Both Bohmian mechanics, a version of quantum mechanics with trajectories, and Feynman's path integral formalism have something to do with particle paths in space and time. The question thus arises how the two ideas relate to each other. In short, the answer is, path integrals provide a re-formulation of Schrödinger's equation, which is half of the defining equations of Bohmian mechanics. I try ...

Both Bohmian mechanics, a version of quantum mechanics with trajectories, and Feynman's path integral formalism have something to do with particle paths in space and time. The question thus arises how the two ideas relate to each other. In short, the answer is, path integrals provide a re-formulation of Schrödinger's equation, which is half of the defining equations of Bohmian mechanics. I try ...

Marco Buratti has conjectured that, given an odd prime p and a multiset L containing p − 1 integers taken from {1, . . . , p−1 2 }, there exists a Hamiltonian path in the complete graph with p vertices whose multiset of edge-lengths is equal to L modulo p. We give a positive answer to this conjecture in the case of multisets of the type {1, 2, 3} by completely classifying such multisets that ar...

Consider a complete probability space (Ω,F ,P;F) equipped with the Þltration F = {Ft; 0 ≤ t <∞}. A stochastic process is a collection of random variables X = {Xt; 0 ≤ t <∞} where, for every t, Xt : Ω → Rd is a random variable. We assume the space Rd is equipped with the usual Borel σ-algebra B(Rd). Every Þxed ω ∈ Ω corresponds to a sample path (or, trajectory), that is, the function t 7→ Xt(ω) ...

In this paper, we will prove that the local time of a Lévy process is a rough path of roughness p a.s. for any 2 < p < 3 under some condition for the Lévy measure. This is a new class of rough path processes. Then for any function g of finite q-variation (1 ≤ q < 3), we establish the integral ∫∞ −∞ g(x)d L x t as a Young integral when 1 ≤ q < 2 and a Lyons’ rough path integral when 2 ≤ q < 3. W...

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We study the design of efficient algorithms for combinatorial pattern matching. More concretely, we study algorithms for tree matching, string matching, and string matching in compressed texts. Tree Matching Survey We begin with a survey on tree matching problems for labeled trees based on deleting, inserting, and relabeling nodes. We review the known results for the tree edit distance problem,...

For two integers n ≥ 3 and 2 ≤ p ≤ n, we denote D(n, p) the digraph obtained from a directed n-cycle by changing the orientations of p − 1 consecutive arcs. In this paper, we show that a family of k-regular (k ≥ 3) bipartite tournament BT4k contains D(4k, p) for all 2 ≤ p ≤ 4k unless BT4k is isomorphic to a digraph D such that (1, 2, 3, ..., 4k, 1) is a Hamilton cycle and (4m+ i− 1, i) ∈ A(D) a...

We improve hardness results for the problem of embedding one finite metric into another with minimum distortion. This problem is equivalent to optimally embedding one weighted graph into another under the shortest path metric. We show that unless P = NP, the minimum distortion of embedding one such graph into another cannot be efficiently approximated within a factor less than 9/4 even when the...