Journal articles: 'Montel theorem'

1

Almira, Jose Maria, and László Székelyhidi. "Local polynomials and the Montel theorem." Aequationes mathematicae 89, no. 2 (October 9, 2014): 329–38. http://dx.doi.org/10.1007/s00010-014-0308-0.

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2

Bär, Christian. "Some Properties of Solutions to Weakly Hypoelliptic Equations." International Journal of Differential Equations 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/526390.

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A linear different operatorLis called weakly hypoelliptic if any local solutionuofLu=0is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and anyLp-solution must vanish.

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Cid, José Ángel, and Rodrigo López Pouso. "A generalization of Montel–Tonelli's Uniqueness Theorem." Journal of Mathematical Analysis and Applications 429, no. 2 (September 2015): 1173–77. http://dx.doi.org/10.1016/j.jmaa.2015.04.064.

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4

Cibulka, Radek, Marián Fabian, and Tomáš Roubal. "An Inverse Mapping Theorem in Fréchet-Montel Spaces." Set-Valued and Variational Analysis 28, no. 1 (February 17, 2020): 195–208. http://dx.doi.org/10.1007/s11228-020-00536-2.

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De Grande-De Kimpe, N., C. Perez-Garcia, and W. H. Schikhof. "Non-Archimedean t-Frames and FM-Spaces." Canadian Mathematical Bulletin 35, no. 4 (December 1, 1992): 475–83. http://dx.doi.org/10.4153/cmb-1992-062-4.

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AbstractWe generalize the notion of t-orthogonality in p-adic Banach spaces by introducing t-frames (§2). This we use to prove that a Fréchet-Montel (FM-)space is of countable type (Theorem 3.1), the non-archimedeancounterpart of a well known theorem in functional analysis over ℝ or ℂ ([6], p. 231). We obtain several characterizations of FM-spaces (Theorem 3.3) and characterize the nuclear spaces among them (§4).

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Almira, J. M., and Kh F. Abu-Helaiel. "A p-adic Montel Theorem and locally polynomial functions." Filomat 28, no. 1 (2014): 159–66. http://dx.doi.org/10.2298/fil1401159a.

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We prove a version of Montel?s Theorem for the case of continuous functions defined over the field Qp of p-adic numbers. In particular, we prove that, if ?m+1 h0 f (x) = 0 for all x ? Qp, and h0 satisfies |h0|p = p?N0, then, for all x0 ? Qp, the restriction of f over the set x0 + pN0Zp coincides with a polynomial px0 (x) = a0(x0) + a1(x0)x +...+ am(x0)xm. Motivated by this result, we compute the general solution of the functional equation with restrictions given by ?m+1 h f (x) = 0 (x ? X and h ? BX(r) = {x ? X : ?x? ? r}), whenever f : X ? Y, X is an ultrametric normed space over a non-Archimedean valued field (K, |?|) of characteristic zero, and Y is a Q-vector space. By obvious reasons, we call these functions uniformly locally polynomial.

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GAUSSIER, HERVÉ, and KANG-TAE KIM. "COMPACTNESS OF CERTAIN FAMILIES OF PSEUDO-HOLOMORPHIC MAPPINGS INTO ${\mathbb C}^n$." International Journal of Mathematics 15, no. 01 (February 2004): 1–12. http://dx.doi.org/10.1142/s0129167x04002168.

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We present a normal family theorem for injective almost holomorphic maps from a manifold with almost complex structures into [Formula: see text]. Our theorem implies a new consequence even for the holomorphic mappings of a complex manifold into [Formula: see text], which can be seen as a generalization of the convergence theorem for Frankel's scaling sequence whose images are not necessarily convex. Moreover, our method is closer in spirit to the circle of ideas centered around the classical Montel theorem.

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8

Bonet, José, and Mikael Lindström. "Spaces of operators between Fréchet spaces." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 1 (January 1994): 133–44. http://dx.doi.org/10.1017/s0305004100071978.

