On the canonical embedding of a JB*‐triple into its bidual

2018 ◽  
Vol 291 (17-18) ◽  
pp. 2578-2584
Author(s):  
José M. Isidro
Keyword(s):  
Canonical Embedding

scholarly journals A COMPARISON OF HOFER'S METRICS ON HAMILTONIAN DIFFEOMORPHISMS AND LAGRANGIAN SUBMANIFOLDS

2003 ◽  
Vol 05 (05) ◽  
pp. 803-811 ◽  
Author(s):  
YARON OSTROVER
Keyword(s):  
Symplectic Manifold ◽  
Isometric Embedding ◽  
Lagrangian Submanifolds ◽  
Canonical Embedding ◽  
Hamiltonian Diffeomorphisms

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold (M,ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π2(M) = 0, the canonical embedding of Ham (M) into ℒ, f ↦ graph (f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

Invariants of a General Branched Cover of P1

2020 ◽  
Author(s):  
Gabriel Bujokas ◽  
Anand Patel
Keyword(s):  
Canonical Embedding ◽  
Branched Cover ◽  
The Ideal

Abstract We investigate the resolution of a general branched cover $\alpha \colon C \to \mathbf{P}^1$ in its relative canonical embedding $C \subset \mathbf{P} E$. We conjecture that the syzygy bundles appearing in the resolution are balanced for a general cover, provided that the genus is sufficiently large compared to the degree. We prove this for the Casnati–Ekedahl bundle, or bundle of quadrics$F$—the 1st bundle appearing in the resolution of the ideal of the relative canonical embedding. Furthermore, we prove the conjecture for all syzygy bundles in the resolution when the genus satisfies $g = 1 \mod d$.

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

scholarly journals K3 curves with index $$k>1$$

2021 ◽  
Author(s):  
Ciro Ciliberto ◽  
Thomas Dedieu
Keyword(s):  
Complete Intersection ◽  
Fano Varieties ◽  
Canonical Embedding ◽  
Geometric Means ◽  
Moduli Stack

AbstractLet $$\mathcal {KC}_g ^k$$ KC g k be the moduli stack of pairs (S, C) with S a K3 surface and $$C\subseteq S$$ C ⊆ S a genus g curve with divisibility k in $$\mathrm {Pic}(S)$$ Pic ( S ) . In this article we study the forgetful map $$c_g^k:(S,C) \mapsto C$$ c g k : ( S , C ) ↦ C from $$\mathcal {KC}_g ^k$$ KC g k to $${\mathcal {M}}_g$$ M g for $$k>1$$ k > 1 . First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when S is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending C in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether $$c_g^k$$ c g k dominates the locus in $${\mathcal {M}}_g$$ M g of k-spin curves with the appropriate number of independent sections. We are able to do this only when S is a complete intersection, and obtain in these cases some classification results for spin curves.

scholarly journals The Operator Space Projective Tensor Product: Embedding into the Second Dual and Ideal Structure

2013 ◽  
Vol 57 (2) ◽  
pp. 505-519 ◽  
Author(s):  
Ranjana Jain ◽  
Ajay Kumar
Keyword(s):  
Tensor Product ◽  
Bilinear Form ◽  
Operator Space ◽  
Ideal Structure ◽  
Operator Spaces ◽  
Canonical Embedding ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Second Dual ◽  
Continuous Inverse

AbstractWe prove that, for operator spaces V and W, the operator space V** ⊗hW** can be completely isometrically embedded into (V ⊗hW)**, ⊗h being the Haagerup tensor product. We also show that, for exact operator spaces V and W, a jointly completely bounded bilinear form on V × W can be extended uniquely to a separately w*-continuous jointly completely bounded bilinear form on V× W**. This paves the way to obtaining a canonical embedding of into with a continuous inverse, where is the operator space projective tensor product. Further, for C*-algebras A and B, we study the (closed) ideal structure of which, in particular, determines the lattice of closed ideals of completely.

scholarly journals The Ring-LWE Problem in Lattice-Based Cryptography: The Case of Twisted Embeddings

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1108
Author(s):  
Jheyne N. Ortiz ◽  
Robson R. de Araujo ◽  
Diego F. Aranha ◽  
Sueli I. R. Costa ◽  
Ricardo Dahab
Keyword(s):  
Public Key Cryptography ◽  
Number Fields ◽  
Canonical Embedding ◽  
Worst Case ◽  
Average Case ◽  
Twist Factor ◽  
Algebraic Lattices ◽  
Cyclotomic Number Fields ◽  
Lattice Based Cryptography ◽  
Respective Number

Several works have characterized weak instances of the Ring-LWE problem by exploring vulnerabilities arising from the use of algebraic structures. Although these weak instances are not addressed by worst-case hardness theorems, enabling other ring instantiations enlarges the scope of possible applications and favors the diversification of security assumptions. In this work, we extend the Ring-LWE problem in lattice-based cryptography to include algebraic lattices, realized through twisted embeddings. We define the class of problems Twisted Ring-LWE, which replaces the canonical embedding by an extended form. By doing so, we allow the Ring-LWE problem to be used over maximal real subfields of cyclotomic number fields. We prove that Twisted Ring-LWE is secure by providing a security reduction from Ring-LWE to Twisted Ring-LWE in both search and decision forms. It is also shown that the twist factor does not affect the asymptotic approximation factors in the worst-case to average-case reductions. Thus, Twisted Ring-LWE maintains the consolidated hardness guarantee of Ring-LWE and increases the existing scope of algebraic lattices that can be considered for cryptographic applications. Additionally, we expand on the results of Ducas and Durmus (Public-Key Cryptography, 2012) on spherical Gaussian distributions to the proposed class of lattices under certain restrictions. As a result, sampling from a spherical Gaussian distribution can be done directly in the respective number field while maintaining its format and standard deviation when seen in Zn via twisted embeddings.

On the iterated normed duals

2021 ◽  
Vol 1 (1) ◽  
pp. 29-46
Author(s):  
Nikica Uglešić
Keyword(s):  
Dual Space ◽  
Canonical Embedding ◽  
Dual Spaces ◽  
The Third ◽  
Direct Sum ◽  
Second Dual ◽  
The Right

Several properties of the normed Hom-functor (dual) D and its iterations Dn are exhibited. For instance, D turns every canonical embedding (into the second dual space) to a retraction (of the third dual onto the first one) having for the right inverse the appropriate canonical embedding (of the first dual space into the third one). Some consequences to the direct-sum presentations and quotients of higher dual spaces are considered.

scholarly journals On the problem of non-smoothness of non-reflexive second conjugate spaces

1975 ◽  
Vol 12 (3) ◽  
pp. 407-416 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Canonical Embedding ◽  
Reflexive Banach Spaces ◽  
Closed Linear Subspace ◽  
Conjugate Space ◽  
The Family

We prove that if E is a Banach space which has a subspace G such that the conjugate space G* contains a proper norm closed linear subspace V of characteristic 1, then E** is not smooth and there exist in πE(E) points of non-smoothness for E**, where πE: E → E** is the canonical embedding. We show that the spaces E having such a subspace G constitute a large proper subfamily of the family of all non-reflexive Banach spaces.

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