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Bhargab B Bhattacharya - One of the best experts on this subject based on the ideXlab platform.
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Canonical Embedding of rectangular duals with applications to vlsi floorplanning
Design Automation Conference, 1992Co-Authors: Susmita Surkolay, Bhargab B BhattacharyaAbstract:The notion of equivalent Embedding of rectangular duals is introduced, leading to a new concept of Canonical Embedding of a rectangular dual; this is a floorplan corresponding to a given neighborhood graph such that the number of directed cycles in its channel digraph is minimum. Strongly maximal rectangular hierarchy (sMRH) in nonslicible floorplans is then defined. The Canonical form of any arbitrary floorplan consists of at most one nonslicing core for each member of sMRH. Such an Embedding therefore represents a floorplan with minimum deviations from a slicing structure. An O(n/sup 2/) algorithm for realizing a Canonical Embedding is also presented. Canonical Embedding lends deep insight to the yet unsolved problem of characterizing inherent nonslicibility and motivates design for slicibility. It also makes determination of safe routing order simple. >
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DAC - Canonical Embedding of rectangular duals with applications to VLSI floorplanning
[1992] Proceedings 29th ACM IEEE Design Automation Conference, 1Co-Authors: Susmita Sur-kolay, Bhargab B BhattacharyaAbstract:The notion of equivalent Embedding of rectangular duals is introduced, leading to a new concept of Canonical Embedding of a rectangular dual; this is a floorplan corresponding to a given neighborhood graph such that the number of directed cycles in its channel digraph is minimum. Strongly maximal rectangular hierarchy (sMRH) in nonslicible floorplans is then defined. The Canonical form of any arbitrary floorplan consists of at most one nonslicing core for each member of sMRH. Such an Embedding therefore represents a floorplan with minimum deviations from a slicing structure. An O(n/sup 2/) algorithm for realizing a Canonical Embedding is also presented. Canonical Embedding lends deep insight to the yet unsolved problem of characterizing inherent nonslicibility and motivates design for slicibility. It also makes determination of safe routing order simple. >
Frederico Xavier - One of the best experts on this subject based on the ideXlab platform.
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A Canonical Embedding of $\textbf{Aut}_{\textbf{hol}}({\bf \mathbb C^n})$
arXiv: Complex Variables, 2020Co-Authors: Francisco Braun, Frederico XavierAbstract:The group $\text{Aut}_{\text{hol}}(\mathbb C^n)$ of self-biholomorphisms of $\mathbb C^n$ consists of affine maps if $n=1$, but in higher dimensions it is a large object that has not been described explicitly. Despite the intricacies involved when $n>1$, surprisingly every $F\in \text{Aut}_{\text{hol}}(\mathbb C^n)$ is uniquely determined inside the group by only two data, of infinitesimal and global nature: the $1$-jet of $F$ at $0$, and the complex Hessian of a certain plurisubharmonic function associated to $F$. If $n=1$ this global datum is zero for all $F$, which is then determined solely by its $1$-jet at $0$, and one recovers $\text{Aut}_{\text{hol}}(\mathbb C)= \text{Aff}(\mathbb C)\cong \mathbb C \times \mathbb C^{*}$. Our main result, formulated as the existence of a Canonical Embedding of $ \text{Aut}_{\text{hol}} ( \mathbb C^n)$, also singles out a natural candidate for moduli space of $ \text{Aut}_{\text{hol}} ( \mathbb C^n)$, for all $n>1$.
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arXiv: Complex Variables, 2020Co-Authors: Francisco Braun, Frederico XavierAbstract:The group $\text{Aut}_{\text{hol}}(\mathbb C^n)$ of self-biholomorphisms of $\mathbb C^n$ consists of affine maps if $n=1$, but in higher dimensions it is a large object that has not been described explicitly. Despite the intricacies involved when $n>1$, surprisingly every $F\in \text{Aut}_{\text{hol}}(\mathbb C^n)$ is uniquely determined inside the group by only two data, of infinitesimal and global nature: the $1$-jet of $F$ at $0$, and the complex Hessian of a certain plurisubharmonic function associated to $F$. If $n=1$ this global datum is zero for all $F$, which is then determined solely by its $1$-jet at $0$, and one recovers $\text{Aut}_{\text{hol}}(\mathbb C)= \text{Aff}(\mathbb C)\cong \mathbb C \times \mathbb C^{*}$. Our main result, formulated as the existence of a Canonical Embedding of $ \text{Aut}_{\text{hol}} ( \mathbb C^n)$, also singles out a natural candidate for moduli space of $ \text{Aut}_{\text{hol}} ( \mathbb C^n)$, for all $n>1$.
