Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.     

An Ore-type analogue of the Sauer-Spencer Theorem

Two graphs G₁ and G₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer proved that if Δ (G₁) Δ (G₂) < 0.5n, then G₁ and G₂ pack. In this note, we study an Ore-type analogue of the Sauer–Spencer Theorem. Let θ(G) = max{d(u) + d(v): uv∈E(G)}. We show that if θ(G₁)Δ(G₂) < n, then G₁ and G₂ pack. We also characterize the pairs (G₁,G₂) of n-vertex graphs satisfying θ(G₁)Δ(G₂) = n that do not pack. This work was supported in part by NSF grant DMS-0400498. The work of the first author was also partly supported by grant 05-01-00816 of the Russian Foundation for Basic Research.     

Action of the Cremona group on a noncommutative ring

The Cremona group acts on the field of two independent commutative variables over complex numbers. We provide a noncommutative algebra that is an analog of a noncommutative field of two independent variables and prove that the Cremona group embeds in a group of outer automorphisms of this algebra. We give two proofs of it, the first proof is technical, the second one is conceptual and proceeds through a construction of a birational invariant of an algebraic variety from its bounded derived category of coherent sheaves.     

Magnetocaloric effect and multifunctional properties of Ni–Mn-based Heusler alloys

The studies of magnetocaloric properties, phase transitions, and phenomena related to magnetic heterogeneity in the vicinity of the martensitic transition (MT) in Ni–Mn–In and Ni–Mn–Ga off-stoichiometric Heusler alloys are summarized. The crystal structure, magnetocaloric effect (MCE), and magnetotransport properties were studied for the following alloys: Ni₅₀Mn_50−xIn_x, Ni_50−xCo_xMn₃₅In₁₅, Ni₅₀Mn_35−xCo_xIn₁₅, Ni₅₀Mn₃₅In₁₄Z (Z=Al, Ge), Ni₅₀Mn₃₅In_15−xSi_x, Ni_50−xCo_xMn_25+yGa_25−y, and Ni_50–xCo_xMn_32−yFeyGa₁₈. It was found that the magnetic entropy change, ΔS, associated with the inverse MCE in the vicinity of the temperature of the magneto-structural transition, TM, persists in a range of (125-5) J/(kg K) for a magnetic field change ΔH=5 T. The corresponding temperature varies with composition from 143 to 400 K. The MT in Ni₅₀Mn_50−xIn_x (x=13.5) results in a transition between two paramagnetic states. Associated with the paramagnetic austenite-paramagnetic martensite transition ΔS=24 J/(kg K) was detected for ΔH=5 T at T=350 K. The variation in composition of Ni₂MnGa can drastically change the magnetic state of the martensitic phase below and in the vicinity of TM. The presence of the martensitic phase with magnetic moment much smaller than that in the austenitic phase above TM leads to the large inverse MCE in the Ni₄₂Co₈Mn_32−yFeyGa₁₈ system. The adiabatic change of temperature (ΔT_ad) in the vicinity of TC and TM of Ni₅₀Mn₃₅In₁₅ and Ni₅₀Mn₃₅In₁₄Z (Z=Al, Ge) was found to be ΔT_ad=−2 K and 2 K for ΔH=1.8 T, respectively. It was observed that |ΔT_ad|≈1 K for ΔH=1 T for both types of transitions. The results on resistivity, magnetoresistance, Hall resistivity in some In-based alloys are discussed.     

Features of X-ray fluorescence determination of rock-forming elements in powder samples of peat sediments

The features of rock-forming elements determination in powder samples of peat sediments are considered. Based on theoretical calculations and experimental data, the mineralogical and particle size effects on the X-ray fluorescence intensity have been estimated. For routine analysis, powder rock samples are usually milled to a particle size of 10–60 μm. Theoretically, estimated variations of rock-forming elements analytical lines intensities in different minerals are up to 30%, and the particle size effect cannot be eliminated simultaneously for all rock-forming elements. Experimental estimation shows that the impossibility of the particle size distribution control can lead to significant variations in the intensities of the analytical lines (3%–18%) that is mainly corresponding to theoretical estimation. Fine milling allowed us to achieve an average particle size of fewer than 20 μm and reduce the measurement uncertainty by 1.5–2 times. 12 samples of peat sediments with organic matter content up to 70 wt% were analyzed by X-ray fluorescence analysis using different sample preparation techniques, as well as by certified methods. The accuracy of X-ray fluorescence analysis of peat sediments prepared as pressed pellets varies from 1.5 to 11 rel% depending on compound contents. It is more than for samples prepared as fused beads, however, the proposed method can be used to determine the variations of rock-forming elements contents in peat sediments where the content variations exceed the measurement uncertainty, and accuracy of analysis is sufficient for further paleoecological reconstructions.     

Density Conditions for Panchromatic Colourings of Hypergraphs

Let be a hypergraph. A panchromatic t-colouring of is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of is . Among other results, it is proved here that if every edge has at least t vertices and whenever , then is panchromatically t-choosable, and this condition is sharp; the minimum such that every t-uniform hypergraph with is panchromatically t-choosable satisfies ; and except possibly when t = 3 or 5, a t-uniform hypergraph is panchromatically t-colourable if whenever , and this condition is sharp. This last result dualizes to a sharp sufficient condition for the chromatic index of a hypergraph to equal its maximum degree. Received November 10, 1998 RID="*" ID="*" This work was carried out while the first author was visiting Nottingham, funded by Visiting Fellowship Research Grant GR/L54585 from the Engineering and Physical Sciences Research Council. The work of this author was also partly supported by grants 96-01-01614 and 97-01-01075 of the Russian Foundation for Fundamental Research.     

Electronic structure of saturated hydrocarbons in the semi-empirical equivalent orbital method

The problem of obtaining the matrix elements of Hartree-Fock Hamiltonians for alkanes using the EO method is considered. It has been shown that the data on the electronic structure of diamond together witht ₁/e splitting in the neopentane photoelectron spectrum are helpful to produce such EO method parameter scale which involves even “through space” interactions. In terms of the EO method the photoelectron spectra of propane, butane, isobutane, and neopentane are interpreted. The valence band structure of polyethylene in analytical form is obtained.     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008