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AbstractMotivated by recent results on the space of compact operators between Banach spaces and by extensions of the Josefson–Nissenzweig theorem to Fréchet spaces, we investigate pairs of Fréchet spaces (E, F) such that every continuous linear map from E into F is Montel, i.e. it maps bounded subsets of E into relatively compact subsets of F. As a consequence of our results we characterize pairs of Köthe echelon spaces (E, F) such that the space of Montel operators from E into F is complemented in the space of all continuous linear maps from E into F.

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9

Lee, Junghun. "An alternative proof of the non-Archimedean Montel theorem for rational dynamics." Proceedings of the Japan Academy, Series A, Mathematical Sciences 92, no. 4 (April 2016): 56–58. http://dx.doi.org/10.3792/pjaa.92.56.

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10

Li, Bao Qin. "A joint theorem generalizing the criteria of Montel and Miranda for normal families." Proceedings of the American Mathematical Society 132, no. 9 (March 25, 2004): 2639–46. http://dx.doi.org/10.1090/s0002-9939-04-07452-0.

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11

BARGMANN, DETLEF. "Simple proofs of some fundamental properties of the Julia set." Ergodic Theory and Dynamical Systems 19, no. 3 (June 1999): 553–58. http://dx.doi.org/10.1017/s0143385799130153.

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Let $f$ be a holomorphic self-map of $\mathbb{C} \backslash \{ 0 \}, \mathbb{C}$, or the extended complex plane $\overline{\mathbb{C}}$ that is neither injective nor constant. This paper gives new and elementary proofs of the well-known fact that the Julia set of $f$ is a non-empty perfect set and coincides with the closure of the set of repelling cycles of $f$. The proofs use Montel–Caratheodory's theorem but do not use results from Nevanlinna theory.

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12

CHARAK, K. S., D. ROCHON, and N. SHARMA. "NORMAL FAMILIES OF BICOMPLEX HOLOMORPHIC FUNCTIONS." Fractals 17, no. 03 (September 2009): 257–68. http://dx.doi.org/10.1142/s0218348x09004314.

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In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice j = 0.

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13

Sharp, Bernice. "The differentiability of convex functions on topological linear spaces." Bulletin of the Australian Mathematical Society 42, no. 2 (October 1990): 201–13. http://dx.doi.org/10.1017/s0004972700028379.

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In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

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14

Sevost’yanov, E. A. "Analog of the Montel Theorem for Mappings of the Sobolev Class with Finite Distortion." Ukrainian Mathematical Journal 67, no. 6 (November 2015): 938–47. http://dx.doi.org/10.1007/s11253-015-1124-y.

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15

Bonet, Jose, and Antonio Galbis. "A note on Taskinen's counterexamples on the problem of topologies of Grothendieck." Proceedings of the Edinburgh Mathematical Society 32, no. 2 (June 1989): 281–83. http://dx.doi.org/10.1017/s0013091500028686.

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By the work of Taskinen (see [4, 5]), we know that there is a Fréchet space E such that Lb(E, l2) is not a (DF)-space. Moreover there is a Fréchet–Montel space F such that is not (DF). In this second example, the duality theorem of Buchwalter (cf. [2, §45.3]) can be applied to obtain that and hence is a (gDF)-space (cf. [1, Ch. 12 or 3, Ch. 8]). The (gDF)-spaces were introduced by several authors to extend the (DF)-spaces of Grothendieck and to provide an adequate frame to consider strict topologies.

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16

Earle, Clifford J. "A Montel Theorem for Holomorphic Functions on Infinite Dimensional Spaces that Omit the Values 0 and 1." Computational Methods and Function Theory 8, no. 1 (June 22, 2007): 195–98. http://dx.doi.org/10.1007/bf03321682.

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17

Leonenko, Nikolai, Enrico Scalas, and Mailan Trinh. "Limit theorems for the fractional nonhomogeneous Poisson process." Journal of Applied Probability 56, no. 01 (March 2019): 246–64. http://dx.doi.org/10.1017/jpr.2019.16.

