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# zbMATH — the first resource for mathematics

## Journal of Applied Probability

 Short Title: J. Appl. Probab. Publisher: Applied Probability Trust, Sheffield ISSN: 0021-9002; 1475-6072/e Online: https://www.cambridge.org/core/journals/journal-of-applied-probability/all-issueshttp://projecteuclid.org/jap Comments: Indexed cover-to-cover
 Documents Indexed: 5,286 Publications (since 1964) References Indexed: 1,487 Publications with 28,057 References.
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#### Latest Issues

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#### Authors

 34 Ross, Sheldon Mark 30 Seneta, Eugene 28 Heyde, Christopher Charles 26 Kella, Offer 24 Daley, Daryl John 24 Gani, Joseph Mark 22 Miyazawa, Masakiyo 22 Stadje, Wolfgang 21 Boxma, Onno Johan 21 Pakes, Anthony G. 20 Ball, Frank G. 20 Ebrahimi, Nader D. 20 Resnick, Sidney Ira 20 Shanthikumar, Jeyaveerasingam George 19 Lefèvre, Claude 19 Neuts, Marcel Fernand 19 Whitt, Ward 17 Shepp, Lawrence Alan 16 Cha, Ji Hwan 16 Pollett, Philip K. 16 Renshaw, Eric 16 Savits, Thomas H. 16 Sudbury, Aidan 15 Brockwell, Peter J. 15 Doney, Ronald Arthur 15 Holst, Lars 15 Phatarfod, Ravindra M. 15 Shaked, Moshe 15 Takács, Lajos 15 Walrand, Jean C. 14 Barbour, Andrew D. 14 Block, Henry W. 14 Bruss, Franz Thomas 14 Cowan, Richard J. 14 Glazebrook, Kevin D. 14 Janson, Svante 14 Klebaner, Fima C. 14 Kyprianou, Andreas E. 13 Asmussen, Søren 13 Borovkov, Konstantin A. 13 Iksanov, Aleksander M. 13 Jagers, Peter 13 Keilson, Julian 13 Langberg, Naftali A. 13 Mahmoud, Hosam M. 13 Ricciardi, Luigi Maria 13 Righter, Rhonda 13 Taylor, Peter Gerrard 13 Teugels, Jozef L. 13 Wolff, Ronald W. 12 Balakrishnan, Narayanaswamy 12 Di Crescenzo, Antonio 12 Haigh, John 12 Mandjes, Michel Robertus Hendrikus 12 McCormick, William P. 12 Peköz, Erol A. 12 Rüschendorf, Ludger 12 Sumita, Ushio 12 Weber, Richard Robert 12 Whittle, Peter 11 Billard, Lynne 11 Britton, Tom 11 Brown, Mark 11 Glynn, Peter W. 11 Kaplan, Norman L. 11 Kijima, Masaaki 11 Kingman, John Frank Charles 11 Klüppelberg, Claudia 11 Möhle, Martin 11 Moran, Patrick Alfred Pierce 11 Natvig, Bent 11 Proschan, Frank 11 Rogers, L. C. G. 11 Tang, Qihe 11 Weiner, Howard Jacob 11 Zacks, Shelemyahu 10 Arnold, Barry Charles 10 Athreya, Krishna Balasundaram 10 Baccelli, François Louis 10 Chen, Anyue 10 Conolly, Brian W. 10 Ekström, Erik 10 Gnedin, Alexander V. 10 Grübel, Rudolf 10 Guillemin, Fabrice M. 10 Hall, Peter Gavin 10 Hannan, Edward James 10 Hines, W. G. S. 10 Hüsler, Jürg 10 Kennedy, Douglas P. 10 Letac, Gérard G. 10 Picard, Philippe 10 Quine, Malcolm P. 10 Sericola, Bruno 10 Sigman, Karl 10 Szekli, Ryszard 10 Zuckerman, Dror 9 Chao, Xiuli 9 Grey, D. R. 9 Itoh, Yoshiaki ...and 4,047 more Authors
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#### Fields

 4,546 Probability theory and stochastic processes (60-XX) 930 Statistics (62-XX) 816 Operations research, mathematical programming (90-XX) 436 Biology and other natural sciences (92-XX) 378 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 161 Computer science (68-XX) 157 Numerical analysis (65-XX) 146 Combinatorics (05-XX) 120 Systems theory; control (93-XX) 69 Statistical mechanics, structure of matter (82-XX) 48 Information and communication theory, circuits (94-XX) 37 Convex and discrete geometry (52-XX) 34 Linear and multilinear algebra; matrix theory (15-XX) 33 History and biography (01-XX) 25 Partial differential equations (35-XX) 23 Calculus of variations and optimal control; optimization (49-XX) 21 Integral transforms, operational calculus (44-XX) 18 Dynamical systems and ergodic theory (37-XX) 17 Special functions (33-XX) 17 Ordinary differential equations (34-XX) 17 Operator theory (47-XX) 16 Real functions (26-XX) 15 Measure and integration (28-XX) 15 Geophysics (86-XX) 14 Difference and functional equations (39-XX) 14 Approximations and expansions (41-XX) 12 Integral equations (45-XX) 11 Number theory (11-XX) 8 Functional analysis (46-XX) 7 Functions of a complex variable (30-XX) 7 Harmonic analysis on Euclidean spaces (42-XX) 4 Mathematical logic and foundations (03-XX) 4 Sequences, series, summability (40-XX) 4 Quantum theory (81-XX) 3 General and overarching topics; collections (00-XX) 3 Nonassociative rings and algebras (17-XX) 3 Differential geometry (53-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Mechanics of particles and systems (70-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 2 Field theory and polynomials (12-XX) 2 Geometry (51-XX) 2 Mechanics of deformable solids (74-XX) 2 Fluid mechanics (76-XX) 2 Optics, electromagnetic theory (78-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Mathematics education (97-XX) 1 Commutative algebra (13-XX) 1 Group theory and generalizations (20-XX) 1 Abstract harmonic analysis (43-XX) 1 General topology (54-XX) 1 Astronomy and astrophysics (85-XX)

