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Randomness and differentiability

Vasco Brattka, Joseph S. Miller, André Nies
2015 Transactions of the American Mathematical Society  
We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if  ...  each nondecreasing computable function [0,1]->R is differentiable at z. (2) We prove that a real number z in [0,1] is weakly 2-random if and only if each almost everywhere differentiable computable function  ...  We would like to thank Santiago Figueira, Jason Rute and Stijn Vermeeren for the careful reading of the paper, and Antonín Kučera for making Demuth's work accessible to us.  ... 
doi:10.1090/tran/6484 fatcat:23xczhbr4redvheyomwtv5hpli

Random fixed points and random differential inclusions

Nikolaos S. Papageorgiou
1988 International Journal of Mathematics and Mathematical Sciences  
Finally we consider a random differential inclusion with upper semicontinuous orientor field and establish the existence of random solutions.  ...  and Reich.  ...  RANDOM DIFFERENTIAL INCLUSIONS. by determining an a priori bound for the random solutions of So let x(.,.) be a random solution.  ... 
doi:10.1155/s0161171288000663 fatcat:wjfdl2pfhfhghkpyzh4de276sq

Randomness and differentiability in higher dimensions [article]

Alex Galicki, Daniel Turetsky
2015 arXiv   pre-print
We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables.  ...  Secondly, we show that weak 2-randomness is equivalent to differentiability of computable a.e. differentiable functions of several variables.  ...  on computable randomness and for valuable discussions.  ... 
arXiv:1410.8578v4 fatcat:4uiyfkgfpncs5lh3z2lgcaqxkm

Differentially- and non-differentially-private random decision trees [article]

Mariusz Bojarski, Anna Choromanska, Krzysztof Choromanski, Yann LeCun
2015 arXiv   pre-print
The random structure of the tree enables us to adapt these methods to a differentially-private setting thus we also propose differentially-private versions of all three schemes.  ...  We give upper-bounds on the generalization error and mathematically explain how the accuracy depends on the number of random decision trees.  ...  (n-dp) and differentially-private (dp) random forest with majority voting (RFMV ) and threshold averaging (RFTA).  ... 
arXiv:1410.6973v2 fatcat:rovqxz4qgvdphbkvndyvu4ebke

Analyzing Random Network Coding with Differential Equations and Differential Inclusions [article]

Dan Zhang, Narayan B. Mandayam
2012 arXiv   pre-print
We develop a framework based on differential equations (DE) and differential inclusions (DI) for analyzing Random Network Coding (RNC), as well as a nonlinear variant referred to as Random Coupon (RC),  ...  The DEDI framework serves as a powerful numerical and analytical tool to study RNC.  ...  CONCLUDING REMARKS We presented the DEDI framework, based on differential equations and/or differential inclusions, for analyzing the rank evolution of RNC and cardinality evolution of RC in a wireless  ... 
arXiv:1004.5108v2 fatcat:32kj4k2yxzbflpj6tgmcuklbsu

Random Lattice Gauge Theories and Differential Forms [article]

F. L. Teixeira
2013 arXiv   pre-print
We provide a brief overview on the application of the exterior calculus of differential forms to the ab initio formulation of field theories on random simplicial lattices.  ...  and a metric-dependent part.  ...  For these, the use of irregular ('random') lattices are often of interest to gain geometrical flexibility.  ... 
arXiv:1304.3485v2 fatcat:g6bnobkxbzgzhiqlbptgpqx7da

Randomized and Rank Based Differential Evolution

Onay Urfalioglu, Orhan Arikan
2009 2009 International Conference on Machine Learning and Applications  
This paper presents a novel extension of DE called Randomized and Rank based Differential Evolution (R2DE) to improve robustness and global convergence speed on multimodal problems by introducing two multiplicative  ...  Within population based EA's, Differential Evolution (DE) is a widely used and successful algorithm.  ...  The proposed method called Randomized and Rank based Differential Evolution (R2DE) integrates two distinct concepts in producing a new population of solution candidates: randomization and the utilization  ... 
doi:10.1109/icmla.2009.29 dblp:conf/icmla/UrfaliogluA09 fatcat:5mocd5d2fvh6vc2eu2pg2ic7nm

Some basic random fixed point theorems with PPF dependence and functional random differential equations

Bapurao C. Dhage
2012 Differential Equations & Applications  
In this paper two basic random fixed point theorems with PPF dependence are proved for random operators in separable Banach spaces with different domain and range spaces.  ...  The obtained abstract results are applied to certain nonlinear functional random differential equations for proving the existence results for random solutions with PPF dependence.  ...  The functional random differential equation (4.3) is not new to the theory of nonlinear functional random differential equations and the existence and uniqueness theorems for FRDE (4.3) are obtained  ... 
doi:10.7153/dea-04-11 fatcat:n6dagpeae5cilfhxzgarzkg4ga

