Статті в журналах: "Multiplicatif gaussien"

1

Lemańczyk, M. "Multiplicative Gaussian cocycles." Aequationes Mathematicae 61, no. 1-2 (February 1, 2001): 162–78. http://dx.doi.org/10.1007/s000100050168.

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2

Robert, Raoul, and Vincent Vargas. "Gaussian multiplicative chaos revisited." Annals of Probability 38, no. 2 (March 2010): 605–31. http://dx.doi.org/10.1214/09-aop490.

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3

Shamov, Alexander. "On Gaussian multiplicative chaos." Journal of Functional Analysis 270, no. 9 (May 2016): 3224–61. http://dx.doi.org/10.1016/j.jfa.2016.03.001.

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4

Lacoin, Hubert, Rémi Rhodes, and Vincent Vargas. "Complex Gaussian Multiplicative Chaos." Communications in Mathematical Physics 337, no. 2 (April 22, 2015): 569–632. http://dx.doi.org/10.1007/s00220-015-2362-4.

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5

KODOGIANNIS, VASSILIS S., MAHDI AMINA, and ILIAS PETROUNIAS. "A CLUSTERING-BASED FUZZY WAVELET NEURAL NETWORK MODEL FOR SHORT-TERM LOAD FORECASTING." International Journal of Neural Systems 23, no. 05 (August 7, 2013): 1350024. http://dx.doi.org/10.1142/s012906571350024x.

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Load forecasting is a critical element of power system operation, involving prediction of the future level of demand to serve as the basis for supply and demand planning. This paper presents the development of a novel clustering-based fuzzy wavelet neural network (CB-FWNN) model and validates its prediction on the short-term electric load forecasting of the Power System of the Greek Island of Crete. The proposed model is obtained from the traditional Takagi–Sugeno–Kang fuzzy system by replacing the THEN part of fuzzy rules with a "multiplication" wavelet neural network (MWNN). Multidimensional Gaussian type of activation functions have been used in the IF part of the fuzzyrules. A Fuzzy Subtractive Clustering scheme is employed as a pre-processing technique to find out the initial set and adequate number of clusters and ultimately the number of multiplication nodes in MWNN, while Gaussian Mixture Models with the Expectation Maximization algorithm are utilized for the definition of the multidimensional Gaussians. The results corresponding to the minimum and maximum power load indicate that the proposed load forecasting model provides significantly accurate forecasts, compared to conventional neural networks models.

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6

Dahab, R., D. Hankerson, F. Hu, M. Long, J. Lopez, and A. Menezes. "Software multiplication using Gaussian normal bases." IEEE Transactions on Computers 55, no. 8 (August 2006): 974–84. http://dx.doi.org/10.1109/tc.2006.132.

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7

Barral, Julien, Xiong Jin, Rémi Rhodes, and Vincent Vargas. "Gaussian Multiplicative Chaos and KPZ Duality." Communications in Mathematical Physics 323, no. 2 (August 3, 2013): 451–85. http://dx.doi.org/10.1007/s00220-013-1769-z.

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8

Safieh, Malek, and Jürgen Freudenberger. "Montgomery Reduction for Gaussian Integers." Cryptography 5, no. 1 (February 1, 2021): 6. http://dx.doi.org/10.3390/cryptography5010006.

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Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

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9

Wang, Kang-Kang, Hui Ye, Ya-Jun Wang, and Ping-Xin Wang. "Impact of Time Delay and Non-Gaussian Noise on Stochastic Resonance and Stability for a Stochastic Metapopulation System Driven by a Multiplicative Periodic Signal." Fluctuation and Noise Letters 18, no. 03 (July 16, 2019): 1950017. http://dx.doi.org/10.1142/s0219477519500172.