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AbstractThe fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

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18

Ermakov, Sergey M., and Maxim G. Smilovitskiy. "Monte-Carlo for solving large linear systems of ordinary diﬀerential equations." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 8, no. 1 (2021): 37–48. http://dx.doi.org/10.21638/spbu01.2021.104.

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Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

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19

Aksoy, A. G., and J. M. Almira. "On Montel and Montel–Popoviciu theorems in several variables." Aequationes mathematicae 89, no. 5 (January 1, 2015): 1335–57. http://dx.doi.org/10.1007/s00010-014-0329-8.

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20

Reuss, Paul. "Cauchy’s theorem and generalization." EPJ Nuclear Sciences & Technologies 4 (2018): 50. http://dx.doi.org/10.1051/epjn/2018010.

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It has already been established that the mean length travelled by a neutral particle in a body containing a diffusing but not absorbing material is independant of its cross section, and consequently equal to the mean chord of the body. An elegant demonstration of this curious feature is presented and analysed thanks to Monte-Carlo simulations.

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21

Burns, Daniel, and Victor Guillemin. "The Tian–Yau–Zelditch Theorem and Toeplitz Operators." Journal of the Institute of Mathematics of Jussieu 10, no. 3 (May 5, 2011): 449–61. http://dx.doi.org/10.1017/s1474748011000016.

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AbstractZelditch's proof of the Tian–Yau–Zelditch Theorem makes use of the Boutet de Monvel–Sjöstrand results on the asymptotic properties of Szegö projectors for strictly pseudoconvex domains. However, as we will show below, the theorem is also closely related to another theorem of Boutet de Monvel's, namely his wave trace formula for Toeplitz operators. Finally, we will derive, for the pseudoconvex manifolds considered by Zelditch in his proof of the Tian–Yau–Zelditch Theorem, an analogue of another result of Boutet de Monvel's, the extendability theorem of Berndtsson for holomorphic functions on Grauert tubes.

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22

Siems, Tobias. "Markov Chain Monte Carlo on finite state spaces." Mathematical Gazette 104, no. 560 (June 18, 2020): 281–87. http://dx.doi.org/10.1017/mag.2020.51.

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We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we prove a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem. Subsequently, we briefly discuss two fundamental MCMC methods, the Gibbs and Metropolis-Hastings sampler. Only very basic knowledge about matrices, convergence of real sequences and probability theory is required.

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23

Silvestre-Alcantara, Whasington, Lutful B. Bhuiyan, Christopher W. Outhwaite, and Douglas Henderson. "A modified Poisson–Boltzmann study of the singlet ion distribution at contact with the electrode for a planar electric double layer." Collection of Czechoslovak Chemical Communications 75, no. 4 (2010): 425–46. http://dx.doi.org/10.1135/cccc2009098.

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The properties of the singlet ion distributions at and around contact in a restricted primitive model double layer are characterized in the modified Poisson–Boltzmann theory. Comparisons are made with the corresponding exact Monte Carlo simulation data, the results from the Gouy–Chapman–Stern theory coupled to an exclusion volume term, and the mean spherical approximation. Particular emphasis is given to the behaviour of the theoretical predictions in relation to the contact value theorem involving the charge profile. The simultaneous behaviour of the coion and counterion contact values is also examined. The performance of the modified Poisson–Boltzmann theory in regard to the contact value theorems is very reasonable with the contact characteristics showing semi-quantitative or better agreement overall with the simulation results. The exclusion-volume-treated Gouy–Chapman– Stern theory reveals a fortuitous cancellation of errors, while the mean spherical approximation is poor.

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Finke, Axel, Arnaud Doucet, and Adam M. Johansen. "Limit theorems for sequential MCMC methods." Advances in Applied Probability 52, no. 2 (June 2020): 377–403. http://dx.doi.org/10.1017/apr.2020.9.

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AbstractBoth sequential Monte Carlo (SMC) methods (a.k.a. ‘particle filters’) and sequential Markov chain Monte Carlo (sequential MCMC) methods constitute classes of algorithms which can be used to approximate expectations with respect to (a sequence of) probability distributions and their normalising constants. While SMC methods sample particles conditionally independently at each time step, sequential MCMC methods sample particles according to a Markov chain Monte Carlo (MCMC) kernel. Introduced over twenty years ago in [6], sequential MCMC methods have attracted renewed interest recently as they empirically outperform SMC methods in some applications. We establish an $\mathbb{L}_r$ -inequality (which implies a strong law of large numbers) and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we also provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.

