Path integrals on finite sets

We construct an analogue of the Feynman path integral for the case of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% aaaeaacaaIXaaabaGaamyAaaaadaWcaaqaaiabgkGi2cqaaiabgkGi% 2kaadshaaaqeduuDJXwAKbYu51MyVXgaiuaacqWFvpGAcaWG0bGaey% ypa0JaamisamaaBaaaleaacaGGOaaabeaakmaaBaaaleaacaGGPaaa% beaakiab-v9aQjaadshaaaa!4A8D!\[ - \frac{1}{i}\frac{\partial }{{\partial t}}\varphi t = H_( _) \varphi t\] in which H ₍₎ is a self-adjoint operator in the space L ²(M)= % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSOaHmkaaa!3744!\[\mathbb{C}\], where _M is a finite set, the paths being functions of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa!375D!\[\mathbb{R}\] with values in M. The path integral is a family of measures F _t,t with values in the operators on L ²(M), or equivalently, a family of complex measures corresponding to matrix coefficients.It is shown that these measures on path space are in some sense dominated by the measure of a Markov process. This implies that F _t,t is concentrated on the set of step functions S[t,t].This allows one to make sense of, and prove, the analogue of Feynman's formula for the propagator of the Hamiltonian H=H ₀+V, where V is a potential, namely the formula: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzamaaCa% aaleqabaGaeyOeI0IaamyAaiaacIcacaWG0bGaai4jaiabgkHiTiaa% dshacaGGPaGaamisaaaakiabg2da9maapebabaGaaeyzamaaCaaale% qabaGaeyOeI0IaamyAamaapedabaGaamOvaiaacIcatCvAUfKttLea% ryqr1ngBPrgaiuGacqWF4baEcaGGOaGaam4CaiaacMcacaGGPaGaae% izaiaabohaaWqaaiaadshaaeaacaWG0bGaai4jaaGdcqGHRiI8aaaa% kiaadAeadaWgaaWcbaGaamiDaiaacEcacaGGSaGaamiDaaqabaGcca% GGOaGaaeizaiab-Hha4jaacMcaaSqaaiaadofacaGGBbGaamiDaiaa% cYcacaWG0bGaai4jaiaac2faaeqaniabgUIiYdaaaa!6410!\[{\text{e}}^{ - i(t' - t)H} = \int_{S[t,t']} {{\text{e}}^{ - i\int_t^{t'} {V(x(s)){\text{ds}}} } F_{t',t} ({\text{d}}x)} \]and the corresponding formulas for the matrix coefficients, in which the integral extends over the paths beginning and ending in the appropriate points. We show that the measures F _t,t are completely determined by these equations and by a certain multiplicative property.The path integral corresponding to a two-particle system without interaction is the direct product of the corresponding path integrals. The propagator for a two-particle system with interaction can be obtained by repeated integration.Finally, we show that the above integral formula can be generalized to the case where the potential is time dependent.     

Analysis of Open Discrete Time Queueing Networks: A Refined Decomposition Approach

This paper deals with approximate analysis methods for open queueing networks. External and internal flows from and to the nodes are characterized by renewal processes with discrete time distributions of their interarrival times. Stationary distributions of the waiting time, the queue size and the interdeparture times are obtained using efficient discrete time algorithms for single server (GI/G/1) and multi-server (GI/D/c) nodes with deterministic service. The network analysis is extended to semi-Markovian representations of each flow among the nodes, which include parameters of the autocorrelation function.     

The Electronic Structure of Azuleno[1,2,3-cd]phenalene and Azuleno[5,6,7-cd]phenalene,a Comparison

The photoelectron (PE.) spectra of azuleno[l, 2, 3-cd]phenalene ( 1 ) and azuleno- [5,6,7-cd]phenalene( 2 ) have been recorded. The first five bands of both compounds could be assigned to transitions corresponding to removal of electrons from 4a₂, 6b₁, 5b₁, 3a₂ and 4b_l orbitals. This assignment is based mainly on a comparison between the observed ionization potentials and orbital energies calculated in a HMO and a PPP model. The UV./VIS. polarized absorption spectrum of 1 in the region 10000–45000 cm^?1 has been measured by means of the stretched film technique. The measurements were performed in polyethylene sheets at 77°K. Several bands could be assigned to π* ← π transitions calculated by a PPP-CI method. A comparison between the electronic structures of 1 and 2 is made by means of a simple HMO diagram.     

