Introduction to Positive Lebesgue

What is/are Positive Lebesgue?

Positive Lebesgue - It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2. ^[1] In a previous article (Wiens, 1991) we established a maximin property, with respect to the power of the test for Lack of Fit, of the absolutely continuous uniform ‘design’ on a design space which is a subset of R q with positive Lebesgue measure. ^[2] It has been previously shown by Londner and Olevskii (Stud Math 255(2):183–191, 2014) that there exists a subset of the circle, of positive Lebesgue measure, so that every set $$\Lambda $$Λ which contains, for arbitrarily large N, an arithmetic progressions of length N and step $$\ell =O\left( N^{\alpha }\right) $$ℓ=ONα, $$\alpha <1$$α<1, cannot be a Riesz sequence in the $$L^{2}$$L2 space over that set. ^[3] Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subset of R with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. ^[4] In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. ^[5] In this paper, we study the problem of how large the Hausdorff dimension of E must be in order for the set of distinct noncongruent k-simplices in E (that is, noncongruent point configurations of k + 1 points from E) to have positive Lebesgue measure. ^[6] It is then proven that, for some set of parameter values having positive Lebesgue measure, the ω -limit set of each point of the trapping region is contained in a invariant set which is just the closure of the unstable manifold of the hyperbolic fixed point of the map. ^[7] The paper is devoted to the special case of the average taxicab distance function given by integration of the taxicab distance on a compact subset of positive Lebesgue measure in the Euclidean coordinate space. ^[8] More precisely, for a given subset of the parameter space, we develop an algorithm that can decide whether the set has a subset of positive Lebesgue measure for which the Markov chain is unstable. ^[9] We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. ^[10] We first show that for a $C^1$ generic diffeomorphism, a Dirac invariant measure whose \emph{statistical basin of attraction} is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. ^[11] For the 5 and 7 body problem, the set of parameters (masses and positions) leading to relative equilibria has positive Lebesgue measure. ^[12] It is shown that if (u, v) is a pair of positive Lebesgue measurable solutions of this integral system, then $$\frac{1}{{p - 1}} + \frac{1}{{q - 1}} = \frac{\lambda }{n}$$1p−1+1q−1=λn, which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations. ^[13] A theorem of Steinhaus states that if $E\subset \mathbb R^d$ has positive Lebesgue measure, then the difference set $E-E$ contains a neighborhood of $0$. ^[14]

positive lebesgue measure

It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2. ^[1] In a previous article (Wiens, 1991) we established a maximin property, with respect to the power of the test for Lack of Fit, of the absolutely continuous uniform ‘design’ on a design space which is a subset of R q with positive Lebesgue measure. ^[2] It has been previously shown by Londner and Olevskii (Stud Math 255(2):183–191, 2014) that there exists a subset of the circle, of positive Lebesgue measure, so that every set $$\Lambda $$Λ which contains, for arbitrarily large N, an arithmetic progressions of length N and step $$\ell =O\left( N^{\alpha }\right) $$ℓ=ONα, $$\alpha <1$$α<1, cannot be a Riesz sequence in the $$L^{2}$$L2 space over that set. ^[3] Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subset of R with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. ^[4] In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. ^[5] In this paper, we study the problem of how large the Hausdorff dimension of E must be in order for the set of distinct noncongruent k-simplices in E (that is, noncongruent point configurations of k + 1 points from E) to have positive Lebesgue measure. ^[6] It is then proven that, for some set of parameter values having positive Lebesgue measure, the ω -limit set of each point of the trapping region is contained in a invariant set which is just the closure of the unstable manifold of the hyperbolic fixed point of the map. ^[7] The paper is devoted to the special case of the average taxicab distance function given by integration of the taxicab distance on a compact subset of positive Lebesgue measure in the Euclidean coordinate space. ^[8] More precisely, for a given subset of the parameter space, we develop an algorithm that can decide whether the set has a subset of positive Lebesgue measure for which the Markov chain is unstable. ^[9] We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. ^[10] We first show that for a $C^1$ generic diffeomorphism, a Dirac invariant measure whose \emph{statistical basin of attraction} is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. ^[11] For the 5 and 7 body problem, the set of parameters (masses and positions) leading to relative equilibria has positive Lebesgue measure. ^[12] A theorem of Steinhaus states that if $E\subset \mathbb R^d$ has positive Lebesgue measure, then the difference set $E-E$ contains a neighborhood of $0$. ^[13]