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41.
Within the concept of projective lattice geometry we are considering the class of stable geometries which have also been introduced in [14]. The investigation of their basic properties will result in fundamental structure theorems which especially give a lattice-geometric characterization of free left modules of rank 6 over proper right Bezout rings of stable rank 2. This yields a proper generalization of previous results of ours. 相似文献
42.
Inclusion-exclusion: Exact and approximate 总被引:1,自引:0,他引:1
It is often required to find the probability of the union of givenn eventsA
1
,...,A
n
. The answer is provided, of course, by the inclusion-exclusion formula: Pr(A
i
)=
i
–
i<j
Pr(A
i
A
j
)±.... Unfortunately, this formula has exponentially many terms, and only rarely does one manage to carry out the exact calculation. From a computational point of view, finding the probability of the union is an intractable, #P-hard problem, even in very restricted cases. This state of affairs makes it reasonable to seek approximate solutions that are computationally feasible. Attempts to find such approximate solutions have a long history starting already with Boole [1]. A recent step in this direction was taken by Linial and Nisan [4] who sought approximations to the probability of the union, given the probabilities of allj-wise intersections of the events forj=1,...k. The developed a method to approximate Pr(A
i
), from the above data with an additive error of exp
. In the present article we develop an expression that can be computed in polynomial time, that, given the sums |S|=j
Pr(
iS
A
i
) forj=1,...k, approximates Pr(A
i
) with an additive error of exp
. This error is optimal, up to the logarithmic factor implicit in the
notation.The problem of enumerating satisfying assignments of a boolean formula in DNF formF=v
l
m
C
i
is an instance of the general problem that had been extensively studied [7]. HereA
i
is the set of assignments that satisfyC
i
, and Pr(
iS
A
i
)=a
S
/2n where
iS
C
i
is satisfied bya
S
assignments. Judging from the general results, it is hard to expect a decent approximation ofF's number of satisfying assignments, without knowledge of the numbersa
S
for, say, all cardinalities
. Quite surprisingly, already the numbersa
S
over |S|log(n+1)uniquely determine the number of satisfying assignments for F.We point out a connection between our work and the edge-reconstruction conjecture. Finally we discuss other special instances of the problem, e.g., computing permanents of 0,1 matrices, evaluating chromatic polynomials of graphs and for families of events whose VC dimension is bounded.Work supported in part by a grant of the Binational Israel-US Science Foundation.Work supported in part by a grant of the Binational Israel-US Science Foundation and by the Israel Science Foundation. 相似文献
43.
The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials (p>1). We study here the algebra of FDA transformations. To every p-form in the FDA, we associate an extended Lie derivative l generating a corresponding gauge transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's. 相似文献
44.
The quantum group IGL
q
(N), the inhomogenization of GL
q
(N), is formulated with
-matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space. 相似文献
45.
We use a quite concrete and simple realization of sl
q
(2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme 1 II 1* as the counterpart of the standard 1 = Sl(2, )/B. 相似文献
46.
W. K. Nicholson 《Periodica Mathematica Hungarica》1993,27(2):85-88
A ringR is called a leftPP-ring if every principal left ideal is projective, equivalently if the left annihilatorl(a) is generated by an idempotent for allaR. These rings seem first to have been discussed by Hattori [2] and examples include (von Neumann) regular rings and domains (possibly noncommutative). In this note we give a new characterization of leftPP-rings, use that to give an elementary proof of a result of Xue [4] characterizing triangularPP-rings, and then determine when the ringT
n
(R) of upper triangular matrices overR is a leftPP-ring. Throughout the paper all rings have a unity and all modules are unitary.This research was supported by NSERC Grant A8075 相似文献
47.
For a fixed setI of positive integers we consider the set of paths (p
o,...,p
k
) of arbitrary length satisfyingp
l
–p
l–1I forl=2,...,k andp
0=1,p
k
=n. Equipping it with the uniform distribution, the random path lengthT
n
is studied. Asymptotic expansions of the moments ofT
n
are derived and its asymptotic normality is proved. The step lengthsp
l
–p
l–1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp
0 andp
k
are not specified. In the special caseI={m+1,m+2,...} (for some fixed m) we derive the exact distribution of a random m-gap subset of {1,...,n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a randomm-gap subset is also presented. 相似文献
48.
Molar excess mixing enthalpies h
E
, Gibbs free energies g
E
and hence entropies s
E
have been obtained using calorimetry and the vapor sorption method at 25°C for hexane isomers+2,2,4,4,6,8,8-heptamethylnonane, a highly branched C
16
. The h
E
and g
E
are negative while Ts
E
are positive, but small. The values are explained by the Prigogine-Flory theory through negative free volume contributions to h
E
and Ts
E
, counterbalanced in the case of Ts
E
by the positive combinatiorial Ts
E
for mixing molecules of different size. No contribution is seen from the interaction between methyl and methylene groups. The excess quantities are also obtained for hexane and heptane isomers mixed with n-hexadecane. Values of h
E
and Ts
E
are now strongly positive, while those of g
E
are only slightly less negative. The interpretation requires two recently advanced contributions in addition to those of the Prigogine-Flory theory: 1) a decrease of order when correlations of orientations between n-C
16
molecules in the pure liquid are replaced in the solution by weaker correlations whose strengths depend on the shapes of the lower alkane isomers. For lower alkane isomers of the same shape, but highly sterically hindered, h
E
and Ts
E
are small, manifesting, 2) a negative contribution, ascribed to a rotational ordering of n-C
16
segments on the sterically-hindered molecule. Enthalpy-entropy compensation is observed for these new contributions, arising from their rapid fall-off with increase of temperature. 相似文献
49.
50.
A new extraction method for ginsenosides from ginseng roots, ginseng leaves and ginseng drug preparations by Sep-Pak C18 cartridges has been studied. Ginsenoside extraction by Sep-Pak cartridges is a rapid, efficient, reproducible method. In addition, the extracts were analyzed by high performance thin layer chromatography (HPTLC) and reverse phase high performance liquid chromatography (HPLC). The major components of ginseng saponins were effectively separated using an ODS-120T column. 相似文献