Laplacian Distribution and Domination

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\frac{\gamma(G)}{m_G[0,1)} \not\in O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$

Remote sensing observatory validation of surface soil moisture using Advanced Microwave Scanning Radiometer E, Common Land Model, and ground based data: Case study in SMEX03 Little River Region, Georgia, U.S.

Optimal soil moisture estimation may be characterized by intercomparisons among remotely sensed measurements, ground‐based measurements, and land surface models. In this study, we compared soil moisture from Advanced Microwave Scanning Radiometer E (AMSR‐E), ground‐based measurements, and a Soil‐Vegetation‐Atmosphere Transfer (SVAT) model for the Soil Moisture Experiments in 2003 (SMEX03) Little River region, Georgia. The Common Land Model (CLM) reasonably replicated soil moisture patterns in dry down and wetting after rainfall though it had modest wet biases (0.001–0.054 m3/m3) as compared to AMSR‐E and ground data. While the AMSR‐E average soil moisture agreed well with the other data sources, it had extremely low temporal variability, especially during the growing season from May to October. The comparison results showed that highest mean absolute error (MAE) and root mean squared error (RMSE) were 0.054 and 0.059 m3/m3 for short and long periods, respectively. Even if CLM and AMSR‐E had complementary strengths, low MAE (0.018–0.054 m3/m3) and RMSE (0.023–0.059 m3/m3) soil moisture errors for CLM and soil moisture low biases (0.003–0.031 m3/m3) for AMSR‐E, care should be taken prior to employing AMSR‐E retrieved soil moisture products directly for hydrological application due to its failure to replicate temporal variability. AMSR‐E error characteristics identified in this study should be used to guide enhancement of retrieval algorithms and improve satellite observations for hydrological sciences

Macro Dark Matter

Dark matter is a vital component of the current best model of our universe, $\Lambda$CDM. There are leading candidates for what the dark matter could be (e.g. weakly-interacting massive particles, or axions), but no compelling observational or experimental evidence exists to support these particular candidates, nor any beyond-the-Standard-Model physics that might produce such candidates. This suggests that other dark matter candidates, including ones that might arise in the Standard Model, should receive increased attention. Here we consider a general class of dark matter candidates with characteristic masses and interaction cross-sections characterized in units of grams and cm$^2$, respectively -- we therefore dub these macroscopic objects as Macros. Such dark matter candidates could potentially be assembled out of Standard Model particles (quarks and leptons) in the early universe. A combination of Earth-based, astrophysical, and cosmological observations constrain a portion of the Macro parameter space. A large region of parameter space remains, most notably for nuclear-dense objects with masses in the range $55 - 10^{17}$ g and $2\times10^{20} - 4\times10^{24}$ g, although the lower mass window is closed for Macros that destabilize ordinary matter.Comment: 13 pages, 1 table, 4 figures. Submitted to MNRAS. v3: corrected small errors and a few points were made more clear, v4: included CMB bounds on dark matter-photon coupling from Wilkinson et al. (2014) and references added. Final revision matches published versio

Relativistic Ritz approach to hydrogen-like atoms I: theoretical considerations

The Rydberg formula along with the Ritz quantum defect ansatz has been a standard theoretical tool used in atomic physics since before the advent of quantum mechanics, yet this approach has remained limited by its non-relativistic foundation. Here I present a long-distance relativistic effective theory describing hydrogen-like systems with arbitrary mass ratios, thereby extending the canonical Ritz-like approach. Fitting the relativistic theory to the hydrogen energy levels predicted by bound-state QED indicates that it is superior to the canonical, nonrelativistic approach. An analytic analysis reveals nonlinear consistency relations within the bound-state QED level predictions that relate higher-order corrections to those at lower order, providing guideposts for future perturbative calculations as well as insights into the asymptotic behavior of Bethe logarithms. Applications of the approach include fitting to atomic spectroscopic data, allowing for the determination the fine-structure constant from large spectral data sets and also to check for internal consistency of the data independently from bound-state QED.Comment: v2: 11 pages of main text, 14 figures, 2 appendice

A perturbative method for resolving contact interactions in quantum mechanics

Long-range effective methods are ubiquitous in physics and in quantum theory, in particular. Furthermore, the reliability of such methods is higher when the nature of short-ranged interactions need not be modeled explicitly. This may be necessary for two reasons: (1) there are interactions that occur over a short range that cannot be accurately modeled with a potential function and/or (2) the entire Hamiltonian loses its reliability when applied at short distances. This work is an investigation of the utility and consequences of omitting a finite region of space from quantum mechanical analysis, accomplished by imposition of an artificial boundary behind which obscured short-ranged physical effects may operate. With this method, a free function of integration that depends on momentum is interpreted as a function encoding information needed to match a long-distance wavefunction to an appropriate state function on the other side of the boundary. Omitting part of the space from analysis implies that the strict unitarity requirement of quantum mechanics must be relaxed, since particles can actually propagate beyond the boundary. Strict orthogonality of eigenmodes and hermiticity of the Hamiltonian must also be relaxed in this method; however, all of these canonical relations are obeyed when averaged over sufficiently long times. What is achieved, therefore, appears to be an effective long-wavelength theory, at least for stationary systems. As examples, the quantum defect theory of the one-dimensional Coulomb interaction is recovered, as well as a new perspective of the inverse-square potential and the free particle, as well as the Wigner time delay associated with contact interactions. Potential applications of this method may include three-dimensional atomic systems and two-dimensional systems, such as graphene.Comment: 13 pages, 7 figures, Phys. Rev. A accepted (12/02/19