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4,225 Publications have been cited 36,106 times in 22,185 Documents Cited by Year
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A new family of life distributions. Zbl 0209.49801
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Limit theorems for continuous-time random walks with infinite mean waiting times. Zbl 1065.60042
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2004
Changes of numéraire, changes of probability measure and option pricing. Zbl 0829.90007
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The value of an Asian option. Zbl 0839.90013
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A versatile Markovian point process. Zbl 0422.60043
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On the genealogy of large populations. Zbl 0516.92011
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Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Zbl 0219.60060
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Imperfect repair. Zbl 0526.60080
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The asymptotic theory of linear time-series models. Zbl 0261.62073
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Subexponential distributions and integrated tails. Zbl 0651.60020
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Estimation for a family of life distributions with applications to fatigue. Zbl 0216.22702
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1969
Product-form queueing networks with negative and positive customers. Zbl 0741.60091
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Non-strong mixing autoregressive processes. Zbl 0552.60049
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1984
Diffusion models in population genetics. Zbl 0134.38103
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1964
Some results for repairable systems with general repair. Zbl 0671.60080
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Age-dependent minimal repair. Zbl 0564.60084
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1985
The general coalescent with asynchronous mergers of ancestral lines. Zbl 0962.92026
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1999
A cluster process representation of a self-exciting process. Zbl 0305.60021
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1974
Martingale convergence in the branching random walk. Zbl 0356.60053
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1977
Weak convergence of probability measures and random functions in the function space D$$[0,\infty)$$. Zbl 0258.60008
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1973
On the eigenvalues of random matrices. Zbl 0807.15015
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Approximations of small jumps of Lévy processes with a view towards simulation. Zbl 0989.60047
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2001
Perburbation theory and finite Markov chains. Zbl 0196.19803
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1968
Inequalities with applications to percolation and reliability. Zbl 0571.60019
van den Berg, J.; Kesten, H.
1985
A logarithmic reduction algorithm for quasi-birth-death processes. Zbl 0789.60055
Latouche, Guy; Ramaswami, V.
1993
An integer-valued p th-order autoregressive structure (INAR(p)) process. Zbl 0704.62081
Alzaid, A. A.; Al-Osh, M.
1990
On quasi-stationary distributions on absorbing discrete-time finite Markov chains. Zbl 0134.34704
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1965
On estimating the diffusion coefficient from discrete observations. Zbl 0796.62070
Florens-Zmirou, Danielle
1993
Stochastic approach to chemical kinetics. Zbl 0231.60090
McQuarrie, Donald A.
1967
Large deviations of heavy-tailed random sums with applications in insurance and finance. Zbl 0903.60021
Klüppelberg, C.; Mikosch, T.
1997
Convolution equivalence and infinite divisibility. Zbl 1051.60019
Pakes, Anthony G.
2004
On quasi-stationary distributions in absorbing continuous-time finite Markov chains. Zbl 0168.16303
Darroch, J. N.; Seneta, E.
1967
Exponential behavior in the presence of dependence in risk theory. Zbl 1097.62110
Albrecher, Hansjörg; Teugels, Jef L.
2006
Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. Zbl 1137.62015
Roberts, Gareth O.; Rosenthal, Jeffrey S.
2007
Optimality of the shortest line discipline. Zbl 0357.60023
Winston, Wayne
1977
On the rate of convergence of normal extremes. Zbl 0403.60024
Hall, Peter
1979
Extreme values of independent stochastic processes. Zbl 0384.60055
Brown, Bruce M.; Resnick, Sidney I.
1977
Limit theorems for iterated random functions. Zbl 1046.60024
Wu, Wei Biao; Shao, Xiaofeng
2004
On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. Zbl 0147.36603
Seneta, E.; Vere-Jones, D.
1966
Diffusion approximations in collective risk theory. Zbl 0191.51202
Iglehart, D. L.
1969
Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules. Zbl 0612.60087
Keilson, J.; Servi, L. D.
1986
Restless bandits: Activity allocation in a changing world. Zbl 0664.90043
Whittle, P.
1988
Precise large deviations for sums of random variables with consistently varying tails. Zbl 1051.60032
Ng, Kai W.; Tang, Qihe; Yan, Jia-An; Yang, Hailiang
2004
A supplement to the strong law of large numbers. Zbl 0305.60008
Heyde, C. C.
1975
Dispersive ordering of distributions. Zbl 0481.60022
Shaked, Moshe
1982
Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov- Smirnov test. Zbl 0225.60037
Durbin, J.
1971
The second-order analysis of stationary point processes. Zbl 0364.60087
Ripley, B. D.
1976
Diffusion with ’collisions’ between particles. Zbl 0139.34804
Harris, T. E.
1965
A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour. Zbl 1068.62093
Klüppelberg, Claudia; Lindner, Alexander; Maller, Ross
2004
Entropy-based measure of uncertainty in past lifetime distributions. Zbl 1003.62087
Di Crescenzo, Antonio; Longobardi, Maria
2002
The strong law of large numbers for extended negatively dependent random variables. Zbl 1213.60058
Chen, Yiqing; Chen, Anyue; Ng, Kai W.
2010
Exact and ordinary lumpability in finite Markov chains. Zbl 0796.60073
Buchholz, Peter
1994
Applications of the hazard rate ordering in reliability and order statistics. Zbl 0793.62053
Boland, Philip J.; El-Neweihi, Emad; Proschan, Frank
1994
Asymptotic behaviour of continuous time, continuous state-space branching processes. Zbl 0301.60060
Grey, D. R.
1974
On the tails of waiting-time distributions. Zbl 0314.60072
Pakes, A. G.
1975
Useful martingales for stochastic storage processes with Lévy input. Zbl 0761.60065
Kella, Offer; Whitt, Ward
1992
The threshold behaviour of epidemic models. Zbl 0519.92023
Ball, Frank
1983
Mixture representations of residual lifetimes of used systems. Zbl 1155.60305
Navarro, Jorge; Balakrishnan, N.; Samaniego, Francisco J.
2008
Chernoff’s theorem in the branching random walk. Zbl 0373.60090
Biggins, J. D.
1977
Multivariate distributions for the life lengths of components of a system sharing a common environment. Zbl 0603.60084
Lindley, Dennis V.; Singpurwalla, Nozer D.
1986
Partial orderings of distributions based on right-spread functions. Zbl 0898.62124
Fernández Ponce, J. M.; Kochar, S. C.; Muñoz Pérez, J.
1998
Analysis of the $$M^ X/G/1$$ queue with $$N$$-policy and multiple vacations. Zbl 0804.60081
Lee, Ho Woo; Lee, Soon Seok; Park, Jeong Ok; Chae, K. C.
1994
Logarithmic asymptotics for steady-state tail probabilities in a single- server queue. Zbl 0805.60093
Glynn, Peter W.; Whitt, Ward
1994
Discrimination between monotonic trends and long-range dependence. Zbl 0623.62085
Künsch, H.
1986
On the prediction of fractional Brownian motion. Zbl 0861.60049
Gripenberg, Gustaf; Norros, Ilkka
1996
The stationary distribution of the infinitely-many neutral alleles diffusion model. Zbl 0356.92012
Watterson, G. A.
1976
Multivariate normal approximations by Stein’s method and size bias couplings. Zbl 0845.60023
Goldstein, Larry; Rinott, Yosef
1996
Non-linear time series regression. Zbl 0227.62052
Hannan, E. J.
1971
The first-passage density of a continuous Gaussian process to a general boundary. Zbl 0576.60029
Durbin, J.
1985
Signatures of indirect majority systems. Zbl 1042.62090
Boland, Philip J.
2001
The total progeny in a branching process and a related random walk. Zbl 0192.54401
Dwass, M.
1969
Multistate reliability models. Zbl 0435.60088
Griffith, William S.
1980
Distributional study of De Finetti’s dividend problem for a general Lévy insurance risk process. Zbl 1137.60047
Kyprianou, A. E.; Palmowski, Z.
2007