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In the present paper, the stability of the population system and the phenomena of the stochastic resonance (SR) for a metapopulation system induced by the terms of time delay, the multiplicative non-Gaussian noise, the additive colored Gaussian noise and a multiplicative periodic signal are investigated in detail. By applying the fast descent method, the unified colored noise approximation and the SR theory, the expressions of the steady-state probability function and the SNR are derived. It is shown that multiplicative non-Gaussian noise, the additive Gaussian noise and time delay can all weaken the stability of the population system, and even result in population extinction. Conversely, the two noise correlation times can both strengthen the stability of the biological system and contribute to group survival. In regard to the SNR for the metapopulation system impacted by the noise terms and time delay, it is revealed that the correlation time of the multiplicative noise can improve effectively the SR effect, while time delay would all along restrain the SR phenomena. On the other hand, although the additive noise and its correlation time can stimulate easily the SR effect, they cannot change the maximum of the SNR. In addition, the departure parameter from the Gaussian noise and the multiplicative noise play the opposite roles in motivating the SR effect in different cases.

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10

Guo, Yong-Feng, Ya-Jun Shen, Bei Xi, and Jian-Guo Tan. "Colored correlated multiplicative and additive Gaussian colored noises-induced transition of a piecewise nonlinear bistable model." Modern Physics Letters B 31, no. 28 (October 10, 2017): 1750256. http://dx.doi.org/10.1142/s0217984917502566.

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In this paper, we investigate the steady-state properties of a piecewise nonlinear bistable model driven by multiplicative and additive Gaussian colored noises with colored cross-correlation. Using the unified colored noise approximation, we derive the analytical expression of the steady-state probability density (SPD) function. Then the effects of colored correlated Gaussian colored noises on SPD are presented. According to the research results, it is found that there appear some new nonlinear phenomena in this system. The multiplicative colored noise intensity, the additive colored noise intensity and the cross-correlation strength between noises can induce the transition. However, the transition cannot be induced by the auto-correlation time of multiplicative and additive Gaussian colored noises as well as the cross-correlation time between noises.

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11

Wang, Kang-Kang, Hui Ye, Ya-Jun Wang, and Ping-Xin Wang. "Time delay and non-Gaussian noise-induced stochastic stability and stochastic resonance for a metapopulation system subjected to a multiplicative periodic signal." Modern Physics Letters B 32, no. 27 (September 27, 2018): 1850327. http://dx.doi.org/10.1142/s021798491850327x.

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In this paper, the stable state transformation and the effect of the stochastic resonance (SR) for a metapopulation system are investigated, which is disturbed by time delay, the multiplicative non-Gaussian noise, the additive colored Gaussian noise and a multiplicative periodic signal. By use of the fast descent method, the approximation of the unified colored noise and the SR theory, the dynamical behaviors for the steady-state probability function and the SNR are analyzed. It is found that non-Gaussian noise, the colored Gaussian noise and time delay can all reduce the stability of the biological system, and even lead to the population extinction. Inversely, the self-correlation times of two noises can both increase the stability of the population system and be in favor of the population reproduction. As regards the SNR for the metapopulation system induced by the noise terms and time delay, it is discovered that time delay and the correlation time of the multiplicative noise can effectively enhance the SR effect, while the multiplicative noise and the correlation time of the additive noise would all the time suppress the SR phenomena. In addition, the additive noise can effectively motivate the SR effect, but not alter the peak value of the SNR. It is worth noting that the departure parameter from the Gaussian noise plays the diametrical roles in stimulating the SR effect in different cases.

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12

Yu, Xingkai, Gumin Jin, and Jianxun Li. "Target Tracking Algorithm for System With Gaussian/Non-Gaussian Multiplicative Noise." IEEE Transactions on Vehicular Technology 69, no. 1 (January 2020): 90–100. http://dx.doi.org/10.1109/tvt.2019.2952368.

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13

Rhodes, Rémi, and Vincent Vargas. "Gaussian multiplicative chaos and applications: A review." Probability Surveys 11 (2014): 315–92. http://dx.doi.org/10.1214/13-ps218.

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14

Jego, Antoine. "Planar Brownian motion and Gaussian multiplicative chaos." Annals of Probability 48, no. 4 (July 2020): 1597–643. http://dx.doi.org/10.1214/19-aop1399.

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15

Barzykin, A. V., and K. Seki. "Stochastic resonance driven by Gaussian multiplicative noise." Europhysics Letters (EPL) 40, no. 2 (October 15, 1997): 117–22. http://dx.doi.org/10.1209/epl/i1997-00433-3.

