Ends,tangles and critical vertex sets

We show that an arbitrary infinite graph G can be compactified by its ends plus its critical vertex sets, where a finite set X of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to X. We further provide a concrete separation system whose ?₀‐tangles are precisely the ends plus critical vertex sets. Our tangle compactification is a quotient of Diestel's (denoted by ), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of and our construction of , we show that G can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's is the finest such compactification, and our is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.     

Removable singularities of plurisubharmonic functions of restricted growth

LetG be a domain in ⁿ ,n2, letA be a connected complex (n–1)-dimensional submanifold ofG, and let be a plurisubharmonic function inGA. We obtain conditions on the growth of that guarantee the local boundedness of at a point a A G and the existence of a plurisubharmonic extension of toG.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 64–72, July, 1998.The author wishes to express her gratitude to Professor E. M. Chirka for numerous useful discussions and for his assistance in writing the present text.     

A Small Maximal Sidon Set

We construct a Sidon set A [1,N] which has (N log N)^1/3 elements and which is maximal in the sense that the inclusion of any other integer from [1, N] destroys the Sidon property.     

Large antipodal families

A family {A _i | i ∈ I} of sets in ℝ ^d is antipodal if for any distinct i, j ∈ I and any p ∈ A _i , q ∈ A _j , there is a linear functional ϕ:ℝ ^d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ _i∈I A _i . We study the existence of antipodal families of large finite or infinite sets in ℝ³. The research was supported by the Hungarian-South African Intergovernmental Scientific and Technological Cooperation Programme, NKTH Grant no. ZA-21/2006 and South African National Research Foundation Grant no. UID 61853, as well as Hungarian National Foundation for Scientific Research Grants no. NK 67867, no. T47102, and no. K72537.     

Permutations with extremal number of fixed points

We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Désarménien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics “DEZ” and “lec.” Explicit formulas for the maximal case are derived by using symmetric function tools.     

1,2,4-triazolium-cation-based energetic salts

3,4,5-Triamino-1,2,4-triazole (guanazine, 1) can be readily methylated with methyl iodide yielding methylguanazinium iodide (2). Salts containing the novel methylguanazinium cation with energetic anions were synthesised by metathesis reactions with silver azide (3), silver nitrate (4), silver perchlorate (5), sodium 5,5'-azotetrazolate (6), silver 5-nitrotetrazolate (7) and silver dinitramide (8), yielding a new family of heterocycle-based salts, which were fully characterised by analytical (mass spectrometry and elemental analysis) and spectroscopic methods (IR, Raman and NMR). In addition, the molecular structures of all compounds were confirmed by X-ray analysis, revealing extensive hydrogen-bonding in the solid state and densities between 1.399 (3) and 1.669 g cm(-3) (5). The hydrogen-bonded ring motifs are discussed in the formalism of graph-set analysis for hydrogen-bond patterns and compared to each other. Preliminary sensitivity testing of the crystalline compounds indicate surprisingly low sensitivities to both friction and impact, the highest friction and shock sensitivity being found for the perchlorate (5, 220 N) and the dinitramide (8, 20 J) salts, respectively. In addition, DSC analysis was used to assess the thermal stabilities of the compounds: 3-6 melt above 200 degrees C with concomitant decomposition, whereas 7 and 8 have clearly defined melting points at 162 and 129 degrees C, respectively, and with decomposition occurring about 30 degrees C above the melting point. Lastly all compounds have positive calculated heats of formation between 336 (4) and 4070 kJ kg(-1) (6) and calculated detonation velocities in the range between 8330 (7) and 8922 m s(-1) (6) making them of interest as new highly energetic materials with low sensitivity.     

Some Properties of a Model Liquid of C 60 Buckyballs

Utilizing the results of published simulations for the liquid C 60 phase using a model of rigid C 60 molecules, it is pointed out that the liquid phase has a critical compressibility ratio Z c = p c /( c k B T c ) in terms of the usual critical thermodynamic variables (critical pressure p c , critical density c and critical temperature T c ), of 0.32. This is to be compared with the value 0.29 for the heavy condensed rare gases Ar, Kr and Xe, in spite of their much lower T c , and with the prediction of 0.27 from Dieterici's phenomenological equation of state. The global shape of the coexistence curve, as embodied in the behaviour of the normalized difference density ( l m g )/ c versus the average density ( l + g )/2 c , where l and g are the liquid and the gas densities, respectively, is also consistent with the shape of the coexistence curve of insulating fluids. Going beyond the assumption of rigid C 60 molecules has interesting consequences on the stability and observability of the liquid phase, and those effects are discussed.     

A systematic theoretical investigation of the valence excited states of the diatomic molecules B2, C2, N2 and O2

A quantitative survey on the performance of multireference (MR), configuration interaction with all singles and doubles (CISD), MRCISD with the Davidson correction and MR-average quadratic coupled cluster (AQCC) methods for a wide range of excited states of the diatomic molecules B₂, C₂, N₂ and O₂ is presented. The spectroscopic constants r _e, ω_e, T _e and D _e for a total of 60 states have been evaluated and critically compared with available experimental data. Basis set extrapolations and size-extensivity corrections are essential for highly accurate results: MR-AQCC mean-errors of 0.001 ?, 10 cm⁻¹, 300 cm⁻¹ and 300 cm⁻¹ have been obtained for r _e, ω_e, T _e and D _e, respectively. Owing to the very systematic behavior of the results depending on the basis set and the choice of method, shortcomings of the calculations, such as Rydberg state coupling or insufficient configuration spaces, can be identified independently of experimental data. On the other hand, significant discrepancies with experiment for states which indicate no shortcomings whatsoever in the theoretical treatment suggest the re-evaluation of experimental results. The broad variety of states included in our survey and the uniform quality of the results indicate that the observed systematics is a general feature of the methods and, hence, is molecule-independent. Received: 12 June 2000 / Accepted: 1 September 2000 / Published online: 21 December 2000     

Smartphones & microfluidics: Marriage for the future

Smartphones have become widely recognized as a very interesting detection and controlling tool in microfluidics. They are portable devices with built‐in cameras and internal microprocessors which carry out image processing. In this case, the external computers are not needed and phones can provide fast and accurate results. Moreover, the connectivity of smartphones gives the possibility to share and provide real‐time results when needed, whether in health diagnostics, environmental monitoring, immunoassays or food safety. Undoubtedly, the marriage of smartphones and microfluidics has a brilliant future in building low cost and easily operable systems for analysis in the field, realizing the idea of people's “smartlife”. The aim of this review is to present and summarize the main advantages and disadvantages of the use of smartphones as well as to take a closer look at some novel achievements published during the last couple of years. In the next paragraphs, readers will find specific uses of a combination of smartphones and microfluidics such as water analysis, health analysis (virus and bacteria detection), and measurement of physical properties or smartphone liquid control in polymer devices.