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25

Agler, Jim, and John E. McCarthy. "Wandering Montel theorems for Hilbert space valued holomorphic functions." Proceedings of the American Mathematical Society 146, no. 10 (May 24, 2018): 4353–67. http://dx.doi.org/10.1090/proc/14086.

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26

MATUTTIS, HANS-GEORG, and NOBUYASU ITO. "NONEXISTENCE OF d-WAVE-SUPERCONDUCTIVITY IN THE QUANTUM MONTE CARLO SIMULATION OF THE HUBBARD MODEL." International Journal of Modern Physics C 16, no. 06 (June 2005): 857–66. http://dx.doi.org/10.1142/s0129183105007571.

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For the existence of d-wave-superconductivity in the Hubbard model, previous quantum Monte Carlo results by other authors, which showed a power law increase of the d-wave susceptibility, seem to contradict a recently published theorem. We show those quantum Monte Carlo calculations were numerically contaminated, analyze the numerical problem and propose a numerically more stable computing scheme.

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27

Fantoni, Riccardo. "Hellmann and Feynman theorem versus diffusion Monte Carlo experiment." Solid State Communications 159 (April 2013): 106–9. http://dx.doi.org/10.1016/j.ssc.2013.01.028.

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SABELFELD, K. K., and I. A. SHALIMOVA. "Mean value theorems in Monte Carlo methods." Russian Journal of Numerical Analysis and Mathematical Modelling 3, no. 3 (1988): 217–30. http://dx.doi.org/10.1515/rnam.1988.3.3.217.

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29

Alekseyev, S. V., Pavel V. Prudnikov, and Vladimir V. Prudnikov. "Ageing Phenomena in Two-Dimensional XY-Model." Solid State Phenomena 190 (June 2012): 3–6. http://dx.doi.org/10.4028/www.scientific.net/ssp.190.3.

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The ageing phenomena in two-dimensional XY model at the low temperatures are investigated by Monte-Carlo method. The two-time correlation function and dynamic susceptibility are measured. Violations of the fluctuation-dissipation theorem are investigated.

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30

Spada, F., M. C. Krol, and P. Stammes. "McSCIA: application of the Equivalence Theorem in a Monte Carlo radiative transfer model for spherical shell atmospheres." Atmospheric Chemistry and Physics Discussions 6, no. 1 (February 15, 2006): 1199–248. http://dx.doi.org/10.5194/acpd-6-1199-2006.

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Abstract. A new multiple-scattering Monte Carlo 3-D radiative transfer model named McSCIA (Monte Carlo for SCIAmachy) is presented. The backward technique is used to efficiently simulate narrow field of view instruments. The McSCIA algorithm has been formulated as a function of the Earth's radius, and can thus perform simulations for both plane-parallel and spherical atmospheres. The latter geometry is essential for the interpretation of limb satellite measurements, as performed by SCIAMACHY on board of ESA's Envisat. The model can simulate UV-vis-NIR radiation. First the ray-tracing algorithm is presented in detail, and then successfully validated against literature references, both in plane-parallel and in spherical geometry. A simple 1-D model is used to explain two different ways of treating absorption. One method uses the single scattering albedo while the other uses the equivalence theorem. The equivalence theorem is based on a separation of absorption and scattering. It is shown that both methods give, in a statistical way, identical results for a wide variety of scenarios. Both absorption methods are included in McSCIA, and it is shown that also for a 3-D case both formulations give identical results. McSCIA limb profiles for atmospheres with and without absorption compare well with the one of the state of the art Monte Carlo radiative transfer model MCC++. A simplification of the photon statistics may lead to very fast calculations of absorption features in the atmosphere. However, these simplifications potentially introduce biases in the results. McSCIA does not use simplifications and is therefore a relatively slow implementation of the equivalence theorem. For the first time, however, the validity of the equivalence theorem is demonstrated in a spherical 3-D radiative transfer model.