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16

Chevillard, Laurent, Rémi Rhodes, and Vincent Vargas. "Gaussian Multiplicative Chaos for Symmetric Isotropic Matrices." Journal of Statistical Physics 150, no. 4 (February 2013): 678–703. http://dx.doi.org/10.1007/s10955-013-0697-9.

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17

Kátai, I., and M. Amer. "Multiplicative functions over the Gaussian integers. II." Acta Mathematica Hungarica 48, no. 3-4 (September 1986): 361–69. http://dx.doi.org/10.1007/bf01951363.

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18

Kátai, I., and M. Amer. "Multiplicative functions over the Gaussian integers. III." Acta Mathematica Hungarica 55, no. 3-4 (September 1990): 315–22. http://dx.doi.org/10.1007/bf01950940.

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19

Wong, Mo Dick. "Universal tail profile of Gaussian multiplicative chaos." Probability Theory and Related Fields 177, no. 3-4 (February 7, 2020): 711–46. http://dx.doi.org/10.1007/s00440-020-00960-3.

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20

Berestycki, Nathanaël, Christian Webb, and Mo Dick Wong. "Random Hermitian matrices and Gaussian multiplicative chaos." Probability Theory and Related Fields 172, no. 1-2 (November 6, 2017): 103–89. http://dx.doi.org/10.1007/s00440-017-0806-9.

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21

Newman, Charles M., and Wei Wu. "Lee–Yang Property and Gaussian Multiplicative Chaos." Communications in Mathematical Physics 369, no. 1 (May 16, 2019): 153–70. http://dx.doi.org/10.1007/s00220-019-03453-0.

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22

Jego, Antoine. "Critical Brownian multiplicative chaos." Probability Theory and Related Fields 180, no. 1-2 (April 9, 2021): 495–552. http://dx.doi.org/10.1007/s00440-021-01051-7.

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Анотація:

AbstractBrownian multiplicative chaos measures, introduced in Jego (Ann Probab 48:1597–1643, 2020), Aïdékon et al. (Ann Probab 48(4):1785–1825, 2020) and Bass et al. (Ann Probab 22:566–625, 1994), are random Borel measures that can be formally defined by exponentiating $$\gamma $$ γ times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter $$\gamma $$ γ is less than 2 were studied. This article considers the critical case where $$\gamma =2$$ γ = 2 , using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta–Heyde normalisation as well as in the derivative martingale normalisation. On the other hand, we construct the critical measure as a limit of subcritical measures. This is the first example of a non-Gaussian critical multiplicative chaos. We are inspired by methods coming from critical Gaussian multiplicative chaos, but there are essential differences, the main one being the lack of Gaussianity which prevents the use of Kahane’s inequality and hence a priori controls. Instead, a continuity lemma is proved which makes it possible to use tools from stochastic calculus as an effective substitute.

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23

Junnila, Janne. "On the Multiplicative Chaos of Non-Gaussian Log-Correlated Fields." International Mathematics Research Notices 2020, no. 19 (August 23, 2018): 6169–96. http://dx.doi.org/10.1093/imrn/rny196.

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Abstract We study non-Gaussian log-correlated multiplicative chaos, where the random field is defined as a sum of independent fields that satisfy suitable moment and regularity conditions. The convergence, existence of moments, and analyticity with respect to the inverse temperature are proven for the resulting chaos in the full subcritical range. These results are generalizations of the corresponding theorems for Gaussian multiplicative chaos. A basic example where our results apply is the non-Gaussian Fourier series $$\sum_{k=1}^\infty \frac{1}{\sqrt{k}}(A_k \cos(2\pi k x) + B_k \sin(2\pi k x)),$$where $A_k$ and $B_k$ are i.i.d. random variables.

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24

Alpay, Daniel, Palle Jorgensen, and Motke Porat. "White noise space analysis and multiplicative change of measures." Journal of Mathematical Physics 63, no. 4 (April 1, 2022): 042102. http://dx.doi.org/10.1063/5.0042756.

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In this paper, we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogs of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations.

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25

Reyhani-Masoleh, Arash, Hayssam El-Razouk, and Amin Monfared. "New Multiplicative Inverse Architectures Using Gaussian Normal Basis." IEEE Transactions on Computers 68, no. 7 (July 1, 2019): 991–1006. http://dx.doi.org/10.1109/tc.2018.2859941.

