Irradiation-Induced Heck Reaction of Unactivated Alkyl Halides at Room Temperature

Article abstract of DOI:10.1021/jacs.7b10009

The palladium-catalyzed Mizoroki-Heck reaction is arguably one of the most significant carbon-carbon bond-construction reactions to be discovered in the last 50 years, with a tremendous number of applications in the production of chemicals. This Nobel-Prize-winning transformation has yet to overcome the obstacle of its general application in a range of alkyl electrophiles, especially tertiary alkyl halides that possess eliminable β-hydrogen atoms. Whereas most palladium-catalyzed cross-coupling reactions utilize the ground-state reactivity of palladium complexes under thermal conditions and generally apply a single ligand system, we report that the palladium-catalyzed Heck reaction proceeds smoothly at room temperature with a variety of tertiary, secondary, and primary alkyl bromides upon irradiation with blue light-emitting diodes in the presence of a dual phosphine ligand system. We rationalize that this unprecedented transformation is achieved by utilizing the photoexcited-state reactivity of the palladium complex to enhance oxidative addition and suppress undesired β-hydride elimination.

Full text of DOI:10.1021/jacs.7b10009

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Article
Irradiation-Induced Heck Reaction of
Unactivated Alkyl Halides at Room Temperature
Guang-Zu Wang, Rui Shang, Wan-Min Cheng, and Yao Fu
J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.7b10009 • Publication Date (Web): 08 Nov 2017
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Journal of the American Chemical Society
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Irradiation-Induced Heck Reaction of Unactivated Alkyl Hal-
ides at Room Temperature
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GuangꢀZu Wang,† Rui Shang,*,†,‡ WanꢀMin Cheng,† and Yao Fu*,†
†Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key Laboratory of Urban Pollutant Conversion,
Anhui Province Key Laboratory of Biomass Clean Energy, iChEM, Department of Chemistry, University of Science and
Technology of China, Hefei 230026, China.
‡Department of Chemistry, School of Science, The University of Tokyo, 7ꢀ3ꢀ1 Hongo, Bunkyoꢀku, Tokyo 113ꢀ 0033, Japan.
Supporting Information Placeholder
ABSTRACT: The palladiumꢀcatalyzed Mizoroki–Heck reaction is arguably one of the most significant carbon–carbon bondꢀconstruction
reactions to be discovered in the last 50 years, with a tremendous number of applications in the production of chemicals. This NobelꢀPrizeꢀ
winning transformation has yet to overcome the obstacle of its general application in a range of alkyl electrophiles, especially tertiary alkyl
halides that possess eliminable βꢀhydrogen atoms. Whereas most palladiumꢀcatalyzed crossꢀcoupling reactions utilize the groundꢀstate
reactivity of palladium complexes under thermal conditions and generally apply a single ligand system, we report that the palladiumꢀ
catalyzed Heck reaction proceeds smoothly at room temperature with a variety of tertiary, secondary, and primary alkyl bromides upon
irradiation with blue lightꢀemitting diodes in the presence of a dual phosphine ligand system. We rationalize that this unprecedented transꢀ
formation is achieved by utilizing the photoexcitedꢀstate reactivity of the palladium complex to enhance oxidative addition and suppress
undesired βꢀhydride elimination.
