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Journal articles on the topic 'Scaling asymptotics'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Paoletti, Roberto. "Local scaling asymptotics in phase space and time in Berezin–Toeplitz quantization." International Journal of Mathematics 25, no. 06 (June 2014): 1450060. http://dx.doi.org/10.1142/s0129167x14500608.

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This paper deals with the local semiclassical asymptotics of a quantum evolution operator in the Berezin–Toeplitz scheme, when both time and phase space variables are subject to appropriate scalings in the neighborhood of the graph of the underlying classical dynamics. Global consequences are then drawn regarding the scaling asymptotics of the trace of the quantum evolution as a function of time.

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2

Riley, N. "Scaling, self-similarity, and intermediate asymptotics." European Journal of Mechanics - B/Fluids 17, no. 3 (May 1998): 389–90. http://dx.doi.org/10.1016/s0997-7546(98)80266-5.

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3

PAOLETTI, ROBERTO. "SCALING ASYMPTOTICS FOR QUANTIZED HAMILTONIAN FLOWS." International Journal of Mathematics 23, no. 10 (October 2012): 1250102. http://dx.doi.org/10.1142/s0129167x12501029.

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In recent years, the near diagonal asymptotics of the equivariant components of the Szegö kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of this theme, here we consider the local scaling asymptotics of the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically how they concentrate on the graph of the underlying classical map.

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4

Hashorva, Enkelejd, and Anthony G. Pakes. "Tail asymptotics under beta random scaling." Journal of Mathematical Analysis and Applications 372, no. 2 (December 2010): 496–514. http://dx.doi.org/10.1016/j.jmaa.2010.07.045.

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5

Szpak, Nikodem. "Asymptotics from Scaling for Nonlinear Wave Equations." Communications in Partial Differential Equations 35, no. 10 (September 2010): 1876–90. http://dx.doi.org/10.1080/03605300903540935.

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6

VENEZIANO, DANIELE. "LARGE DEVIATIONS OF MULTIFRACTAL MEASURES." Fractals 10, no. 01 (March 2002): 117–29. http://dx.doi.org/10.1142/s0218348x02000872.

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We analyze the extremes of stationary multifractal measures using large deviation theory. We consider various cases involving discrete multiplicative cascades: scalar or vector cascades with dependent or independent generators, bare or dressed measures, and marginal (single-point) or joint (multi-point) extremes. In each case, we obtain the scaling behavior of the probability of large deviations as the resolution of the cascade diverges. Existing rough exponential limits for scalar cascades are confirmed, whereas for other cases our scaling relationships differ from previously published results. For scalar cascades, we refine the rough limits by obtaining the asymptotic pre-factor to the exponential term. Based on these refined asymptotics, we propose a variant to the Probability Distribution/Multiple Scaling (PDMS) technique to estimate the co-dimension function c(γ).

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7

PAOLETTI, ROBERTO. "LOCAL TRACE FORMULAE AND SCALING ASYMPTOTICS IN TOEPLITZ QUANTIZATION." International Journal of Geometric Methods in Modern Physics 07, no. 03 (May 2010): 379–403. http://dx.doi.org/10.1142/s021988781000435x.

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A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal.

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8

Ignat, Tatiana I. "Asymptotics for nonlocal evolution problems by scaling arguments." Differential Equations & Applications, no. 4 (2013): 613–26. http://dx.doi.org/10.7153/dea-05-36.

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9

Calka, Pierre, and J. E. Yukich. "Variance asymptotics and scaling limits for Gaussian polytopes." Probability Theory and Related Fields 163, no. 1-2 (November 14, 2014): 259–301. http://dx.doi.org/10.1007/s00440-014-0592-6.

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10

Calka, Pierre, and J. E. Yukich. "Variance asymptotics and scaling limits for random polytopes." Advances in Mathematics 304 (January 2017): 1–55. http://dx.doi.org/10.1016/j.aim.2016.08.006.