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26

Forkel, Johannes, and Jonathan P. Keating. "The classical compact groups and Gaussian multiplicative chaos." Nonlinearity 34, no. 9 (July 22, 2021): 6050–119. http://dx.doi.org/10.1088/1361-6544/ac1164.

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27

Wang, Hui, Athanasios Tsiairis, and Jinqiao Duan. "Bifurcation in Mean Phase Portraits for Stochastic Dynamical Systems with Multiplicative Gaussian Noise." International Journal of Bifurcation and Chaos 30, no. 11 (September 15, 2020): 2050216. http://dx.doi.org/10.1142/s0218127420502168.

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We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e. saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts.

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28

Safieh, Malek, Johann-Philipp Thiers, and Jürgen Freudenberger. "A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers." Electronics 9, no. 12 (December 2, 2020): 2050. http://dx.doi.org/10.3390/electronics9122050.

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This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.

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29

Guo, Yong-Feng, Bei Xi, Fang Wei, and Jian-Guo Tan. "The mean first-passage time in simplified FitzHugh–Nagumo neural model driven by correlated non-Gaussian noise and Gaussian noise." Modern Physics Letters B 32, no. 28 (October 4, 2018): 1850339. http://dx.doi.org/10.1142/s0217984918503396.

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In this paper, the mean first-passage time (MFPT) in simplified FitzHugh–Nagumo (FHN) neural model driven by correlated multiplicative non-Gaussian noise and additive Gaussian white noise is studied. Firstly, using the path integral approach and the unified colored-noise approximation (UCNA), the analytical expression of the stationary probability distribution (SPD) is derived, and the validity of the approximation method employed in the derivation is checked by performing numerical simulation. Secondly, the expression of the MFPT of the system is obtained by applying the definition and the steepest-descent method. Finally, the effects of the multiplicative noise intensity D, the additive noise intensity Q, the noise correlation time [Formula: see text], the cross-correlation strength [Formula: see text] and the non-Gaussian noise deviation parameter q on the MFPT are discussed.

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30

Mutothya, Nicholas Mwilu, Yong Xu, Yongge Li, Ralf Metzler, and Nicholas Muthama Mutua. "First passage dynamics of stochastic motion in heterogeneous media driven by correlated white Gaussian and coloured non-Gaussian noises." Journal of Physics: Complexity 2, no. 4 (November 23, 2021): 045012. http://dx.doi.org/10.1088/2632-072x/ac35b5.

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Abstract We study the first passage dynamics for a diffusing particle experiencing a spatially varying diffusion coefficient while driven by correlated additive Gaussian white noise and multiplicative coloured non-Gaussian noise. We consider three functional forms for position dependence of the diffusion coefficient: power-law, exponential, and logarithmic. The coloured non-Gaussian noise is distributed according to Tsallis’ q-distribution. Tracks of the non-Markovian systems are numerically simulated by using the fourth-order Runge–Kutta algorithm and the first passage times (FPTs) are recorded. The FPT density is determined along with the mean FPT (MFPT). Effects of the noise intensity and self-correlation of the multiplicative noise, the intensity of the additive noise, the cross-correlation strength, and the non-extensivity parameter on the MFPT are discussed.

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31

Sura, Philip, Matthew Newman, Cécile Penland, and Prashant Sardeshmukh. "Multiplicative Noise and Non-Gaussianity: A Paradigm for Atmospheric Regimes?" Journal of the Atmospheric Sciences 62, no. 5 (May 1, 2005): 1391–409. http://dx.doi.org/10.1175/jas3408.1.

