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Q I Minju - One of the best experts on this subject based on the ideXlab platform.
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the natural disposition theorem on n dimension Measurable Function
Journal of Anhui University, 2009Co-Authors: Q I MinjuAbstract:The definition of the entire dense spot of Lebesgue Measurable set was generalized from 1-dimension to n-dimension,so did the concept of the degree and the relative degree of the discontinuity of Function,according to the definition of the entire dense spot,the almost every point of n-dimension Lebesgue Measurable set being the entire dense spot was directly proved by Vitali cover and Lusin theorem,that n-dimension Lebesgue Measurable Function was almost everywhere equal to an almost everywhere continuous Function,which was gotten by the conception of the degree and the relative degree of the discontinuity of Function.
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the re understanding on Lebesgue non Measurable set
Journal of Shanghai Dianji University, 2007Co-Authors: Q I MinjuAbstract:A Lebesgue non-Measurable set with characteristic Function that is somewhat similar to Dirchlet Function is given,according to the properties of the essentialization theorem on Lebesgue Measurable Function.
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the re understanding on Lebesgue Measurable Function
Journal of Shanghai Dianji University, 2007Co-Authors: Q I MinjuAbstract:Based on that Lebesgue Measurable Function is almost everywhere equal to the almost everywhere continuous Function on ,an equivalent definition and several properties on Lebesgue Measurable Function are given,and so is the tentative plan on Lebesgue Measurable Function to enter the mathematics teaching of the engineering course.
N Nadirashvili - One of the best experts on this subject based on the ideXlab platform.
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on the restricted mean value property for Measurable Functions
1994Co-Authors: W Hansen, N NadirashviliAbstract:Let U be a domain in ℝ d , d ≥ 1, and r > 0 a real Function on U such that B(x,r(x)):= {y ∈ ℝd: ‖x - y‖ < r(x)} ⊂ U for every x ∈ U. A Lebesgue Measurable Function f on U which is bounded by some harmonic Function h ≥ 0 on U is called r-median if $$f\left( x \right) = \int {fd{\lambda _{B\left( {x,r\left( x \right)} \right)}}} $$ for every x ∈ U (⋋ Lebesgue measure on ℝ d ⋋ B = ⋋(B) -11 B ⋋ for every ball B).
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a converse to the mean value theorem for harmonic Functions
Acta Mathematica, 1993Co-Authors: Wolfhard Hansen, N NadirashviliAbstract:(A Lebesgue measure on Rd). The converse question to what extent this restricted mean value property implies harmonicity has a long history (we axe indebted to I. Netuka for valuable hints). Volterra [26] and Kellogg [20] noted first that a continuous Function f on the closure U of U satisfying (*) is harmonic on U. At least if U is regular there is a very elementary proof for this fact (see Burckel [7]): Let g be the difference between f and the solution of the Dirichlet problem with boundary value f . If g#0 , say a=sup g(U)> 0, choose x6{g=a} having minimal distance to the boundary. Then (*) leads to an immediate contradiction. In fact, for continuous Functions on U the question is settled for arbitrary harmonic spaces and arbitrary representing measures #x # ~ for harmonic Functions. If f is bounded on U and Borel Measurable the answer may be negative unless restrictions on the radius r(x) of the balls B ~ are imposed (Veech [23]): Let U = ] I , 1[, / (0 )=0 , f = i on ] -1 ,0[ , f= l on ]0, 1[, 0~B x for x?t0 (similarly in R d, d/>2)! There are various positive results, sometimes under restrictions on U, but always under restrictions on the Function x~-+r(x) (Feller [9], Akcoglu and Sharpe [1], Baxter [2] and [3], Heath [17], Veech [23] and [24]). For example Heath [17] showed for arbitrary U that a bounded Lebesgue Measurable Function on U having the restricted mean value property (.) is harmonic provided that , for some e>0, cd(x, CU)
Shi Rui - One of the best experts on this subject based on the ideXlab platform.
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On a class of operators in the hyperfinite $\mathrm{II}_1$ factor
Mathematica Scandinavica, 2017Co-Authors: Zhu Zhangsheng, Fang Junsheng, Shi RuiAbstract:Let $R$ be the hyperfinite $\mathrm {II}_1$ factor and let $u$, $v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2\pi i\theta } uv$ for an irrational number $\theta$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue Measurable Function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra
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On a class of operators in the hyperfinite ${\rm II}_1$ factor
2014Co-Authors: Zhu Zhangsheng, Fang Junsheng, Shi RuiAbstract:Let $R$ be the hyperfinite ${\rm II}_1$ factor and let $u,v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2\pi i\theta} uv$ for an irrational number $\theta$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue Measurable Function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra.Comment: 20pages. arXiv admin note: text overlap with arXiv:0708.1968 by other author
Pereira F.l. - One of the best experts on this subject based on the ideXlab platform.
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Conditions for the absence of jumps of the solution to the adjoint system of the maximum principle for optimal control problems with state constraints
'Pleiades Publishing Ltd', 2020Co-Authors: Arutyunov A.v., Karamzin D.y., Pereira F.l.Abstract:Properties of Lagrange multipliers from the Pontryagin maximum principle for problems with state constraints are investigated. Sufficient conditions for the continuity of the solution of the adjoint equation depending on how the extremal trajectory approaches the state constraint boundary are obtained. The proof uses the notion of closure with respect to measure of a Lebesgue Measurable Function and the Carathéodory theorem. © 2016, Pleiades Publishing, Ltd
F.l. Pereira - One of the best experts on this subject based on the ideXlab platform.
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Conditions for the absence of jumps of the solution to the adjoint system of the maximum principle for optimal control problems with state constraints
2016Co-Authors: Aram V. Arutyunov, D. Yu. Karamzin, F.l. PereiraAbstract:Properties of Lagrange multipliers from the Pontryagin maximum principle for problems with state constraints are investigated. Sufficient conditions for the continuity of the solution of the adjoint equation depending on how the extremal trajectory approaches the state constraint boundary are obtained. The proof uses the notion of closure with respect to measure of a Lebesgue Measurable Function and the Caratheodory theorem.