Asymptotic Distance Research Articles

Control charts for high-dimensional time series with estimated in-control parameters

In this article, we study the effect of misspecification caused by fitting the target process in the Phase I analysis of the monitoring procedure on the behavior of several types of multivariate exponentially weighted moving average (MEWMA) control charts in the high-dimensional setting. In particular, the classical MEWMA control charts, whose control statistics are based on the exact and asymptotic Mahalanobis distance, are considered together with the novel approaches where the Euclidean distance and the diagonalized Euclidean distance are employed in the construction of control statistics. The high-dimensional distributions of the control statistics are deduced at each time. These results are later used to assess the performance of the considered control charts under misspecification. Both theoretical and empirical findings lead to the conclusion that the control charts based on the Euclidean distance and the diagonalized Euclidean distance are robust to misspecification for moderate dimensions of the data-generating model, whereas they tend to overestimate the in-control average run lengths (ARLs) in the case of larger dimensions. On the other hand, the control schemes based on the Mahalanobis distance are considerably affected by the estimation of the parameters of the target process, and their application results in drastically smaller values of the ARLs, especially when the dimension of the data-generating model is large.

Conditioning in Tropical Probability Theory.

We define a natural operation of conditioning of tropical diagrams of probability spaces and show that it is Lipschitz continuous with respect to the asymptotic entropy distance.

Optimal estimate of field concentration between multiscale nearly-touching inclusions for 3-D Helmholtz system

We are concerned with the field concentration between two nearly-touching inclusions with high-contrast material parameters, which is a central topic in the theory of composite materials. The degree of concentration is characterised by the blowup rate of the gradient of the underlying field. In this paper, we derive optimal gradient estimates for the wave field of the 3-D Helmholtz system in the quasi-static regime. There are two salient features of our results that are new to the literature. First, we cover all the possible scenarios that the size of the inclusions are in different scales in terms of the asymptotic distance parameter $ \epsilon $, which can be used to characterise the curvature effects of the shape of the inclusions on the field concentration. Second, our estimates can not only recover the known results in the literature for the static case, but can also reveal the interesting frequency effect on the field concentration. In fact, a novel phenomenon is shown that even if the static part vanishes, field blowup can still occur due to the (low) frequency effect.

Wiener, edge-Wiener, and vertex-edge-Wiener index of Basilica graphs

We determine the exact value of the Wiener index, the edge-Wiener index, and the vertex-edge-Wiener index of the Basilica graphs, i.e., the sequence of finite Schreier graphs associated with the action of the Basilica group on the rooted binary tree. Moreover, we give a formula for the total distance of every vertex in the Basilica graphs, and we are able to make it explicit for some special vertices. We finally introduce the notions of asymptotic Wiener index and asymptotic total distance, which are compatible with that of convergence of the sequence of finite Basilica graphs to an infinite orbital limit graph in the Gromov–Hausdorff topology: the asymptotic values are explicitly computed.

Conservation Implications of Spatiotemporal Variation in the Terrestrial Ecology of Western Spadefoots

