Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

Quasi-Particle Dual Mappings on \widetilde{\mathbb{H}}_v Carrier Spaces for Dynamical Structures of S 2n -Invariant-Based, Integer-Rank Tensorial Sets: A Democratic-Recoupling S 2n -Compatible Overview of v-Auxiliary Labelling in Superboson Algebras of Uniform Multispin NMR Ensembles

The independent cardinalities of invariants (SIs) of the dual group and its (2n)-fold uniform spin ensembles are derived via S _2n -decompositional approaches. The fundamental terms arise from character theory, whereas the (2i < 2n) subsidary statistical subsets arise from arguments based on time-reversal symmetry (TRV) in the context of democratic recoupling – an approach consistent with a projective view and also with the nature of (Weyl) (I ? I)₁???() _i ???(I ? I) _n single pairwise exchange yielding TRV spin symmetry. In contrast to traditional |D ⁰(U)|((?SU(2))⁽²ⁿ⁾) formalisms, democratic sampling and recoupling now play important roles. As derived via superboson algebra, the dual tensorial properties of [A...]_(2n) uniform 2n-fold NMR spin systems are seen also as fundamental in defining aspects of quantum computing via NMR analogues. Representational aspects of (quasiparticle) pattern algebras are given for superbosons over dual projective carrier subspaces of Liouville space. Hence an independent Lie-algebraic proof is derived of the fundamental “sign” structure of superbosons acting on { $\sum_v \oplus_{\tilde\lambda}\widetilde{\mathbb{H}}_v^(\tilde\lambda)|[\tilde\lambda]$ ${\mathcal{S}}_n$ ;vaux. SU(2) × ${\mathcal{S}}_n$ } carrier subspaces. This constitutes a direct extension of earlier work [Physica A 198 (1993) 245]. The explicit SI-based tensorial v-auxiliary labels of the latter are defined via the dual action $\tilde U \times {\mathcal{P}}:\tilde {\mathbb{H}} \to \tilde {\mathbb{H}}\left\{ { \cdot \left| {\tilde U \in SU\left( 2 \right),} \right.{\mathcal{P}}\left( {\tilde \Gamma } \right) \in {\mathcal{S}}_n } \right\}$ for the dual group and [A...] _n (S _n ) identical spin ensembles involving high degeneracies. Motivation for the work arose from certain fundamental quantum questions for dual group actions on multiple invariant-based systems, themselves subject to projective-compatible, S _n democratic recoupling. to graph-based results for such specialised uniform systems. Attention is drawn to various correspondences to geometric, quasiparticle (or superboson) and Lie-algebraic concepts for these systems, as being of topical interest both to quantum physics and to computing.     

nBody democratic recoupling and role of Schur Fn. products on Sn, a NMR dual-group view pertinent to coherence-transfer

For automorphic spin symmetries over isotopomer networks defining both NMR coherence transfer and cage-cluster ro-vibrational (RV) weighting phenomena, the role of Schur functions (SF) and their SF products (SFP) (restricted to S_n) is re-evaluated, beyond that typical of atomic physics. It is seen now as being equally applicable to molecular physics. Our special focus here is on the SU2 × S_n dual group for its importance in replacing (Jucys) recoupling schemata by generalised S_n democracy invariants, which ensure the retention of simple reducibility (SR) of carrier spaces for superboson mappings, under U × P(Γ) actions over Liouville space, i.e., as in Physica A 198 (1993) 245. SFP mapping onto S_n and simple Young rule SF mappings onto {[λ]}(S_n) sets are developed in the high n-index, weak-branching limit and utilise known similarities in the algorithms for the Young and Littlewood–Richardson rules. Over discrete k rank, from k=n (down), the Σ_ν{/T ^k(ν)/x007D;k-set orders are related to the ⊵ ordered bipartite χ _12n ^[λ] (S _2n characters. This revised version was published online in July 2006 with corrections to the Cover Date.     

