Two principles for extending probability measures

A principle for proving the existence of an extension of a probability measure and a principle which answers the question of uniqueness of the extension are given. It turns out that these principles can be applied in a great number of situations treated in the literature.     

Existence results for hemivariational inequalities with measures

We prove existence results for multivalued quasilinear elliptic problems of hemivariational inequality type with measure data right-hand sides. In case of L ¹-data, we study existence and enclosure behaviors of solutions by an appropriate sub-supersolution approach. The proofs of our results are based on general existence theory for multivalued pseudomonotone operators, and approximation-, truncation-, and special test function techniques.     

Two measures for proving Gentzen's Hauptsatz without mix

 This paper presents a cut-elimination procedure for classical and intuitionistic logic, in which cut is eliminated directly, without introducing the mix rule. The well-known problem of cut eliminations, when in the derivation the contractions of the cut-formulae are above the premisses of the cut, will be solved by new transformations of the derivation. Received: 1 June 2001 / Published online: 5 November 2002 Mathematics Subject Classification (2000): 03F05 Key words or phrases: Systems of sequents – Cut-elimination theorem     

Two inverse results

Let A be a finite subset of a group G ₀ with |A ^?1 A|≤2|A?2. We show that there are an element α∈A and a non-null proper subgroup H of G such that one of the following holds:

• x ^?1 Hy?A ^?1 A, for all x,y∈A not both in Hα
• x Hy ^?1?AA ^?1, for all x,y∈A not both in αH

where G is the subgroup generated by A ^?1 A. Assuming that A ^?1 A≠G and that $\left| {A^{ - 1} A} \right| < \tfrac{{5|A|}} {3} $ , we show that there are a normal subgroup K of G and a subgroup H with K?H?A ^?1 A and 2|K|≥|H| such that $A^{ - 1} AK = KA^{ - 1} A = A^{ - 1} Aand6|K| \geqslant |A^{ - 1} A| = 3|H|$ .     

Two Lil-type results for the Brownian bridge

The necessary and sufficient condition for a function to be upper class relative to a Brownian motion process was obtained by T. Sirao and T. Nisida (1952, Nagoya Math. J. 4, 97–101) and a related result on first passage distribution was obtained by P. K. Sen and M. J. Wichura (in press, Estimation of First Crossing Time Distribution for Brownian Motion Processes Relative to Upper Class Boundaries). Corresponding results for the Brownian bridge are discussed.     

有关单机两代理排序问题的两个结果

研究了两个单机两代理排序问题. 在第一个两代理排序问题中, 代理A的目标函数为极小化所有工件的加权完工时间总和, 代理B的目标函数为极小化最大工件费用. 在第二个两代理排序问题中, 代理A的目标函数为极小化所有工件的加权完工时间总和, 代理B的目标函数为极小化所有工件的最大完工时间. 证明了第一个问题是强NP-难的, 改进了已有的一般意义NP-难的结果; 对第二个问题给出了一个与现有的动态规划算法不同的动态规划算法.     

Diagonalization and representation results for nonpositive sesquilinear form measures

We study decompositions of operator measures and more general sesquilinear form measures E into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent E as a trace class valued measure of bounded variation on a new Hilbert space related to E. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain properties, but this choice itself is not canonical. We present relations to Naimark type dilations and direct integrals.     

Distribution results for the occupation measures of continuous Gaussian fields

For a d-dimensional random field X(t) define the occupation measure corresponding to the level α by the Lebesgue measure of that portion of the unit cube over which X(t)?α. Denoting this by M[X, α], it is shown that for sample continuous Gaussian fields

$\text{P}\text{{M[X,α]>β}=}\text{exp}\text{{?}\text{1}\text{2}\text{α}{}^2\text{k}{}_{\mathrm{\beta }}\text{(1+0(1))}}$

as α→∞, for a particular functional kβ. This result is applied to a variety of fields related to the planar Brownian motion, and for each such field we obtain bounds for kβ.     

Two examples on atomic doubling measures

It is shown that there is a compact set X⊂[0,1] with dimHX=1 on which all doubling measures are purely atomic. It is also shown that there is a compact set X⊂[0,1] with a dense set of isolated points and dimHX=0 on which no doubling measures are purely atomic.     

Two equivalent measures on weighted hypergraphs

Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2^N, and real edge-weights w(e) for e∈E. Given a convex n-gon P in the plane with vertices x₀,x₁,…,x_n-1 which are arranged in this order clockwisely, let each node i∈N correspond to the vertex x_i and define the area A_P(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0?i<j<k?n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size c_H(i,j,k) in H is defined as the sum of weights of edges e∈E such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:

•

A_P(H)?A_P(H^′) for all convex n-gons P.

•

c_H(i,j,k)?c_H^′(i,j,k) for all convex three-cuts C(i,j,k).

From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H^′ satisfy “A_P(H)?A_P(H^′) for all convex n-gons P” is immediately obtained.     

Two results in negation-free logic

Negation-free propositional logic (or first-order logic) is clearly less expressive than the corresponding full system with negation. However, we present two complexity results for logic without negation that are no different from those for the original system. First, the problem of determining logical implication between sentences composed solely of conjunctions and disjunctions is shown to be as difficult as that between arbitrary sentences. Second, we show that the problem of determining a minimum satisfying assignment for a propositional formula in negation-free conjunctive normal form, even with no more than two disjuncts per clause, is NP-complete. We also show that unless P = NP, no polynomial time approximation scheme can exist for this problem.     

Two results on arithmetical functions

Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type $${\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n)\qquad {\rm and} {\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n),$$ where χ is a suitable multiplicative function, f₁,…, f _r are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by $ \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} $ complex.     

Asymptotic results for conditional measures of association of a random sum

Asymptotic results are obtained for several conditional measures of association. The chosen random variables are the first two order statistics and the total sum within a random sum. Many of the results have confirmed the “one-jump” property of the risk model. Non-trivial limits are obtained when the dependence among the first two order statistics is considered. Our results help in understanding the extreme behaviour of well-known reinsurance treaties that involve only few large claims. Interestingly, the Pearson product-moment correlation coefficient between the first two order statistics provides an alternative procedure to estimate the tail index of the underlying distribution.     

New results on measures of maximal entropy

It has recently been demonstrated that there are strongly irreducible subshifts of finite type with more than one measure of maximal entropy. Here we obtain a number of results concerning the uniqueness of the measure of maximal entropy. In addition, we construct for anyd≥2 andk a strongly irreducible subshift of finite type ind dimensions with exactlyk ergodic (extremal) measures of maximal entropy. Ford≥3, we construct a strongly irreducible subshift of finite type ind dimensions with a continuum of ergodic measures of maximal entropy. Research supported in part by AFOSR grant # 91-0215 and NSF grant # DMS-9103738. Research supported by the Swedish National Science Foundation.