In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a ...concurrent vector field on submanifolds in Riemannian manifolds, is proved that a k-Yamabe soliton (Mn,g,vT,λ) on a hypersurface in the Euclidean space Rn+1 is contained either in a hypersphere or a hyperplane. We provide an example to support this study and all of the results in this paper can be implemented to Yamabe solitons for k-curvature with k=1.
To any completely integrable second-order system of real or complex partial differential equations:
y
x
k
1
x
k
2
=
F
k
1
,
k
2
x
1
,
⋯
,
x
n
,
y
,
y
x
1
,
…
,
y
x
n
with
1
⩽
k
1
,
k
2
⩽
n
and with
F
...k
1
,
k
2
=
F
k
2
,
k
1
in
n
⩾
2
̲
independent variables
(
x
1
,
…
,
x
n
)
and in one dependent variable
y
, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials to the associated submanifold of solutions, what the vanishing of the Hachtroudi curvature gives can be precisely translated to characterize when both families of Segre varieties and of conjugate Segre varieties associated to a Levi nondegenerate real analytic hypersurface
M
in
C
n
(
n
⩾
3
) can be straightened to be affine complex (conjugate) lines. In continuation to a previous paper devoted to the quite distinct
C
2
-case, this then characterizes in an effective way those hypersurfaces of
C
n
+
1
in higher complex dimension
n
+
1
⩾
3
that are locally biholomorphic to a piece of the
(
2
n
+
1
)
-dimensional Heisenberg quadric, without any special assumption on their defining equations.
In a seminal paper from 1995, Arya et al. Euclidean spanners: Short, thin, and lanky, in Proceedings of the 27th Annual ACM Symposium on Theory of Computing, ACM, New York, 1995, pp. 489--498 devised ...a construction that, for any set $S$ of $n$ points in $\mathbb R^d$ and any $\epsilon > 0$, provides a $(1+\epsilon)$-spanner with diameter $O(\log n)$, weight $O(\log^2 n) \cdot w(MST(S))$, and constant maximum degree. Another construction from the same work provides a $(1+\epsilon)$-spanner with $O(n)$ edges and diameter $O(\alpha(n))$, where $\alpha$ stands for the inverse Ackermann function. There are also a few other known constructions of $(1+\epsilon)$-spanners. Das and Narasimhan A fast algorithm for constructing sparse Euclidean spanners, in Proceedings of the 10th Annual ACM Symposium on Computational Geometry (SOCG), ACM, New York, 1994, pp. 132--139 devised a construction with constant maximum degree and weight $O(w(MST(S)))$, but the diameter may be arbitrarily large. In another construction by Arya et al., there is diameter $O(\log n)$ and weight $O(\log n) \cdot w(MST(S))$, but this construction may have arbitrarily large maximum degree. While these constructions address some important practical scenarios, they fail to address situations in which we are prepared to compromise on one of the parameters but cannot afford for this parameter to be arbitrarily large. In this paper we devise a novel unified construction that trades gracefully among the maximum degree, diameter, and weight. For a positive integer $k$ our construction provides a $(1+\epsilon)$-spanner with maximum degree $O(k)$, diameter $O(\log_k n + \alpha(k))$, weight $O(k \cdot \log_k n \cdot \log n) \cdot w(MST(S))$, and $O(n)$ edges. Note that for $k = O(1)$ this gives rise to maximum degree $O(1)$, diameter $O(\log n)$, and weight $O(\log^2 n) \cdot w(MST(S))$, which is one of the aforementioned results of Arya et al. For $k= n^{1/\alpha(n)}$ this gives rise to diameter $O(\alpha(n))$, weight $O(n^{1/\alpha(n)} \cdot \log n \cdot \alpha(n)) \cdot w(MST(S))$, and maximum degree $O(n^{1/\alpha(n)})$. In the corresponding result from Arya et al., the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our bound of $O(\log_k n + \alpha(k))$ on the diameter is optimal under the constraints that the maximum degree is $O(k)$ and the number of edges is $O(n)$. Similarly to the bound of Arya et al., our bound on the weight is optimal up to a factor of $\log n$. Our construction also provides a similar trade-off in the complementary range of parameters, i.e., when the weight should be smaller than $\log^2 n$, but the diameter is allowed to grow beyond $\log n$. Moreover, all our results apply to doubling metrics. En route to these results we devise optimal constructions of 1-spanners for general tree metrics, and we employ them to build our Euclidean spanners. Subsequent papers have utilized our constructions of 1-spanners for tree metrics to resolve a long-standing conjecture of Arya et al. PUBLICATION ABSTRACT
In this work, we consider
M
=
(
B
r
3
,
g
¯
)
as the Euclidean three-ball with radius
r
equipped with the metric
g
¯
=
e
2
h
,
conformal to the Euclidean metric, where the function
h
=
h
(
x
)
...depends only on the distance of
x
to the center of
B
r
3
. We show that if a free boundary CMC surface
Σ
in
M
satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of
Σ
, the positional conformal vector field
x
and its potential function
σ
,
then either
Σ
is a disk or
Σ
is an annulus rotationally symmetric. These results extend to the CMC case and to many other different conformally Euclidean spaces the main result obtained by Li and Xiong (J Geom Anal 28(4):3171–3182, 2018).
We propose an embedding-based framework for subsequence matching in time-series databases that improves the efficiency of processing subsequence matching queries under the Dynamic Time Warping (DTW) ...distance measure. This framework partially reduces subsequence matching to vector matching, using an embedding that maps each query sequence to a vector and each database time series into a sequence of vectors. The database embedding is computed offline, as a preprocessing step. At runtime, given a query object, an embedding of that object is computed online. Relatively few areas of interest are efficiently identified in the database sequences by comparing the embedding of the query with the database vectors. Those areas of interest are then fully explored using the exact DTW-based subsequence matching algorithm. We apply the proposed framework to define two specific methods. The first method focuses on time-series subsequence matching under unconstrained Dynamic Time Warping. The second method targets subsequence matching under constrained Dynamic Time Warping (cDTW), where warping paths are not allowed to stray too much off the diagonal. In our experiments, good trade-offs between retrieval accuracy and retrieval efficiency are obtained for both methods, and the results are competitive with respect to current state-of-the-art methods.
Noncommutative Euclidean spaces Dubois-Violette, Michel; Landi, Giovanni
Journal of geometry and physics,
08/2018, Volume:
130
Journal Article
Peer reviewed
Open access
We give a definition of noncommutative finite-dimensional Euclidean spaces Rn. We then remind our definition of noncommutative products of Euclidean spaces RN1 and RN2 which produces noncommutative ...Euclidean spaces RN1+N2. We solve completely the conditions defining the noncommutative products of the Euclidean spaces RN1 and RN2 and prove that the corresponding noncommutative unit spheres SN1+N2−1 are noncommutative spherical manifolds. We then apply these concepts to define “noncommutative” quaternionic planes and noncommutative quaternionic tori on which acts the classical quaternionic torus TH2=U1(H)×U1(H).
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