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31

Spada, F., M. C. Krol, and P. Stammes. "McSCIA: application of the Equivalence Theorem in a Monte Carlo radiative transfer model for spherical shell atmospheres." Atmospheric Chemistry and Physics 6, no. 12 (October 25, 2006): 4823–42. http://dx.doi.org/10.5194/acp-6-4823-2006.

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Abstract. A new multiple-scattering Monte Carlo 3-D radiative transfer model named McSCIA (Monte Carlo for SCIAmachy) is presented. The backward technique is used to efficiently simulate narrow field of view instruments. The McSCIA algorithm has been formulated as a function of the Earth's radius, and can thus perform simulations for both plane-parallel and spherical atmospheres. The latter geometry is essential for the interpretation of limb satellite measurements, as performed by SCIAMACHY on board of ESA's Envisat. The model can simulate UV-vis-NIR radiation. First the ray-tracing algorithm is presented in detail, and then successfully validated against literature references, both in plane-parallel and in spherical geometry. A simple 1-D model is used to explain two different ways of treating absorption. One method uses the single scattering albedo while the other uses the equivalence theorem. The equivalence theorem is based on a separation of absorption and scattering. It is shown that both methods give, in a statistical way, identical results for a wide variety of scenarios. Both absorption methods are included in McSCIA, and it is shown that also for a 3-D case both formulations give identical results. McSCIA limb profiles for atmospheres with and without absorption compare well with the one of the state of the art Monte Carlo radiative transfer model MCC++. A simplification of the photon statistics may lead to very fast calculations of absorption features in the atmosphere. However, these simplifications potentially introduce biases in the results. McSCIA does not use simplifications and is therefore a relatively slow implementation of the equivalence theorem.

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32

Ben Alaya, Mohamed, and Ahmed Kebaier. "Central limit theorem for the multilevel Monte Carlo Euler method." Annals of Applied Probability 25, no. 1 (February 2015): 211–34. http://dx.doi.org/10.1214/13-aap993.

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Hoel, Håkon, and Sebastian Krumscheid. "Central limit theorems for multilevel Monte Carlo methods." Journal of Complexity 54 (October 2019): 101407. http://dx.doi.org/10.1016/j.jco.2019.05.001.

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Tu, Zhenhan, and Shasha Zhang. "Montel-type theorems in several complex variables with continuously moving targets." Chinese Annals of Mathematics, Series B 31, no. 3 (February 2, 2010): 373–84. http://dx.doi.org/10.1007/s11401-009-0009-5.

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35

Schwick, Wilhelm. "A New Joint Proof of the Theorems of Picard and Montel." Results in Mathematics 21, no. 3-4 (May 1992): 403–7. http://dx.doi.org/10.1007/bf03323097.

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Chauvin, B., and Rouault. "A stochastic simulation for solving scalar reaction–diffusion equations." Advances in Applied Probability 22, no. 01 (March 1990): 88–100. http://dx.doi.org/10.1017/s0001867800019340.

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A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

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Chauvin, B., and Rouault. "A stochastic simulation for solving scalar reaction–diffusion equations." Advances in Applied Probability 22, no. 1 (March 1990): 88–100. http://dx.doi.org/10.2307/1427598.

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A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

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Khoshkholgh, Sarouyeh, Andrea Zunino, and Klaus Mosegaard. "Informed proposal Monte Carlo." Geophysical Journal International 226, no. 2 (April 29, 2021): 1239–48. http://dx.doi.org/10.1093/gji/ggab173.

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SUMMARY Any search or sampling algorithm for solution of inverse problems needs guidance to be efficient. Many algorithms collect and apply information about the problem on the fly, and much improvement has been made in this way. However, as a consequence of the No-Free-Lunch Theorem, the only way we can ensure a significantly better performance of search and sampling algorithms is to build in as much external information about the problem as possible. In the special case of Markov Chain Monte Carlo (MCMC) sampling we review how this is done through the choice of proposal distribution, and we show how this way of adding more information about the problem can be made particularly efficient when based on an approximate physics model of the problem. A highly non-linear inverse scattering problem with a high-dimensional model space serves as an illustration of the gain of efficiency through this approach.

