The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in 1/N . We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power series of a knot. We give examples that illustrate our method and its results.
To study electrostatic actuation, researchers commonly use a setup proposed by G. I. Taylor in [Proc. R. Soc. Lond. Ser. A, 306 (1968), pp. 423–434]. It consists of soap film held at a distance h above a rigid plate so that when a voltage difference is applied between the two components, the top film deflects towards the bottom plate. The most striking feature of this system is when the voltage difference exceeds a critical value V*, the electrostatic forces dominate the surface forces and the soap film gets ‘pulled-into’ or collapses onto the bottom plate. This so-called ‘pull-in’ instability is a ubiquitous feature of electrostatic actuation and as a result, has been the subject of many studies. Recently, Siddique et al. [J. Electrostatics, 69 (2011), pp. 1–6] measured the value of V* as a function of the separation distance and found that the standard prediction breaks down as h increases. Here, we continue the work done in [N. D. Brubaker and J. A. Pelesko, European J. Appl. Math., 22 (2011), pp. 455–470] by investigating the cause of this discrepancy. Specifically, we model the effect of gravity on the generalized version of Taylor's model and study whether it provides the proper correction to the predicted value of V*. In doing so, we derive two nonlinear eigenvalue value problems and investigate their solutions sets.
We consider a supersymmetric hybrid inflation scenario in which the U (1) R-symmetry is explicitly broken by Planck scale suppressed operators in the superpotential. We provide an example with minimal Kähler potential, with the R-symmetry breaking term relevant during inflation being αS 4 , where S denotes the well-known gauge singlet inflaton superfield. The inflationary potential takes into account the radiative and supergravity corrections, as well as the soft supersymmetry breaking terms. For successful inflation, with the scalar spectral index in the currently preferred range, n s ≈ 0.97 ± 0.010, |α| 10 −7 . The tensor to scalar ratio r 10 −4 , while |dn s /d ln k| ∼ O(10 −3 ) − O(10 −4 ).PACS numbers: 98.80.Cq INTRODUCTIONModern cosmology has seen rapid developments due to experiments such as COBE and WMAP. Augmenting their unprecedented successes will be Planck, which may for the first time yield direct evidence of inflation. At the same time, there are enormous strides being made in particle physics, with the LHC having made perhaps the greatest single discovery in the field in decades, while the testing of supersymmetry (SUSY) is highly anticipated. For the first time particle and cosmological models can be tested with precision, and the deep connections between these two fields motivate us to consider the effects that particle physics considerations have on inflationary models. This has led to models such as, among others, SUSY hybrid inflation.The standard version of SUSY hybrid inflation [1-3] remains one of the most successful and well-motivated inflationary models. It is the most general non-trivial model one can write with a gauge singlet field S and supermultiplets Φ andΦ that respects U (1) R , such that the latter two fields belong to non-trivial representations of the gauge group G. It has a connection to particle physics in that within it grand unified theories (GUTs) are naturally incorporated. In this model the gauge group G is broken to a subgroup H at the end of inflation, and the energy scale at which this occurs is related to local temperature anisotropies in the cosmic microwave background radiation [1]; this scale happens to be close to the GUT scale, indicating that G may be related to a GUT. In addition, supergravity (SUGRA) corrections remain under control [4] because this model can yield solutions within the WMAP nine-year 2σ bounds without trans-Planckian inflaton field values. In this model, where only (minimal) SUGRA and radiative corrections are added to the global potential, a numerical lower bound on the scalar spectral index, n s ≈ 0.985, develops. This is somewhat disfavored, although it is within the 2σ bounds, n s = 0.971 ± 0.010 [5]. However, the addition of a linear soft-SUSY breaking term [6] reduces n s significantly [7]; this term is also important in explaining the MSSM µ-problem [8][9][10]. It has been shown that small gravity waves (tensor-to-scalar ratio r) are produced with minimal Kähler (retaining the lowest-order SUGRA term), radiative corrections, and positi...
We further develop the theoretical framework for the quantum trellis decoder proposed by Ollivier and Tillich in 2006. We show that while conceptually different, certain results from the theory of trellis decoding for classical linear block codes have quantum analogies. We compare and contrast the two theories throughout. The decoder works for any stabilizer code S and separates into two parts: an expensive, one-time, offline computation which builds a compact, graphical representation of the normalizer of the code, S ⊥ , and a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. Mathematical results about the trellises are obtained for prime-dimension qudit codes . We demonstrate the effectiveness of this approach numerically by natively decoding the rotated surface code and the 4.8.8/6.6.6 color code families on qubits for moderate distances without any concept of homology. Code capacity thresholds for all three families are found to be between 10-11% for distances up to 17-21 under a Z-only noise model. Computational limitations and open problems are discussed. We additionally use the developed framework to promote a new, quantitative metric describing the fundamental difficulty of decoding a given stabilizer code. We apply this to show that, for example, the color code families are more difficult to decode than the rotated surface code family. We also investigate CSS codes under this framework and use this metric to quantitatively discuss the computational savings of decoding X-and Z-errors independently.
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