Queues with negative arrivals. Zbl 0744.60110
Gelenbe, Erol; Glynn, Peter; Sigman, Karl
1991
A Pollaczek-Khintchine formula for $$M/G/1$$ queues with disasters. Zbl 0867.60082
Jain, Gautam; Sigman, Karl
1996
On the optimal assignment of customers to parallel servers. Zbl 0378.60095
Weber, Richard R.
1978
Aspects of positive ageing. Zbl 0612.60072
Deshpande, Jayant V.; Kochar, Subhash C.; Singh, Harshinder
1986
Pseudo-conservation laws in cyclic-service systems. Zbl 0639.60087
Boxma, O. J.; Groenendijk, W. P.
1987
Some new results on stochastic comparisons of parallel systems. Zbl 0995.62104
Khaledi, Baha-Eldin; Kochar, Subhash
2000
On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. Zbl 1054.60012
Cai, Jun; Tang, Qihe
2004
The M/M/1 queue with mass exodus and mass arrivals when empty. Zbl 0876.60079
Chen, Anyue; Renshaw, Eric
1997
Longest common subsequences of two random sequences. Zbl 0313.60008
Chvatal, Vaclav; Sankoff, David
1975
An upper bound for the probability of a union. Zbl 0349.60007
Hunter, David
1976
An explicit solution to an optimal stopping problem with regime switching. Zbl 0988.60038
Guo, Xin
2001
A threshold AR(1) model. Zbl 0541.62073
Petruccelli, Joseph D.; Woolford, Samuel W.
1984
Marked point processes as limits of Markovian arrival streams. Zbl 0778.60035
Asmussen, Søren; Koole, Ger
1993
New partial ordering of survival functions based on the notion of uncertainty. Zbl 0817.62082
1995
Multivariate Hawkes processes: an application to financial data. Zbl 1242.62093
Embrechts, Paul; Liniger, Thomas; Lin, Lu
2011
Approximations of boundary crossing probabilities for a Brownian motion. Zbl 0978.60092
Novikov, Alex; Frishling, Volf; Kordzakhia, Nino
1999
First-passage-time density and moments of the Ornstein-Uhlenbeck process. Zbl 0651.60080
Ricciardi, Luigi M.; Sato, Shunsuke
1988
Sensitivity and convergence of uniformly ergodic Markov chains. Zbl 1092.60027
Mitrophanov, A. Yu.
2005
Hazard rate ordering of order statistics and systems. Zbl 1111.62098
Navarro, Jorge; Shaked, Moshe
2006
Dispersive distributions and the connection between dispersivity and strong unimodality. Zbl 0453.60023
Lewis, Toby; Thompson, J. W.
1981
G-networks with triggered customer movement. Zbl 0781.60088
Gelenbe, Erol
1993
Saddlepoint expansions for conditional distributions. Zbl 0638.62018
Skovgaard, Ib M.
1987
Martingales versus PDEs in finance: an equivalence result with examples. Zbl 0996.91069
Heath, David; Schweizer, Martin
2000
A quantum-mechanical central limit theorem. Zbl 0224.60049
Cushen, C. D.; Hudson, R. L.
1971
Evolutionarily stable strategies with two types of player. Zbl 0398.90120
Taylor, Peter D.
1979
Consistency of Hill’s estimator for dependent data. Zbl 0836.60020
Resnick, Sidney; Stărică, Cătălin
1995
Diffusion approximations for randomly arriving expert opinions in a financial market with Gaussian drift. Zbl 1458.91208
Sass, Jörn; Westphal, Dorothee; Wunderlich, Ralf
2021
A product form for the general stochastic matching model. Zbl 07364095
Moyal, Pascal; Bušić, Ana; Mairesse, Jean
2021
Branching processes in a random environment with immigration stopped at zero. Zbl 1434.60240
Dyakonova, Elena; Li, Doudou; Vatutin, Vladimir; Zhang, Mei
2020
Simultaneous ruin probability for two-dimensional Brownian risk model. Zbl 1464.60031
Dȩbicki, Krzysztof; Hashorva, Enkelejd; Michna, Zbigniew
2020
Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model. Zbl 1443.60068
Nguyen, Dang H.; Nguyen, Nhu N.; Yin, George
2020
A unifying approach to branching processes in a varying environment. Zbl 1434.60242
Kersting, Götz
2020
Tails of exit times from unstable equilibria on the line. Zbl 1464.60057
Bakhtin, Yuri; Pajor-Gyulai, Zsolt
2020
Stochastic comparisons of largest-order statistics for proportional reversed hazard rate model and applications. Zbl 1457.60026
Li, Lu; Wu, Qinyu; Mao, Tiantian
2020
Decorrelation of a class of Gibbs particle processes and asymptotic properties of $$U$$-statistics. Zbl 1457.60066
Beneš, Viktor; Hofer-Temmel, Christoph; Last, Günter; Večeřa, Jakub
2020
Maximizing the $$p$$th moment of the exit time of planar Brownian motion from a given domain. Zbl 1457.60125
Boudabra, Maher; Markowsky, Greg
2020
Weak convergence of random processes with immigration at random times. Zbl 1434.60068
Dong, Congzao; Iksanov, Alexander
2020
A functional limit theorem for general shot noise processes. Zbl 1434.60106
Iksanov, Alexander; Rashytov, Bohdan
2020
Strong convergence of multivariate maxima. Zbl 1462.60068
Falk, Michael; Padoan, Simone A.; Rizzelli, Stefano
2020
On the extension of signature-based representations for coherent systems with dependent non-exchangeable components. Zbl 1464.60015
Navarro, Jorge; Fernández-Sánchez, Juan
2020
Asymptotic results on Hoppe trees and their variations. Zbl 1444.05041
Hiesmayr, Ella; Işlak, Ümit
2020
Robust Wasserstein profile inference and applications to machine learning. Zbl 1436.62336
Blanchet, Jose; Kang, Yang; Murthy, Karthyek
2019
Exit problems for general draw-down times of spectrally negative Lévy processes. Zbl 1415.60048
Li, Bo; Vu, Nhat Linh; Zhou, Xiaowen
2019
Asymptotic behaviour of randomised fractional volatility models. Zbl 1465.60071
Horvath, Blanka; Jacquier, Antoine; Lacombe, Chloé
2019
Limit theorems for the fractional nonhomogeneous Poisson process. Zbl 1418.60025
Leonenko, Nikolai; Scalas, Enrico; Trinh, Mailan
2019
Relational exchangeability. Zbl 1418.60027
Crane, Harry; Dempsey, Walter
2019
Inhomogeneous phase-type distributions and heavy tails. Zbl 1428.62103
2019
On first exit times and their means for Brownian bridges. Zbl 1436.60075
Geiss, Christel; Luoto, Antti; Salminen, Paavo
2019
First passage upwards for state-dependent-killed spectrally negative Lévy processes. Zbl 1415.60050
Vidmar, Matija
2019
Is the Sibuya distribution a progeny? Zbl 1418.60112
Letac, Gérard
2019
Exact simulation of generalised Vervaat perpetuities. Zbl 1418.60040
Dassios, Angelos; Qu, Yan; Lim, Jia Wei
2019
Absolute regularity of semi-contractive GARCH-type processes. Zbl 1418.60029
Doukhan, Paul; Neumann, Michael H.
2019
A particle system approach to aggregation phenomena. Zbl 1416.35139
Flandoli, Franco; Leocata, Marta
2019
Degree correlations in scale-free random graph models. Zbl 1421.05082
Stegehuis, Clara
2019
A martingale approach for asset allocation with derivative security and hidden economic risk. Zbl 1425.91408
Siu, Tak Kuen; Zhu, Jinxia; Yang, Hailiang
2019
Transient and slim versus recurrent and fat: random walks and the trees they grow. Zbl 1436.60084
Iacobelli, Giulio; Figueiredo, Daniel R.; Neglia, Giovanni
2019
Branching Brownian motion with spatially homogeneous and point-catalytic branching. Zbl 1436.60077
Bocharov, Sergey; Wang, Li
2019
On the existence of sure profits via flash strategies. Zbl 1422.91650
Fontana, Claudio; Pelger, Markus; Platen, Eckhard
2019
The $$N$$-star network evolution model. Zbl 1416.05258
Fazekas, István; Noszály, Csaba; Perecsényi, Attila
2019
On the rate of convergence for the length of the longest common subsequences in hidden Markov models. Zbl 1415.60013
Houdré, C.; Kerchev, G.
2019
On association and other forms of positive dependence for Feller processes. Zbl 1415.60022
Tu, Eddie
2019
On negative association of some finite point processes on general state spaces. Zbl 1418.60042
Last, Günter; Szekli, Ryszard
2019
Preservation of the mean residual life order for coherent and mixed systems. Zbl 1418.60013
Lindqvist, Bo H.; Samaniego, Francisco J.; Wang, Nana
2019
Central limit theorem for mean and variogram estimators in Lévy-based models. Zbl 1418.60017
Rønn-Nielsen, Anders; Vedel Jensen, Eva B.
2019
Cramér type moderate deviations for random fields. Zbl 1418.60018
Beknazaryan, Aleksandr; Sang, Hailin; Xiao, Yimin
2019
Discrete-type approximations for non-Markovian optimal stopping problems. I. Zbl 1427.60069
Leão, Dorival; Ohashi, Alberto; Russo, Francesco
2019
Runoff on rooted trees. Zbl 1427.60203
Jones, Owen Dafydd
2019
Skeletal stochastic differential equations for continuous-state branching processes. Zbl 1427.60176
Fekete, D.; Fontbona, J.; Kyprianou, A. E.
2019
Monte Carlo integration with a growing number of control variates. Zbl 1427.60048
Portier, François; Segers, Johan
2019
On maxima of stationary fields. Zbl 1427.60099
Soja-Kukieła, N.
2019
Exponential convergence rate of ruin probabilities for level-dependent Lévy-driven risk processes. Zbl 1432.91099
Goffard, Pierre-Olivier; Sarantsev, Andrey
2019
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text:
# Definition:Lipschitz Equivalence/Metrics

