scholarly journals Variance of the volume of random real algebraic submanifolds

2018 ◽  
Vol 371 (6) ◽  
pp. 4129-4192 ◽  
Author(s):  
Thomas Letendre
Keyword(s):  
Random Real

scholarly journals Random real trees

2006 ◽  
Vol 15 (1) ◽  
pp. 35-62 ◽  
Author(s):  
Jean-François Le Gall
Keyword(s):  
Random Real

scholarly journals Generalizing random real forcing for inaccessible cardinals

2019 ◽  
Vol 234 (2) ◽  
pp. 547-580
Author(s):  
Shani Cohen ◽  
Saharon Shelah
Keyword(s):  
Random Real

scholarly journals Denjoy, Demuth and density

2014 ◽  
Vol 14 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Laurent Bienvenu ◽  
Rupert Hölzl ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Computable Function ◽  
Density Theorem ◽  
Positive Density ◽  
Random Real ◽  
Open Problems ◽  
Covering Problems ◽  
Closed Class ◽  
Random Reals ◽  
Upper And Lower Derivatives ◽  
Computable Functions

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class 𝒞 ⊆ [0, 1] containing z has positive density at z. Under the stronger assumption that z is not LR-hard, we show that every such class has density one at z. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Löf random reals and K-trivial sets: the noncupping and covering problems. We say that f : [0, 1] → ℝ satisfies the Denjoy alternative at z ∈ [0, 1] if either the derivative f′(z) exists, or the upper and lower derivatives at z are +∞ and -∞, respectively. The Denjoy–Young–Saks theorem states that every function f : [0, 1] → ℝ satisfies the Denjoy alternative at almost every z ∈ [0, 1]. We answer a question posed by Kučera in 2004 by showing that a real z is computably random if and only if every computable function f satisfies the Denjoy alternative at z. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real zDA-random if every Markov computable function satisfies the Denjoy alternative at z. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin.24(3) (1983) 391–406) by showing that every Turing incomplete Martin-Löf random real is DA-random. The proof involves the notion of nonporosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Löf randomness.

A distributed clustering process

1991 ◽  
Vol 28 (04) ◽  
pp. 737-750 ◽  
Author(s):  
E. G. Coffman ◽  
P.-J. Courtois ◽  
E. N. Gilbert ◽  
Ph. Piret
Keyword(s):  
Probability Distributions ◽  
Distributed Clustering ◽  
Random Real ◽  
Real Numbers ◽  
Flow Process ◽  
Integer Points ◽  
The Laplace Transform ◽  
The Given ◽  
General Graphs ◽  
Total Resource

The points of a graph G will form clusters as a result of a flow process. Initially, points i of G own resources xi which are i.i.d. random real numbers. Afterwards, resources flow between points, but always from a point to a neighbor that has accumulated a larger total resource. Thus points with small resource tend to lose it and points with large resource tend to gain. Eventually the flow stops with only two kinds of points, nulls with no resource left and absorbers with such large resource that no neighbor can take it. The final resource at an absorber is a sum of certain initial resources xi , and the corresponding points i form one cluster. Analytical results are obtainable when G is the chain of integer points on the line. Probability distributions are derived for the distance between consecutive absorbers and the size of a cluster. Indeed these distributions do not involve the given distribution for the xi. The Laplace transform of the distribution of final resources at absorbers is derived but the distribution itself is obtained by a simulation. For general graphs G only partial results are obtained.

RELATIVIZING CHAITIN'S HALTING PROBABILITY

2005 ◽  
Vol 05 (02) ◽  
pp. 167-192 ◽  
Author(s):  
ROD DOWNEY ◽  
DENIS R. HIRSCHFELDT ◽  
JOSEPH S. MILLER ◽  
ANDRÉ NIES
Keyword(s):  
Computability Theory ◽  
Algorithmic Randomness ◽  
Jump Operator ◽  
Random Real ◽  
Primary Object ◽  
Object Of Study ◽  
Free Machine ◽  
Computable Functions ◽  
Omega Operator

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness.

scholarly journals Large level crossings of a random polynomial

1988 ◽  
Vol 1 (4) ◽  
pp. 259-269 ◽  
Author(s):  
Kambiz Farahmand
Keyword(s):  
Random Polynomial ◽  
Random Real ◽  
Expected Number ◽  
Level Crossings ◽  
Large Level ◽  
Real Value ◽  
Normally Distributed

We know the expected number of times that a polynomial of degree n with independent random real coefficients asymptotically crosses the level K, when K is any real value such that (K2/n)→0 as n→∞. The present paper shows that, when K is allowed to be large, this expected number of crossings reduces to only one. The coefficients of the polynomial are assumed to be normally distributed. It is shown that it is sufficient to let K≥exp(nf) where f is any function of n such that f→∞ as n→∞.

Adding one random real

1996 ◽  
Vol 61 (1) ◽  
pp. 80-90 ◽  
Author(s):  
Tomek Bartoszyński ◽  
Andrzej Rosłanowski ◽  
Saharon Shelah
Keyword(s):  
Measure Zero ◽  
Continuous Functions ◽  
Random Real ◽  
Cardinal Invariants

AbstractWe study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.

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