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11

Shen, Jianhong, and Gilbert Strang. "Asymptotics of Daubechies Filters, Scaling Functions, and Wavelets." Applied and Computational Harmonic Analysis 5, no. 3 (July 1998): 312–31. http://dx.doi.org/10.1006/acha.1997.0234.

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12

Shiffman, Bernard, Steve Zelditch, and Qi Zhong. "Random zeros on complex manifolds: conditional expectations." Journal of the Institute of Mathematics of Jussieu 10, no. 3 (March 11, 2011): 753–83. http://dx.doi.org/10.1017/s1474748011000041.

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AbstractWe study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros but we show that it has quite a different small distance behaviour. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension 1. To prove this, we give universal scaling asymptotics for around p. The key tool is the conditional Szegő kernel and its scaling asymptotics.

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13

Iksanov, Alexander, Alexander Marynych, and Matthias Meiners. "Asymptotics of random processes with immigration I: Scaling limits." Bernoulli 23, no. 2 (May 2017): 1233–78. http://dx.doi.org/10.3150/15-bej776.

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14

Barenblatt, G. I., and A. J. Chorin. "New Perspectives in Turbulence: Scaling Laws, Asymptotics, and Intermittency." SIAM Review 40, no. 2 (January 1998): 265–91. http://dx.doi.org/10.1137/s0036144597320047.

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15

Paoletti, Roberto. "Equivariant local scaling asymptotics for smoothed Töplitz spectral projectors." Journal of Functional Analysis 269, no. 7 (October 2015): 2254–301. http://dx.doi.org/10.1016/j.jfa.2015.03.007.

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16

Camosso, Simone. "Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions." Annali di Matematica Pura ed Applicata (1923 -) 195, no. 6 (February 1, 2016): 2027–59. http://dx.doi.org/10.1007/s10231-016-0552-0.

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17

Odnobokov, N. Yu. "Asymptotics of stationary measure under scaling in stochastic exchange processes." Moscow University Mathematics Bulletin 68, no. 1 (February 2013): 32–36. http://dx.doi.org/10.3103/s0027132213010063.

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18

Paoletti, Roberto. "A note on scaling asymptotics for Bohr-Sommerfeld Lagrangian submanifolds." Proceedings of the American Mathematical Society 136, no. 11 (June 2, 2008): 4011–17. http://dx.doi.org/10.1090/s0002-9939-08-09410-0.

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19

L'vov, Victor S., and Itamar Procaccia. "“Intermittency” in Hydrodynamic Turbulence as Intermediate Asymptotics to Kolmogorov Scaling." Physical Review Letters 74, no. 14 (April 3, 1995): 2690–93. http://dx.doi.org/10.1103/physrevlett.74.2690.

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20

Barenblatt, G. I. "Intermediate asymptotics, scaling laws and renormalization group in continuum mechanics." Meccanica 28, no. 3 (September 1993): 177–83. http://dx.doi.org/10.1007/bf00989119.

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21

HONDA, K. "SCALING THEORY ON GROWING ROUGH SURFACES." Fractals 04, no. 03 (September 1996): 331–37. http://dx.doi.org/10.1142/s0218348x96000443.

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We study the dynamics of growth processes of rough surfaces based on mathematical models such as the Kardar-Parisi-Zhang equation. The white-noise assumption in the KPZ equation is, however, noted to fail for higher dimensional cases. A careful continuum limit leads to a smooth surface solution for the cases. We develop the scaling theory to give an insight into the problem, by means of the intermediate asymptotics of the second kind, which is a very useful notion for the purpose. We find the roughness exponent and the dynamic exponent as functions of the substrate dimensionality for a model equation applicable to quenched disorder systems.

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22

NICODEMUS, ROLF, S. GROSSMANN, and M. HOLTHAUS. "The background flow method. Part 2. Asymptotic theory of dissipation bounds." Journal of Fluid Mechanics 363 (May 25, 1998): 301–23. http://dx.doi.org/10.1017/s0022112098001177.

