On Sufficient Conditions for Convergence of Two-Dimensional Continued Fractions

By the method of majorant fractions and equivalent transformations, the analogies of leszyski–Pringsheim criteria for two-dimensional continued fractions are obtained.     

Geometry of a crossed product

We introduce a continuous dimension function on the Grothendieck group over the crossed product -algebra . The function has an elegant geometry: on every minimal flow it takes the value of the ``rotation number" of ; such a problem was posed in 1936 by A. Weil.

     

三元重心Thiele型混合有理插值

有理插值比多项式插值有更好的近似,但有理插值一般很难控制极点的产生.基于Thiele型连分式插值与重心有理插值,构造三元重心Thiele型混合有理插值,当选取适当的权后能避免部分极点的产生.文章最后通过数值例子验证了这种方法的正确性和有效性.     

Metrical diophantine approximation for continued fraction like maps of the interval

We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

     

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction.

Let the continued fraction expansion of any irrational number be denoted by and let the -th convergent of this continued fraction expansion be denoted by . Let

where . Let . It is shown that if , then the Rogers-Ramanujan continued fraction diverges at . is an uncountable set of measure zero. It is also shown that there is an uncountable set of points such that if , then does not converge generally.

It is further shown that does not converge generally for 1$">. However we show that does converge generally if is a primitive -th root of unity, for some . Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

     

Components of the Fundamental Category II

In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, vol. 2286, Cancun, Mexico, pp. 16–37, Springer, Berlin Heidelberg New York, 2002; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003) of a partially ordered topological space (Nachbin, Topology and Order, Van Nostrand, Princeton, 1965; Johnstone, Stone Spaces, Cambridge University Press, Cambridge, MA, 1982), as arising in e.g. concurrency theory (Fajstrup et al., Theor. Comp. Sci. 357: 241–278, 2006), initiated in (Fajstrup et al., APCS, 12(1): 81–108, 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of Fajstrup et al. (APCS, 12(1): 81–108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 95–146, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, Cahiers Topologie Géom. Différentielle Catég., 44: 281–316, 2003; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003), we show in this paper a similar theorem for component categories (conjectured in Fajstrup et al. (APCS, 12(1): 81–108, 2004). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models (Goubault and Haucourt, A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005 – Concurrency Theory: 16th International Conference, San Francisco, USA, August 23–26. Lecture Notes in Computer Science, vol. 3653, pp. 503–517, Springer, Berlin Heidelberg New York, 2005), corresponding to partially ordered subspaces of R ⁿ minus isothetic hyperrectangles. In this last case again, we conjecture (and give some hints) that component categories enjoy some nice adjunction relations directly with the fundamental category.      

Photoionization spectra of even-parity states of Sm atom with multistep excitation

Two-color stepwise excitation and photoionization schemes are adopted to study the spectra of bound even-parity high-lying states of the Sm atom with three different excitation paths via the 4f⁶6s6p ⁷D_J (J=1, 2 and 3) intermediate states. In order to obtain the information of these high-lying states, the Sm atom in these high-lying states is photoionized with an extra photon. Among 231 states detected in the energy region between 35,545 and 44,225 cm⁻¹, 108 states are newly discovered, while the rest can be identified as the same with the literature. In most cases, comparisons of the spectra corresponding to the three different excitation paths may partially determine the total angular momentum of the observed peaks with the selection rules. In addition, the relative intensities of all related transition lines are given.     

GRAFTING OF PEROXIDE-INITIATED MALEIC ANHYDRIDE ON SPHERICAL PE／PP IN-REACTOR BLEND GRANULES

Spherical polyethylene/polypropylene (PE/PP) in-reactor blend granules with various ethylene/propylene molar ratios and high porosity were synthesized using a high yield TiCl4/MgCl2 supported catalyst. A solution of benzoyl peroxide (BPO)/maleic anhydride (MAH)/xylene (interfacial reagent) or BPO/MAH/St (comonomer) was absorbed onto the PE/PP inreactor blend granules, and solid phase gratt polymerization of MAH on PE/PP was conducted. The amount of grafted MAH on PE/PP was measured through chemical titration. The results showed that solid phase graft polymerization of MAH in PE/PP in-reactor blend granules produced graft copolymer with high amount of grafted MAH, and the amount of grafted MAH was raised slightly when St was introduced as comonomer. The graft in-reactor blend was fractionated into five fractions through temperature-gradient extraction fractionation (TGEF), and the fractions were analyzed by FTIR. The results revealed that MAH is mainly grafted on the PE segments, whereas MAH was predominantly grafted on the PP segments when St was present in the graft polymerization system. In addition, the final product is still in the form of regular spherical granules, which is beneficial for industrial processing.     

Quality test of a mechanical switching valve for two-dimensional open tubular gas chromatography

A commercially available mechanical micro switching valve has been tested for inertness. The test mixtures used were a modified Grob test and a solution consisting of eight typical aroma compounds, both at the 2-ng level. Cryogenic focusing of the sample after chromatography including one passage through the valve revealed that irreversible interaction of some test substances with valve components was rather small. A composite main peak of a chromatogram from a fruit juice aroma exhibiting a “mixed odor” was selectively transferred (heart-cutting) to a second column and resolved into three peaks. GC-GC/sniffing assigned a minor peak to a substance formerly hidden under the composite peak. Separate heating of the valve during temperature programming of the column was found to be unnecessary.