Joint Extension of States of Subsystems for a CAR System

 The problem of existence and uniqueness of a state of a joint system with given restrictions to subsystems is studied for a Fermion system, where a novel feature is non-commutativity between algebras of subsystems. For an arbitrary (finite or infinite) number of given subsystems, a product state extension is shown to exist if and only if all states of subsystems except at most one are even (with respect to the Fermion number). If the states of all subsystems are pure, then the same condition is shown to be necessary and sufficient for the existence of any joint extension. If the condition holds, the unique product state extension is the only joint extension. For a pair of subsystems, with one of the given subsystem states pure, a necessary and sufficient condition for the existence of a joint extension and the form of all joint extensions (unique for almost all cases) are given. For a pair of subsystems with non-pure subsystem states, some classes of examples of joint extensions are given where non-uniqueness of joint extensions prevails. Received: 17 May 2002 / Accepted: 16 January 2003 Published online: 17 April 2003 Communicated by D. Buchholz and K.Fredenhagen     

Scaling analysis of cotton cellulose/LiCl·DMAc solution using light scattering and rheological measurements

Semidilute solution of cotton lint (CC1) in 8 wt % LiCl/N,N‐dimethylacetamide was investigated using static light scattering (SLS) and rheological measurements. The reduced osmotic modulus estimated by SLS measurements for CC1 solutions are proportional to c^1.16 in the semidilute region. From the exponent of 1.16, de Gennes' scaling theory derives the relationship between radius of gyration, R_g, and molecular weight, M_w, of CC1 as R_g ∝ M^0.62 This corresponds to the Mark‐Houwink‐Sakurada exponent of 0.86. This exponent is very close to that estimated from scaling analysis of zero shear rate viscosity, that is 0.85. Apparent radius of gyration, R_g,app, estimated by SLS measurements for CC1 solutions are proportional to c^?0.5 in the semidilute region. R_g,app indicates the mesh size of polymer entanglement in the semidilute region. On the assumption of the Gaussian behavior of CC1 molecule in the semidilute region, the exponent of ?0.5 gives the relationship between the molar mass between entanglements, M_e, and c as following relationship: M_e ∝ c^?1. This agrees with the concentration dependence on plateau modulus estimated from the dynamic viscoelastic measurements. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2155–2160, 2006     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.     

Characteristics of the biodegradability and physical properties of stereocomplexes between poly(L‐lactide) and poly(D‐lactide) copolymers

Random and block copolymerizations of L ‐ or D ‐lactide with ε‐caprolactone (CL) were performed with a novel anionic initiator, (C₅Me₅)₂SmMe(THF), and they resulted in partial epimerization, generating D ,L ‐ or meso‐lactide polymers with enhanced biodegradability. A blend of PLLA‐r‐PCL [82/18; PLLA = poly(L ‐LA) and PCL = poly(ε‐caprolactone)] and PDLA‐r‐PCL [79/21; PDLA = poly(D ‐LA)] prepared by the solution‐casting method generated a stereocomplex, the melting temperature of which was about 40 °C higher than that of the nonblended copolymers. A blend of PLLA‐b‐PCL (85/15) and PDLA‐b‐PCL (82/18) showed a lower elongation at break and a remarkably higher tensile modulus than stereocomplexes of PLLA‐r‐PCL/PDLA‐r‐PCL and PLLA/PDLA. The biodegradability of a blend of PLLA‐r‐PCL (65/35) and PDLA‐r‐PCL (66/34) with proteinase K was higher than that of PLLA‐b‐PCL (47/53) and PDLA‐b‐PCL (45/55), the degradability of which was higher than that of a PLLA/PDLA blend. A blend film of PLLA‐r‐PDLLA (69/31)/PDLA‐r‐PDLLA (68/32) exhibited higher degradability than a film of PLLA/PDLLA [PDLLA = poly(D ,L ‐LA)]. A stereocomplex of PLLA‐r‐PCL‐r‐PDMO [80/18/2; PDMO = poly(L ‐3,D ,L ‐6‐dimethyl‐2,5‐morpholinedion)] with PDLA‐r‐PCL‐r‐PDMO (81/17/2) showed higher degradability than PLLA‐r‐PDMO (98/2)/PDLA‐r‐PDMO (98/2) and PLLA‐r‐PCL (82/18)/PDLA‐r‐PCL (79/21) blends. The tensile modulus of a blend of PLLA‐r‐PCL‐r‐PDMO and PDLA‐r‐PCL‐r‐PDMO was much higher than that of a blend of PLLA‐r‐PDMO and PDLA‐r‐PDMO. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 438–454, 2005     

Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy

Conoscopic studies have been made on the helix unwinding process in the chiral smectic C phase of MHPOBC. The unwinding process is rather unique in the sense that abrupt changes of the biaxiality and apparent tilt angle are followed by further gradual changes before the complete helix unwinding. The process is qualitatively explained by simulating the conoscopic images using the 4 ×4 matrix method with consideration of model molecular distributions, by taking account of both the ferroelectric and dielectric coupling between molecules and a field. The transmittance loss due to selective reflection was also measured under the application of a stepwise d.c. field. The shift of the dip position due to the loss, toward a longer wavelength region, was observed at low field. In contrast, the emergence of a transmittance loss at the same wavelength as that of an unperturbed state was observed. This phenomenon is explained by the ferroelectric deformation of a helix with the pitch unchanged.     

Liquid-cooled ceramic Nd:YAG split-disk amplifier for high-average-power laser

A liquid-cooled ceramic Nd:YAG split-disk laser amplifier has been designed for a high-average-power laser system. The maximum average output power of 17.1 W (1.71 J at 10 Hz) and the beam size of far-field pattern with 1.5 times diffraction limit was obtained. The combination of liquid cooling for disk laser material and wavefront compensation with a phase conjugation mirror was demonstrated. The wavefront degradation by liquid cooling was observed and measured with Michelson interferometer and far-field patterns.     

Yang–Mills Theory over Compact Symplectic Manifolds

In this paper, Yang–Mills theory over a compactKähler manifold is naturally extended to a compactsymplectic manifold. The relation betweenthe Yang–Mills equation and symplecticstructure is explicitly clarified, and the moduli spaceof Yang–Mills connections over a compactsymplectic manifold is constructed. Furthermore, theabsolute minima of the Yang–Mills functional arecharacterized, and finite dimensionality ofthe moduli space of the minimizers of the Yang–Millsfunctional is shown.     

Compactness under constructive scrutiny

How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop‐style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by‐product, the fan theorem for detachable bars of the complete binary fan proves to be necessary for the unit interval possessing the Heine‐Borel property for coverings by countably many possibly empty open balls. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)