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We study analytically the asymptotics of the upper bound on energy dissipation for the two-dimensional plane Couette flow considered numerically in Part 1 of this work, in order to identify the mechanisms underlying the variational approach. With the help of shape functions that specify the variational profiles either in the interior or in the boundary layers, it becomes possible to quantitatively explain all numerically observed features, from the occurrence of two branches of minimizing wavenumbers to the asymptotic parameter scaling with the Reynolds number. In addition, we derive a new variational principle for the asymptotic bound on the dissipation rate. The analysis of this principle reveals that the best possible bound can only be attained if the variational profiles allow the shape of the boundary layers to change with increasing Reynolds number.

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23

Trofimova, A. A., and A. M. Povolotsky. "Crossover scaling functions in the asymmetric avalanche process." Journal of Physics A: Mathematical and Theoretical 55, no. 2 (December 20, 2021): 025202. http://dx.doi.org/10.1088/1751-8121/ac3ebb.

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Abstract We consider the particle current in the asymmetric avalanche process on a ring. It is known to exhibit a transition from the intermittent to continuous flow at the critical density of particles. The exact expressions for the first two scaled cumulants of the particle current are obtained in the large time limit t → ∞ via the Bethe ansatz and a perturbative solution of the TQ-equation. The results are presented in an integral form suitable for the asymptotic analysis in the large system size limit N → ∞. In this limit the first cumulant, the average current per site or the average velocity of the associated interface, is asymptotically finite below the critical density and grows linearly and exponentially times power law prefactor at the critical density and above, respectively. The scaled second cumulant per site, i.e. the diffusion coefficient or the scaled variance of the associated interface height, shows the O(N −1/2) decay expected for models in the Kardar–Parisi–Zhang universality class below the critical density, while it is growing as O(N 3/2) and exponentially times power law prefactor at the critical point and above. Also, we identify the crossover regime and obtain the scaling functions for the uniform asymptotics unifying the three regimes. These functions are compared to the scaling functions describing crossover of the cumulants of the avalanche size, obtained as statistics of the first return area under the time space trajectory of the Vasicek random process.

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24

Paoletti, Roberto. "Local trace formulae and scaling asymptotics for general quantized Hamiltonian flows." Journal of Mathematical Physics 53, no. 2 (February 2012): 023501. http://dx.doi.org/10.1063/1.3679660.

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25

Hamm, Andreas. "The Influence of Noise on Fractals." Zeitschrift für Naturforschung A 49, no. 12 (December 1, 1994): 1238–40. http://dx.doi.org/10.1515/zna-1994-1224.

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Abstract The weak-noise asymptotics of the blurring effect of noise on fractals can be described by scaling laws. It does not only depend on the geometric properties of the fractals but also on their generating dynamics. This is illustrated with the example of the Feigenbaum attractor.

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26

Chamecki, Marcelo, Nelson L. Dias, Scott T. Salesky, and Ying Pan. "Scaling Laws for the Longitudinal Structure Function in the Atmospheric Surface Layer." Journal of the Atmospheric Sciences 74, no. 4 (April 1, 2017): 1127–47. http://dx.doi.org/10.1175/jas-d-16-0228.1.

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Abstract Scaling laws for the longitudinal structure function in the atmospheric surface layer (ASL) are studied using dimensional analysis and matched asymptotics. Theoretical predictions show that the logarithmic scaling for the scales larger than those of the inertial subrange recently proposed for neutral wall-bounded flows also holds for the shear-dominated ASL composed of weakly unstable, neutral, and all stable conditions (as long as continuous turbulence exists). A 2/3 power law is obtained for buoyancy-dominated ASLs. Data from the Advection Horizontal Array Turbulence Study (AHATS) field experiment confirm these scalings, and they also show that the length scale formed by the friction velocity and the turbulent kinetic energy dissipation rate consistently outperforms the distance from the ground z as the relevant scale in all cases regardless of stability. With this new length scale, the production range of the longitudinal structure function collapses for all measurement heights and stability conditions. A new variable to characterize atmospheric stability emerges from the theory: namely, the ratio between the buoyancy flux and the TKE dissipation rate.

