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.     

Thioacetylacetone: structural and vibrational assignments.

Thioacetylacetone and its variously deuterated isotopomers have been investigated using electronic and vibrational spectroscopy combined with quantum chemical calculations. Thioacetylacetone is known for its photochromic properties, but the structures of the initial and final forms have been the subject of a long debate. Analysis of the IR spectra recorded in low-temperature argon and xenon matrices, room-temperature solutions, and in the gas phase has allowed us to establish the nature of the photochromic species and of its precursor. Similar to the case of another beta-thioxoketone, monothiodibenzoylmethane, the photo-product has been assigned to the nonchelated SH exo-rotamer of the (Z)-enethiol tautomeric form, whereas the dominant ground-state species corresponds to the chelated (Z)-enol tautomeric form. Detailed vibrational assignments have been proposed for both forms based on quantum chemical calculations and polarization experiments. In the case of the chelated (Z)-enol species prevailing in the ground state, a second-order perturbative anharmonic analysis at the B3LYP/cc-pVTZ level indicated strong anharmonic effects associated with the intramolecular hydrogen bond, leading to a shift of more than 600 cm-1 of the wavenumber of the OH-stretching vibration. A small fraction of the SH endo-rotameric chelated (Z)-enethiol form was also detected under unperturbed conditions. The (Z)-enethiol form can be converted into the (Z)-enol form by irradiation at 290 nm.     

高岭石吸附乙烯和苯的Delft分子力学研究

运用Ｄｅｌｆｔ分子力学（ＤＭＭ）程序及其粘土和共轭烯烃力场，计算研究了高岭石对乙烯和苯的吸附作用，探讨了吸附对粘土晶体和有机分子的结构、电荷分布和能量的影响，求得了高岭石吸附乙烯和苯的吸附热等重要物理量．     

Transannular reactions of two parallel 1,3-butadiynes: syntheses,structures, and reactions of 1-azacyclotetradeca-3,5,10,12-tetrayne derivatives

The synthesis of 1-alkyl and 1-aryl-1-azacyclotetradeca-3,5,10,12-tetraynes was achieved in a stepwise approach. The key intermediate was 1,13-dibromotrideca-2,4,9,11-tetrayne (18). Reaction with methyl- (19 a), ethyl- (19 b), isopropyl- (19 c), n-butyl- (19 d), and tert-butylamine (19 e) as well as aniline (19 f) and p-methoxyaniline (19 g) gave the corresponding 14-membered tetraynes 20 a-20 g. The ring inversion process of 20 b was studied by variable temperature (1)H NMR spectroscopy. From these measurements a value of 10.6 kcal mol(-1) was calculated for DeltaG(not equal). X-ray investigations on single crystals of 20 b, 20 c, and 20 f revealed the axial position for the substituent at each nitrogen atom. For 20 b we encountered the chair conformation, for 20 c both chair and boat conformations, and for 20 f the boat conformation in the solid state. The reaction of 20 c with concentrated HCl in ethanol yielded 2,10-dichloro-6-isopropyl-6-azatricyclo[9.3.0.0(4,8)]tetradeca-1(11),2,4(8),9-tetraene (25 c). Compound 25 c was oxidized by 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) to 27 c. The structure of the latter was confirmed by X-ray investigations. The reaction of 20 c in aqueous HCl lead to the formation of 10-chloro-2-isopropyl-1,3,4,6,7,8-hexahydro-2H-benzo[g]isoquinolin-9-one (37 c). The structure of 37 c was verified by X-ray studies on single crystals.     

Synthesis of <Emphasis Type="Italic">N</Emphasis>-1-(Indanyloxymethyl) and <Emphasis Type="Italic">N</Emphasis>-1-(4-Hydroxybut-2-enyloxymethyl) Analogues of the HIV Drugs Emivirine and GCA-186

Summary. A series of Emivirine and GCA-186 analogues substituted at N-1 with indan-1-yloxymethyl (6a–6c) and indan-2-yloxymethyl (6d–6f) were synthesized by reaction of the corresponding bis(indanyloxy)methans with uracils having 5-ethyl or 5-isopropyl and 6-benzyl or 6-(3,5-dimethylbenzyl) substituents. A route to the corresponding N-1 substituted 4-hydroxybut-2-enyloxymethyl analogue was also devised. All newly synthesized compounds showed potent activity against wild-type HIV-1, the most active compound being 5-ethyl-1-(indan-1-yloxymethyl)-6-(3,5-dimethylbenzyl)uracil (6b), which was 50-fold more active than Emivirine.Present address: Chemistry Department, Faculty of Science, Tanta University, Tanta, EgyptA research center funded by The Danish National Research Foundation for studies on nucleic acid chemical biology     

The application of "click chemistry" for the decoration of 2(1H)-pyrazinone scaffold: generation of templates

The "click chemistry" approach has been explored on the 2-(1H)-pyrazinone scaffold for the generation of pharmacologically interesting heterocyclic moieties. Huisgen 1,3-dipolar cycloaddition has been evaluated as the key step for the construction of the 1,2,3-triazole ring at the C-3 position of 2-(1H)-pyrazinones. Two different pathways have been successfully evaluated: (1) via C-C or C-O linkage of the acetylenic part to the C-3 position of the 2-(1H)-pyrazinone scaffold or (2) via azide introduction in the C-3 position. The subsequent application of "click chemistry" resulted in the formation of hitherto unknown skeletons. Microwave irradiation has successfully been applied in different steps of the sequence.     

Changes in calmodulin main-chain dynamics upon ligand binding revealed by cross-correlated NMR relaxation measurements

The fast dynamics of protein backbones are often investigated by nuclear magnetic relaxation experiments that report on the degree of spatial restriction of the amide bond vector. By comparing calmodulin in the peptide-bound and peptide-free states with these classical methods, we observe little difference in the dynamics of the polypeptide main chain (average order parameter decrease of 0.01 unit upon binding). However, when using NMR methods that monitor the mobility of the CO-Calpha bond vector, we reveal a significant reduction of dynamics of the protein main chain (average order parameter decrease of 0.048 units). Previous investigations have suggested that the side-chain dynamics is reduced by an average of 0.07 order parameter units upon ligand binding (Lee, A. L.; Kinnear, S. A.; Wand, A. J. Nat. Struct. Biol. 2000, 7, 72-77). The current findings suggest that the change of the CO-Calpha bond vector dynamics is intermediate between the changes in NH and side-chain dynamics and report a previously undetected loss of main-chain entropy. Weak site-to-site correlations between the different motional indicators are also observed.