Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and 19 A∗ = A−{0}. A graph G is A-connected if G has an orientation D(G) such that for every 20 function b : V (G) → A satisfying Pv∈V (G) b(v) = 0, there is a function f : E(G) → A∗ 21 such that for each vertex v ∈ V (G), the total amount of f values on the edges directed 22 out from v minus the total amount of f va...

A few years ago, Grimmett, Kesten and Zhang proved that for supercritical bond percolation on Z3, simple random walk on the infinite cluster is a.s. transient. We generalize this result to a class of wedges in Z3 including, for any ε ε (0,1), the wedge Wε={(x,y,z) ε Z3 : x ≥ 0, |z| ≤ x ε} which can be thought of as representing a (2 + ε)-dimensional lattice. Our proof builds on recent work of B...

The Satisfiability Modulo Theories solver Z3 [10] is used in several program analysis and verification tools at Microsoft Research. Some of these tools require bit-precise reasoning for accurately modeling machine arithmetic instructions. But this alone is rarely sufficient, and an integration with other theories is required. The Pex tool [20] performs program exploration of .NET programs by ge...

In this work, the Z3-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given. E-mail: [email protected]

The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prov...

We recently defined cancer metastatic chemoprevention as utilizing safe and effective molecules to comprehensively prevent the spark of activation-adhesion-extravasation-proliferation metastatic cascade caused by circulating tumor cells (CTCs). The strategy focuses on preventing the most important starting point of the cascade. We identified an extract from a well-known medical plant Murraya pa...

In this paper, we investigate various kinds of extensions of twin-good rings. Moreover, we prove that every element of an abelian neat ring R is twin-good if and only if R has no factor ring isomorphic to‌ Z2  or Z3. The main result of [24] states some conditions that any right self-injective ring R is twin-good. We extend this result to any regular Baer ring R by proving that every elemen...

Satisfiability Modulo Theories (SMT) problem is a decision problem for logical first order formulas with respect to combinations of background theories such as: arithmetic, bit-vectors, arrays, and uninterpreted functions. Z3 is a new and efficient SMT Solver freely available from Microsoft Research. It is used in various software verification and analysis applications.

A flurry of recent theoretical activity has produced specific model system realizations of fractionalized phases in two dimensions. Essentially all of the fractionalized states constructed so far are Z2 states. 7 On a formal level, these realizations employ the following route to Z2 fractionalization: Strong local correlations lead to a U~1! gauge theory as a low-energy description; this gauge ...