Demonstration of the origins of cooperativity within SU2×L 6 mapping over\left\{ {I\tilde H_\upsilon } \right\} Liouvillian carrier subspaces characterising the MQ-NMR cluster [A]6(L 6) and its\{ |kq\upsilon :[\tilde \lambda ] > > \} tensorial bases

The fundamental mappings over carrier subspace and substructures associated with \(\{ |kq\upsilon > > \} \) augmented spin algebras of Liouville space, and their mapping onto a subduced symmetry, are derived for [A]₆(L ₆) spin clusters within the combinatorial context of Rota-Cayley algebra over a field. Use of suitable lexical sets of combinatorialp-tuples (number partitions) over {|IM(M ₁?M _n )>}M, followed by the subsequent use ofL _n inner tensor product (ITP) algebra, allows the substructure of Liouville space to be derived. For SU2×L ₆ mapping over the simply-reducible \(\left\{ {I\tilde H_\upsilon } \right\}\) carrier subspaces, the \(D^k \left( {\tilde U} \right) \times \tilde \Gamma ^{\left[ {\tilde \lambda } \right]} \left( \upsilon \right)\) (L ₆) dual irreps, also arise as a consequence of the Liouville space recoupling termsv≡{k ₁?k _n } being distinct labels for \(\left\{ {I\tilde H_\upsilon } \right\}\) which are themselves amenible to combinatorial analysis within the concept of Rota-Cayley algebra. Hence, theL _n -induced symmetry aspects of multiquantum NMR density matrix formalisms and their dual \(\{ |kq\upsilon :[\tilde \lambda ] > > \} \) tensorial bases of spin cluster problems are derived and the nature of the cooperative, aspect between the individual symmetries comprising the duality is demonstrated, i.e. in the context of the operator bases of Liouville space. These practical arguments correlate, well with those based on an augmented boson pattern algebra derived from a Heisenburg algebra for superoperators, ?_±,?₀. An earlier, treatment of conventional Hilbert space SU2×L ₆ dualitycould only be realised in terms of standard SU2 boson algebra. Since the recoupling Rota-‘field’v for Liouville space is an explicit aspect of the dual mapping, a direct demonstration of cooperativity exists.     

Comprehensive automation of the solid phase extraction gas chromatographic mass spectrometric analysis (SPE-GC/MS) of opioids,cocaine, and metabolites from serum and other matrices

The analysis of opioids, cocaine, and metabolites from blood serum is a routine task in forensic laboratories. Commonly, the employed methods include many manual or partly automated steps like protein precipitation, dilution, solid phase extraction, evaporation, and derivatization preceding a gas chromatography (GC)/mass spectrometry (MS) or liquid chromatography (LC)/MS analysis. In this study, a comprehensively automated method was developed from a validated, partly automated routine method. This was possible by replicating method parameters on the automated system. Only marginal optimization of parameters was necessary. The automation relying on an x-y-z robot after manual protein precipitation includes the solid phase extraction, evaporation of the eluate, derivatization (silylation with N-methyl-N-trimethylsilyltrifluoroacetamide, MSTFA), and injection into a GC/MS. A quantitative analysis of almost 170 authentic serum samples and more than 50 authentic samples of other matrices like urine, different tissues, and heart blood on cocaine, benzoylecgonine, methadone, morphine, codeine, 6-monoacetylmorphine, dihydrocodeine, and 7-aminoflunitrazepam was conducted with both methods proving that the analytical results are equivalent even near the limits of quantification (low ng/ml range). To our best knowledge, this application is the first one reported in the literature employing this sample preparation system.     

Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.     

On nonoscillating integrals for computing inhomogeneous Airy functions

Integral representations are considered of solutions of the inhomogeneous Airy differential equation . The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of . In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

     

Effects of fuel blending on first stage and overall ignition processes

The effects of blending ratio on mixtures of an alcohol-to-jet (ATJ) fuel and a conventional petroleum-derived fuel on first stage ignition and overall ignition delay are examined at engine-relevant ambient conditions. Experiments are conducted in a high-temperature pressure vessel that maintains a small flow of dry air at the desired temperature (825 K and 900 K) and pressure (6 MPa and 9 MPa) for fuel injections from a custom single-hole, axially-oriented injector, representing medium (7.5 mg) and high (10 mg) engine loading. Formaldehyde, imaged using planar laser-induced fluorescence, is measured at discrete time steps throughout the first and second stage ignition process and is used as a marker of unburned short-chain hydrocarbons formed after the initial breakdown of the fuel. The formaldehyde images are used to calculate the first stage ignition delay for each ambient and fuel loading condition. Chemiluminescence imaging of excited hydroxyl radical at 75 kHz is used to determine the overall ignition delay. At all conditions, increased volume fraction of ATJ resulted in longer, but non-linearly increasing, overall ignition delay. Across all of the blends, first stage ignition delay accounted for about 15% of the increase in overall ignition delay compared to the military's aviation kerosene, F-24, which is Jet A with additives, while extended first stage ignition duration accounted for 85% of the increase. It is observed that blends consisting of 0–60% by volume of the low cetane number ATJ fuel produced nearly identical first stage ignition delays. These results will inform the development of ignition models that can capture the non-linear effects of fuel blending on ignition processes.