## Definition

### Definition 1

Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.

Let $\exists h, k \in \R_{>0}$ such that:

$\forall x, y \in A: h \map {d_2} {x, y} \le \map {d_1} {x, y} \le k \map {d_2} {x, y}$

Then $d_1$ and $d_2$ are described as Lipschitz equivalent.

### Definition 2

Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.

Let $\exists K_1, K_2 \in \R_{>0}$ such that:

$(1): \quad \forall x, y \in A: \map {d_2} {x, y} \le K_1 \map {d_1} {x, y}$
$(2): \quad \forall x, y \in A: \map {d_1} {x, y} \le K_2 \map {d_2} {x, y}$

Then $d_1$ and $d_2$ are described as Lipschitz equivalent.

## Terminology

Despite the close connection with the concept of Lipschitz continuity, this concept is rarely seen in mainstream mathematics, and appears not to have a well-established name.

The name Lipschitz equivalence appears in 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces:

There does not appear to be a standard name for this; the name we use is reasonably appropriate ...

## Source of Name

This entry was named for Rudolf Lipschitz.

format: plain


text:
### Solve the following : Calculate of Q3, D5, and P35 for the given data Income (in lakhs ) 1 2 3 4 5 6 No. of family 2 5 20 25 15 12 - Economics

#### QUESTION

Solve the following :

Calculate of Q3, D5, and P35 for the given data

 Income (in lakhs ₹) 1 2 3 4 5 6 No. of family 2 5 20 25 15 12

#### SOLUTION

Arrange the data of the income in ascending order.
(i.e. from lowest number to highest as shown below) and the out the cumulative frequency.

 Income (in lakhs RS) no. of family cumulative frequency(cf) 1 2 2 2 5 7 3 20 27 4 25 52 5 15 67 6 12 79 N = 79

Q3 = Size of 3${\left(\frac{n+1}{4}\right)}^{\text{th observation}}$

= size of 3${\left(\frac{79+1}{4}\right)}^{\text{th observation}}$

= size of 3${\left(\frac{80}{4}\right)}^{\text{th observation}}$

= size of 3 ${\left(20\right)}^{\text{th observation}}$

= size of 60th observation  lies in cf 67

Hence, quartile value = Rs .5 lakhs Q3 = Rs. 5 Lakhs

D= size of 5${\left(\frac{n+1}{10}\right)}^{\text{th observation}}$

=size of 5${\left(\frac{79+1}{10}\right)}^{\text{th observation}}$

= size of 5${\left(\frac{80}{10}\right)}^{\text{th observation}}$

= size of 5${\left(8\right)}^{\text{th observation}}$

= size of ${\left(40\right)}^{\text{th observation}}$

Size of 40th observation lies in cf 52 , hence decile value =Rs. 4 Lakhs

D5 =Rs. 4 Lakhs

P35 =Size of 35${\left(\frac{n+1}{100}\right)}^{\text{th observation}}$

P35 =Size of 35${\left(\frac{79+1}{100}\right)}^{\text{th observation}}$

P35 =Size of 35${\left(\frac{80}{100}\right)}^{\text{th observation}}$

=Size of 35 ${\left(0.80\right)}^{\text{th observation}}$

=Size of 28th observation lies in cf 52

Hence, percentile value = Rs.4 lakhs P35 = Rs. 4 lakhs.


text:
# Math Help - proof by induction

1. ## proof by induction

Say I am given that $w^r$ means the string is spelled backwards. Ex:

$(hello)^R$ = olleh

The definition is:

1. w is a string of length 0 then w^R = empty set
2. w = ua, where $a \in \Sigma$ and $u \in \Sigma ^ \ast$ , is a string of length n+1>0 then $w^r = au^R$

prove the following:

a. $(xy)^R = y^Rx^R$

2. Originally Posted by EquinoX
Say I am given that $w^r$ means the string is spelled backwards. Ex:

$(hello)^R$ = olleh

The definition is:

1. w is a string of length 0 then w^R = empty set
2. w = ua, where $a \in \Sigma$ and $u \in \Sigma ^ \ast$ , is a string of length n+1>0 then $w^r = au^R$

prove the following:

a. $(xy)^R = y^Rx^R$
Use induction on the length of y. Condition 2. of the definition gives the base case, where y has length 1. If y has length n+1 then y=za, where z has length n. Use the inductive hypothesis to show that $(xy)^r = (xza)^r = a(xz)^r = \ldots$ (you take it from there).

Originally Posted by EquinoX
nobody are willing to help me?
If you post a query at 5 a.m. (UK time), don't get ratty if I don't reply by 7 a.m.

3. well, actually I actually I already got to that part and that's the point where I am stuck at...

4. Originally Posted by EquinoX
well, actually I actually I already got to that part and that's the point where I am stuck at...
$(xy)^r = (xza)^r = a(xz)^r$
${\color{white}(xy)^r = (xza)^r} = az^r x^r$ (by the inductive hypothesis)
${\color{white}(xy)^r = (xza)^r} = (za)^r x^r$ (by 2.)
${\color{white}(xy)^r = (xza)^r} = y^r x^r$.

That's the inductive step.

5. what if it's $(w^R)^R = w$

I think I am supposed to use the answer from the previous part to answer this right?

6. can anyone give me some hints?

7. Hello,

Use the induction. If $w=xa$ where $a\in\Sigma,\, (x^R)^R=x$, the inductive step is like this:
$(w^R)^R=((xa)^R)^R=(a^Rx^R)^R=(ax^R)^R=(x^R)^Ra^R= xa=w$.

Bye.