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27

Abate, Joseph, and Ward Whitt. "Limits and Approximations for the Busy-Period Distribution in Single-Server Queues." Probability in the Engineering and Informational Sciences 9, no. 4 (October 1995): 581–602. http://dx.doi.org/10.1017/s0269964800004071.

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Limit theorems are established and relatively simple closed-form approximations are developed for the busy-period distribution in single-server queues. For the M/G/l queue, the complementary busy-period c.d.f. is shown to be asymptotically equivalent as t → ∞ to a scaled version of the heavy-traffic limit (obtained as p → 1), where the scaling parameters are based on the asymptotics as t → ∞. We call this the asymptotic normal approximation, because it involves the standard normal c.d.f. and density. The asymptotic normal approximation is asymptotically correct as t → ∞ for each fixed p and as p → 1 for each fixed t and yields remarkably good approximations for times not too small, whereas the direct heavy-traffic (p → 1) and asymptotic (t → ∞) limits do not yield such good approximations. Indeed, even the approximation based on three terms of the standard asymptotic expansion does not perform well unless t is very large. As a basis for generating corresponding approximations for the busy-period distribution in more general models, we also establish a more general heavy-traffic limit theorem.

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28

Duffield, N. G. "Economies of scale in queues with sources having power-law large deviation scalings." Journal of Applied Probability 33, no. 3 (September 1996): 840–57. http://dx.doi.org/10.2307/3215363.

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We analyse the queue QL at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[Wt/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions limL→xL–1 log P[QL > Lb] = – I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings v(t) = tv, a(t) = ta (such as occur in FBM) we analyse the asymptotics of the shape function limb→xb–u/a(I(b) – δbv/a) = vu for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[QL > Lb] ≈ exp[−δLbv/a] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone, and also perturbed by an Ornstein–Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.

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29

Duffield, N. G. "Economies of scale in queues with sources having power-law large deviation scalings." Journal of Applied Probability 33, no. 03 (September 1996): 840–57. http://dx.doi.org/10.1017/s0021900200100257.

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We analyse the queue QL at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[W t/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions lim L→x L –1 log P[QL > Lb] = – I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings v(t) = tv, a(t) = ta (such as occur in FBM) we analyse the asymptotics of the shape function lim b→x b –u/a (I(b) – δbv/a ) = vu for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[QL > Lb] ≈ exp[−δLbv/a ] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone, and also perturbed by an Ornstein–Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.

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30

Dębicki, Krzysztof, and Michel Mandjes. "Exact overflow asymptotics for queues with many Gaussian inputs." Journal of Applied Probability 40, no. 3 (September 2003): 704–20. http://dx.doi.org/10.1239/jap/1059060897.

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In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.

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31

Dębicki, Krzysztof, and Michel Mandjes. "Exact overflow asymptotics for queues with many Gaussian inputs." Journal of Applied Probability 40, no. 03 (September 2003): 704–20. http://dx.doi.org/10.1017/s0021900200019653.

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In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.

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32

Cirillo, Emilio N. M., Ida de Bonis, Adrian Muntean, and Omar Richardson. "Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstructure." Meccanica 55, no. 11 (October 14, 2020): 2159–78. http://dx.doi.org/10.1007/s11012-020-01253-8.

