JJ-holomorphic curves from closed JJ-anti-invariant forms

We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher dimensional analogue is established. We also show the dimension of closed $J$-anti-invariant $2$-forms on an almost complex $4$-manifold is a birational invariant, in the sense that it is invariant under degree one pseudoholomorphic maps.Comment: 28 page

Topological-Like Features in Diagrammatical Quantum Circuits

In this paper, we revisit topological-like features in the extended Temperley--Lieb diagrammatical representation for quantum circuits including the teleportation, dense coding and entanglement swapping. We perform these quantum circuits and derive characteristic equations for them with the help of topological-like operations. Furthermore, we comment on known diagrammatical approaches to quantum information phenomena from the perspectives of both tensor categories and topological quantum field theories. Moreover, we remark on the proposal for categorical quantum physics and information to be described by dagger ribbon categories.Comment: v1: Latex. v2,v3: original proposals in v1 are stated in a physical style. This manuscript is a formal written version of Y. Zhang's talk at the workshop "Cats, Kets and Cloisters", Computing Laboratory, Oxford University, July 17-23, 200

Low-Profile Spoof Surface Plasmon Polaritons Traveling-Wave Antenna for Endfire Radiation

This paper proposes a low-profile and highly efficient endfire radiating travelling-wave antenna based on spoof surface plasmon polaritons (SSPPs) transmission line. The aperture is approximately $0.32\lambda_0\times0.01\lambda_0$ where $\lambda_0$ is the space wavelength at the operational frequency 8 GHz. This antenna provides an endfire radiation beam within 7.5-8.5 GHz. The maximum gain and total efficiency reaches 9.2 dBi and $96\%$, respectively. In addition to the endfire operation, it also provides a beam scanning functionality within 9-12 GHz. Measurement results are finally given to validate the proposed SSPPs antenna

Permutation and Its Partial Transpose

Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang--Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the "PPT" algebra which guides the construction of multipartite symmetric states. The virtual knot theory having permutation as a virtual crossing provides a topological language describing quantum computation having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley--Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang--Baxter equations; and virtual Temperley--Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the "ABPK" diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies nontrivial unitary braid representations with universal quantum gates, and derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.Comment: 42 pages, 22 figures, late

Virtual Extension of Temperley--Lieb Algebra

The virtual knot theory is a new interesting subject in the recent study of low dimensional topology. In this paper, we explore the algebraic structure underlying the virtual braid group and call it the virtual Temperley--Lieb algebra which is an extension of the Temperley--Lieb algebra by adding the group algebra of the symmetrical group. We make a connection clear between the Brauer algebra and virtual Temperley--Lieb algebra, and show the algebra generated by permutation and its partial transpose to be an example for the virtual Temperley--Lieb algebra and its important quotients.Comment: 10 pages, late

Universal Quantum Gate, Yang--Baxterization and Hamiltonian

It is fundamental to view unitary braiding operators describing topological entanglements as universal quantum gates for quantum computation. This paper derives a unitary solution of the Quantum Yang--Baxter equation via Yang--Baxterization and constructs the Hamiltonian responsible for the time-evolution of the unitary braiding operator.Comment: v1, 11 pages; v2, 11 pages; v3, 12 pages, additional comments; v4, 10 pages; v5, published versio

Linear stability analysis of the Hele-Shaw cell with lifting plates

The first stages of finger formation in a Hele-Shaw cell with lifting plates are investigated by means of linear stability analysis. The equation of motion for the pressure field (growth law) results to be that of the directional solidification problem in some unsteady state. At the beginning of lifting the square of the wavenumber of the dominant mode results to be proportional to the lifting rate (in qualitative agreement with the experimental data), to the square of the length of the cell occupied by the more viscous fluid, and inversely proportional to the cube of the cell gap. This dependence on the cell parameters is significantly different of that found in the standard cell.Comment: 5 pages, RevTeX, 3 postscript files include

Importance Sampling of Word Patterns in DNA and Protein Sequences

Monte Carlo methods can provide accurate p-value estimates of word counting test statistics and are easy to implement. They are especially attractive when an asymptotic theory is absent or when either the search sequence or the word pattern is too short for the application of asymptotic formulae. Naive direct Monte Carlo is undesirable for the estimation of small probabilities because the associated rare events of interest are seldom generated. We propose instead efficient importance sampling algorithms that use controlled insertion of the desired word patterns on randomly generated sequences. The implementation is illustrated on word patterns of biological interest: Palindromes and inverted repeats, patterns arising from position specific weight matrices and co-occurrences of pairs of motifs

Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

The celebrated Cheeger's Inequality establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this paper we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is non-linear, we have to exploit other properties of the diffusion process to recover a spectral property concerning the "second eigenvalue" of the resulting Laplacian. Moreover, we show that higher order spectral properties cannot hold in general using the current framework. We consider a stochastic diffusion process, in which each vertex also experiences Brownian noise from outside the system. We show a relationship between the second eigenvalue and the convergence behavior of the process. We show that various hypergraph parameters like multi-way expansion and diameter can be bounded using this operator's spectral properties. Since higher order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher order Cheeger-like inequalities. For any positive integer $k$, we give a polynomial time algorithm to compute an $O(\log r)$-approximation to the $k$-th procedural minimizer, where $r$ is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis for constant values of $k$.Comment: A preliminary version of this paper appeared in STOC 2015 [Louis] (arXiv:1408.2425 [cs.DM]) and the current paper is the result of a merge with [Chan, Tang, Zhang] (arXiv:1510.01520 [cs.DM]

Online dynamic mode decomposition for time-varying systems

Dynamic mode decomposition (DMD) is a popular technique for modal decomposition, flow analysis, and reduced-order modeling. In situations where a system is time varying, one would like to update the system's description online as time evolves. This work provides an efficient method for computing DMD in real time, updating the approximation of a system's dynamics as new data becomes available. The algorithm does not require storage of past data, and computes the exact DMD matrix using rank-1 updates. A weighting factor that places less weight on older data can be incorporated in a straightforward manner, making the method particularly well suited to time-varying systems. A variant of the method may also be applied to online computation of "windowed DMD", in which only the most recent data are used. The efficiency of the method is compared against several existing DMD algorithms: for problems in which the state dimension is less than about~200, the proposed algorithm is the most efficient for real-time computation, and it can be orders of magnitude more efficient than the standard DMD algorithm. The method is demonstrated on several examples, including a time-varying linear system and a more complex example using data from a wind tunnel experiment. In particular, we show that the method is effective at capturing the dynamics of surface pressure measurements in the flow over a flat plate with an unsteady separation bubble.Comment: 22 pages, 7 figures, 1 tabl