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.     

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 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.     

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.     

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.     

Two new associated Laguerre integral results

New associated Laguerre integral results are obtained by eliminating an unnecessary restriction in [1]. The expression used to obtain these new results is compared to that given in [2] and shown to be different and of greater generality. Finally, some physical moments are discussed that require such integrals.     

Two results on maximum nonlinear functions

Maximum nonlinear functions are widely used in cryptography because the coordinate functions F _β (x) := tr(β F(x)), , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions , gcd(k, m) = 1, and the Kasami power functions , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.      

Two hardness results for core stability in hedonic coalition formation games

We establish NP-completeness of two problems on core stable coalitions in hedonic games. In the first problem every player has only two acceptable coalitions in his preference list, and in the second problem the preference structures arise from the distances in an underlying metric space.     

Two local dissimilarity measures for weighted graphs with application to protein interaction networks

We extend the Czekanowski-Dice dissimilarity measure, classically used to cluster the vertices of unweighted graphs, to weighted ones. The first proposed formula corresponds to edges weighted by a probability of existence. The second one is adapted to edges weighted by intensity or strength. We show on simulated graphs that the class identification process is improved by computing weighted compared to unweighted edges. Finally, an application to a drosophila protein network illustrates the fact that using these new formulas improves the ’biological accuracy’ of partitioning.