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Randomness and differentiability
2015
Transactions of the American Mathematical Society
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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
1988
International Journal of Mathematics and Mathematical Sciences
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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]
2015
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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]
2012
arXiv
pre-print
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2010 pre-print arXiv:1004.5108v1 fatcat:ekvunxdaxrchvovobyzqajc4qe
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]
2013
arXiv
pre-print
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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
2009
2009 International Conference on Machine Learning and Applications
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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
2012
Differential Equations & Applications
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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
2011
IEEE Transactions on Information Theory
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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
2013
Journal of Dynamics and Differential Equations
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2012 pre-print arXiv:1206.2329v1 fatcat:sfcuvwyelnbplbpfskuofveesqOther Versions
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]
2020
arXiv
pre-print
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2020 pre-print arXiv:2006.10926v1 fatcat:m2q2j4dcrbgo5kjfed3hhv4uga
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]
2020
arXiv
pre-print
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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]
2013
Lecture Notes in Computer Science
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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
2013
Proceedings of the American Mathematical Society
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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
1975
Proceedings of the IEEE
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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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