Breathing solitary-pulse pairs in a linearly coupled system

It is shown that pairs of solitary pulses (SPs) in a linearly-coupled system with opposite group-velocity dispersions form robust breathing bound states. The system can be realized by temporal-modulation coupling of SPs with different carrier frequencies propagating in the same medium, or by coupling of SPs in a dual-core waveguide. Broad SP pairs are produced in a virtually exact form by means of the variational approximation. Strong nonlinearity tends to destroy the periodic evolution of the SP pairs.Comment: Optics Letters, in pres

Span programs and quantum algorithms for st-connectivity and claw detection

We introduce a span program that decides st-connectivity, and generalize the span program to develop quantum algorithms for several graph problems. First, we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries to the n x n adjacency matrix to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also show that if T is a path, a star with two subdivided legs, or a subdivision of a claw, its presence as a subgraph in the input graph G can be detected with O(n) quantum queries to the adjacency matrix. Under the promise that G either contains T as a subgraph or does not contain T as a minor, we give O(n)-query quantum algorithms for detecting T either a triangle or a subdivision of a star. All these algorithms can be implemented time efficiently and, except for the triangle-detection algorithm, in logarithmic space. One of the main techniques is to modify the st-connectivity span program to drop along the way "breadcrumbs," which must be retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure

DNA-Mediated Electrochemistry

The base pair stack of DNA has been demonstrated as a medium for long-range charge transport chemistry both in solution and at DNA-modified surfaces. This chemistry is exquisitely sensitive to structural perturbations in the base pair stack as occur with lesions, single base mismatches, and protein binding. We have exploited this sensitivity for the development of reliable electrochemical assays based on DNA charge transport at self-assembled DNA monolayers. Here, we discuss the characteristic features, applications, and advantages of DNA-mediated electrochemistry

Compressibility and probabilistic proofs

We consider several examples of probabilistic existence proofs using compressibility arguments, including some results that involve Lov\'asz local lemma.Comment: Invited talk for CiE 2017 (full version