Exact Decay Rate Research Articles

An estimate from below for the Buffon needle probability of the four-corner Cantor set

Let $\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\K_n = \Cant_n \times \Cant_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\K_n$ is essentially the average length of the projections of $\K_n$, also known as the Favard length of $\K_n$. A classical theorem of Besicovitch implies that the Favard length of $\K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp(- c\log_* n)$, due to Peres and Solomyak. ($\log_* n$ is the number of times one needs to take log to obtain a number less than 1 starting from $n$). In Nazarov-Peres-Volberg paper (arxiv:math 0801.2942) the power estimate from above was obtained. The exponent in this paper was less than 1/6 but could have been slightly improved. On the other hand, a simple estimate shows that from below we have the estimate $\frac{c}{n}$. Here we apply the idea from papers of Nets Katz (MRL (1996), 527-536) and Bateman-Katz (arXiv:math/0609187v1 2006) to show that the estimate from below can be in fact improved to $c \frac{\log n}{n}$. This is in drastic difference from the case of {\em random} Cantor sets studied by Peres and Solomyak in Pacific J. Math. 204 (2002), 473-496.

Asymptotic expansion for the widths of resonances in Born–Oppenheimer approximation

We develop a microlocal study for scalar classical analytic pseudo-differential operators of principal type. Next, we reduce a system of pseudo-differential operators to the normal form. Then we obtain asymptotic expansion and the exact exponential decay rate for the width of resonances arising from microlocal tunneling effect.

Relations between invasion percolation and critical percolation in two dimensions

We study invasion percolation in two dimensions. We compare connectivity properties of the origin’s invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any k≥1, the k-point function of the first so-called pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint pc-open clusters. Further, for k>1, we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular.

New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory

There is a unique Lorentz-violating modification of the Maxwell theory of photons, which maintains gauge invariance, CPT, and renormalizability. Restricting the modified-Maxwell theory to the isotropic sector and adding a standard spin-one-half Dirac particle p^\pm with minimal coupling to the nonstandard photon \widetilde{\gamma}, the resulting modified-quantum-electrodynamics model involves a single dimensionless "deformation parameter," \widetilde{\kappa}_{tr}. The exact tree-level decay rates for two processes have been calculated: vacuum Cherenkov radiation p^\pm \to p^\pm \widetilde{\gamma} for the case of positive \widetilde{\kappa}_{tr} and photon decay \widetilde{\gamma} \to p^+ p^- for the case of negative \widetilde{\kappa}_{tr}. From the inferred absence of these decays for a particular high-quality ultrahigh-energy-cosmic-ray event detected at the Pierre Auger Observatory and an excess of TeV gamma-ray events observed by the High Energy Stereoscopic System telescopes, a two-sided bound on \widetilde{\kappa}_{tr} is obtained, which improves by eight orders of magnitude upon the best direct laboratory bound. The implications of this result are briefly discussed.

Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

This paper studies the geometric decay property of the joint queue-length distribution {p(n 1,n 2)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c 1,c 2,d 1 and d 2, an upper bound $\overline{\eta}(c_{1},c_{2})$ of the decay rate is derived in the sense $$\exp\Bigl\{\limsup_{n\rightarrow\infty}n^{-1}\log p(c_{1}n+d_{1},c_{2}n+d_{2})\Bigr\}\leq\overline{\eta}(c_{1},c_{2})<1.$$ It is shown that the upper bound coincides with the exact decay rate in most systems for which the exact decay rate is known. Moreover, as a function of c 1 and c 2, $\overline{\eta }(c_{1},c_{2})$ takes one of eight types, and the types explain some curious properties reported in Fujimoto and Takahashi (J. Oper. Res. Soc. Jpn. 39:525---540 [1996]).

Existence of solutions of the very fast diffusion equation in bounded and unbounded domain

We prove the existence of a unique solution of the following Neumann problem \(u_t = (\phi_m(u))_{xx}\), u > 0, in (a, b) × (0, T), u(x, 0) = u0(x) ≥ 0 in (a, b), \((\phi_m(u))_x(a,t) = g(t)\) and \((\phi_m(u))_x(b,t)=-f(t) \forall 0 0: \int_{-a}^bu_0dx > \int_0^{t'}(f+g)\,dt\}\)m≤ 0, \(0 \le f, g \in L_{loc}^{\infty}([0,\infty))\), \(0\le u_0\in L^{\infty}(a,b)\) and the case −1 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in \(\mathbb{R} \times (0,T)\) for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity.

