Abstract The classical Cauchy surface area formula states that the surface area of the boundary ∂ K = Σ $\partial K=\Sigma $ of any n-dimensional convex body in the n-dimensional Euclidean space R n ...$\mathbb{R}^{n}$ can be obtained by the average of the projected areas of Σ along all directions in S n − 1 $\mathbb{S}^{n-1}$ . In this note, we generalize the formula to the boundary of arbitrary n-dimensional submanifold in R n $\mathbb{R}^{n}$ by introducing a natural notion of projected areas along any direction in S n − 1 $\mathbb{S}^{n-1}$ . This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013) by using a tubular neighborhood. We also define the projected r-volumes of Σ onto any r-dimensional subspaces and obtain a recursive formula for mean projected r-volumes of Σ.
The classical Cauchy surface area formula states that the surface area of the boundary
∂
K
=
Σ
of any
n
-dimensional convex body in the
n
-dimensional Euclidean space
R
n
can be obtained by the ...average of the projected areas of Σ along all directions in
S
n
−
1
. In this note, we generalize the formula to the boundary of arbitrary
n
-dimensional submanifold in
R
n
by introducing a natural notion of projected areas along any direction in
S
n
−
1
. This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83,
2013
) by using a tubular neighborhood. We also define the projected
r
-volumes of Σ onto any
r
-dimensional subspaces and obtain a recursive formula for mean projected
r
-volumes of Σ.
We circumvent two gaps in the Gromoll–Walschap classification of metric fibrations from Euclidean spaces by presenting an alternative for the respective part of the proof. Combining it with the work ...of Florit–Goertsches–Lytchak–Töben, the classification of Riemannian foliations on Euclidean spaces is completed.
The connection between Gaussian random fields and Markov random fields has been well-established in Euclidean spaces, with Matérn covariance functions playing a pivotal role. In this paper, we ...explore the extension of this link to circular spaces and uncover different results. It is known that Matérn covariance functions are not always positive definite on the circle; however, the circular Matérn covariance functions are shown to be valid on the circle and are the focus of this paper. For these circular Matérn random fields on the circle, we show that the corresponding Markov random fields can be obtained explicitly on equidistance grids. Consequently, the equivalence between the circular Matérn random fields and Markov random fields is then exact and this marks a departure from the Euclidean space counterpart, where only approximations are achieved. Moreover, the key motivation in Euclidean spaces for establishing such link relies on the assumption that the corresponding Markov random field is sparse. We show that such sparsity does not hold in general on the circle. In addition, for the sparse Markov random field on the circle, we derive its corresponding Gaussian random field.
In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the ...Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.
We consider the classic Facility Location,
k
-Median, and
k
-Means problems in metric spaces of doubling dimension
d
. We give nearly linear-time approximation schemes for each problem. The ...complexity of our algorithms is
Õ(2
(1/ε)
O(d2)
n)
, making a significant improvement over the state-of-the-art algorithms that run in time
n
(d/ε)
O(d)
.
Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting
k
-Median and
k
-Means and efficient bicriteria approximation schemes for
k
-Median with outliers,
k
-Means with outliers and
k
-Center.
We consider the problem of minimizing the sum of non-smooth convex functions in non-Euclidean spaces, e.g., probability simplex, via only local computation and communication on an undirected graph. ...We propose two algorithms motivated by mass–spring–damper dynamics. The first algorithm uses an explicit update that computes subgradients and Bregman projections only, and matches the convergence behavior of centralized mirror descent. The second algorithm uses an implicit update that solves a penalized subproblem at each iteration, and achieves the iteration complexity of O(1∕K). The results are also demonstrated via numerical examples.
In this work, we study a linear operator f on a pre-Euclidean space V by using properties of a corresponding graph. Given a basis B of V, we present a decomposition of V as an orthogonal direct sum ...of certain linear subspaces {Ui}i∈I, each one admitting a basis inherited from B, in such way that f=∑i∈Ifi. Each fi is a linear operator satisfying certain conditions with respect to Ui. Considering this new hypothesis, we assure the existence of an isomorphism between the graphs of f relative to two different bases. We also study the minimality of V by using the graph of f relative to B.
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