A. Polishchuk - One of the best experts on this subject based on the ideXlab platform.
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Triple Massey products on curves, Fay’s trisecant identity and tangents to the Canonical Embedding, preprint math.AG/0107194
2012Co-Authors: A. PolishchukAbstract:Fay’s trisecant identity is an important special identity satisfied by theta functions on the Jacobian of a curve C (see e.g. [1] for an account of its relation with Schottky problem). Here is a list of some works containing its proof: [2], [3], [4], [5], [10], [18], [19], (characteristic zero); [11] (arbitrary characteristic). In this paper we give a new proof of this identity (valid in arbitrary characteristic). It turns out that Fay’s trisecant identity follows from the A∞-identity satisfied by certain triple Massey products on C. However, we should stress that our proof does not dependent on the general theory of Massey products. We compute explicitly the triple products we need, and the reader can take the answer as a definition. The main identity between these triple products is an easy consequence of the residue theorem. The trisecant identity follows immediately once we express our Massey products in terms of theta functions using the Riemann’s theorem. Looking at similar Massey products associated with vector bundles, we recover the matrix analogue of the trisecant identity (involving the so called Cauchy-Szegö kernels) obtained in [2] and [6],[7]. We observe that this matrix trisecant identity can be conveniently written using the notion of quasideterminant introduced by I. Gelfand and V. Retakh (see [8]). On the other hand, identities with some more special triple Massey products lead to a relation between tangents to the Canonical Embedding of C at triples of points. As a corollary, we obtain a formula for the tangent line to a Canonically embedded curve at a given point. In the case when C is an elliptic curve, the trisecant identity is equivalent to the associative Yang-Baxter equation satisfied by the Kronecker function (see [17]). In this case the Massey products we consider are the same as in loc.cit.. Using the homological mirror symmetry for an elliptic curve one can express these products in terms of indefinite theta series (see [15]). The category of coherent sheaves on a curve C of higher genus can be considered as a subcategory in the coherent sheaves on the Jacobian J of C. It would be interesting to study implications for our Massey products of the homological mirror symmetry for J
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Triple Massey products on curves, Fay's trisecant identity and tangents to the Canonical Embedding
arXiv: Algebraic Geometry, 2001Co-Authors: A. PolishchukAbstract:We show that Fay's trisecant identity follows from the A_{infinity}-constraint between certain triple Massey products in the derived category of coherent sheaves on a curve. We also deduce the matrix analogue of this identity that can be conveniently formulated using quasideterminants of matrices with non-commuting entries. On the other hand, looking at more special triple Massey products we derive a formula for the tangent line to a Canonically embedded curve at a given point.
Susmita Surkolay - One of the best experts on this subject based on the ideXlab platform.
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Canonical Embedding of rectangular duals with applications to vlsi floorplanning
Design Automation Conference, 1992Co-Authors: Susmita Surkolay, Bhargab B BhattacharyaAbstract:The notion of equivalent Embedding of rectangular duals is introduced, leading to a new concept of Canonical Embedding of a rectangular dual; this is a floorplan corresponding to a given neighborhood graph such that the number of directed cycles in its channel digraph is minimum. Strongly maximal rectangular hierarchy (sMRH) in nonslicible floorplans is then defined. The Canonical form of any arbitrary floorplan consists of at most one nonslicing core for each member of sMRH. Such an Embedding therefore represents a floorplan with minimum deviations from a slicing structure. An O(n/sup 2/) algorithm for realizing a Canonical Embedding is also presented. Canonical Embedding lends deep insight to the yet unsolved problem of characterizing inherent nonslicibility and motivates design for slicibility. It also makes determination of safe routing order simple. >
Susmita Sur-kolay - One of the best experts on this subject based on the ideXlab platform.
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DAC - Canonical Embedding of rectangular duals with applications to VLSI floorplanning
[1992] Proceedings 29th ACM IEEE Design Automation Conference, 1Co-Authors: Susmita Sur-kolay, Bhargab B BhattacharyaAbstract:The notion of equivalent Embedding of rectangular duals is introduced, leading to a new concept of Canonical Embedding of a rectangular dual; this is a floorplan corresponding to a given neighborhood graph such that the number of directed cycles in its channel digraph is minimum. Strongly maximal rectangular hierarchy (sMRH) in nonslicible floorplans is then defined. The Canonical form of any arbitrary floorplan consists of at most one nonslicing core for each member of sMRH. Such an Embedding therefore represents a floorplan with minimum deviations from a slicing structure. An O(n/sup 2/) algorithm for realizing a Canonical Embedding is also presented. Canonical Embedding lends deep insight to the yet unsolved problem of characterizing inherent nonslicibility and motivates design for slicibility. It also makes determination of safe routing order simple. >