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Abstract Atmospheric circulation statistics are not strictly Gaussian. Small bumps and other deviations from Gaussian probability distributions are often interpreted as implying the existence of distinct and persistent nonlinear circulation regimes associated with higher-than-average levels of predictability. In this paper it is shown that such deviations from Gaussianity can, however, also result from linear stochastically perturbed dynamics with multiplicative noise statistics. Such systems can be associated with much lower levels of predictability. Multiplicative noise is often identified with state-dependent variations of stochastic feedbacks from unresolved system components, and may be treated as stochastic perturbations of system parameters. It is shown that including such perturbations in the damping of large-scale linear Rossby waves can lead to deviations from Gaussianity very similar to those observed in the joint probability distribution of the first two principal components (PCs) of weekly averaged 750-hPa streamfunction data for the past 52 winters. A closer examination of the Fokker–Planck probability budget in the plane spanned by these two PCs shows that the observed deviations from Gaussianity can be modeled with multiplicative noise, and are unlikely the results of slow nonlinear interactions of the two PCs. It is concluded that the observed non-Gaussian probability distributions do not necessarily imply the existence of persistent nonlinear circulation regimes, and are consistent with those expected for a linear system perturbed by multiplicative noise.

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32

Peters, John M., and Sergey Kravtsov. "Origin of Non-Gaussian Regimes and Predictability in an Atmospheric Model." Journal of the Atmospheric Sciences 69, no. 8 (August 1, 2012): 2587–99. http://dx.doi.org/10.1175/jas-d-11-0316.1.

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Abstract This study details properties of non-Gaussian regimes and state-dependent ensemble spreads of trajectories in a reduced phase space of an idealized three-level quasigeostrophic (QG3) dynamical model. Methodologically, experiments using two empirical stochastic models of the QG3 time series disentangle the causes of state-dependent persistence properties and nonuniform self-forecast skill of the QG3 model. One reduced model is a standard linear inverse model (LIM) forced by state-independent, additive noise. This model has a linear deterministic operator resulting in a phase-space velocity field with uniform divergence. The other, more general nonlinear stochastic model (NSM) includes a nonlinear propagator and is driven by state-dependent, multiplicative noise. This NSM is found to capture well the full QG3 model trajectory behavior in the reduced phase space, including the non-Gaussian features of the QG3 probability density function and phase-space distribution of the trajectory spreading rates. Two versions of the NSM—one with a LIM-based drift tensor and QG3-derived multiplicative noise and another with the QG3-derived drift tensor and additive noise—allow the authors to determine relative contributions of the mean drift and multiplicative noise to non-Gaussian regimes and predictability in the QG3 model. In particular, while the regimes arise predominantly because of the nonlinear component of the mean phase-space tendencies, relative predictability of the regimes depends on both the phase-space structure of multiplicative noise and the degree of local convergence of mean phase-space tendencies.

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33

Duplantier, Bertrand, Rémi Rhodes, Scott Sheffield, and Vincent Vargas. "Critical Gaussian multiplicative chaos: Convergence of the derivative martingale." Annals of Probability 42, no. 5 (September 2014): 1769–808. http://dx.doi.org/10.1214/13-aop890.

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34

Duncan, T. E., and B. Pasik-Duncan. "DISTRIBUTED PARAMETER SYSTEMS WITH A MULTIPLICATIVE FRACTIONAL GAUSSIAN NOISE." IFAC Proceedings Volumes 38, no. 1 (2005): 35–38. http://dx.doi.org/10.3182/20050703-6-cz-1902.00866.

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35

Kátai, I., and M. Amer. "Multiplicative functions over the Gaussian integers with regularity properties." Acta Mathematica Hungarica 48, no. 1-2 (March 1986): 187–92. http://dx.doi.org/10.1007/bf01949063.

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36

Duplantier, Bertrand, Rémi Rhodes, Scott Sheffield, and Vincent Vargas. "Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation." Communications in Mathematical Physics 330, no. 1 (April 4, 2014): 283–330. http://dx.doi.org/10.1007/s00220-014-2000-6.

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37

Camia, Federico, Alberto Gandolfi, Giovanni Peccati, and Tulasi Ram Reddy. "Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos." Communications in Mathematical Physics 381, no. 3 (February 2021): 889–945. http://dx.doi.org/10.1007/s00220-020-03932-9.

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AbstractWe study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia et al. (Nucl Phys B 902:483–507, 2016), we take a (massless or massive) Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. For domains conformally equivalent to a disk, the sum diverges logarithmically due to the small loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener–Itô chaos expansion of linear functionals of vertex-like fields. Our methods apply to other variants of the model in which, for example, Brownian loops are replaced by disks.