Empfindliche Methode zur Bestimmung geringer Schwefelgehalte in Wolframoxiden und Wolframpulvern

Zusammenfassung Der Schwefel in Wolframoxiden und Wolframpulvern wird im Wasserstoffstrom bei 800°C in H₂S übergeführt, der in Cadmium-Zinkacetat-Lösung absorbiert und anschließend jodometrisch titriert wird. Die Anwesenheit von WO₃ ermöglicht eine quantitative Umwandlung von Sulfatschwefel in H₂S. Die Standardabweichung beträgt für ein Wolframoxid mit 310 ppm S±13 ppmS und für ein Wolfram-Metallpulver mit 45 ppm S±1 ppm S.

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Sulfur is converted into hydrogen sulphide by hydrogenation at 800°C. This H₂S is absorbed in a cadmium-zinc acetate solution and titrated iodometrically. The presence of WO₃ makes possible a quantitative conversion of sulphate sulphur into H₂S. The standard deviation is ±13 ppm S for WO₃ containing 310 ppm S, and ±1 ppm S for metallic tungsten powder containing 45 ppm S.

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Herrn Prof. Dr.E. Hayek zum 60. Geburtstag gewidmet.     

Beitrag zur Frage der qualitativen und quantitativen Bestimmung einiger Pyridinderivate mit Hilfe der IR-Absorptionsspektroskopie

Zusammenfassung Es wird die Möglichkeit einer qualitativen und quantitativen Reinheitsprüfung mit Mitteln der IR-Absorptionsspektroskopie an einer Anzahl von Pyridinderivaten untersucht. Zu diesem Zweck werden die IR-Absorptionsspektren der reinen Substanzen, sowie von binären und ternären Gemischen aufgenommen. Die Spektren werden aus-gewertet und die Nachweisgrenzen angegeben.Dem Verband der Chemischen Industrie und der Deutschen Forschungsgemeinschaft danken wir für die Unterstützung dieser Arbeit.     

Electron-rich radicals by neutralizationreionization mass spectrometry. Generation, dissociations and energetics of the hydrogen atom adduct to acetamide

Protonated acetamide exists as two planar conformers, the more stable anti-form (anti-1(+)) and the syn-form (syn-1(+)), DeltaG(degree) (298) (anti-->syn) = 10.8 kJ mol(-1). Collisional neutralization of 1(+) produces 1-hydroxy-1-amino-1-ethyl radicals (anti-1 and syn-1) which in part survive for 3.7 micros. The major dissociation of 1 is loss of the hydroxyl hydrogen atom (approximately 95%) which is accompanied by loss of one of the methyl hydrogen atoms (approximately 3%) and loss of the methyl group (approximately 2%). The most favorable dissociation of the OH bond is calculated to be only 34 kJ mol(1) endothermic but requires 88 kJ mol(-1) in the transition state. Other dissociations of 1, e.g., loss of one of the amide hydrogens, methyl hydrogens, and loss of ammonia are calculated to proceed through higher- energy transition states and are not kinetically competitive if proceeding from the ground doublet electronic state of 1. The unimolecular dissociation of 1 following collisional electron transfer is promoted by large Franck-Condon effects that result in 8090 kJ mol(-1) vibrational excitation in the radicals. Radicals 1 are calculated to exoergically abstract hydrogen atoms from acetamide in water, but not in the gas phase. The different reactivity is due to solvent effects that favor the products, (.)CH(2)CONH(2) and CH(3)CH(OH)NH(2), over the reactants.     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.     

A moment problem associated to rational Szegő functions

Leta ₁,...,a _p be distinct points in the finite complex plane ?, such that |a _j|>1,j=1,..., p and let \(b_j = 1/\bar \alpha _j ,\) j=1,..., p. Let μ₀, μ _π ^(j) , ν _π ^(j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [?π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ?^* outside {a ₁,...,a _p,b ₁,...,b _p}.