ABSTRACTConservation of species reliant on ephemeral resources can be especially challenging in the face of a changing climate. Western spadefoots (Spea hammondii) are small burrowing anurans that breed in ephemeral pools, but adults spend the majority of their lives underground in adjacent terrestrial habitat. Western spadefoots are of conservation concern throughout their range because of habitat loss, but little is known about the activity patterns and ecology of their terrestrial life stage. We conducted a radio‐telemetry study of adult western spadefoots at 2 sites in southern California, USA, from December 2018 to November 2019 to characterize their survival, behavior, and movements from breeding through aestivation to refine conservation and management for the species. Western spadefoot survival varied seasonally, with risk of mortality higher in the active season than during aestivation. The probability of movement between successive observations was higher during the winter and spring and when atmospheric moisture was high and soil water content at 10‐cm depth was low. The amount of rain between observations had the strongest effect on the probability of movement between observations; for every 20 mm of rainfall between observations, western spadefoots were 2.4 times more likely to move. When movements occurred, movement rates were highest when both relative humidity and soil water content at 10‐cm depth were high. The conditions under which western spadefoots were likely active on the surface, likely to have moved, and moved at the highest rates are conditions that reduce the risk of desiccation of surface‐active spadefoots. Western spadefoot home range areas varied between study sites and were mostly <1 ha, although 1 individual's home range area was >6 ha. Western spadefoots rapidly dispersed from the breeding pools, and asymptotic distances from the breeding pool were generally reached by June. The asymptotic distance from the breeding pool varied between sites, with the 95th percentile of the posterior predictive distribution reaching 486 m at 1 site and 187 m at the other. Western spadefoots did not select most habitat components disproportionately to their availability, but at Crystal Cove State Park, they avoided most evaluated vegetation types (graminoids, forbs, and shrubs). Spatial variation was evident in most evaluated western spadefoot behaviors; context‐dependent behavior suggests that site‐specific management is likely necessary for western spadefoots. Furthermore, comparison with an earlier study of western spadefoots at Crystal Cove State Park indicated substantial temporal variation in western spadefoot behavior. Therefore, basing management decisions on short‐term studies might fail to meet conservation objectives. Better understanding the influences of spatial context and climatic variation on western spadefoot behavior will improve conservation efforts for this species. © 2021 This article is a U.S. Government work and is in the public domain in the USA. The Journal of Wildlife Management published by Wiley Periodicals LLC on behalf of The Wildlife Society.

Assessing parameters for ring polymer molecular dynamics simulations at low temperatures: DH + H chemical reaction

Ring polymer molecular dynamics (RPMD) is an accurate method for calculating thermal chemical reaction rates. It has recently been discovered that low-temperature calculations are strongly affected by the simulation parameters. Here, for the thermally activated reaction DH + H → D + H2, we calculate the RPMD rate constants at T=50, 100, and 300 K and demonstrate that for T≥100 K the standard input parameters yield accurate results, but at low temperatures (e.g., 50 K) one must increase the asymptotic distance and force constant, and decrease the umbrella integration step.

The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal

Let K n ≔ { Q n : Q n ( z ) = ∑ k = 0 n a k z k , a k ∈ C , | a k | = 1 } . \begin{equation*} {\mathcal {K}}_n \coloneq \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \quad a_k \in {\mathbb {C}}\,, \quad |a_k| = 1 \right \}\,. \end{equation*} A sequence ( P n ) (P_n) of polynomials P n ∈ K n P_n \!\in \! {\mathcal {K}}_n is called ultraflat if ( n + 1 ) − 1 / 2 | P n ( e i t ) | (n + 1)^{-1/2}|P_n(e^{it})| converge to 1 1 uniformly in t ∈ R t \!\in \! {\mathbb {R}} . In this paper we prove that 1 2 π ∫ 0 2 π | ( P n − P n ∗ ) ( e i t ) | q d t ∼ 2 q Γ ( q + 1 2 ) Γ ( q 2 + 1 ) π n q / 2 \begin{equation*} \frac {1}{2\pi } \int _0^{2\pi }{\left | (P_n - P_n^*)(e^{it}) \right |^q \, dt} \sim \frac {{2}^q \Gamma \left (\frac {q+1}{2} \right )}{\Gamma \left (\frac q2 + 1 \right ) \sqrt {\pi }} \,\, n^{q/2} \end{equation*} for every ultraflat sequence ( P n ) (P_n) of polynomials P n ∈ K n P_n \in {\mathcal {K}}_n and for every q ∈ ( 0 , ∞ ) q \in (0,\infty ) , where P n ∗ P_n^* is the conjugate reciprocal polynomial associated with P n P_n , Γ \Gamma is the usual gamma function, and the ∼ \sim symbol means that the ratio of the left and right hand sides converges to 1 1 as n → ∞ n \rightarrow \infty . Another highlight of the paper states that 1 2 π ∫ 0 2 π | ( P n ′ − P n ∗ ′ ) ( e i t ) | 2 d t ∼ 2 n 3 3 \begin{equation*} \frac {1}{2\pi }\int _0^{2\pi }{\left | (P_n^\prime - P_n^{*\prime })(e^{it}) \right |^2 \, dt} \sim \frac {2n^3}{3} \end{equation*} for every ultraflat sequence ( P n ) (P_n) of polynomials P n ∈ K n P_n \in {\mathcal {K}}_n . We prove a few other new results and reprove some interesting old results as well.