On exact convergence rates for solutions of linear systems of Volterra difference equations†

The asymptotic behaviour of the solution of general linear Volterra non-convolution difference equations on a finite dimensional space, is investigated. It is proved under appropriate assumptions that the solution converges to a limit, which is in general non-trivial. These results are then used to obtain the exact rate of decay of solutions of a class of convolution Volterra difference equations, which have no characteristic roots. In particular, we obtain the exact rate of convergence of the solution of equations whose kernel does not converge exponentially. A useful formula for the weighted limit of a discrete convolution is also obtained.

Quantifying the Mass Balance of Ice Caps on Severnaya Zemlya, Russian High Arctic. III: Sensitivity of Ice Caps in Severnaya Zemlya to Future Climate Change

A coupled surface mass balance and ice-flow model was used to predict the response of three ice caps on Severnaya Zemlya, Russian Arctic, to the present climate and to future climate changes as postulated by the Intergovernmental Panel on Climate Change (IPCC). Ice cap boundary conditions are derived from recent airborne geophysical surveying (Dowdeswell et al., 2002), and model inputs are constructed from available climate data. Model results indicate that, currently, the state of balance of ice caps on Severnaya Zemlya is dependent on their size. For small ice caps, such as Pioneer Ice Cap (area 199 km2), mass balance is extremely negative. Under current climate conditions, these relatively small ice caps are predicted to disappear within ∼1000 years. For larger ice caps, however, such as the Academy of Sciences Ice Cap (area 5586 km2), the accumulation zone is much larger, which results in these ice caps being approximately in balance today, but still susceptible to decay in future climate scenarios. When climate conditions are changed in the model, as predicted by the IPCC, the mass balance of all ice caps in Severnaya Zemlya is predicted to become negative within a 100 years or so. Although it is difficult to say with certainty the exact rate of decay, it is likely that ice loss from Severnaya Zemlya will contribute, over a period of a few hundred years, a rise in sea level of the order of a few centimeters.

On exact rates of decay of solutions of linear systems of Volterra equations with delay

This paper considers the resolvent of a finite-dimensional linear convolution Volterra integral equation. The main results give conditions which ensure that the exact rate of decay of the resolvent can be determined using a positive weight function related to the kernel. The decay rates can be exponential or subexponential. Many other related results on exact rates of exponential and subexponential decay of solutions of Volterra integro-differential equations are given. We also present an application to a linear compartmental system with discrete and continuous lags.

Sample-path large deviations for tandem and priority queues with Gaussian inputs

This paper considers Gaussian flows multiplexed in a queueing network. A single node being a useful but often incomplete setting, we examine more advanced models. We focus on a (two-node) tandem queue, fed by a large number of Gaussian inputs. With service rates and buffer sizes at both nodes scaled appropriately, Schilder’s sample-path large-deviations theorem can be applied to calculate the asymptotics of the overflow probability of the second queue. More specifically, we derive a lower bound on the exponential decay rate of this overflow probability and present an explicit condition for the lower bound to match the exact decay rate. Examples show that this condition holds for a broad range of frequently used Gaussian inputs. The last part of the paper concentrates on a model for a single node, equipped with a priority scheduling policy. We show that the analysis of the tandem queue directly carries over to this priority queueing system.

Greybody factors of charged dilaton black holes in 2 + 1 dimensions

We have studied scalar perturbations of charged dilaton black holes in 2+1 dimensions. The black hole considered here is a solution to the low-energy string theory in 2+1 dimensions. The exact decay rates and the grey body factors for the massless minimally coupled scalar is computed for both the charged and the uncharged dilaton black holes. The charged and the uncharged black hole show similar behavior for grey body factors, reflection coefficients and decay rates.

UPPER BOUND FOR THE DECAY RATE OF THE MARGINAL QUEUE-LENGTH DISTRIBUTION IN A TWO-NODE MARKOVIAN QUEUEING SYSTEM(&lt;Special Issue&gt;Network Design, Control and Optimization)

We study a geometric decay property for two-node queueing networks, not restricted to ones having acyclic configuration. We take a matrix-analytic approach, and prove the geometric decay property of the marginal queue-length distributions by giving an upper bound of the exact decay rate for each node. The upper bound coincides with the exact decay rate for Jackson networks and MAP/M/1→/M/1 tandem queues.

