A recurrent graph G has the infinite collision property if two independent random walks on G, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in Z with d ≥ 19 and the ...

Do patents provide critical incentives to encourage investment in innovation? Or, instead, do patents impose legal risks and burdens on innovators that discourage innovation, as some critics now claim? This paper reviews empirical economic evidence on how well patents perform as a property system. Do Patents Perform Like Property? 1

In this paper, we define the primitive/Seifert-fibered property for a knot in S . If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of examples of primitive/Seifert-fibered knots. By analyzing the twisted torus knots, ...

Two known results in multiple testing, one relating to the directional error control of augmented step-down procedure proved by Shaffer (1980) and the other on the monotonicity of the critical values of step-up procedure proved by Dalal and Mallows (1992), are revisited and improved in this article. Shaffer’s result on the directional error control property of augmented step-down procedure with...

Recent work shows that global motion sensitivity, a property of extrastriate cortex, can be altered by early visual deprivation, while binocularity, a property of primary visual cortex, is still plastic. This contradicts the hypothesis that critical periods end later at higher levels of the system.

We study the geometry of certain one-dimensional maps as dynamical systems. We prove the property of bounded and bounded nearby geometry of certain C one-dimensional maps with finitely many critical points. This property enables us to give the quasίsymmetric classification of geometrically finite one-dimensional maps.

Let p(z) be a monic cubic complex polynomial with distinct roots and distinct critical points. We say a critical point has the Voronoi property if it lies in the Voronoi cell of a root θ, V (θ), i.e. the set of points that are closer to θ than to the other roots. We prove at least one critical point has the Voronoi property and characterize the cases when both satisfy this property. It is known...

In this paper, some properties of the periodic shadowing are presented. It is shown that a homeomorphism has the periodic shadowing property if and only if so does every lift of it to the universal covering space. Also, it is proved that continuous mappings on a compact metric space with the periodic shadowing and the average shadowing property also have the shadowing property and then are chao...

A necessary and sufficient condition for the resolution of the weak extension problem is given. This criterion is applied to also give a criterion for the solvability of the classical Bochner’s extension problem in the L–category. The solution of the L–extension problem by Bochner [1] giving the relation between the order of the operator, the dimension, and index p, for which the L–extension pr...