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91.
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
92.
For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|c
t
·n
2–1/t
which do not contain a complete bipartite graghK
t,t
!+1 This establishes the exact order of magnitude of the Turán numbers ex (n, K
t,s
) for any fixedt and allst!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.Research supported in part by NSF Grants DMS-8707320 and DMS-9102866.Research supported in part by Hungarian National Foundation for Scientific Research Grant 相似文献
93.
V. Suresh 《K-Theory》1996,10(6):597-610
Let X be a smooth projective surface over a number field k. Let (CH0(X)) denote the Chow group of zero-cyles modulo rational equivalence on X. Let CH0(X) be the subgroup of CH
0(X) consisting of classes which vanish when going over to an arbitrary completion of k. Bloch put forward a conjecture asserting that this group is isomorphic to the Tate-Shafarevich group of a certain Galois module atttached to X. In this paper, we disprove this general conjecture. We produce a conic bundle X over an elliptic curve, for which the group (CH0(X) is not zero, but the Galois-theoretic Tate-Shafarevich group vanishes. 相似文献
94.
Let be a product order on [n]
p
i.e. for A, B [n]
p
, 1a
1<a
2<...<a
p
º-n and 1<-b
1<b
2<...<b
p
<-n we have AB iff a
i<-b
i for all i=1, 2,..., p. For a linear extension < of (ordering [n]
p
as
) let F
<
[n],p
(m) count the number of A
i
's, i<-m such that 1A
i. Clearly,
for every m and <, where <l denotes the lexicographic order on [n]
p
. In this note we prove that the colexicographical order, <c, provides a corresponding lower bound i.e. that
holds for any linear extension < of .This project together with [2] was initiated by the first author and continued in colaboration with the second author. After the death of the first author the work was continued and finalized by the second and the third author.Research supported by NSF grant DMS 9011850. 相似文献
95.
Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z
k
-coloring f: P Z
k
there exists either a chain or an antichain A, |A|=n and
aA
f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)2–c(k) p(n, k) (n+k–2)2+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing). 相似文献
96.
Extending previous work on the geometric characterization of separability in the autonomous case, necessary and sufficient conditions are established for the complete separability of a system of time-dependent second-order ordinary differential equations. In deriving this result, extensive use is made of the theory of derivations of scalar and vector-valued forms along the projection :J
1
EE of the first jet bundle of a fibre bundleE . Two illustrative examples are discussed, which fully demonstrate all aspects of the constructive nature of the theory. 相似文献
97.
One more model of aq-harmonic oscillator based on theq-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form ofq-creation andq-annihilation operators,q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed. 相似文献
98.
The transformation of amorphous thin alloy films of Co–Zr to the crystalline state was observed with an electron microscope in the diffraction mode. This investigation elucidates the crystallization temperatures,T
x
, of thin films as opposed to theT
x
presented for thicker samples; usually melt-quenched in the thickness range of 20–50 m. TheT
x
of the thin films are compared also with theT
x
for sputtered alloys of Co–Zr at 5 m. 相似文献
99.
J. Czarnowska 《Periodica Mathematica Hungarica》1993,26(2):101-110
To 80th birthday of Paul Erds 相似文献
100.
Israel Gohberg Naum Krupnik Ilya Spitkovsky 《Integral Equations and Operator Theory》1993,17(3):322-337
The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceL
p
(,) with an arbitrary weight is studied. The contour consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper. 相似文献