End effects for pre-stressed and pre-polarized piezoelectric solids in anti-plane shear

This paper is concerned with investigation of the influence of pre-stress and pre-polarization on the decay of Saint-Venant end effects in piezoelectric solids. This issue is examined within the framework of incremental anti-plane shear states for prestressed and prepolarized solids. Attention is confined to a specific class of materials, namely those of hexagonal 6mm or 4mm symmetry. It is shown that the mechanical and electrical fields uncouple in this case, and the governing system of coupled second-order linear partial differential equations uncouple to a single homogeneous second-order equation with constant coefficients for the displacement and a Poisson equation for the electric potential. The exact decay rate for mechanical and electrical end effects is determined and the influence of pre-stress and pre-polarization is assessed. Motivation for this work arises from the current rapid developments in smart materials technology.

An Explicit Unconditionally Stable Numerical Method for Solving Damped Nonlinear Schrödinger Equations with a Focusing Nonlinearity

This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrodinger equations (NLSs). The method is explicit, unconditionally stable, and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing NLSs in two dimensions with a linear, cubic, or quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter ${\delta}$ is larger than a threshold value ${\delta}_{\rm th}$. We note that our method can also be applied to solve the three-dimensional Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC).

Delayed Nickel Decay in Gamma‐Ray Bursts

Recently observed emission lines in the X-ray afterglow of � -ray bursts suggest that iron-group elements are either produced in the � -ray burst or are present nearby. If this material is the product of a thermonuclear burn, then such material would be expected to be rich in nickel-56. If the nickel remains partially ionized, this prevents the electron capture reaction normally associated with the decay of nickel-56, dramatically increasing the decay timescale. Here we examine the consequences of rapid ejection of a fraction of a solar mass of iron-group material from the center of a collapsar/hypernova. The exact rate of decay then depends on the details of the ionization and, therefore, the ejection process. Future observations of iron, nickel, and cobalt lines can be used to diagnose the origin of these elements and to better understand the astrophysical site of � ray bursts. In this model, the X-ray lines of these iron-group elements could be detected in suspected hypernovae that did not produce an observable � -ray burst due to beaming. Subject headings: gamma rays: bursts — line: profiles — nuclear reactions, nucleosynthesis, abundances — supernovae: general

First-order chiral phase transition in high-energy collisions: Can nucleation prevent spinodal decomposition?

We discuss homogeneous nucleation in a first-order chiral phase transition within an effective field theory approach to low-energy QCD. Exact decay rates and bubble profiles are obtained numerically and compared to analytic results obtained with the thin-wall approximation. The thin-wall approximation overestimates the nucleation rate for any degree of supercooling. The time scale for critical thermal fluctuations is calculated and compared to typical expansion times for high-energy hadronic or heavy-ion collisions. We find that significant supercooling is possible, and the relevant mechanism for phase conversion might be that of spinodal decomposition. Some potential experimental signatures of supercooling, such as an increase in the correlation length of the scalar condensate, are also discussed.

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ϕ. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal \({\bar 4}\) crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d0,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d0. In the limit as d0→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal \({\bar 4}\) class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.

LOW TEMPERATURE PROPERTIES OF THE BLUME–EMERY–GRIFFITHS (BEG) MODEL IN THE REGION WITH AN INFINITE NUMBER OF GROUND STATE CONFIGURATIONS

For the low temperature Blume–Emery–Griffiths Zd, d ≥2, lattice model taking site spin values 0, +1, -1 we construct, using a polymer expansion, two pure states in the parameter region [Formula: see text] where there are an infinite number of configurations with minimal energy. Each state is invariant under translation by two lattice spacings and the two states are related by a unit translation. Using analyticity techniques we show that the truncated n-point function decays exponentially with an n-independent lower bound on the decay rate. For the truncated two-point function, we find the exact exponential decay rate in the limit β→∞.

On the decay rates of buffers in continuous flow lines

Consider a tandem system of machines separated by infinitely large buffers. The machines process a continuous flow of products, possibly at different speeds. The life and repair times of the machines are assumed to be exponential. We claim that the overflow probability of each buffer has an exponential decay, and provide an algorithm to determine the exact decay rates in terms of the speeds and the failure and repair rates of the machines. These decay rates provide useful qualitative insight into the behavior of the flow line. In the derivation of the algorithm we use the theory of Large Deviations.

Black hole emission rates and the AdS/CFT correspondence

We study the emission rates of scalar, spinor and vector particles from a 5 dimensional black hole for arbitrary partial waves. The solution is lifted to 6 dimensions, and the near horizon $ BTZ \times S^3$ geometry of the black hole solution is probed to determine the greybody factors. We show that the exact decay rates can be reproduced from a $(1+1)$-dimensional conformal field theory which lies on the boundary of the near horizon geometry. The AdS/CFT correspondence is used to determine the dimension of the CFT operators corresponding to the bulk fields. These operators couple to plane waves incident on the CFT from infinity to produce emission in the bulk.