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Abstract We study the upscaling of a system of many interacting particles through a heterogenous thin elongated obstacle as modeled via a two-dimensional diffusion problem with a one-directional nonlinear convective drift. Assuming that the obstacle can be described well by a thin composite strip with periodically placed microstructures, we aim at deriving the upscaled model equations as well as the effective transport coefficients for suitable scalings in terms of both the inherent thickness at the strip and the typical length scales of the microscopic heterogeneities. Aiming at computable scenarios, we consider that the heterogeneity of the strip is made of an array of periodically arranged impenetrable solid rectangles and identify two scaling regimes what concerns the small asymptotics parameter for the upscaling procedure: the characteristic size of the microstructure is either significantly smaller than the thickness of the thin obstacle or it is of the same order of magnitude. We scale up the diffusion–polynomial drift model and list computable formulas for the effective diffusion and drift tensorial coefficients for both scaling regimes. Our upscaling procedure combines ideas of two-scale asymptotics homogenization with dimension reduction arguments. Consequences of these results for the construction of more general transmission boundary conditions are discussed. We illustrate numerically the concentration profile of the chemical species passing through the upscaled strip in the finite thickness regime and point out that trapping of concentration inside the strip is likely to occur in at least two conceptually different transport situations: (i) full diffusion/dispersion matrix and nonlinear horizontal drift, and (ii) diagonal diffusion matrix and oblique nonlinear drift.

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33

Canzani, Yaiza, and Boris Hanin. "$$C^\infty $$ C ∞ Scaling Asymptotics for the Spectral Projector of the Laplacian." Journal of Geometric Analysis 28, no. 1 (May 27, 2017): 111–22. http://dx.doi.org/10.1007/s12220-017-9812-5.

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34

Paoletti, Roberto. "Local Scaling Asymptotics for the Gutzwiller Trace Formula in Berezin–Toeplitz Quantization." Journal of Geometric Analysis 28, no. 2 (June 15, 2017): 1548–96. http://dx.doi.org/10.1007/s12220-017-9878-0.

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35

van der Hofstad, Remco, and Harsha Honnappa. "Large deviations of bivariate Gaussian extrema." Queueing Systems 93, no. 3-4 (October 15, 2019): 333–49. http://dx.doi.org/10.1007/s11134-019-09632-z.

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Abstract We establish sharp tail asymptotics for componentwise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different versus the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.

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36

Likhanov, Nikolay, and Ravi R. Mazumdar. "Cell loss asymptotics for buffers fed with a large number of independent stationary sources." Journal of Applied Probability 36, no. 1 (March 1999): 86–96. http://dx.doi.org/10.1239/jap/1032374231.

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In this paper we derive asymptotically exact expressions for buffer overflow probabilities and cell loss probabilities for a finite buffer which is fed by a large number of independent and stationary sources. The technique is based on scaling, measure change and local limit theorems and extends the recent results of Courcoubetis and Weber on buffer overflow asymptotics. We discuss the cases when the buffers are of the same order as the transmission bandwidth as well as the case of small buffers. Moreover we show that the results hold for a wide variety of traffic sources including ON/OFF sources with heavy-tailed distributed ON periods, which are typical candidates for so-called ‘self-similar’ inputs, showing that the asymptotic cell loss probability behaves in much the same manner for such sources as for the Markovian type of sources, which has important implications for statistical multiplexing. Numerical validation of the results against simulations are also reported.

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Likhanov, Nikolay, and Ravi R. Mazumdar. "Cell loss asymptotics for buffers fed with a large number of independent stationary sources." Journal of Applied Probability 36, no. 01 (March 1999): 86–96. http://dx.doi.org/10.1017/s0021900200016867.

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In this paper we derive asymptotically exact expressions for buffer overflow probabilities and cell loss probabilities for a finite buffer which is fed by a large number of independent and stationary sources. The technique is based on scaling, measure change and local limit theorems and extends the recent results of Courcoubetis and Weber on buffer overflow asymptotics. We discuss the cases when the buffers are of the same order as the transmission bandwidth as well as the case of small buffers. Moreover we show that the results hold for a wide variety of traffic sources including ON/OFF sources with heavy-tailed distributed ON periods, which are typical candidates for so-called ‘self-similar’ inputs, showing that the asymptotic cell loss probability behaves in much the same manner for such sources as for the Markovian type of sources, which has important implications for statistical multiplexing. Numerical validation of the results against simulations are also reported.

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38

Goddard, J. V., and S. Naire. "The spreading and stability of a surfactant-laden drop on an inclined prewetted substrate." Journal of Fluid Mechanics 772 (May 7, 2015): 535–68. http://dx.doi.org/10.1017/jfm.2015.212.

