On the spectrum of the sum of generators for a finitely generated group

Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π. For the universal representationπ _un of Γ, we prove in particular the following results. First, the spectrum Sp(π _un (h)) contains the complex numberz of modulus one iff Sp(π _un (h)) is invariant under multiplication byz, iff there exists a character such that η(S)={z}. Second, forS ⁻¹=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π _un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Etude des conditions d'accès au triazino-1,2,4-[4,5-b] indazole

1,2,4-Triazino[4,5-b]indazol-1(2H)one and its derivatives were prepared by transposition of 3-[2-(-1,3,4-oxadiazolyl)]indazole or by ring closure of indazole ethoxymethylidenehydrazides. The synthesis of 1,2,3,4-tetrahydro-l,2,4-triazino[4,5-b]indazole-1,4-dione was achieved by cyclising the N-carbethoxyhydrazide of indazole-3-carboxylic acid and the synthesis of 1,2,4-triazino-[4,5-b]indazol-4(3H)one was made by cyclising the N-carbethoxy-hydrazone of indazole-3-carboxaldehyde. The Oxydation of 1,2,4-triazino[4,5-b]indazole-l(2H)thione gave 1,2,4-triazino-[4,5-b]indazoles. Nmr spectral data are reported.     

Composition and major odorous compounds of the essential oil of Bifora radians,an aldehyde-producing weed

The composition of the essential oil of Bifora radians, an aldehyde-producing weed, has been investigated by capillary gas chromatography, coupled gas chromatography – mass spectrometry, on-line catalytic hydrogenation and coupled gas chromatography – infrared spectrometry. The nineteen compounds identified included eighteen aldehydes: seven alkanals (C6, C9, C10, C11, C12, C13, and C14), ten alkenals, including five (E)-2-alkenals (C12, C13, C14, C15, and C16), and one (E,E)-2,4-alkadienal (C13). Typical Bifora odors were attributed to three major (E)-2-alkenals, C12, C13, and C14.     

Complex strict convexity of certain Banach spaces

Istratescu's characterization of complex strict convex (csc) Banach spaces is used to show that a modulared sum of a sequence ofcsc Banach spaces is again acsc Banach space. The equivalence of the Strong Maximum Modulus Property and complex strict convexity is used to show thatL ¹(,X) iscsc whenX is (real) strictly convex and thatl ¹(X _n) iscsc if and only if eachX _n iscsc.     

Improvement in the synthesis of metallophthalocyanines using microwave irradiation

A successful application of microwave irradiation, in which phthalocyanines were synthesized under solventless conditions from 1,2-phthalonitrile or phthalic anhydride and urea in the presence of metal templates is described. It was found that in comparison with conventional heating, the microwave process is a very useful alternative for cyclotetramerization processes because of reduction of the reaction time, better yield, and easy-to-perform procedure.     

Quantitation of diethylene glycol monoethyl ether in citrus oils

The use of diethylene glycol monoethyl ether as a flavor solvent is restricted. Since existing methods of quantitation are tedious or require sophisticated multi-dimensional GC instrumentation, a simple method has been developed using low cost GC-MS. Monitoring of ions 31 and 45 enables quantitation of DEGEE in essential oils at levels down to 1 μg/g with accuracy equal to or better than 15%.     

An Absence of Non-specific Immune Response towards para-Sulphonato-calix[n]arenes

The effects of para-Sulphonato-calix[4]arene, para-Sulphonato-calix[6]arene and para-Sulphonato-calix[8]arene on the activation of NADPH oxidase in neutrophils has been studied. All three molecules do not induce NADPH oxidase activation, and hence do not stimulate neutrophils. Measurement of cell viability demonstrates that these three water-soluble calix[n]arene derivatives are not cytotoxic.     

13C NMR of indazoles

The carbon-13 chemical shifts and some selected coupling constants of 183 indazoles are reported. The main conclusions of the original references are briefly summarized.Published in Khimiya Geterotsiklicheskikh Soedinenii, No. 9, pp. 1159–1179, September, 1995.