8. Originally Posted by Opalg
$(xy)^r = (xza)^r = a(xz)^r$
${\color{white}(xy)^r = (xza)^r} = az^r x^r$ (by the inductive hypothesis)
${\color{white}(xy)^r = (xza)^r} = (za)^r x^r$ (by 2.)
${\color{white}(xy)^r = (xza)^r} = y^r x^r$.

That's the inductive step.

Hmm.. I am asked to do this using the answer from part a. Not really sure if your answer really uses of the techniques in part a..

9. Originally Posted by wisterville
Hello,

Use the induction. If $w=xa$ where $a\in\Sigma,\, (x^R)^R=x$, the inductive step is like this:
$(w^R)^R=((xa)^R)^R=(a^Rx^R)^R=(ax^R)^R=(x^R)^Ra^R= xa=w$.

Bye.
One thing I don't understand, you mentioned w = xa, and $a\in\Sigma$ so what is x here? is $x\in (\Sigma) ^ *$ ?? Please do correct me if I am wrong

10. Hello,

Originally Posted by EquinoX
One thing I don't understand, you mentioned w = xa, and $a\in\Sigma$ so what is x here? is $x\in (\Sigma) ^ *$ ?? Please do correct me if I am wrong
Yes, you guessed right. I gave you hints, it's up to you to make it into a rigorous proof. You should know what "a proof by induction" looks like: the base case and the inductive step (assuming that the assertion holds when the length is k-1 and prove it for k. As I wrote (x^R)^R=x, you see that I assumed that the length of x is k-1 and so on...)

Bye.

11. $(x^R)^R = x$, is just the inductive hypothesis that you're assuming to help you solve the induction right?

you assume that a string x with a length of n holds the property $(x^R)^R = x$, so therefore in your inductive proof that you show on your first post is the induction for n+1 and so on?

12. Hello,

Originally Posted by EquinoX
$(x^R)^R = x$, is just the inductive hypothesis that you're assuming to help you solve the induction right?

you assume that a string x with a length of n holds the property $(x^R)^R = x$, so therefore in your inductive proof that you show on your first post is the induction for n+1 and so on?
Yes, you got it right.

Bye.

13. Originally Posted by wisterville
Hello,

Yes, you got it right.

Bye.

one last question, what would be the base case here? Is the base case for part a and b the same? From my point of view, we break w = ua. then assume that n = 0, and therefore u is a string of length 1 and so is a. So (au)^R is equal to au^R.

Please do correct me, as that's the way I see it

14. Hello,

It seems the base case is intended to be a string of length 0. (cf. Cond.1)

a. Prove (xy)^R=y^Rx^R where the length of y is 0.
Use y=y^R=empty set.

b. Prove (w^R)^R=w where the length of w is 0.
Use w=w^R=empty set.

The base case is not exactly "the same" since you prove something different. But you use the same fact (that is, Cond.1)...

Bye.


text:
#### A general framework for handling commitment in online throughput maximization

##### Lin Chen, Franziska Eberle, Nicole Megow, Kevin Schewior, Cliff Stein

We study a fundamental online job admission problem where jobs with deadlines arrive online over time at their release dates, and the task is to determine a preemptive single-server schedule which maximizes the number of jobs that complete on time. To circumvent known impossibility results, we make a standard slackness assumption by which the feasible time window for scheduling a job is at least $1+\varepsilon$ times its processing time, for some $\varepsilon>0$. We quantify the impact that different provider commitment requirements have on the performance of online algorithms. Our main contribution is one universal algorithmic framework for online job admission both with and without commitments. Without commitment, our algorithm with a competitive ratio of $O(1/\varepsilon)$ is the best possible (deterministic) for this problem. For commitment models, we give the first non-trivial performance bounds. If the commitment decisions must be made before a job's slack becomes less than a $\delta$-fraction of its size, we prove a competitive ratio of $O(\varepsilon/((\varepsilon-\delta)\delta^2))$, for $0<\delta<\varepsilon$. When a provider must commit upon starting a job, our bound is $O(1/\varepsilon^2)$. Finally, we observe that for scheduling with commitment the restriction to the `unweighted' throughput model is essential; if jobs have individual weights, we rule out competitive deterministic algorithms.

arrow_drop_up


text:
1. Sep 20, 2008

herbyoung

1. The problem statement, all variables and given/known data

Let I be the interval I=[0,infinity). Let f: I to R be uniformly continuous. Show there exist positive constants A and B such that |f(x)|<=Ax+B for all x that belongs to I.

2. The attempt at a solution

2. Sep 20, 2008

CompuChip

Let I be the interval I = [0, infinity). Let f: I to R be uniformly continuous.

Suppose that for all positive A and B, there is an x = x(A, B) in I such that |f(x)| > Ax + B.

All you have to derive a contradiction with is the uniform continuity of f. Can you write this out in terms of f(x) (i.e. the definition)?

3. Mar 3, 2011

mariama1

I am interesting to know how we can prove this .

4. Mar 3, 2011

CompuChip

I didn't solve it myself at the time, but some quick scribbles show that if you use the definition and the reverse triangle inequality it should work out.
How far did you get?

5. Mar 3, 2011

Staff: Mentor

We don't provide complete answers here, only hints. If you're trying to do this problem yourself, please start a new thread instead of hijacking an old one.


text:
## Friday, April 27, 2012

### tacoli: a simple logging format

A post on Hacker News entitled Log Everything As JSON. Make Your Life Easier reminded me of my private logging strategy which has the following properties:

1. Easy to parse and analyze with Unix command-line tools such as grep, cut, sort, uniq, and wc

2. Easy to parse and analyze in code using Perl, Ruby, or Go

3. Compact

4. Easily expandable and lacking the ambiguity of simple delimited log formats

I call it tacoli (which stands for Tabs, Colons and Lines).  Here are the tacoli logging rules: Each log entry is a single line that starts with the date/time; the second entry on the line is a string called the 'generator' which indicates where the log line came from (such as the program or module); all the other entries have the format "key: value"; and entries are tab-delimited and no tabs are allowed in keys, values or the generator name.

That's it.  Here's an example log line from Apache in this format:

22/Apr/2012:06:29:07 +0000      apache  ip: 18.12.25.55 method: GET     uri: /example.html code:301        size:305        referer: http://blog.jgc.org/2009/08/geek-weekend-day-1-bletchley-park.html        agent: Mozilla/5.0 (Windows NT 6.1; WOW64) AppleWebKit/535.19 (KHTML, like Gecko) Chrome/18.0.1025.162

Note that it's easy to make Apache output this format just by using tabs and adding the appropriate key: to each field in the LogFormat.  No special logger module required.  In fact, anything that can 'printf' a string can create tacoli lines trivially.

It's trivial to parse in code, all you need is 'split' to break on the tabs, and then split again to break the key name from the value.  No specialized JSON (or other parser) required.

It's trivial to extend without breaking any tools.  Just add a new field (anywhere on the line) with a new key.

It's simple to work with using Unix tools.  Since the format is 'one log entry per line' it works well with wc -l to count instances of anything and it interfaces with all the other Unix tools that expect to work with lines (and even in code the line oriented nature is helpful since getting a complete entry is a single line read).

If you want to extract a single field from each line of the log file then it's easy to do with grep.  Here's an example that extracts all the lines that have an ip entry and just extracts that

grep -Po "\tip: [^\t]+" access.log

The key name can be trivially removed using cut

grep -Po "\tip: [^\t]+" access.log | cut -d: -f2-

and the output can be fed into the other Unix tools.  Also, if you know that your log file format hasn't changed you can still use the positional information to simplify parsing and fall back to cut.
It isn't quite as compact as a log file format that only uses position to indicate meaning, but compression largely overcomes that problem and key names can be chosen to be short and unique.