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We consider a viscous drop, loaded with an insoluble surfactant, spreading over an inclined plane that is covered initially with a thin surfactant-free liquid film. Lubrication theory is employed to model the flow using coupled nonlinear evolution equations for the film thickness and surfactant concentration. Exploiting high-resolution numerical simulations, we describe the late-time multi-region asymptotic structure of the spatially one-dimensional spreading flow. A simplified differential–algebraic equation model is derived for key variables characterising the spreading process, using which the late-time spreading and thinning rates are determined. Focusing on the neighbourhood of the drop’s leading-edge effective contact line, we then examine the stability of this region to small-amplitude disturbances with transverse variation. A dispersion relationship is described using long-wavelength asymptotics and numerical simulations, which reveals physical mechanisms and new scaling properties of the instability.

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39

Milišić, Vuk, and Christian Schmeiser. "Asymptotic limits for a nonlinear integro-differential equation modelling leukocytes’ rolling on arterial walls." Nonlinearity 35, no. 2 (December 16, 2021): 843–69. http://dx.doi.org/10.1088/1361-6544/ac3eb5.

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Abstract We consider a nonlinear integro-differential model describing z, the position of the cell center on the real line presented in Grec et al (2018 J. Theor. Biol. 452 35–46). We introduce a new ɛ-scaling and we prove rigorously the asymptotics when ɛ goes to zero. We show that this scaling characterizes the long-time behavior of the solutions of our problem in the cinematic regime (i.e. the velocity z ˙ tends to a limit). The convergence results are first given when ψ, the elastic energy associated to linkages, is convex and regular (the second order derivative of ψ is bounded). In the absence of blood flow, when ψ, is quadratic, we compute the final position z ∞ to which we prove that z tends. We then build a rigorous mathematical framework for ψ being convex but only Lipschitz. We extend convergence results with respect to ɛ to the case when ψ′ admits a finite number of jumps. In the last part, we show that in the constant force case [see model 3 in Grec et al (2018 J. Theor. Biol. 452 35–46), i.e. ψ is the absolute value)] we solve explicitly the problem and recover the above asymptotic results.

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40

DEHESA, J. S., A. MARTÍNEZ-FINKELSHTEIN, and V. N. SOROKIN. "SHORT-WAVE ASYMPTOTICS OF THE INFORMATION ENTROPY OF A CIRCULAR MEMBRANE." International Journal of Bifurcation and Chaos 12, no. 11 (November 2002): 2387–92. http://dx.doi.org/10.1142/s0218127402005935.

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The spreading of the position and momentum probability distributions for the stable free oscillations of a circular membrane of radius l is analyzed by means of the associated Boltzmann–Shannon information entropies in the correspondence principle limit (n → ∞, m fixed), where the numbers (n, m), n ∈ ℕ and m ∈ ℤ, uniquely characterize an oscillation of this two-dimensional system. This is done by solving the short-wave asymptotics of the physical entropies in the two complementary spaces, which boils down to the calculation of the asymptotic behavior of certain entropic integrals of Bessel functions. It is rigorously shown that the position and momentum entropies behave as 2 ln (l) + ln (4π) - 2 and ln (n) - 2 ln (l) + ln (2π3) when n → ∞, respectively. So the total entropy sum has a logarithmic dependence on n and it does not depend on the membrane radius. The former indicates that the ordering of short-wavelength oscillations is exactly identical for the entropic sum and the single-particle energy. The latter holds for all oscillations of the membrane because of the uniform scaling invariance of the entropy sum.

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41

Antonov, N. V., A. N. Vasil'ev, and A. S. Stepanenko. "Scaling function ??0 asymptotics of the correlation function in theO n ?4 model." Theoretical and Mathematical Physics 88, no. 1 (July 1991): 779–81. http://dx.doi.org/10.1007/bf01016349.