Analyzing Random Network Coding With Differential Equations and Differential Inclusions

Dan Zhang, Narayan B. Mandayam
2011 IEEE Transactions on Information Theory  
We develop a framework based on differential equations (DE) and differential inclusions (DI) for analyzing Random Network Coding (RNC) in an arbitrary wireless network.  ...  We also briefly discuss its application in MAC and PHY adaptation and the extension to Random Coupon Selection.  ...  Extension to Random Coupon Selection Random Coupon Selection (RC) [11] is another transmission scheme based on randomized operations.  ... 
doi:10.1109/tit.2011.2170107 fatcat:bs2lvgub25gytgyhgjtreoaw7q

Random Attractors for Degenerate Stochastic Partial Differential Equations

Benjamin Gess
2013 Journal of Dynamics and Differential Equations  
We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian  ...  Applications include stochastic generalized porous media equations, stochastic generalized degenerate p-Laplace equations and stochastic reaction diffusion equations.  ...  In order to analyze the random differential equations obtained by stationary transformations based on the nonlinear Ornstein-Uhlenbeck processes constructed above, we will need some growth properties for  ... 
doi:10.1007/s10884-013-9294-5 fatcat:oz6ayttz3ra3zg6lfof5msr4dy

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators [article]

Sixian Jin, Kei Kobayashi
2020 arXiv   pre-print
SDEs to be considered are unique in two different aspects: i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; ii) the standard  ...  The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator  ...  Stochastic differential equations involving a random time change and associated L p bounds Throughout the rest of the paper, let (F t ) t≥0 be a filtration on the probability space (Ω, F, P) satisfying  ... 
arXiv:2006.10926v2 fatcat:yofa7fu3mbewrkcpqkzz7oir7e

Differential Privacy at Risk: Bridging Randomness and Privacy Budget [article]

Ashish Dandekar, Debabrota Basu, Stephane Bressan
2020 arXiv   pre-print
We analyse roles of the sources of randomness, namely the explicit randomness induced by the noise distribution and the implicit randomness induced by the data-generation distribution, that are involved  ...  The calibration of noise for a privacy-preserving mechanism depends on the sensitivity of the query and the prescribed privacy level.  ...  Acknowledgements We want convey a special thanks to Pierre Senellart at DI,École Normale Supérieure, Paris for his careful reading of our drafts and thoughtful interventions.  ... 
arXiv:2003.00973v2 fatcat:gdt3xf6ho5bgrjvsy5p5x7vssq

Random Projections, Graph Sparsification, and Differential Privacy [chapter]

Jalaj Upadhyay
2013 Lecture Notes in Computer Science  
This paper initiates the study of preserving differential privacy (DP) when the data-set is sparse.  ...  In certain sense, our approach is complementary to the Randomized sanitization for answering cut queries [17]: we use graph sparsification, while Randomized sanitization uses graph densification.  ...  The author would also like to thank the attendees of the C&O reading group and the members of CrySP and CACR for the useful discussions during the author's informal presentations.  ... 
doi:10.1007/978-3-642-42033-7_15 fatcat:faaothvrvrez3c6zjglvwnfxby

Schnorr randomness and the Lebesgue differentiation theorem

Noopur Pathak, Cristóbal Rojas, Stephen G. Simpson
2013 Proceedings of the American Mathematical Society  
Using this correspondence, we prove that a point x ∈ [0, 1] d is Schnorr random if and only if the Lebesgue Differentiation Theorem holds at x for all L 1 -computable functions f ∈ L 1 ([0, 1] d ).  ...  We exhibit a close correspondence between L 1 -computable functions and Schnorr tests.  ...  function f : [0, 1] → R of bounded variation is differentiable at x. (3) x is weakly 2-random if and only if every almost everywhere differentiable computable function f : [0, 1] → R is differentiable  ... 
doi:10.1090/s0002-9939-2013-11710-7 fatcat:ss3rivyrjfei5ex3ezsaulduoe

Random differential equations in science and engineering

R.F. Pawula, N.J. Bershad
1975 Proceedings of the IEEE  
In addition to covering the basics of E = ZR, with all the permutations and combinations, the book introduces, develops, and uses the concepts of modeling, arrays, matrices, etc., which too often remain  ...  In conjunction with Chapters 3 and 4, dc theory is developed by describing the current/voltage rela-  ...  Chapter 9 briefly discusses moment stability and Lyapunov stability of the solutions to random differential equations.  ... 
doi:10.1109/proc.1975.9959 fatcat:3mjxnhbaqzbajf5gwaqkfqovpy
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