If you enjoyed this blog post, you might enjoy my travel book for people interested in science and technology: The Geek Atlas. Signed copies of The Geek Atlas are available.

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# line emission spectrum example

Emission-line spectra Low-density clouds of gas floating in space will emit emission lines if they are excited by energy from nearby stars. When an absorption spectrum and emission spectrum are combined, you get a continuous spectrum with all the wavelengths. (a) When an electric discharge is passed through hydrogen gas at low pressure, a bluish light is emitted. Truly white light, like that emitted by some stars and moons is a perfect example of continuous spectrum. Hydrogen spectrum is an example of line emission spectrum or atomic emission spectrum. Other articles where Line spectrum is discussed: spectroscopy: Basic atomic structure: The emission and absorption spectra of the elements depend on the electronic structure of the atom. (b) This light shows discontinuous line spectrum of several isolated sharp lines through prism. Every element has a unique atomic emission spectrum, as shown by the examples of helium $$\left( \ce{He} \right)$$ and iron $$\left( \ce{Fe} \right)$$. There are two types of line spectrum: an emission spectrum and an absorption spectrum. This hypothetical element shall have 4 energy levels. Line spectrum can therefore be found in both absorption and emission spectra which are exact opposites of each other. Take for example an element, which will be named X. Classical theory was unable to explain the existence of atomic emission spectra, also known as line-emission spectra. The line emission (atomic) spectrum of an element is unique. In a very simplistic way you can consider a gas of atoms of a substance illuminated by light. Each line represents a unique wavelength, and the entire thing is unique to that particular element. No ads = no money for us = no free stuff for you! The emission of photons through electronic relaxations in atoms produces a line emission spectrum. Spectroscopy is the study of emission and absorption spectra. Line Spectrum. Examples Line emission spectrum of H violet n6-n2 blue ng-> nz green n4 n2 Yed ng n2 Calculate the energy of the light emitted when an e- from n=4 to n= 2 goe What is the wavelength of this light ? An emission line will appear in a spectrum if the source emits specific wavelengths of radiation. It is a sequence of lines of different color/frequency that represents a kind of "photograph" of the structure of an atom. Line emission spectra (atomic spectra) can be used to identify the presence of an element in a sample because no two elements produce the same line emission (atomic) spectrum. The first is also called a bright line spectrum and consists of a few brightly colored lines against a dark background. Explain why the line spectrum doÄs not contain a liné Corresponding to â¦ This means that the gas is receiving energy carried by the photons of light (as in E=hf). Each element has a specific set of energy levels, or allowed quantities of energy that electrons can absorb. 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text:
# How do you write word phrases for these expressions: w-5?

Sep 20, 2017

#### Answer:

Find the difference between a number and $5$

#### Explanation:

The operation being done is subtraction.

In words this is described by the word 'difference.

"What is the difference between a number, $w \mathmr{and} 5$?"


text:
# Laplace approximations in inference

## Lightweight uncertainties, especially for heavy neural nets

Second mode? I see no second mode.

Approximating probability distributions by a Gaussian with the same mode. Thanks to limit theorems this is not always a terrible idea, especially since Neural networks seem pretty keen to converge to Gaussians in various senses.

Specifically, locally approximates the posterior using a Gaussian $p(\theta \mid \mathcal{D}) \approx \mathcal{N}\left(\theta_{\mathrm{MAP}}, \Sigma\right).$

This makes it look like a special case of approximate method of moments, or integral probability metrics. Is that the case?

I am particularly keen to see it work for probabilistic neural nets. Such an approach is classic for neural nets . There are many variants of this technique for different assumptions. Laplace approximations have the attractive feature of providing estimates for forward and inverse problems .

The basic idea is that we hold $$x \in \mathbb{R}^{n}$$ fixed and use the Jacobian matrix $$J(x):=\left.\nabla_{\theta} f(x ; \theta)\right|_{\theta_{\mathrm{MAP}}} \in \mathbb{R}^{d \times k}$$, to the network as $f(x ; \theta) \approx f\left(x ; \theta_{\mathrm{MAP}}\right)+J(x)^{\top}\left(\theta-\theta_{\mathrm{MAP}}\right)$ where the variance is now justifed as a Taylor expansion. Under this approximation, since $$\theta$$ is a posteriori distributed as Gaussian $$\mathcal{N}\left(\theta_{\mathrm{MAP}}, \Sigma\right)$$, it follows that the marginal distribution over the network output $$f(x)$$ is also Gaussian, given by $p(f(x) \mid x, \mathcal{D}) \sim \mathcal{N}\left(f\left(x ; \theta_{\mathrm{MAP}}\right), J(x)^{\top} \Sigma J(x)\right).$ For more on this, see . It is essentially a gratis Laplace approximation in the sense that if I have fit the networks I can already calculate those Jacobians so I am probably 1 line of code away from getting some kind of uncertainty estimate. However, I have no particular guarantees to hope that it is well calibrated, because the simplifications were chosen a priori and might not be appropriate to the combination of model and data that I actually have.

Recent work here ties many of these ideas together:

## Learnable Laplace approximations

Agustinus Kristiadi and team have created various methods for low-overhead neural uncertainty quantification via Laplace approximation that have greater flexibility for adaptively choosing the type and manner of approximation. See, e.g. Painless Uncertainty for Deep Learning and their papers .

Kristiadi, Hein, and Hennig (2021) which generalises to learnable uncertainty to, for example, allow the distribution to reflect uncertainty about datapoints drawn far from the training distribution. They define an augmented Learnable Uncertainty Laplace Approximation (LULA) network $$\tilde{f}$$ with more parameters $$\tilde{\theta}=\theta_{\mathrm{MAP}}, \hat{\theta}.$$

Let $$f: \mathbb{R}^{n} \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$$ be an $$L$$-layer neural network with a MAP-trained parameters $$\theta_{\text {MAP }}$$ and let $$\widetilde{f}: \mathbb{R}^{n} \times \mathbb{R}^{\widetilde{d}} \rightarrow \mathbb{R}^{k}$$ along with $$\widetilde{\theta}_{\text {MAP }}$$ be obtained by adding LULA units. Let $$q(\widetilde{\theta}):=\mathcal{N}\left(\tilde{\theta}_{\mathrm{MAP}}, \widetilde{\Sigma}\right)$$ be the Laplace-approximated posterior and $$p\left(y \mid x, \mathcal{D} ; \widetilde{\theta}_{\mathrm{MAP}}\right)$$ be the (approximate) predictive distribution under the LA. Furthermore, let us denote the dataset sampled i.i.d. from the data distribution as $$\mathcal{D}_{\text {in }}$$ and that from some outlier distribution as $$\mathcal{D}_{\text {out }}$$, and let $$H$$ be the entropy functional. We construct the following loss function to induce high uncertainty on outliers while maintaining high confidence over the data (inliers): $\begin{array}{rl} \mathcal{L}_{\text {LULA }}\left(\widetilde{\theta}_{\text {MAP }}\right)&:=\frac{1}{\left|\mathcal{D}_{\text {in }}\right|} \sum_{x_{\text {in }} \in \mathcal{D}_{\text {in }}} H\left[p\left(y \mid x_{\text {in }}, \mathcal{D} ; \widetilde{\theta}_{\text {MAP }}\right)\right] \\ &-\frac{1}{\left|\mathcal{D}_{\text {out }}\right|} \sum_{x_{\text {out }} \in \mathcal{D}_{\text {out }}} H\left[p\left(y \mid x_{\text {out }}, \mathcal{D} ; \widetilde{\theta}_{\text {MAP }}\right)\right] \end{array}$ and minimize it w.r.t. the free parameters $$\widehat{\theta}$$.

