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Minimizing movements for dislocation dynamics with a mean curvature term

Nicolas Forcadel, Aurélien Monteillet (2009)

ESAIM: Control, Optimisation and Calculus of Variations

We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution...

Minkowski sums of Cantor-type sets

Kazimierz Nikodem, Zsolt Páles (2010)

Colloquium Mathematicae

The classical Steinhaus theorem on the Minkowski sum of the Cantor set is generalized to a large class of fractals determined by Hutchinson-type operators. Numerous examples illustrating the results obtained and an application to t-convex functions are presented.

Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model

Florian Fuchs, Robert Stelzer (2013)

ESAIM: Probability and Statistics

We consider strictly stationary infinitely divisible processes and first extend the mixing conditions given in Maruyama [Theory Probab. Appl. 15 (1970) 1–22] and Rosiński and Żak [Stoc. Proc. Appl. 61 (1996) 277–288] from the univariate to the d-dimensional case. Thereafter, we show that multivariate Lévy-driven mixed moving average processes satisfy these conditions and hence a wide range of well-known processes such as superpositions of Ornstein − Uhlenbeck (supOU) processes or (fractionally integrated)...

Mixing on rank-one transformations

Darren Creutz, Cesar E. Silva (2010)

Studia Mathematica

We prove that mixing on rank-one transformations is equivalent to "the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums". In particular, all polynomial staircase transformations are mixing.

Mixing properties of nearly maximal entropy measures for ℤ d shifts of finite type

E. Robinson, Ayşe Şahin (2000)

Colloquium Mathematicae

We prove that for a certain class of ℤ d shifts of finite type with positive topological entropy there is always an invariant measure, with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.

Möbius fitting aggregation operators

Anna Kolesárová (2002)

Kybernetika

Standard Möbius transform evaluation formula for the Choquet integral is associated with the 𝐦𝐢𝐧 -aggregation. However, several other aggregation operators replacing 𝐦𝐢𝐧 operator can be applied, which leads to a new construction method for aggregation operators. All binary operators applicable in this approach are characterized by the 1-Lipschitz property. Among ternary aggregation operators all 3-copulas are shown to be fitting and moreover, all fitting weighted means are characterized. This new method...

Mod 2 normal numbers and skew products

Geon Ho Choe, Toshihiro Hamachi, Hitoshi Nakada (2004)

Studia Mathematica

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by d ₙ ( x ) : = ∑ i = 1 n 1 E ( 2 i - 1 x ) ( m o d 2 ) , where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if N - 1 ∑ n = 1 N d ₙ ( x ) converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that N - 1 ∑ n = 1 N d ₙ ( x ) converges a.e. and the limit equals 1/3 or 2/3 depending on x.

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