Effect of distance-dependent dispersivity on density-driven flow in porous media

In this study, the effect of distance-dependent dispersion coefficients on density-driven flow is investigated. The linear asymptotic model, which assumes that dispersivities increase linearly with distance from the source of contamination and reach asymptotic values at a large asymptotic distance, is employed. An in-house numerical model is adapted to handle distance-dependent dispersion. The effect of asymptotic-dispersion on aquifer contamination is analyzed for two tests: (i) a seawater intrusion problem in a coastal aquifer and (ii) a leachate transport problem from a surface deposit site. Global Sensitivity Analysis (GSA) combined with the Polynomial Chaos Expansion (PCE) surrogate modeling is conducted to assess the influence of the dispersion coefficients on the contamination plume for both configurations.For the seawater intrusion problem, the results show that the length of the toe is mainly controlled by the asymptotic transverse dispersivity whereas the spread of the concentration is sensitive to the asymptotic longitudinal dispersivity and the asymptotic dispersivity distance. The latter is the most important parameter controlling the amount of salt which intrudes into the aquifer. For the leachate transport problem, the results show that the asymptotic longitudinal dispersivity coefficient does not affect the concentration distribution. The asymptotic dispersivity distance has a strong effect on the total amount of contaminant that enters the aquifer. This effect can be three times more important than the effect of the asymptotic transverse dispersivity. These findings are likely to be helpful for the investigation and management of density-driven flow problems.

Analytical study of D-dimensional fractional Klein–Gordon equation with a fractional vector plus a scalar potential

D-dimensional fractional Klein–Gordon equation with fractional vector and scalar potential has been studied. Both fractional potentials are taken as attractive Coulomb-type with different multiplicative parameters, namely v and s. Jumarie-type definitions for fractional calculus have been used. We have succeeded in achieving Whittaker-type classical differential equation in fractional mode for the required eigenfunction. Fractional Whittaker equation has been manipulated using the behaviour of the eigenfunction at asymptotic distance and origin. This manipulation delivers fractional-type confluent hypergeometric equation to solve. Power series method has been employed to do the task. All the obtained results agree with the existing results in literature when fractional parameter $$\alpha $$ is unity. Finally, we furnish numerical results with a few eigenfunction graphs for different spatial dimensions and fractional parameters.

Determination of the Scattering Amplitude in a Limited Area of Calculation

An algorithm based on direct discretization of a two-dimensional equation is proposed for numerically solving the problem of two-dimensional scattering. The asymptotic distance is reduced by presenting the solution as the sum of free waves. The usefulness of the developed algorithm is demonstrated by solving the problem of scattering on a nonspherical core.

The eventually distance minimizing rays in moduli spaces

The eventually distance minimizing ray (EDM ray) in moduli spaces of the Riemann surfaces of analytic finite type with 3g + n − 3 > 0 is studied, which was introduced by Farb and Masur [5]. The asymptotic distance of EDM rays in a moduli space and the distance of end points of EDM rays in the boundary of the moduli space in the augmented moduli space are discussed in this article. A relation between the asymptotic distance of EDM rays and the distance of their end points is established. It is proved also that the distance of end points of two EDM rays is equal to that of end points of two Strebel rays in the Teichmüller space of a covering Riemann surface which are leftings of some representatives of the EDM rays. Meanwhile, simpler proofs for some known results are given.

On the energy landscape of spherical spin glasses

We investigate the energy landscape of the spherical mixed even p-spin model near its maximum energy. We relate the distance between pairs of near maxima to the support of the Parisi measure at zero temperature. We then provide an algebraic relation that characterizes one-step replica symmetric breaking Parisi measures. For these measures, we show that any two nonparallel spin configurations around the maximum energy are asymptotically orthogonal to each other. In sharp contrast, we study models with full replica symmetry breaking and show that all possible values of the asymptotic distance are attained near the maximum energy.

Asymptotic separation between solutions of Caputo fractional stochastic differential equations

ABSTRACTUsing a temporally weighted norm, we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show that the asymptotic distance between two distinct solutions is greater than as t → ∞ for any ϵ > 0. As a consequence, the mean square Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear Caputo fractional stochastic differential equation is always non-negative.

Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family

Let Φ be a quasi-periodically forced quadratic map, where the rotation constant ω is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under Φ) attracting graph of a nowhere continuous measurable function ψ from the circle to .This paper investigates how a smooth attractor degenerates into a strange one, as a parameter approaches a critical value , and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attractor collides with a repeller.Our results show that the asymptotic minimum distance between the two colliding invariant curves decreases linearly in the parameter , as approaches the critical parameter value from below.Furthermore, we have been able to show that the asymptotic growth of the supremum of the derivative of the attracting graph is asymptotically bounded from both sides by a constant times the reciprocal of the square root of the minimum distance above.

Fast construction of the exchange operator in an atom-centred basis with concentric atomic density fitting

ABSTRACTAn algorithm is presented for computing the Hartree–Fock exchange matrix using concentric atomic density fitting with data and instruction count complexities. The algorithm exploits the asymptotic distance dependence of the three-centre Coulomb integrals along with the rapid decay of the density matrix to accelerate the construction of the exchange matrix. The new algorithm is tested with computations on systems with up to 1536 atoms and a quadruple-zeta basis set (up to 15585 basis functions). Our method handles screening of high angular momentum contributions in a particularly efficient manner, allowing the use of larger basis sets for large molecules without a prohibitive increase in cost.

Long-range interactions of hydrogen atoms in excited states. I.2S−1Sinteractions and Dirac-δperturbations

The theory of the long-range interaction of metastable excited atomic states with ground-state atoms is analyzed. We show that the long-range interaction is essentially modified when quasi-degenerate states are available for virtual transitions. A discrepancy in the literature regarding the van der Waals coefficient C_6(2S;1S) describing the interaction of metastable atomic hydrogen (2S state) with a ground-state hydrogen atom is resolved. In the the van der Waals range a_0 << R << a_0/alpha, where a_0= hbar/(alpha m c) is the Bohr radius and alpha is the fine structure constant, one finds the symmetry-dependent result E_{2S;1S}(R) ~ (-176.75 +/- 27.98) E_h (a_0/R)^6 (E_h denotes the Hartree energy). In the Casimir-Polder range a_0/alpha << R << hbar c/L, where L = E(2S_{1/2})-E(2P_{1/2}) is the Lamb shift energy, one finds E_{2S;1S}(R) ~ (-121.50 +/- 46.61) E_h (a_0/R)^6. In the the Lamb shift range R >> hbar c/L, we find an oscillatory tail with a negligible interaction energy below 10^(-36) Hz. Dirac-delta perturbations to the interaction are also evaluated and results are given for all asymptotic distance ranges; these effects describe the hyperfine modification of the interaction, or, expressed differently, the shift of the hydrogen 2S hyperfine frequency due to interactions with neighboring 1S atoms. The 2S hyperfine frequency has recently been measured very accurately in atomic beam experiments.

The difference between two random mixed quantum states: exact and asymptotic spectral analysis

We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlson’s theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.

Long module skew codes are good

Module skew codes are one sided modules for (a quotient of) a skew polynomial ring where multiplication is twisted by an automorphism of the Galois group of the alphabet field. We prove that long module skew codes over a fixed finite field are asymptotically good by using a non-constructive counting argument. We show that for fixed alphabet size, and automorphism order and large length their asymptotic rate and relative distance satisfy a modified Varshamov–Gilbert bound.

Randomly Punctured LDPC Codes

In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binary-input additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results.

Casimir effect for a stack of conductive planes

The Casimir interaction in a stack of equally spaced infinitely thin layers is investigated within the zero-frequency mode summation method. The response properties are considered to be described by a constant conductivity or by a Drude-Lorentz model with a finite set of oscillators consistent with the optical characteristics for graphite. It is found that the asymptotic distance dependence is affected significantly by the specific response. While the energy is $\sim 1/d^3$ for the constant conductivity model, the energy exhibits fractional dependence $\sim 1/d^{5/2}$ for the Drude-Lorentz description. The Casimir force on a plane is also strongly dependent upon the particular plane location in the stack. Furthermore, the calculated Casimir energy within the Drude-Lorentz model yields results in good agreement with measured cohesion energy in graphite.