I am assuming that by the entropy functional they mean the entropy of the normal distribution, $H(\mathcal{N}(\mu, \sigma)) = {\frac {1}{2}}\ln \left((2\pi \mathrm {e} )^{k}\det \left({\boldsymbol {\Sigma }}\right)\right)$ but this looks expensive due to that determinant calculation in a (large) $$d\times d$$ matrix. Or possibly they mean some general entropy with respect to some density $$p$$ $H(p)=\mathbb{E}_{p}\left[-\log p( x)\right]$ which I suppose one could estimate as $H(p)=\frac{1}{N}\sum_{i=1}^N \left[-\log p(x_i)\right]$ without taking that normal Laplace approximation at this step, if we could find the density, and assuming the $$x_i$$ were drawn from it.

The result is a slightly weird hybrid fitting procedure that requires two loss functions and which feels a little ad hoc, but maybe it works?

## By stochastic weight averaging

A Bayesian extension of a gradient descent trick called Stochastic Weight Averaging .

## For model selection

We can estimate marginal likelihood with respect to hyperparameters by Laplace approximation, apparently. See Immer et al. (2021).

## In function spaces

Where the Laplace approximations are Gaussian Processes over some index space. TBC. See Piterbarg and Fatalov (1995); Wacker (2017); Alexanderian et al. (2016); Alexanderian (2021) and possibly Solin and Särkkä (2020) and possibly the INLA stuff.

## INLA

Integrated nested Laplace approximation . TBC.

## Generalized Gauss-Newton and linearization

The generalized Gauss-Newton approximation (GGN) . replaces an expensive second order derivative by a product of first order derivatives. I am not sure if the Kronecker-factored Approximate Curvature (K-FAC) is a special case of GGN or a synonym.

Various papers analyze this .

## References

Alexanderian, Alen. 2021. arXiv:2005.12998 [Math], January.
Alexanderian, Alen, Noemi Petra, Georg Stadler, and Omar Ghattas. 2016. SIAM Journal on Scientific Computing 38 (1): A243–72.
Bishop, Christopher M. 2006. Pattern Recognition and Machine Learning. Information Science and Statistics. New York: Springer.
Breslow, N. E., and D. G. Clayton. 1993. Journal of the American Statistical Association 88 (421): 9–25.
Daxberger, Erik, Agustinus Kristiadi, Alexander Immer, Runa Eschenhagen, Matthias Bauer, and Philipp Hennig. 2021. In arXiv:2106.14806 [Cs, Stat].
Flaxman, Seth, Andrew Gordon Wilson, Daniel B Neill, Hannes Nickisch, and Alexander J Smola. 2015. “Fast Kronecker Inference in Gaussian Processes with Non-Gaussian Likelihoods.” In, 10.
Foong, Andrew Y. K., Yingzhen Li, José Miguel Hernández-Lobato, and Richard E. Turner. 2019. arXiv:1906.11537 [Cs, Stat], June.
Gorad, Ajinkya, Zheng Zhao, and Simo särkkä. 2020. “Parameter Estimation in Non-Linear State-Space Models by Automatic Differentiation of Non-Linear Kalman Filters.” In, 6.
Huggins, Jonathan H., Trevor Campbell, Mikołaj Kasprzak, and Tamara Broderick. 2018. arXiv:1809.09505 [Cs, Math, Stat], September.
Immer, Alexander, Matthias Bauer, Vincent Fortuin, Gunnar Rätsch, and Mohammad Emtiyaz Khan. 2021. arXiv:2104.04975 [Cs, Stat], June.
Immer, Alexander, Maciej Korzepa, and Matthias Bauer. 2021. In International Conference on Artificial Intelligence and Statistics, 703–11. PMLR.
Ingebrigtsen, Rikke, Finn Lindgren, and Ingelin Steinsland. 2014. Spatial Statistics, Spatial Statistics Miami, 8 (May): 20–38.
Izmailov, Pavel, Wesley J. Maddox, Polina Kirichenko, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson. 2020. In Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, 1169–79. PMLR.
Izmailov, Pavel, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson. 2018. March.
Khan, Mohammad Emtiyaz, Alexander Immer, Ehsan Abedi, and Maciej Korzepa. 2020. arXiv:1906.01930 [Cs, Stat], July.
Kristiadi, Agustinus, Matthias Hein, and Philipp Hennig. 2020. In ICML 2020.
———. 2021. In Uncertainty in Artificial Intelligence.
Lindgren, Finn, and Håvard Rue. 2015. Journal of Statistical Software 63 (i19): 1–25.
Long, Quan, Marco Scavino, Raúl Tempone, and Suojin Wang. 2013. Computer Methods in Applied Mechanics and Engineering 259 (June): 24–39.
MacKay, David J C. 2002. Information Theory, Inference & Learning Algorithms. Cambridge University Press.
MacKay, David J. C. 1992. Neural Computation 4 (3): 448–72.
Maddox, Wesley, Timur Garipov, Pavel Izmailov, Dmitry Vetrov, and Andrew Gordon Wilson. 2019. February.
Martens, James, and Roger Grosse. 2015. In Proceedings of the 32nd International Conference on Machine Learning, 2408–17. PMLR.
Opitz, Thomas, Raphaël Huser, Haakon Bakka, and Håvard Rue. 2018. Extremes 21 (3): 441–62.
Petersen, Kaare Brandt, and Michael Syskind Pedersen. 2012.
Piterbarg, V. I., and V. R. Fatalov. 1995. Russian Mathematical Surveys 50 (6): 1151.
Ritter, Hippolyt, Aleksandar Botev, and David Barber. 2018. In.
Rue, Håvard, Andrea Riebler, Sigrunn H. Sørbye, Janine B. Illian, Daniel P. Simpson, and Finn K. Lindgren. 2016. arXiv:1604.00860 [Stat], September.
Saumard, Adrien, and Jon A. Wellner. 2014. arXiv:1404.5886 [Math, Stat], April.
Snoek, Jasper, Oren Rippel, Kevin Swersky, Ryan Kiros, Nadathur Satish, Narayanan Sundaram, Md Mostofa Ali Patwary, Prabhat, and Ryan P. Adams. 2015. arXiv:1502.05700 [Stat], July.
Solin, Arno, and Simo Särkkä. 2020. Statistics and Computing 30 (2): 419–46.
Tang, Yanbo, and Nancy Reid. 2021. arXiv:2107.10885 [Math, Stat], July.
Wacker, Philipp. 2017. arXiv:1701.07989 [Math], April.
Wilson, Andrew Gordon, and Pavel Izmailov. 2020. February.

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text:
### On Optimal Systems

David Blackwell
Source: Ann. Math. Statist. Volume 25, Number 2 (1954), 394-397.

