The lower density conjecture for harmonic measure

In this paper, we establish the lower density conjecture for harmonic measure in simply connected plane domains.     

Projection estimates for harmonic measure

Stochastic proofs of the Beurling projection theorem and the Hall projection theorem for harmonic measure are given. Somed-dimensional versions (for alld>1) which follow from this     

A uniqueness result for harmonic functions

Let , , and suppose is harmonic in and on the closure of . If the gradient of vanishes continuously on a subset of of positive -dimensional Lebesgue measure and satisfies certain regularity conditions, then must be identically constant.

     

Functions harmonic for a Markov process

The compactness property of a family of functions harmonic for a Markov process is studied and, in particular, an inequality of Harnack type is derived. It is shown that under broad conditions the property that a function be locally harmonic implies that it is harmonic.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 587–596, April, 1973.     

The harmonic space associated with a “reasonable” standard process

Ohne Zusammenfassung     

A partial regularity result for harmonic maps into a Finsler manifold

In this paper, we consider the energy of maps from an Euclidean space into a Finsler space and study the partial regularity of energy minimizing maps. We show that the -dimensional Hausdorff measure of the singular set of every energy minimizing map is 0 for some , when m=3,4. Received: 6 June 2001 / Accepted: 10 July 2001 / Published online: 12 October 2001     

Matrix random products with singular harmonic measure

Any Zariski dense countable subgroup of SL(d, mathbb R){SL(d, mathbb {R})} is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in mathbb R^d{mathbb {R}^d} in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and arbitrarily long Lyapunov vector.     

The number of extremal components of a rigid measure

The Littlewood-Richardson rule can be expressed in terms of measures, and the fact that the Littlewood-Richardson coefficient is one amounts to a rigidity property of some measure. We show that the number of extremal components of such a rigid measure can be related to easily calculated geometric data. We recover, in particular, a characterization of those extremal measures whose (appropriately defined) duals are extremal as well. This result is instrumental in writing explicit solutions of Schubert intersection problems in the rigid case.     

Explicit relations for the reduced module and harmonic measure

Exact relations are established for the reduced modules of the families of curves separating certain connected subsets in arbitrary connected domains. New explicit expressions for the harmonic measure of a connected boundary subset are obtained as corollaries.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 576–578, April, 1993.     

k-途径正则有向图

In this paper, we define a class of strongly connected digraph, called the k-walk- regular digraph, study some properties of it, provide its some algebraic characterization and point out that the 0-walk-regular digraph is the same as the walk-regular digraph discussed by Liu and Lin in 2010 and the D-walk-regular digraph is identical with the weakly distance-regular digraph defined by Comellas et al in 2004.     

The harmonic consistency index for the analytic hierarchy process

A new consistency measure, the harmonic consistency index, is obtained for any positive reciprocal matrix in the analytic hierarchy process. We show how this index varies with changes in any matrix element. A tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9, as is often recommended. Using simulation, the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret. In addition, new properties of the column sums of reciprocal matrices are obtained.     

Absolutely continuous measure for a jump-type Fleming-Viot process

In this paper, we prove that the random measure of the one-dimensional jump-type Fleming-Viot process is absolutely continuous with respect to the Lebesgue measure in R, provided the mutation operator satisfies certain regularity conditions. This result is an important step towards the representation of the Fleming-Viot process with jumps in terms of the solution of a stochastic partial differential equation.