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38

Fletcher, Steven J., and Andrew S. Jones. "Multiplicative and Additive Incremental Variational Data Assimilation for Mixed Lognormal–Gaussian Errors." Monthly Weather Review 142, no. 7 (June 27, 2014): 2521–44. http://dx.doi.org/10.1175/mwr-d-13-00136.1.

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Abstract An advance that made Gaussian-based three- and four-dimensional variational data assimilation (3D- and 4DVAR, respectively) operationally viable for numerical weather prediction was the introduction of the incremental formulation. This reduces the computational costs of the variational methods by searching for a small increment to a background state whose evolution is approximately linear. In this paper, incremental formulations for 3D- and 4DVAR with lognormal and mixed lognormal–Gaussian-distributed background and observation errors are presented. As the lognormal distribution has geometric properties, a geometric version for the tangent linear model (TLM) is proven that enables the linearization of the observational component of the cost functions with respect to a geometric increment. This is combined with the additive TLM for the mixed distribution–based cost function. Results using the mixed incremental scheme with the Lorenz’63 model are presented for different observational error variances, observation set sizes, and assimilation window lengths. It is shown that for sparse accurate observations the scheme has a relative error of ±0.5% for an assimilation window of 100 time steps. This improves to ±0.3% with more frequent observations. The distributions of the analysis errors are presented that appear to approximate a lognormal distribution with a mode at 1, which, given that the background and observational errors are unbiased in Gaussian space, shows that the scheme is approximating a mode and not a median. The mixed approach is also compared against a Gaussian-only incremental scheme where it is shown that as the z-component observational errors become more lognormal, the mixed approach appears to be more accurate than the Gaussian approach.

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39

Liu, Jinhua, Jiawen Huang, and Yuanyuan Huang. "Multiplicative Watermarking Method with the Visual Saliency Model Using Contourlet Transform." Security and Communication Networks 2021 (October 7, 2021): 1–12. http://dx.doi.org/10.1155/2021/1325573.

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We have proposed an image adaptive watermarking method by using contourlet transform. Firstly, we have selected high-energy image blocks as the watermark embedding space through segmenting the original image into nonoverlapping blocks and designed a watermark embedded strength factor by taking advantage of the human visual saliency model. To achieve dynamic adjustability of the multiplicative watermark embedding parameter, the relationship between watermark embedded strength factor and watermarked image quality is developed through experiments with the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM), respectively. Secondly, to detect the watermark information, the generalized Gaussian distribution (GGD) has been utilized to model the contourlet coefficients. Furthermore, positions of the blocks selected, watermark embedding factor, and watermark size have been used as side information for watermark decoding. Finally, several experiments have been conducted on eight images, and the results prove the effectiveness of the proposed watermarking approach. Concretely, our watermarking method has good imperceptibility and strong robustness when against Gaussian noise, JPEG compression, scaling, rotation, median filtering, and Gaussian filtering attack.

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40

Thanh, Nguyen Van. "Internal stabilization of stochastic 3D Navier–Stokes–Voigt equations with linearly multiplicative Gaussian noise." Random Operators and Stochastic Equations 27, no. 3 (September 1, 2019): 153–60. http://dx.doi.org/10.1515/rose-2019-2013.

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Abstract We show that the 3D stochastic Navier–Stokes–Voigt equations with linearly multiplicative Gaussian noise can be stabilized in probability by linear internal feedback controllers with support large enough.

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41

Osoba, Osonde, and Bart Kosko. "The Noisy Expectation-Maximization Algorithm for Multiplicative Noise Injection." Fluctuation and Noise Letters 15, no. 01 (March 2016): 1650007. http://dx.doi.org/10.1142/s0219477516500073.

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We generalize the noisy expectation-maximization (NEM) algorithm to allow arbitrary modes of noise injection besides just adding noise to the data. The noise must still satisfy a NEM positivity condition. This generalization includes the important special case of multiplicative noise injection. A generalized NEM theorem shows that all measurable modes of injecting noise will speed the average convergence of the EM algorithm if the noise satisfies a generalized NEM positivity condition. This noise-benefit condition has a simple quadratic form for Gaussian and Cauchy mixture models in the case of multiplicative noise injection. Simulations show a multiplicative-noise EM speed-up of more than [Formula: see text] in a simple Gaussian mixture model. Injecting blind noise only slowed convergence. A related theorem gives a sufficient condition for an average EM noise benefit for arbitrary modes of noise injection if the data model comes from the general exponential family of probability density functions. A final theorem shows that injected noise slows EM convergence on average if the NEM inequalities reverse and the noise satisfies a negativity condition.