difficulty in reconciling suppression of βꢀH elimination after oxiꢀ
dative addition of alkyl halides, while necessarily utilizing βꢀH
1.INTRODUCTION
Palladiumꢀcatalyzed crossꢀcouplings constitute a fundamental
carbon–carbon bondꢀformation methodology in modern organic
synthesis.1 These reactions have proven to be highly efficient for
the coupling of unsaturated carbon electrophiles and nucleophiles
to meet the requirements of various synthetic tasks.2 Although
palladiumꢀcatalyzed crossꢀcoupling reactions are highly advanced,
work has mainly focused on the development and application of
the groundꢀstate reactivity of palladium species through the Pd(0)ꢀ
Pd(II) catalytic cycle.3 While these reactions are highly efficient
for coupling C(sp2)ꢀhybridized carbon electrophiles, couplings
with a C(sp3)ꢀhybridized electrophile4 are less efficient because of
the inherent instability of alkyl palladium species,5 interference of
βꢀhydride (βꢀH) elimination,6 and difficulties associated with the
oxidative addition of the alkyl electrophile (Fig. 1a).7 Elegant
contributions have been presented that use alkyl electrophiles as
amenable substrates in various palladiumꢀcatalyzed crossꢀ
couplings by suppressing undesired βꢀH elimination through the
application of new ligand designs,8 or through the use of firstꢀ
rowꢀtransition metals, such as nickel, which is less likely to inꢀ
duce βꢀH elimination.9 Although being developed for years, Heck
reactions on an sp3ꢀhybridized CꢀX center remains challenging
because, unlike other crossꢀcoupling reactions, βꢀH elimination is
required as an elementary step in the Heck reaction.10 Thus, simpꢀ
ly suppressing βꢀH elimination in alkyl Heck reaction by ligand
design is less feasible. Pdꢀcatalyzed Heck reactions have been
established by using unactivated primary and second alkyl halides
by Fu,11 Alexanian,12 and Zhou;13 however, the scope remains
limited. No precedent has been achieved for tertiary alkyl halides
possessing eliminable βꢀH. This lack of progress stems from the
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Figure 1. Problems facing the palladium-catalyzed Heck reaction
lectively. Under the optimized conditions, we did not detect any
under thermal conditions and a hypothetical solution. a, The Heck
reaction developed under thermal conditions was found to be less effective
for coupling alkyl electrophiles possessing eliminable βꢀhydrogen, espeꢀ
cially for tertiary alkyl electrophiles. b, Irradiation facilitated oxidative
addition of alkyl electrophiles with a Pd(0) complex through singleꢀ
electron transfer (SET). c, Geometric change of a Pd(II) complex reported
under irradiation with destabilization to induce radical reactivity. d, Possiꢀ
ble solution that could enable efficient Heck reaction using alkyl electroꢀ
philes by merging radical and organometallic reactivity of palladium
species under irradiation excited conditions
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product of Suzuki coupling or oxidative Heck reaction of aryl
boronate.21 The reaction conditions resemble those of widely apꢀ
plied Pdꢀcatalyzed Heck reactions with the only difference being
irradiation with blue LEDs.
Table 1. Irradiation-induced Palladium-catalyzed Mizoro-
ki–Heck Reaction Using tert-Butyl Bromide orthogonal
with Suzuki–Miyaura coupling and studies on reaction
parametersa
9
elimination as a productive step to deliver the desired olefin prodꢀ
uct. We envisioned that photoirradiation of a Pd(0) complex mayꢀ
facilitate a singleꢀelectronꢀtransfer process to activate alkyl halꢀ
ides (Fig. 1b). The use of irradiation with a xenon lamp to faciliꢀ
tate radicalꢀtype oxidative addition has been reported by Ryu et
al.14 In addition, Pd(II) complexes can undergo geometric exꢀ
change from square planar to tetrahedron under irradiation.15 This
geometric change is caused by the formation of an excitedꢀstate
Pd(II) complex involving outer orbitals that possess higher energy
and may induce radical reactivity of the alkylꢀPd(II) species (Fig.
1c).16 This photoexcitation effect may suppress undesired βꢀH
elimination by preventing bonding of the alkyl radical to palladiꢀ
um, especially for tertiary alkyl radicals with large steric bulkiꢀ
ness. Thus, the following steps will proceed through a radical
insertion process, in contrast to a pure organometallic insertion
pathway, to generate an insertion product, which then undergoes
βꢀH elimination (Fig. 1d). With this hypothesis in mind, we tested
palladiumꢀcatalyzed Heck reactions of unactivated alkyl halides
under various irradiation conditions. Herein, we reveal that, by
using blue lightꢀemitting diodes (LEDs) irradiation17 and applying
a dual phosphine ligand system, palladiumꢀcatalyzed Heck reacꢀ
tions of unactivated tertiary, secondary, and primary alkyl broꢀ
mides with styreneꢀtype substrates proceeds efficiently at room
temperature. The discovery solves the longꢀstanding problem of
using unactivated tertiary alkyl halides in palladiumꢀcatalyzed
Heck reactions, and makes it feasible to construct a quaternary
carbon center connected to a double bond, which is easily funcꢀ
tionalized for complex molecule synthesis. Although palladium
catalysis has been studied for decades, most of the contributions
have been restricted to the framework of groundꢀstate reactivity.