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Kolokoltsov, Vassili, Feng Lin, and Aleksandar Mijatović. "Monte carlo estimation of the solution of fractional partial differential equations." Fractional Calculus and Applied Analysis 24, no. 1 (January 29, 2021): 278–306. http://dx.doi.org/10.1515/fca-2021-0012.

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Abstract The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

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Bierkens, Joris, and Andrew Duncan. "Limit theorems for the zig-zag process." Advances in Applied Probability 49, no. 3 (September 2017): 791–825. http://dx.doi.org/10.1017/apr.2017.22.

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AbstractMarkov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkenset al.(2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.

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Del Moral, P. "A uniform convergence theorem for the numerical solving of the nonlinear filtering problem." Journal of Applied Probability 35, no. 04 (December 1998): 873–84. http://dx.doi.org/10.1017/s0021900200016570.

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The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

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Del Moral, P. "A uniform convergence theorem for the numerical solving of the nonlinear filtering problem." Journal of Applied Probability 35, no. 4 (December 1998): 873–84. http://dx.doi.org/10.1239/jap/1032438382.

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The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

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Alaya, Mohamed Ben, and Gilles Pagès. "Rate of convergence for computing expectations of stopping functionals of an α-mixing process." Advances in Applied Probability 30, no. 02 (June 1998): 425–48. http://dx.doi.org/10.1017/s0001867800047364.

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The shift method consists in computing the expectation of an integrable functional F defined on the probability space ((ℝ d ) N , B(ℝ d )⊗N , μ⊗N ) (μ is a probability measure on ℝ d ) using Birkhoff's Pointwise Ergodic Theorem, i.e. as n → ∞, where θ denotes the canonical shift operator. When F lies in L 2( F T , μ⊗N ) for some integrable enough stopping time T, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributions P on ((ℝ d ) N , B(ℝ d )⊗N ), namely when the canonical process (X n ) n∊N is P-stationary, α-mixing and fulfils Ibragimov's assumption for some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distribution P ν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.

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44

Alaya, Mohamed Ben, and Gilles Pagès. "Rate of convergence for computing expectations of stopping functionals of an α-mixing process." Advances in Applied Probability 30, no. 2 (June 1998): 425–48. http://dx.doi.org/10.1239/aap/1035228077.

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The shift method consists in computing the expectation of an integrable functional F defined on the probability space ((ℝd)N, B(ℝd)⊗N, μ⊗N) (μ is a probability measure on ℝd) using Birkhoff's Pointwise Ergodic Theorem, i.e. as n → ∞, where θ denotes the canonical shift operator. When F lies in L2(FT, μ⊗N) for some integrable enough stopping time T, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributions P on ((ℝd)N, B(ℝd)⊗N), namely when the canonical process (Xn)n∊N is P-stationary, α-mixing and fulfils Ibragimov's assumption for some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distribution Pν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.

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Pan, Feng Ping, Hong Kai Liao, Jia Luo, and Xi Zhang. "Optimal PI Controller Tuning Based on ITAE Criterion for Low-Order Systems with Large Time Delay." Applied Mechanics and Materials 411-414 (September 2013): 1716–19. http://dx.doi.org/10.4028/www.scientific.net/amm.411-414.1716.

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For low order process with large time delay, a kind of optimal PI controller tuning method is proposed based on generalized Hermite-Biehler theorem and Genetic Algorithm. Firstly, the stable region of PI controller is obtained by using the generalized Hermite-Biehler theorem. Then the optimum parameters are selected from this region based on ITAE criterion and genetic algorithm. A tuning formula is obtained by nonlinear fitting of optimization result, which has the capability to cover the variety of normalized time delays up to 100. Simulation of Monte-Carlo stochastic experiment indicates that the proposed method has good performance robustness when parameter uncertainty occurs, compared with other four PI tuning methods.

