A criterion for global hyperbolicity of left-invariant Lorentz metrics on Lie groups

We show that every left-invariant Lorentz metric on a non-abelian simply connected Lie group is globally hyperbolic whenever its restriction to the commutator ideal of the Lie algebra is positive definite. We also show that a left-invariant Lorentz metric on the three-dimensional Heisenberg group is globally hyperbolic if and only if its restriction to the center of the Lie algebra is positive definite or degenerate.     

Reductions of 3-connected graphs with minimum degree at least four

We show that if G is a 3-connected graph of minimum degree at least 4 and with |V (G)| ≥ 7 then one of the following is true: (1) G has an edge e such that G/e is a 3-connected graph of minimum degree at least 4; (2) G has two edges uv and xy with ux, vy, vx ∈ E(G) such that the graph G/uv/xy obtained by contraction of edges uv and xy in G is a 3-connected graph of minimum degree at least 4; (3) G has a vertex x with N(x) = {x₁, x₂, x₃, x₄} and x₁x₂, x₃x₄ ∈ E(G) such that the graph (G ? x)/x₁x₂/x₃x₄ obtained by contraction of edges x₁x₂ and x₃x₄ in G – x is a 3-connected graph of minimum degree at least 4.

Each of the three reductions is necessary: there exists an infinite family of 3- connected graphs of minimum degree not less than 4 such that only one of the three reductions may be performed for the members of the family and not the two other reductions.     

Hyperharmonic series involving Hurwitz zeta function

We show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. These results enable us to reformulate Euler's formula involving the Hurwitz zeta function. In additon, we improve Conway and Guy's formula for hyperharmonic numbers.     

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was:

Theorem. (See Borwein, Choi, Yazdani, 2001.) Letf(z)=±z±z²±?±zN−1, and ζ a primitive Nth root of unity. If N is an odd positive integer then

     

Unprovability threshold for the planar graph minor theorem

This note is part of the implementation of a programme in foundations of mathematics to find exact threshold versions of all mathematical unprovability results known so far, a programme initiated by Weiermann. Here we find the exact versions of unprovability of the finite graph minor theorem with growth rate condition restricted to planar graphs, connected planar graphs and graphs embeddable into a given surface, assuming an unproved conjecture (*): ‘there is a number a>0 such that for all k≥3, and all n≥1, the proportion of connected graphs among unlabelled planar graphs of size n omitting the k-element circle as minor is greater than a’. Let γ be the unlabelled planar growth constant (27.2269≤γ<30.061). Let P(c) be the following first-order arithmetical statement with real parameter c: “for every K there is N such that whenever G₁,G₂,…,G_N are unlabelled planar graphs with |G_i|<K+c⋅log₂i then for some i<j≤N, G_i is isomorphic to a minor of G_j”. Then

1.

for every , P(c) is provable in IΔ₀+exp;

2.

for every , P(c) is unprovable in .

We also give proofs of some upper and lower bounds for unprovability thresholds in the general case of the finite graph minor theorem.     

A determinantal approach to Appell polynomials

A new definition by means of a determinantal form for Appell (1880) [1] polynomials is given. General properties, some of them new, are proved by using elementary linear algebra tools. Finally classic and non-classic examples are considered and the coefficients, calculated by an ad hoc Mathematica code, for particular sequences of Appell polynomials are given.     

Transport properties of single crystal

Transport properties of BaNi₂P₂ single crystals prepared by high-pressure synthesis method have been investigated. The temperature dependence of the resistivity is that of a typical metal with the anisotropy ratio ρ/ρ of 6.3 and suggests that electron–phonon interaction dominates the scattering mechanism. We have also found that the Hall effect and the magnetoresistance can be explained by a two-carrier model which is consistent with a multiple-band structure with both hole and electron characters.     

Efficient Building Blocks for Solid-Phase Peptide Synthesis of Spin Labeled Peptides for Electron Paramagnetic Resonance and Dynamic Nuclear Polarization Applications

Specific spin labeling allows the site-selective investigation of biomolecules by EPR and DNP enhanced NMR spectroscopy. A novel spin labeling strategy for commercially available Fmoc-amino acids is developed. In this approach, the PROXYL spin label is covalently attached to the hydroxyl side chain of three amino acids hydroxyproline (Hyp), serine (Ser) and tyrosine (Tyr) by a simple three-step synthesis route. The obtained PROXYL containing building-blocks are N-terminally protected by the Fmoc-protection group, which makes them applicable for the use in solid-phase peptide synthesis (SPPS). This approach allows the insertion of the spin label at any desired position during SPPS, which makes it more versatile than the widely used post synthetic spin labeling strategies. For the final building-blocks, the radical activity is proven by EPR. DNP enhanced solid-state NMR experiments employing these building-blocks in a TCE solution show enhancement factors of up to 26 for ¹H and ¹³C (¹H→¹³C cross-polarization). To proof the viability of the presented building-blocks for insertion of the spin label during SPPS the penta-peptide Acetyl-Gly-Ser(PROXYL)-Gly-Gly-Gly was synthesized employing the spin labeled Ser building-block. This peptide could successfully be isolated and the spin label activity proved by EPR and DNP NMR measurements, showing enhancement factors of 12.1±0.1 for ¹H and 13.9±0.5 for ¹³C (direct polarization).     

Complete Minors In <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">s,s</Emphasis></Subscript>-Free Graphs

We prove that for a fixed integer s2 every K _s,s -free graph of average degree at least r contains a K _p minor where . A well-known conjecture on the existence of dense K _s,s -free graphs would imply that the value of the exponent is best possible. Our result implies Hadwigers conjecture for K _s,s -free graphs whose chromatic number is sufficiently large compared with s.     

Integer parts of powers of rational numbers

We prove that the sequence [ξ(5/4)ⁿ], n=1,2, . . . , where ξ is an arbitrary positive number, contains infinitely many composite numbers. A corresponding result for the sequences [(3/2)ⁿ] and [(4/3)ⁿ],n=1,2, . . . , was obtained by Forman and Shapiro in 1967. Furthermore, it is shown that there are infinitely many positive integers n such that ([ξ(5/4)ⁿ],6006)>1, where 6006=2·3·7·11·13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to ξ(5/3)ⁿ and to ξ(7/5)ⁿ, n∈ℕ. The corresponding sets of possible divisors are also described.