Upper Bound for Isometric Embeddings
نویسنده
چکیده
The isometric embeddings 2;K → p;K (m ≥ 2, p ∈ 2N) over a field K ∈ {R,C,H} are considered, and an upper bound for the minimal n is proved. In the commutative case (K = H) the bound was obtained by Delbaen, Jarchow and Pe lczyński (1998) in a different way. Let K be one of three fields R,C,H (real, complex or quaternion). Let K be the K-linear space consisting of columns x = [ξi] n 1 , ξi ∈ K, with the right (for definiteness) multiplication by scalars α ∈ K. The normed space p;K is K provided with the norm ‖x‖p = ( n ∑ k=1 |ξi| )1/p , 1 ≤ p < ∞. For p = 2 this space is Euclidean, ‖x‖2 = √ 〈x, x〉, where the inner product 〈x, y〉 of x and a vector y = [ηi]1 is 〈x, y〉 = n ∑
منابع مشابه
The Spectrum and Isometric Embeddings of Surfaces of Revolution For Gus and Sonia
A sharp upper bound on the first S invariant eigenvalue of the Laplacian for S invariant metrics on S is used to find obstructions to the existence of S equivariant isometric embeddings of such metrics in (R, can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in (R, can). This leads to ge...
متن کاملThe Spectrum and Isometric Embeddings of Surfaces of Revolution
An upper bound on the first S invariant eigenvalue of the Laplacian for S invariant metrics on S is used to find obstructions to the existence of isometric embeddings of such metrics in (R, can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the surface of revolution cannot be isometrically embedded in (R, can). This leads to a generalization of a...
متن کاملLower bounds for projective designs, cubature formulas and related isometric embeddings
Yudin’s lower bound [21] for the spherical designs is generalized to the cubature formulas on the projective spaces over a field K ⊂ {R, C, H} and thus to isometric embeddings l 2;K → l p;K with p ∈ 2N. For large p and in some other situations this is essentially better than those known before. AMS Classification: 46B04, 05B30
متن کاملImproved Linear Embeddings via Lagrange Duality
Near isometric orthogonal embeddings to lower dimensions are a fundamental tool in data science and machine learning. In this paper, we present the construction of such embeddings that minimizes the maximum distortion for a given set of points. We formulate the problem as a non convex constrained optimization problem. We first construct a primal relaxation and then use the theory of Lagrange du...
متن کاملHamming dimension of a graph - The case of Sierpiński graphs
The Hamming dimension of a graph G is introduced as the largest dimension of a Hamming graph into which G embeds as an irredundant induced subgraph. An upper bound is proved for the Hamming dimension of Sierpiński graphs S k , k ≥ 3. The Hamming dimension of S 3 grows as 3. Several explicit embeddings are constructed along the way, in particular into products of generalized Sierpiński triangle ...
متن کامل