On upper semicontinuity of the Allen–Cahn twisted eigenvalues

2021 ◽  
pp. 1-12
Author(s):  
Krutika Tawri
Keyword(s):  
Minimal Surface ◽  
Critical Points ◽  
Upper Bound ◽  
Jacobi Operator ◽  
Upper Semicontinuity ◽  
Asymptotic Limit ◽  
Local Minimizers ◽  
Zero Sets ◽  
Asymptotic Upper Bound

We give an asymptotic upper bound for the kth twisted eigenvalue of the linearized Allen–Cahn operator in terms of the kth eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic limit of the zero sets of the Allen–Cahn critical points. We use an argument based on the notion of second inner variation developed in Le (On the second inner variations of Allen–Cahn type energies and applications to local minimizers. J. Math. Pures Appl. (9) 103 (2015) 1317–1345).

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

scholarly journals Structure and Stability of Filamentary Clouds Supported by Lateral Magnetic Field

2015 ◽  
Vol 11 (S315) ◽  
Author(s):  
Tomoyuki Hanawa ◽  
Kohji Tomisakar
Keyword(s):  
Magnetic Field ◽  
Magnetic Flux ◽  
Upper Bound ◽  
Unit Length ◽  
Flux Ratio ◽  
Critical Value ◽  
Asymptotic Limit ◽  
Analytical Models ◽  
Line Density ◽  
Magnetic Tension

AbstractWe have constructed two types of analytical models for an isothermal filamentary cloud supported mainly by magnetic tension. The first one describes an isolated cloud while the second considers filamentary clouds spaced periodically. The filamentary clouds are assumed to be highly flattened in both the models. The former is proved to be the asymptotic limit of the latter in which each filamentary cloud is much thinner than the distance to the neighboring filaments. These models show that the mass to flux ratio is crucial for the magnetohydrodynamical equilibrium. The upper bound for the line density, i.e., the mass per unit length, is proportional to the magnetic flux. The mass to flux ratio is slightly larger than the critical value, ($2 \pi \sqrt{G}$)−1, in the first model and lower in the second model. The first model is unstable against fragmentation and the wavelength of the fastest growing mode is several times longer than the cloud diameter. The second model is likely to be unstable only when the mass to flux ratio is supercritical.

scholarly journals Rigidity and stability of Einstein metrics for quadratic curvature functionals

2015 ◽  
Vol 2015 (700) ◽  
Author(s):  
Matthew J. Gursky ◽  
Jeff A. Viaclovsky
Keyword(s):  
Critical Points ◽  
Moduli Space ◽  
Einstein Metrics ◽  
Riemannian Metrics ◽  
Local Minimizers ◽  
Lagrange Equations ◽  
Critical Metrics ◽  
Local Properties ◽  
Space Of Riemannian Metrics ◽  
Curvature Functionals

AbstractWe investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to “gauge” the Euler–Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffeomorphism and scaling). A corollary is a local “reverse Bishop's inequality” for such metrics. In particular, any metric

Convergence to stationarity in the Moran model

2000 ◽  
Vol 37 (03) ◽  
pp. 705-717 ◽  
Author(s):  
Peter Donnelly ◽  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Upper Bound ◽  
Separation Distance ◽  
Fixed Size ◽  
Moran Model ◽  
Variation Distance ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
Asymptotic Upper Bound

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A 1 and A 2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t)) t≥0 which counts the number of individuals of type A 1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t ∗ = Nγ-1logN + cN the separation distance between the law of X(t ∗) and its stationary distribution converges to 1 - exp(-γe-γc ) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

Asymptotic upper bound for multiplier design

1996 ◽  
Vol 32 (5) ◽  
pp. 420 ◽  
Author(s):  
R.G.E. Pinch
Keyword(s):  
Upper Bound ◽  
Asymptotic Upper Bound

Convergence to stationarity in the Moran model

2000 ◽  
Vol 37 (3) ◽  
pp. 705-717 ◽  
Author(s):  
Peter Donnelly ◽  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Upper Bound ◽  
Separation Distance ◽  
Fixed Size ◽  
Moran Model ◽  
Variation Distance ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
Asymptotic Upper Bound

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A1 and A2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t))t≥0 which counts the number of individuals of type A1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t∗ = Nγ-1logN + cN the separation distance between the law of X(t∗) and its stationary distribution converges to 1 - exp(-γe-γc) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

Sign in / Sign up

Export Citation Format

Share Document