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42

Fong, Silas, and Vincent Tan. "Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel." Entropy 19, no. 7 (July 15, 2017): 364. http://dx.doi.org/10.3390/e19070364.

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43

Bakhtin, Yuri, and Andrzej Święch. "Scaling limits for conditional diffusion exit problems and asymptotics for nonlinear elliptic equations." Transactions of the American Mathematical Society 368, no. 9 (December 22, 2015): 6487–517. http://dx.doi.org/10.1090/tran/6574.

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44

Hnatich, M., and D. Horváth. "Modified Self-Scaling Relation for the Inertial and Low Energy Containing Scales of Decaying Turbulence." International Journal of Modern Physics B 12, no. 04 (February 10, 1998): 405–31. http://dx.doi.org/10.1142/s0217979298000272.

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The limits of a new form of scaling, named Extended Self Similarity (ESS) originally suggested [R. Benzi et al., Phys. Rev.E48 (1993), 29] for the inertial, dissipation and transition scales are discussed. A modification of the ESS concept is put forward using the model of decaying turbulence at high Reynolds numbers [L. Ts. Adzhemyan et al., Czech. J. Phys.45 (1995), 517]. In this model the statistical description is simplified by the hypotheses of homogeneity, isotropy, incompressibility and self-similarity, for the power law stage of decay the presence of a single scaling length — Karman scale — is assumed within the energy containing range. The second and third structure functions of the velocity field [S2(r) and S3(r)] have been calculated using the well-known connections between the mean energy spectrum and S2(r), and between mean spectral transfer and third structure function S3(r). Both structure functions have been investigated in the inertial and low enery containing ranges, then expressed in the form involving the leading Kolmogorov's K41 asymptotics [S2(r)∝ r2/3, S3(r)∝ r] and its asymptotical corrections. These corrections allow to determine corrections to the original ESS form [Formula: see text] (for K41) and to find out the modified variant of the ESS.

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45

DREHER, FABIAN, and MARC KESSEBÖHMER. "Escape rates for special flows and their higher order asymptotics." Ergodic Theory and Dynamical Systems 39, no. 06 (September 25, 2017): 1501–30. http://dx.doi.org/10.1017/etds.2017.76.

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In this paper escape rates and local escape rates for special flows are studied. In a general context the first result is that the escape rate depends monotonically on the ceiling function and fulfils certain scaling, invariance and continuity properties. In the context of metric spaces local escape rates are considered. If the base transformation is ergodic and exhibits an exponential convergence in probability of ergodic sums, then the local escape rate with respect to the flow is just the local escape rate with respect to the base transformation, divided by the integral of the ceiling function. Also, a reformulation with respect to induced pressure is presented. Finally, under additional regularity conditions higher order asymptotics for the local escape rate are established.

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46

Galaktionov, Victor A. "Critical global asymptotics in higher-order semilinear parabolic equations." International Journal of Mathematics and Mathematical Sciences 2003, no. 60 (2003): 3809–25. http://dx.doi.org/10.1155/s0161171203210176.

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We consider a higher-order semilinear parabolic equationut=−(−Δ)mu−g(x,u)inℝN×ℝ+,m>1. The nonlinear term is homogeneous:g(x,su)≡|s|p−1sg(x,u)andg(sx,u)≡|s|Qg(x,u)for anys∈ℝ, with exponentsP>1, andQ>−2m. We also assume thatgsatisfies necessary coercivity and monotonicity conditions for global existence of solutions with sufficiently small initial data. The equation is invariant under a group of scaling transformations. We show that there exists a critical exponentP=1+(2m+Q)/Nsuch that the asymptotic behavior ast→∞of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solutionb(x,t)=t−N/2mf(xt−1/2m)of the parabolic operator∂/∂t+(−Δ)m, so that fort≫1,u(x,t)=C0(ln t)−N/(2m+Q)[b(x,t)+o(1)], whereC0is a constant depending onm,N, andQonly.