This discovery reveals that screening of irradiation sources (such
as LEDs) as a parameter in palladium catalysis enables access to
excitedꢀstate reactivity,18 offers new opportunities to discover
novel reactions, and can help solve challenging problems that
remain in traditional groundꢀstate palladiumꢀcatalyzed crossꢀ
coupling reactions.
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a. Yields and conversions determined by GC analysis using an internal
standard. dba = ((1E,4E)ꢀ1,5ꢀdiphenylpentaꢀ1,4ꢀdienꢀ3ꢀone), dppf = (1,1ꢀ
bis(diphenyphosphino)ferrocene),
bis(diphenylphosphino)xanthene), CyXantphos
Xantphos
=
=
(9,9ꢀdimethylꢀ4,5ꢀ
((9,9ꢀdimethylꢀ9Hꢀ
xantheneꢀ4,5ꢀdiyl)bis(dicyclohexylphosphine)), NiXantphos
bis(diphenylphosphino)phenoxazine), dppp
=
(4,6ꢀ
(1,3ꢀ
=
bis(diphenylphosphino)propane), BINAP = (2,2ꢀbis(diphenylphosphino)ꢀ
1,1ꢀbinaphthyl), dtbpy = (4,4ꢀdiꢀ tertꢀbutylꢀ2,2ꢀbipyridine).
Studies of the reaction parameters revealed that the alkyl Heck
reaction under irradiation conditions has distinct controlling fac
tors compared with the reaction conducted under thermal condiꢀ
tions. The outcome of the reaction is highly dependent on the
palladium source, with only Pd(PPh3)4 and Pd(PPh3)2Cl2 being
effective. Other palladium sources such as Pd2(dba)3, PdCl2,
Pd(OAc)2, and Pd(dppf)Cl2 were entirely ineffective (entry 2–5).
It was soon realized that the PPh3 and Xantphos both play an esꢀ
sential role in this reaction. The use of PCy3 significantly reduced
the yield of the desired product and no reaction occurred when
tri(2ꢀfuranyl) phosphine, with an adjacent coordinating oxygen,
was used (entries 6 and 7). From these results we can conclude
that the monodentate phosphine ligand is not just a sacrificial
reagent to reduce Pd(II) because the use of PCy3, which has a
higher reducing ability, gave poor performance. Testing triꢀ
phenylphosphine in combination with other bidentate phosphine
ligands revealed that the structure of the bidentate ligand also
significantly affects the efficiency of the reaction. The reaction
did not proceed at all in the absence of Xantphos (entry 8). Only
bisphosphine ligands with a conjugated backbone similar to that
of Xantphos were effective (entries 9–13). Although the reasons
2.RESULT AND DISCUSSION
The optimized reaction conditions used to crossꢀcouple
4,4,5,5ꢀtetramethylꢀ2ꢀ(4ꢀvinylphenyl)ꢀ1,3,2ꢀdioxaborolane with
tertꢀbutyl bromide are shown in Table 1. tertꢀButyl bromide is one
of the most challenging substrates for transitionꢀmetalꢀcatalyzed
crossꢀcouplings because it contains an abundance of hydrogen
atoms that can easily undergo βꢀelimination and because its high
steric bulk hinders oxidative addition reactions. Nevertheless, this
substrate has recently been applied in nickelꢀcatalyzed Kumada
couplings19 and Suzuki couplings,20 wherein βꢀH elimination was
suppressed because of the intrinsically lower electronegativity of
the Ni(II) compared with that of the Pd(II).9 The use of tertꢀbutyl
bromide as a coupling partner in palladiumꢀcatalyzed Heck couꢀ
pling is unprecedented. The optimized reaction shows that a simꢀ
ple catalytic system composed of commercially available
Pd(PPh3)2Cl2
(5
mol%),
4,5ꢀbis(diphenylphosphino)ꢀ9,9ꢀ
dimethylxanthene (Xantphos, 6 mol%), and a mild inorganic base
(potassium acetate), under irradiation with blue LEDs at room
temperature enables efficient Heck coupling regioꢀ and stereoseꢀ
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for the observed ligand dependence are not clear at present, we
trophiles showed that alkyl iodide was less effective because of
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speculate that the conjugated backbone of the bisphosphine ligand
influences the degree of photoexcitation and thus the reactivity on
the palladium center. The essential role of irradiation was estabꢀ
lished in further control experiments (entries 14 and 15) in which
the same reaction system did not deliver any detectable product
under thermal conditions at 100 °C. One intriguing aspect of this
reaction is that
the occurrence of side reactions, and alkyl chloride was not actiꢀ
vated (entries 21 and 22). We also tested several photoredox cataꢀ
lysts,22 for this reaction instead of the palladium/phosphine system.