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GUENTNER, ERIK, and NIGEL HIGSON. "A NOTE ON TOEPLITZ OPERATORS." International Journal of Mathematics 07, no. 04 (August 1996): 501–13. http://dx.doi.org/10.1142/s0129167x9600027x.

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We study Toeplitz operators on Bergman spaces using techniques from the analysis of Dirac-type operators on complete Riemannian manifolds, and prove an index theorem of Boutet de Monvel from this point of view. Our approach is similar to that of Baum and Douglas [2], but we replace boundary value theory for the Dolbeaut operator with much simpler estimates on complete manifolds.

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Thrapsaniotis, E. G. "Monte Carlo methods via the use of the central-limit theorem." Europhysics Letters (EPL) 63, no. 4 (August 2003): 479–84. http://dx.doi.org/10.1209/epl/i2003-00491-5.

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Georgiev, Danko. "Monte Carlo simulation of quantum Zeno effect in the brain." International Journal of Modern Physics B 29, no. 07 (March 2, 2015): 1550039. http://dx.doi.org/10.1142/s0217979215500393.

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Environmental decoherence appears to be the biggest obstacle for successful construction of quantum mind theories. Nevertheless, the quantum physicist Henry Stapp promoted the view that the mind could utilize quantum Zeno effect to influence brain dynamics and that the efficacy of such mental efforts would not be undermined by environmental decoherence of the brain. To address the physical plausibility of Stapp's claim, we modeled the brain using quantum tunneling of an electron in a multiple-well structure such as the voltage sensor in neuronal ion channels and performed Monte Carlo simulations of quantum Zeno effect exerted by the mind upon the brain in the presence or absence of environmental decoherence. The simulations unambiguously showed that the quantum Zeno effect breaks down for timescales greater than the brain decoherence time. To generalize the Monte Carlo simulation results for any n-level quantum system, we further analyzed the change of brain entropy due to the mind probing actions and proved a theorem according to which local projections cannot decrease the von Neumann entropy of the unconditional brain density matrix. The latter theorem establishes that Stapp's model is physically implausible but leaves a door open for future development of quantum mind theories provided the brain has a decoherence-free subspace.

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ABDALLA, AREEG, and JAMES BUCKLEY. "MONTE CARLO METHODS IN FUZZY GAME THEORY." New Mathematics and Natural Computation 03, no. 02 (July 2007): 259–69. http://dx.doi.org/10.1142/s1793005707000768.

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In this paper, we consider a two-person zero-sum game with fuzzy payoffs and fuzzy mixed strategies for both players. We define the fuzzy value of the game for both players [Formula: see text] and also define an optimal fuzzy mixed strategy for both players. We then employ our fuzzy Monte Carlo method to produce approximate solutions, to an example fuzzy game, for the fuzzy values [Formula: see text] for Player I and [Formula: see text] for Player II; and also approximate solutions for the optimal fuzzy mixed strategies for both players. We then look at [Formula: see text] and [Formula: see text] to see if there is a Minimax theorem [Formula: see text] for this fuzzy game.

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Yoder, P. D., U. Krumbein, K. Gärtner, N. Sasaki, and W. Fichtner. "Statistical Enhancement of Terminal Current Estimation for Monte Carlo Device Simulation." VLSI Design 6, no. 1-4 (January 1, 1998): 303–6. http://dx.doi.org/10.1155/1998/34726.

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We present a new generalized Ramo-Shockley theorem (GRST) to evaluate contact currents, applicable to classical moment-based simulation techniques, as well as semiclassical Monte Carlo and quantum mechanical transport simulation, which remains valid for inhomogeneous media, explicitly accounts for generation/recombination processes, and clearly distinguishes between electron, hole, and displacement current contributions to contact current. We then show how this formalism may be applied to Monte Carlo simulation to obtain equations for minimum-variance estimators of steady-state contact current, making use of information gathered from all particles within the device. Finally, by means of an example, we demonstrate this technique’s performance in acceleration of convergence time.

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Journal articles on the topic 'Montel theorem'

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