Our approach in that paper was to first prove an analogous result for planar Brownian motion and then to use strong approximation. The goal of this paper, which is purely expository, is to show how to prove (1.1) using only random walk methods. We also go beyond (1.1) and study the size of the set of ‘frequent points’, i.e. those points in Z2 which are visited an unusually large number of times...

Proof. Fix a pair of neighbouring datasets X ' X ′ and let E ⊆ R be any measurable set. Let ΘX,X′ be as in the statement and write ΘX,X′ = R \ΘX,X′ . Using the assumptions on ΘX,X′ we see that P[θX ∈ E] = P[θX ∈ E ∩ΘX,X′ ] + P[θX ∈ E ∩ΘX,X′ ] ≤ eP[θX′ ∈ E ∩ΘX,X′ ] + δ ≤ eP[θX′ ∈ E] + δ . Now we proceed with the proof of Lemma 1. Let X ' X ′ be two neighbouring datasets and let us write Z1 = ZX ...

Cyclic codes over R have been introduced recently. In this paper, we study the cyclic codes over R and their Z2 image. Making use of algebraic structure, we find the some good Z2 codes of length 28.

I discuss the correspondence between realistic four dimensional free fermionic models and Z2 × Z2 orbifold compactification. I discuss the properties of the Z2 × Z2 orbifold that are reflected in the realistic free fermionic models. I argue that the properties of the realistic free fermionic models arise due to the underlying Z2×Z2 orbifold compactification with nontrivial background fields. I ...

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In this paper, we study the algebraic structure of Z2[u]Z2[u, v]-additive codes which are Z2[u, v]-submodules where u 2 = v2 = 0 and uv = vu. In particular, we determine a Gray map from Z2[u]Z2[u, v] to Z 2α+8β 2 and study generator and parity check matrices for these codes. Further we study the structure of Z2[u]Z2[u, v]additive cyclic codes and constacyclic codes.

We calculate Betti numbers of the framed moduli space of instantons on Ĉ/Z2, under the assumption that the corresponding torsion free sheaves E have vanishing properties(Hom(E, E(−l∞)) = Ext (E, E(−l∞)) = 0). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers o...

We discuss examples of D-branes probing toric singularities, and the computation of their world-volume gauge theories from the geometric data of the singularities. We consider several such examples of D-branes on partial resolutions of the orbifoldsC3/Z2 × Z2,C 3/Z2 × Z3 and C4/Z2 × Z2 × Z2. Email: [email protected]

In recent years, methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this Letter it is shown that the latter method is actually related to (and may be derived from) a particular case of the former if one regards dual of Lie algebroids as special Poisson manifolds. The core of the proo...