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42

Wu, Weisan. "The Discrete Gaussian Expectation Maximization (Gradient) Algorithm for Differential Privacy." Computational Intelligence and Neuroscience 2021 (December 30, 2021): 1–13. http://dx.doi.org/10.1155/2021/7962489.

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In this paper, we give a modified gradient EM algorithm; it can protect the privacy of sensitive data by adding discrete Gaussian mechanism noise. Specifically, it makes the high-dimensional data easier to process mainly by scaling, truncating, noise multiplication, and smoothing steps on the data. Since the variance of discrete Gaussian is smaller than that of the continuous Gaussian, the difference privacy of data can be guaranteed more effectively by adding the noise of the discrete Gaussian mechanism. Finally, the standard gradient EM algorithm, clipped algorithm, and our algorithm (DG-EM) are compared with the GMM model. The experiments show that our algorithm can effectively protect high-dimensional sensitive data.

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43

Saha, Surajit, Suvajit Pal, Jayanta Ganguly, and Manas Ghosh. "Exploring Optical Dielectric Function of Impurity Doped Quantum Dots in Presence of Gaussian White Noise." Journal of Advanced Physics 6, no. 1 (March 1, 2017): 48–55. http://dx.doi.org/10.1166/jap.2017.1294.

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We investigate the total optical dielectric function (TODF) of impurity doped quantum dot (QD) in presence and absence of noise. Noise invoked is Gaussian white noise and the QD is doped with Gaussian impurity. Noise has been introduced to the system additively and multiplicatively. The TODF profiles have been monitored as a function of incident photon energy for different values of several important parameters. Moreover, the role of mode of application of noise (additive/multiplicative) on the TODF profiles has also been meticulously analyzed. We have found that the shift of TODF peak position and change in TODF peak height sensitively depend on presence/absence of noise and also on its mode of application. Introduction of multiplicative noise causes greater deviation of TODF profiles from that of noise-free condition than using additive noise.

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44

Shen, Feng, and Dingjie Xu. "Direct sequence code acquisition in multiplicative and non-Gaussian noises." International Journal of Electronics 96, no. 5 (May 2009): 479–89. http://dx.doi.org/10.1080/00207210802696134.

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45

Guotong Zhou and G. B. Giannakis. "Harmonics in Gaussian multiplicative and additive noise: Cramer-Rao bounds." IEEE Transactions on Signal Processing 43, no. 5 (May 1995): 1217–31. http://dx.doi.org/10.1109/78.382405.

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46

Yuren Zhou, Xinsheng Lai, Yuanxiang Li, and Wenyong Dong. "Ant Colony Optimization With Combining Gaussian Eliminations for Matrix Multiplication." IEEE Transactions on Cybernetics 43, no. 1 (February 2013): 347–57. http://dx.doi.org/10.1109/tsmcb.2012.2207717.

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47

Vitrenko, A. N. "Exactly solvable nonlinear model with two multiplicative Gaussian colored noises." Physica A: Statistical Mechanics and its Applications 359 (January 2006): 65–74. http://dx.doi.org/10.1016/j.physa.2005.04.036.

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48

Falconer, Kenneth, and Xiong Jin. "Exact dimensionality and projection properties of Gaussian multiplicative chaos measures." Transactions of the American Mathematical Society 372, no. 4 (May 23, 2019): 2921–57. http://dx.doi.org/10.1090/tran/7776.

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49

Remy, Guillaume, and Tunan Zhu. "The distribution of Gaussian multiplicative chaos on the unit interval." Annals of Probability 48, no. 2 (March 2020): 872–915. http://dx.doi.org/10.1214/19-aop1377.

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50

Barbu, Viorel, and Michael Röckner. "Nonlinear Fokker–Planck equations driven by Gaussian linear multiplicative noise." Journal of Differential Equations 265, no. 10 (November 2018): 4993–5030. http://dx.doi.org/10.1016/j.jde.2018.06.026.

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