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CHAVES, MANUELA, and VICTOR A. GALAKTIONOV. "LYAPUNOV FUNCTIONALS IN SINGULAR LIMITS FOR PERTURBED QUASILINEAR DEGENERATE PARABOLIC EQUATIONS." Analysis and Applications 01, no. 04 (October 2003): 351–85. http://dx.doi.org/10.1142/s0219530503000193.

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As a key example, we study the asymptotic behaviour near finite focusing time t=T of radial solutions of the porous medium equation with absorption [Formula: see text] with bounded compactly supported initial data u(x,0)=u0(|x|), and exponents m>1 and p>pc, where pc=pc(m,N)∈(-m,0) is a critical exponent. We show that under certain assumptions, the behaviour of the solution as t→T- near the origin is described by self-similar Graveleau solutions of the porous medium equation ut=Δum. In the rescaled variables, we deal with an exponential non-autonomous perturbation of a quasilinear parabolic equation, which is shown to admit an approximate Lyapunov functional. The result is optimal, and in the critical case p=pc an extra ln (T-t) scaling of the Graveleau asymptotics is shown to occur. Other types of self-similar and non self-similar focusing patterns are discussed.

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Laarhoven, Thijs. "Approximate Voronoi cells for lattices, revisited." Journal of Mathematical Cryptology 15, no. 1 (November 17, 2020): 60–71. http://dx.doi.org/10.1515/jmc-2020-0074.

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AbstractWe revisit the approximate Voronoi cells approach for solving the closest vector problem with preprocessing (CVPP) on high-dimensional lattices, and settle the open problem of Doulgerakis–Laarhoven–De Weger [PQCrypto, 2019] of determining exact asymptotics on the volume of these Voronoi cells under the Gaussian heuristic. As a result, we obtain improved upper bounds on the time complexity of the randomized iterative slicer when using less than $2^{0.076d + o(d)}$ memory, and we show how to obtain time–memory trade-offs even when using less than $2^{0.048d + o(d)}$ memory. We also settle the open problem of obtaining a continuous trade-off between the size of the advice and the query time complexity, as the time complexity with subexponential advice in our approach scales as $d^{d/2 + o(d)}$ matching worst-case enumeration bounds, and achieving the same asymptotic scaling as average-case enumeration algorithms for the closest vector problem.

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Yu, Jianduo, Chuanzhong Li, Mengkun Zhu, and Yang Chen. "Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalues." Journal of Mathematical Physics 63, no. 6 (June 1, 2022): 063504. http://dx.doi.org/10.1063/5.0062949.

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We discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight [Formula: see text]. Based on the ladder operator approach, two auxiliary quantities are defined. We show that the auxiliary quantities and the recurrence coefficients satisfy some equations with the aid of three compatibility conditions, which will be used to derive the Riccati equations and Painlevé III. We show that the Hankel determinant has an integral representation involving a particular σ-form of Painlevé III and to calculate the asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s = (2 n + 1 + λ) t is fixed, where λ is a parameter with λ ≔ ( α ∓ 1)/2. The asymptotic behaviors of the Hankel determinant for large s and small s are obtained, and Dyson’s constant is recovered here. They have generalized the results in the literature [Min et al., Nucl. Phys. B 936, 169–188 (2018)] where α = 0. By combining the Coulomb fluid method with the orthogonality principle, we obtain the asymptotic expansions of the recurrence coefficients, which are applied to derive the relationship between second order differential equations satisfied by our monic orthogonal polynomials and the double-confluent Heun equations as well as to calculate the smallest eigenvalue of the large Hankel matrices generated by the above weight. In particular, when α = t = 0, the asymptotic behavior of the smallest eigenvalue for the classical Gaussian weight exp(− z2) [Szegö, Trans. Am. Math. Soc. 40, 450–461 (1936)] is recovered.

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Mestel, B. D., and A. H. Osbaldestin. "Asymptotics of scaling parameters for period-doubling in unimodal maps with asymmetric critical points." Journal of Mathematical Physics 41, no. 7 (July 2000): 4732–46. http://dx.doi.org/10.1063/1.533398.

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