Under these conditions, none of the desired product was detected,
which shows that a catalytic cycle of singleꢀelectron redox is less
feasible (see Supporting Information, Sꢀ12, for detailed parameter
studies). Using K2CO3 as base with one equivalent amount of
water showed comparable efficiency with KOAc. The additional
one equivalent amount of water in amide solvent is supposed to
facilitate reduction of Pd(II) precatalyst to Pd(0) (see Supporting
Information for detailed optimization of bases).
Table 2. Reaction Scope with Respect to Alkyl Bromidea
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Subsequent studies revealed that the scope of the reaction with
respect to the alkyl bromide is very broad (Table 2). Besides unꢀ
precedented coupling partners of tertiary alkyl bromides, primary
and secondary alkyl bromides were also amenable to the reaction
at room
Table 3. Reaction Scope with Respect to Alkenesa
a. Yields of isolated products. Ratio of stereoisomers was determined by
1HꢀNMR analysis. The reactions were conducted in front of a cooling fan
in a room of constant temperature (T = 30 ± 2 °C ).
a. Yields of isolated products. Ratio of stereoisomers was determined by
1HꢀNMR analysis. The reactions were conducted in front of a cooling fan
in a room of constant temperature (T = 30 ± 2 °C ).
temperature. Alkyl bromides derived from complex nature prodꢀ
ucts are suitable substrates under mild reaction conditions, as
demonstrated by the use of epiandrosterone (12), cholesterol (13),
and testosterone (28). It is notable that although the stereoisomerꢀ
ically pure axial alkyl bromide was used (compounds 12 and 13),
the reaction delivered a diastereomeric mixture of olefination at
both axial and equatorial positions on the cyclohexyl ring, which
suggests that the alkyl radical is formed. For cholesterol substrate
13, the minor diastereoisomer was isolated and crystallized to
obtain a single crystal. Xꢀray diffraction analysis (CCDCꢀ
1560078) unambiguously confirmed its stereostructure. The mild
reaction conditions enabled a broad range of functional groups to
be tolerated, including ester (1, 19), silyl ether (3), alcohol (4),
alkyl chloride (5), ether (6), cyano (7), amide (10), ketone (12),
the blue LEDs used had a wavelength of only around 460 nm. We
wondered whether the energy of the light source would signifiꢀ
cantly affect the outcome of the reaction and therefore applied
other light sources with different irradiation wavelength. The use
of 36 W green LEDs (wavelength 520–525 nm) of lower excitaꢀ
tion energy instead of blue LEDs resulted in a clear drop in yield
(entry 16). Purple LEDs (wavelength 395–405 nm), with higher
irradiation energy, were also effective (entry 18), but the use of
highꢀenergy ultraviolet (UV) light (254 nm) converted the styrene
into other byproducts (e.g., oligomerization or polymerization)
without giving the desired product. White LEDs were much less
effective (entry 17). The reaction stopped completely when it was
exposed to ambient air (entry 20). Testing a range of alkyl elecꢀ
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equiv.), tꢀBuBr (1.0 equiv.), and styrene (1.0 equiv.) in DMA after 2 h
under argon atmosphere at room temperature.
boronate (17), aldehyde (18), and acetal (20) groups. All reactions
delivered the products with excellent Eꢀstereoselectivity, as comꢀ
monly observed for palladiumꢀcatalyzed aryl Heck reactions unꢀ
der thermal conditions. In previous reports by Alexanian12 and
Zhou13 of palladiumꢀcatalyzed intermolecular Heck reactions
under thermal conditions, only primary and secondary alkyl halꢀ
ides could be applied, and the reactions delivered the olefin prodꢀ
uct as a mixture of Z/E isomers. The stereoselectivity of this reacꢀ
tion using tertiary alkyl bromide is remarkable because only the
Eꢀisomer was observed in all cases. Although it is not yet clear
whether this stereoselectivity is related to the irradiation, it is
reasonable to expect that the very mild reaction conditions may
contribute to the high stereoselectivity in the βꢀH elimination step
to deliver the thermodynamically favored stereoisomer. The reacꢀ
tion can be performed on a gram scale, as demonstrated for comꢀ
pounds 10 and 12 (Table 2). Notably, the pyridine group in 10
was tolerated under the reaction conditions, which suggests that
the formation of a carbonium ion is unlikely in this reaction (Taꢀ
ble 2). The basic nitrogen atom would be expected to react with a
carbonium ion to disturb the desired reaction.