#### Abstract

For any sequence $x_1, x_2, \cdots$ of chance variables satisfying $| x_n | \leqq 1$ and $E(x_n\| x_1, \cdots, x_{n-1} \leqq -u(\max | x_n\| x_1, \cdots, x_{n-1})$, where $u$ is a fixed constant, $0 < u < 1$, and for any positive number $t$, $\mathrm{Pr} \{ \underset{n}\sup (x_1 + \cdots + x_n) \geqq t\} \leqq \big(\frac{1 - u}{1 + u}\big)^t.$ Equality holds for integral $t$ when $x_1, x_2, \cdots$ are independent with $\mathrm{Pr} \{x_n = 1\} = (1 - u)/2, \quad \mathrm{Pr} \{x_n = -1\} = (1 + u)/2.$ This has a simple interpretation in terms of gambling systems, and yields a new proof of Levy's extension of the strong law of large numbers to dependent variables [2], with an improved estimate for the rate of convergence.

First Page:
Full-text: Open access


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x2
Introduction
=

# Least Common Multiples

In a way, least common multiples are the opposite of greatest common divisors. Where divisors are always smaller than their parent numbers, multiples are always greater.


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# What is this flange on the engine?

Flying home from the Denver ComiCon, on a 737, I saw an angled flange on the engine. Could you describe its purpose?

• Jun 20 '14 at 4:16
• Good thing the photos are copyright. You wouldn't want someone steeling those. Nov 21 '14 at 23:56
• @kjmccarx I posted that because I improperly failed to accredit photos in another stack exchange and was flagged for copyright violation. I have cleaned up all of my questions and answers so that any photos I used are either my own, or copyright as creative commons and attributed correctly. Nov 22 '14 at 0:03
• @CGCampbell Fair enough. I was just giving you a hard time. Always better to be safe than sorry. Nov 22 '14 at 0:10
• You know what? This question is 5+ years old, they're my photos, they're in the PD, I attributed them... I'm good with it. Dec 7 '19 at 2:03

This is called a "strake" or "chine" (Aerodynamic fan cowl strake/chine, Nacelle strake/chine).
They allow the aircraft to generate more lift at lower speeds, which entails such positive consequences as lower stall speeds, lower landing speeds, lower take-off speeds and shorter runways.

Strakes should be used if the nacelles are mounted closely under the wing of the aircraft, see also Why don't all aircraft use/need nacelle chines?. Nacelles are usually mounted like that in order to

With an optimal arrangement and at high angles of attack, such vortex generators, which are known as nacelle strakes or chines, generate a powerful vortex that flows over the wing, where on a slat in front of said wing it delays airflow separation until the aircraft flies at greater angles of attack.

The idea dates back to 1971 (US Patent 3744745: Liftvanes):

in operational conditions wherein high angles of attack are encountered, such as in landing or takeoff, the vanes oppose the strong upwash around the nacelle, reducing the flow separation on its upper areas, and providing a strong downwash marked by marginal trailing vortices

DC-10 wind tunnel tests showed a significant loss in maximum lift coefficient in the flap deflected configurations, with landing slat extension, compared to predictions. This resulted in a stall speed increase of about 5 knots in the approach configuration. The initial wing stall occured behind the nacelles and forward of the inboard ailerons. The problem was traced by flow visualization techniques to the effects of the nacelle wake at high angles of attack and the absence of the slat in the vicinity of the nacelle pylons. The solution was developed in the NASA Ames Research Center 12 ft. pressurized tunnel and turned out to be a pair of strakes mounted forward on each side of the nacelles in planes about 45 degrees above the horizontal. The final strake shape was optimized in flight tests. The strakes are simply large vortex generators. The vortices mix the nacelle boundary layer air with the free stream and reduce the momentum loss in the wake. The vortices then pass just over the upper surface of the wing, continuing this mixing process. The counterrotating vortices also create a downwash over the wing region unprotected by the slat, further reducing the premature stall. The effect of the strakes is to reduce the required takeoff and landing field lengths by about 6%, a very large effect.

• Very nice, thorough answer with great references. Interesting stuff too! Jun 20 '14 at 12:08
• So what happens to airflow during cruise? Do these strakes have very little effect or drag for most of the flight? If they did, I would think they would be made in-flight adjustable. Jun 20 '14 at 14:04
• @PhilPerry It's claimed that the chines provide these benefits "without having a negative influence on the drag at cruising", and that "The longitudinal inclination of the vortex generator is selected such that the drag is as low as possible during cruising."
– user2168
Jun 20 '14 at 16:11
• @PhilPerry But, there are retractable chines (The Boeing Company: Retractable nacelle chine. US Patent 8087617). They say that there is some unwanted drag associated with fixed chines.
– user2168
Jun 20 '14 at 16:16
• @PhilPerry When cruising, the angle of attack is lower, the chine is more-or-less aligned with the airflow, and it does not generate large vortices. This is all made visible by condensation when flying on a humid day. In a sense, the reduction of angle of the attack in cruise is the adjustment you are thinking of. May 5 '15 at 15:20

Adding to user2168's answer, here are some pictures visualizing the principle of nacelle chines, also called nacell strakes or aerodynamic fan cowl strakes.
Please note that the vortices are always there, but they can only be seen in special conditions, see the end of this answer for an explanation.

As was stated above, some conditions have to be true if vortices are to be seen:

• To be more precise, the humidity, pressure and temperature of the ambient air has to "match" the pressure drop induced by the nacelle chines/strakes (this pressure drop is due to the flow acceleration, i.e. a higher flow velocity, caused by the nacelle chines/strakes). For passenger aircraft, this is usually the case on humid days not too far above the ground.

• Let's assume this is true. In addition, we assume that the air is incompressible, which is usually true for velocities $\mathit{Ma} \leq 0.3$ - which is the case for take-off and landing.
Now, the increase in flow velocity $c$ caused by the nacelle chines/strakes leads to the decrease of static pressure $p$, which leads to the decrease of static temperature $T$ (the total/stagnation pressure $p_0 \approx p_t$ and total/stagnation temperature $T_0 \approx T_t$ approximately stay constant, $p_0 \approx p_t \approx \mathit{const}$, $T_0 \approx T_t \approx \mathit{const}$).
If the temperature drops below the local dew point, condensation occurs: the gaseous water vapor transforms into liquid water droplet clouds - this is what can be seen in pictures.
As a side note: During condensation, the "enthalpy / latent heat of condensation" is released.

• Similar effects and explanations can be found e.g. in How does this vortex form inside a jet engine?, Why does condensation form on the wing especially during take-off and landing? and What caused a fluctuating cloud to form in a jet engine intake on a humid day?.

Image sources:

I do not believe the teaching in the patent is correct.

Instead of the downwash from the strake suppressing the wing upwash (and remember, the other side of the vortex adds to the wing upwash!) we believe that the vortex is well organized and can withstand the adverse pressure gradient of the wing better than the 'junk' wake of the nacelle without the strake.

The size and location of the strake determine when the vortex will burst, and hence govern when the inboard wings stalls. Ideally you would like this to be at the same time as the outboard wing. Too late and you get unacceptable pitch up. Too early and you lose low speed performance (higher stall speeds equated to higher speed schedules and longer takeoff field lengths).

A side note, the outboard strake is of little to no value at all, which is why Boeing airplanes only have strakes on the inboard side.

• hi and welcome. this is a comment on another answer (that includes a patent), on its own it's not clear how your answer answers the question itself.
– ymb1
Jan 14 '20 at 17:06
• @Dougball The answer contains valuable information. Please edit the answer so that it forms a comprehensive answer to the question. Jan 15 '20 at 3:13