General measure theory

Federer, Herbert

doi:10.1007/978-3-642-62010-2_3

Cited by 967 publications

(1,762 citation statements)

References 0 publications

Supporting

Mentioning

1,735

Contrasting

Unclassified

Order By: Relevance

“…In the smooth case, the volume of Vε(B) is thus a polynomial in ε, and its coefficients are multiples of the curvature measures of B. H. Federer [9] actually showed that the volume of Vε(B) is always a polynomial in ε for ε < ρ, even if the boundary of V is not smooth. The coefficients of this polynomial thus provide a way to generalize the definition of curvature measures as soon as ρ is strictly positive.…”

Section: A First Approachmentioning

confidence: 99%

Restricted delaunay triangulations and normal cycle

Cohen-Steiner

Morvan

2003

Proceedings of the Nineteenth Annual Symposium on Computational Geometry

343

176

View full text Add to dashboard Cite

We address the problem of curvature estimation from sampled smooth surfaces. Building upon the theory of normal cycles, we derive a definition of the curvature tensor for polyhedral surfaces. This definition consists in a very simple and new formula. When applied to a polyhedral approximation of a smooth surface, it yields an efficient and reliable curvature estimation algorithm. Moreover, we bound the difference between the estimated curvature and the one of the smooth surface in the case of restricted Delaunay triangulations.

show abstract

Section: A First Approachmentioning

confidence: 99%

Restricted delaunay triangulations and normal cycle

Cohen-Steiner

Morvan

2003

Proceedings of the Nineteenth Annual Symposium on Computational Geometry

343

176

View full text Add to dashboard Cite

show abstract

“…Therefore, if N supports no pseudo-holomorphic S 2 's, then we can apply the Federer's dimension reduction argument (cf. [10]) to prove Theorem 1.6. Assume that (N, J, h) doesn't support any pseudo-holomorphic S 2 .…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…We can follow the Federer's dimension reduction argument (cf. [10,31]) to obtain the result. For simplicity, we only indicate that (1):…”

Section: Blow-up Analysis For J-holomorphicmentioning

confidence: 99%

Regularity and Blow-Up Analysis for J-Holomorphic Maps

Wang

2003

Commun. Contemp. Math.

View full text Add to dashboard Cite

If u ∈ H 1 (M, N ) is a weakly J-holomorphic map from a compact without boundary almost hermitian manifold (M, j, g) into another compact without boundary almost hermitian manifold (N, J, h Let (M, j, g) (respectively (N, J, h)) be a smooth hermitian almost complex manifold with dimension 2m (respectively 2n). Assume further that(N, h) is compact without boundary and isometrically embedded into some euclidean R k via the Nash's embedding theorem. Denote the Sobolev spaceis said to be weakly (j, J)-holomorphic map (or J-holomorphic map for abbreviation) if du preserves the almost complex structures in the sense: In this paper we are interested in regularity for weakly J-holomorphic Now we state our first theoremOur idea to prove theorem A follows from the two new observations: (1) Under the assumption that u(B r (x)) is contained in a coordinate chart U of N , we can use the local coordinate frame on U to express J as SO (2n) For m ≥ 2, observe that M 2,2m−2 is a scaling-invariant subspace of where F t : M → M is a parameter family of diffeomorphisms generated by X. It follows from Proposition 3.2 that any stationary J-holomorphic map satisfies the energy monotonicity inequality: there is a C 0 = C 0 (M, j, g) > 0such that e C0r r 2−2mBr (x)for any x ∈ M and 0 < r ≤ R ≤ R 0 = R 0 (M, j, g). A direct consequence of (1.5) is: for any x ∈ M and 0 < r ≤ R 0 ,Hence theorem A yields the partial regularity for stationary J-holomorphic maps, which is an analogy to the partial regularity for stationary harmonic maps. More precisely,Based on both the energy monotonicity inequality (1.5) and the small energy regularity theorem A, we find that the blow-up techniques for stationary harmonic maps developed by Lin [L] can be modified to study the convergence issues for sequences of stationary J-holomorphic maps.From now on, we call a nonconstant smooth J-holomorphic map ω :as a pseudo-holomorphic S 2 , here j 0 is the standard complex structure on S 2 . We prove stationary J-holomorphic maps which converges weakly to u ∈ H 1 (M, N ). Then u is a weakly J-holomorphic map and there exists a closedas convergence of Radon measures, for some nonnegative Radon measure νThe main difference between our proof of theorem D and §2 of [L] is that we need to verify that the concentration set Σ is j-holomorphic (2m − 2)-rectifiable set. Once we achieve this, then both the conformality and the removablity of isolated singularity for pseudo-holomorphic curves (cf. [Y] [PW]) guarantee that the restriction of a bubble on (T x Σ) ⊥ can be lifted to be a pseudo-holomorphic S 2 .It is a very important problem to quantify the density function θ for the defect measure ν in the content of blow-up analysis for stationary harmonic The ideas to prove theorem E are based on the observations that on The paper is written as follows. In §2, we prove theorem A and C. In §3, we prove theorem D, E and F. In §4, we discuss the relationship between J-holomorphic maps and harmonic maps in the case either (M, j, g) and (N, J, h) are almost Kähler manifo...

show abstract

“…For classical bosons, there is yet another phase transition of the second kind, similar to that occurring for ideal gases, at the temperature that can be calculated from the relation fvd~ f dp = N, (8) e(p2/(2m)+W(z,T)-I~o)/(kT)-I where /Jo = minz W(x, T) and V is the volume of the system. (1) becomes the Vlasov-Poisson-BoItzmann equation, is the phase transition temperature.…”

Section: + --0 Xer 3 Per 3 M Ozmentioning

confidence: 99%

Phase transition for classical bosons, fermions, and ordinary classical particles

Маслов

1998

Math Notes

View full text Add to dashboard Cite

On one hand, this means that the original mapping in the theorem above possesses the differentiability property, and on the other hand, this leads to the natural question of the extendability of this theorem to infinite-dimensional mappings.Remark 3. The following close (but weaker) statement was obtained in the recent paper [10]: if f is a function measurable with respect to a Gaussian measure p on a separable Banaeh space X defined on a #-measurable set A and satisfying there the Lipschitz condition along the Cameron-Martin space E, then there exists a p-measurable function on X satisfying the same condition and coinciding with f #almost everywhere. This statement follows from the theorem above, since f has a Borel modification satisfying the same condition on a Boret subset B C A with p(B) = #(A). Our theorem implies that the same is true if X is a locally convex space with the Radon Gaussian measure p (in this case, according to [11], the measure # is supported by a Suslin linear subspace).In the preceding paper, for classical fermions we considered the case of an attracting interaction potential V(xi-xj) attaining its minimum at the point xi = x I 9 However, actual interaction potentials V(z) > -A, A > 0, are repulsive if the particles are at the same point and attain their minimum at some point a :~ 0.

show abstract

General measure theory

Cited by 967 publications

References 0 publications

Restricted delaunay triangulations and normal cycle

Restricted delaunay triangulations and normal cycle

Regularity and Blow-Up Analysis for J-Holomorphic Maps

Phase transition for classical bosons, fermions, and ordinary classical particles

Contact Info

Product

Resources

About