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The scope of the reaction with respect to styreneꢀcoupling partner
is summarized in Table 3. Both electronꢀdeficient styrene (1, 24,
27, 32) and electronꢀrich styrene (22, 30) derivatives are amenable
substrates. Notably, 2ꢀvinylpyridine also reacted with tertꢀbutyl
bromide efficiently (27). Pentafluorostyrene also reacted as an
amenable substrate (32). The good reactivity of these electronꢀ
deficient substrates suggests that a mechanism that proceeds
through generation of a benzylic carbenium ion followed byproꢀ
ton elimination is not likely, and supports a βꢀH elimination
pathway. 4ꢀDiphenylaminostyrene (30) and 1,1ꢀdiphenylethene
(33), which are often used as radical scavengers in freeꢀradical
processes, also reacted well. The alkyl radical may thus only exist
as a palladiumꢀintercepted radical, and not as a free alkyl radical.
Styrene substrates, acrylamide substrate (36), and 9ꢀvinylꢀ9Hꢀ
carbazole (37) also reacted with tertiary alkyl bromide to deliver
the desired products in moderate yields. For reasons that are not
yet clear, the reaction was not effective with acrylate esters. We
also substantially tested other alkene substrates, such as αꢀ
aliphatic alkene, internal alkene, and 1,3ꢀdiene. However, these
alkene substrates gave either very low yield (<10%) or were unreꢀ
active under the optimized reaction condition.
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Scheme 1. Mechanistic studies
As noted in our working hypothesis, the reaction may proceed
through a singleꢀelectronꢀtransfer oxidative addition to generate a
palladiumꢀintercepted alkyl radical under irradiation, and the irraꢀ
diation may excite the palladium complex to prevent recombinaꢀ
tion of the alkyl radical to form a Pd(II)ꢀalkyl complex that unꢀ
dergoes βꢀH elimination.23 We conducted radical clock experiꢀ
ments,24 which are often used to probe the radical reactivity of
metal alkyl species (Scheme. 1a). The results showed that radical
rearranged products were formed exclusively and confirmed that
the oxidative addition under irradiation forms an alkyl palladium
species with distinct radical properties. In addition, competition
experiments between primary and tertiary alkyl bromide examinꢀ
ing the kinetics of oxidative addition were conducted (Scheme.
1b). The results clearly demonstrated that oxidative addition of
alkyl halides to deliver the thermodynamically more stable alkyl
radical proceeds faster. The results of the competition experiment
also supports the conclusion that oxidative addition under blue
LEDs irradiation proceeds through a singleꢀelectronꢀtransfer
mechanism to cleave the C–Br bond. Previous studies have shown
that for SN2ꢀtype oxidative addition of alkyl bromide to a phosꢀ
phineꢀligated Pd(0) complex,25 the rate of secondary alkyl oxidaꢀ
tive addition is much slower than for a primary alkyl bromide, and
the oxidative addition does not proceed with tertiary alkyl halides
because of steric hindrance.26 We also conducted intramolecular
kinetic isotope effect (KIE) experiments using βꢀDꢀstyrene as
substrate (Scheme. 1c). The small observed KIE (1.4) suggests
that a massꢀtransfer process to form the benzylic brominated
product followed by E2 elimination to deliver the alkene product
is unlikely because this would be expected to result in a large
KIE.27 Furthermore, the introduction of benzyl bromide poisons
the reaction system. The addition of 50 mol% (1ꢀbromoethyl)
benzene stopped the desired reaction completely. Combined with
the observed insensitivity of the reaction towards the electronic
nature of the styrene, these results support the conclusion that a βꢀ
H elimination mechanism delivers the product instead of an ionic
or massꢀtransfer mechanism. To gain information on the valance
state of the palladium species in this catalytic system, Xꢀray phoꢀ
toelectron spectroscopy (XPS) measurements (Figure S1) were
also conducted. XPS measurements of the reaction mixture reꢀ
vealed three peaks corresponding to Pd(II), Pd(I), and Pd(0), sugꢀ
gesting the reaction proceeds through a Pd(0)ꢀPd(I)ꢀPd(II) mechaꢀ
nism. Given that the reaction proceeds through Pd(I), which is
prone to dimerize in the presence of a bisphosphine ligand,28 the
a, Radical clock experiments revealed the radical properties of the alkylꢀ
metal species. b, Kinetic experiments between tertiary and primary alkyl
bromides. c, An intramolecular kinetic isotope experiment. d, UV/Vis
absorption spectrum of the reaction mixture, Pd(II), and Pd(0) phosphine
complexes. Standard conditions: Pd(PPh3)2Cl2 (1.0 equiv.), Xantphos (1.0
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beneficial effect of the monophosphine ligand may originate in its
coordination to Pd(I), thereby preventing the formation of the
ASSOCIATED CONTENT
1
2
Supporting Information
dimer complex with consequent disruption of the catalytic cycle.
