min) of the minimal relation T _min associated with this system in ${L^2_\Delta(\mathcal{I})}$ satisfy ${n_-(T_{\rm min})\leq n_+(T_{\rm min})}$ . We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size ${N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}$ . Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform ${V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}$ where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension ${\tilde{T}}$ of T _min induced by the boundary problem is unitarily equivalent to the multiplication operator in L ²(Σ). Hence the multiplicity of spectrum of ${\tilde{T}}$ does not exceed N _Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.     

The Cyclomatic Number of a Graph and its Independence Polynomial at −1

If by s _k is denoted the number of independent sets of cardinality k in a graph G, then ${I(G;x)=s_{0}+s_{1}x+\cdots+s_{\alpha}x^{\alpha}}$ is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97–106, 1983), where α = α(G) is the size of a maximum independent set. The inequality |I (G; ?1)| ≤ 2 ^ν(G), where ν(G) is the cyclomatic number of G, is due to (Engström in Eur. J. Comb. 30:429–438, 2009) and (Levit and Mandrescu in Discret. Math. 311:1204–1206, 2011). For ν(G) ≤ 1 it means that ${I(G;-1)\in\{-2,-1,0,1,2\}.}$ In this paper we prove that if G is a unicyclic well-covered graph different from C ₃, then ${I(G;-1)\in\{-1,0,1\},}$ while if G is a connected well-covered graph of girth ≥ 6, non-isomorphic to C ₇ or K ₂ (e.g., every well-covered tree ≠ K ₂), then I (G; ?1) = 0. Further, we demonstrate that the bounds {?2 ^ν(G), 2 ^ν(G)} are sharp for I (G; ?1), and investigate other values of I (G; ?1) belonging to the interval [?2 ^ν(G), 2 ^ν(G)].     

Transition Probability of Brownian Motion in the Octant and its Application to Default Modelling

ABSTRACT

We derive a semi-analytical formula for the transition probability of three-dimensional Brownian motion in the positive octant with absorption at the boundaries. Separation of variables in spherical coordinates leads to an eigenvalue problem for the resulting boundary value problem in the two angular components. The main theoretical result is a solution to the original problem expressed as an expansion into special functions and an eigenvalue which has to be chosen to allow a matching of the boundary condition. We discuss and test several computational methods to solve a finite-dimensional approximation to this nonlinear eigenvalue problem. Finally, we apply our results to the computation of default probabilities and credit valuation adjustments in a structural credit model with mutual liabilities.     

GaCl(3)-catalyzed insertion of diazene derivatives into the cyclopropane ring

GaCl3 has been found to efficiently catalyze the formal cycloadditions of diazene derivatives 6 onto 2-arylcyclopropane-1,1-dicarboxylates 5, giving rise to the pyrazolidine derivatives 7. The insertion into the cyclopropane ring proceeds with complete regioselectivity to furnish 5-arylpyrazolidine-1,2,3,3-tetracarboxylates exclusively; however, the cis-configured azo compound N-phenyltriazolinedione gave the two possible regioisomeric pyrazolidine derivatives in ratios varying from 1:1.5 to 1:3. Conceivable mechanisms of these transformations are being discussed.     

Unsymmetrical Ni(II)-imidoylamidine complexes derived from a novel oxime-mediated single-pot reaction of nitriles

The neutral bisimidoylamidine (or 1,3,5-triazapentadiene) Ni(II) complexes [Ni{HN=C(R)N=C(C(6)H(4)CO)N}(2)] {R=Me, Et, nPr, iPr, (CH(2))(3)Cl}, bearing an iminoisoindolinone moiety, have been generated by a novel 2-propanone oxime-mediated single-pot reaction of phthalonitrile and nickel(II) acetate in solution of the corresponding alkylnitrile (RCN). Single-crystal X-ray diffraction analyses indicate a square-planar geometry of the imidoylamidine core. The synthesised compounds represent rare specimens of genuine unsymmetrical imidoylamidine complexes as well as their first examples generated from nitriles in a single-pot reaction.     

Satellites of Xe transitions induced by infrared active vibrational modes of CF4 and C2F6 molecules

Absorption and luminescence excitation spectra of Xe/CF(4) mixtures were studied in the vacuum UV region at high resolution using tunable synchrotron radiation. Pressure-broadened resonance bands and bands associated with dipole-forbidden states of the Xe atom due to collision-induced breakdown of the optical selection rules are reported. The spectra display in addition numerous satellite bands corresponding to transitions to vibrationally excited states of a Xe-CF(4) collisional complex. These satellites are located at energies of Xe atom transition increased by one quantum energy in the IR active v(3) vibrational mode of CF(4) (v(3) = 1281 cm(-1)). Satellites of both resonance and dipole-forbidden transitions were observed. Satellites of low lying resonance states are spectrally broad bands closely resembling in shape their parent pressure-broadened resonance bands. In contrast, satellites of dipole-forbidden states and of high lying resonance states are spectrally narrow bands (FWHM ～10 cm(-1)). The satellites of dipole-forbidden states are orders of magnitude stronger than transitions to their parent states due to collision-induced breakdown of the optical selection rules. These satellites are attributed to a coupling of dipole-forbidden and resonance states induced by the electric field of the transient CF(4) (v(3) = 0 ? v(3) = 1) dipole. Similar satellites are present in spectra of Xe/C(2)F(6) mixtures where these bands are induced by the IR active v(10) mode of C(2)F(6). Transitions to vibrationally excited states of Xe-CF(4)(C(2)F(6)) collision pairs were also observed in two-photon LIF spectra.