Furthermore, we cannot rule out the possibility that the PPh3 conꢀ
tributes to facilitate oxidative addition and to suppress βꢀH elimiꢀ
nation through its labile coordination with the palladium intermeꢀ
diate. Beside the dual phosphine ligand system, the essential irraꢀ
diation effect that is sensitive to the energy of incident light was
also very intriguing. The reaction entirely stopped after the irradiꢀ
ation source was removed. We can exclude the possibility that
irradiation just initiates a radical chain process. UV/Vis measureꢀ
ments were conducted with a mixture of stoichiometric amounts
of palladium salts and ligands (Scheme. 1d). It was observed that
the reaction mixture exhibits a clear absorption peak centered at
350 nm with absorption onset that overlaps with the blue LEDs
wavelength (460 nm). This absorption onset explains the best
performance of the blue LEDs among other irradiation sources.
Removing styrene from the reaction mixture did not change the
absorption wavelength or intensity, whereas removing tertꢀbutyl
bromide and removing Xantphos caused a reduction in the absorpꢀ
tion intensity and a hypsochromic shift of about 5 nm. An imꢀ
portant observation is that when Pd(PPh3)2Cl2 was removed from
the reaction mixture, the absorption peak at around 350 nm disapꢀ
peared, revealing that the absorption peak of λmax = 350 nm with
an onset around 460 nm originates from the palladium intermediꢀ
ate form in the reaction mixture. To verify this conclusion, we
also measured the absorption spectrum of Pd(PPh3)2Cl2 and an
absorption peak around 350 nm was observed. The measurement
of UV/Vis absorption of Pd(PPh3)4 gives a broadened absorption
band extending from the visible region to the UV region, revealꢀ
ing that irradiation by blue LEDs can also excite the Pd(0) phosꢀ
phine complex, which may facilitate singleꢀelectron oxidative
addition. Irradiation with suitable wavelengths is crucial for the
reaction as demonstrated in optimization studies because lowꢀ
energy irradiation (such as green LEDs) does not excite the pallaꢀ
dium complex, and higher energy irradiation (in the UV region)
may cause decomposition of the intermediate and destroy the
desired reaction process. We may conceive that an irradiation
wavelength between that of purple and blue light may offer the
best efficiency for this reaction, although this is currently limited
by the irradiation sources available. Based on the absorption specꢀ
trum, these studies suggest that irradiation at different waveꢀ
lengths should be considered as a parameter when optimizing
palladiumꢀcatalyzed reactions induced by irradiation.
This material is available free of charge via the Internet at
http://pubs.acs.org.
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Experimental details and characterization data for all prodꢀ
ucts (PDF)
AUTHOR INFORMATION
Corresponding Author
*Eꢀmail: rui@chem.s.uꢀtokyo.ac.jp, fuyao@ustc.edu.cn
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Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENT
This work was supported by NSFC (21325208, 21572212),
MOST (2017YFA0303500), CAS (XDB20000000), FRFCU and
PCSIRT. This paper is dedicated to Professor Teruaki Mukaiyama
in celebration of his 90th birthday (Sotsuju). Dedicated to Profesꢀ
sor ChenꢀHo Tung on the occasion of his 80th birthday.
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