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Cristinel Mardare - One of the best experts on this subject based on the ideXlab platform.
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A New Approach to the Fundamental Theorem of Surface Theory
Archive for Rational Mechanics and Analysis, 2008Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices ( a _ αβ ) of order two and a field of symmetric matrices ( b _ αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of $${\mathbb{R}}^{2}$$ , then there exists an immersion $${\bf \theta}:\omega \to {\mathbb{R}}^{3}$$ such that these fields are the first and second fundamental forms of the surface $${\bf \theta}(\omega)$$ , and this surface is unique up to proper isometries in $${\mathbb{R}}^3$$ . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a _ αβ and b _ αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the Matrix equation $$\partial{\bf A}_2-\partial_2{\bf A}_1+{\bf A}_1{\bf A}_2-{\bf A}_2{\bf A}_1={\bf 0}\,{\rm in}\,\omega,$$ where A _1 and A _2 are Antisymmetric Matrix fields of order three that are functions of the fields ( a _ αβ ) and ( b _ αβ ), the field ( a _ αβ ) appearing in particular through the square root U of the Matrix field $${\bf C} = \left(\begin{array}{lll} a_{11} & a_{12} & 0\\ a_{21} & a_{22} & 0\\ 0 & 0 & 1\end{array}\right).$$ The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization $${\bf \nabla}{\bf \Theta}={\bf RU}$$ of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension $${\bf \Theta}$$ of the unknown immersion $${\bf \theta}$$ . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental theorem of surface theory set out by S. M ardare [20–22], the unknown immersion $${\bf \theta}: \omega \to {\mathbb{R}}^3$$ is found in the present approach to exist in function spaces “with little regularity”, such as $$W^{2,p}_{\rm loc}(\omega;{\mathbb{R}}^3)$$ , p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells where rotation fields are introduced as bona fide unknowns.
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A new approach to the fundamental theorem of surface theory
Archive for Rational Mechanics and Analysis, 2008Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a_{αβ}) of order two and a field of symmetric matrices (b_{αβ}) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of R^2, then there exists an immersion θ : ω → R^3 such that these fields are the first and second fundamental forms of the surface θ(ω), and this surface is unique up to proper isometries in R^3. The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a_{αβ} and b_{αβ}, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the Matrix equation ∂_1A_2−∂_2A_1+A_1A_2−A_2A_1=0 inω, where A_1 and A_2 are Antisymmetric Matrix fields of order three that are functions of the fields (a_{αβ}) and (b_{αβ}), the field (a_{αβ}) appearing in particular through the square root U of the Matrix field a_{11} a_{12} 0 C = a_{21} a_{22} 0 0 0 1 The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization ∇Θ = RU of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension Θ of the unknown immersion θ. In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental theorem of surface theory set out by S. Mardare [2007], the unknown immersion θ : ω → R^3 is found in the present approach in function spaces “with little regularity”, such as W^{2,p}_loc(ω;R^3), p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells where rotation fields are introduced as bona fide unknowns.
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Differential Geometry New compatibility conditions for the fundamental theorem of surface theory
2007Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (aαβ ) of order two and a field of symmetric matrices (bαβ ) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ : ω → R 3 such that these fields are the first and second fundamental forms of the surface θ (ω) and this surface is unique up to proper isometries in R 3 . In this Note, we identify new compatibility conditions, expressed again in terms of the functions aαβ and bαβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂1A2 − ∂2A1 + A1A2 − A2A1 = 0 in ω, where A1 and A2 are Antisymmetric Matrix fields of order three that are functions of the fields (aαβ ) and (bαβ ) ,t he fi eld(aαβ ) appearing in particular through its square root. The unknown immersion θ : ω → R 3 is found in the present approach in function
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New compatibility conditions for the fundamental theorem of surface theory
Comptes Rendus Mathématique, 2007Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a_{αβ}) of order two and a field of symmetric matrices (b_{αβ}) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R^2, then there exists an immersion θ : ω → R^3 such that these fields are the first and second fundamental forms of the surface θ(ω) and this surface is unique up to proper isometries in R^3. In this Note, we identify new compatibility conditions, expressed again in terms of the functions a_{αβ} and b_{αβ}, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂_1A_2−∂_2A_1+A_1A_2−A_2A_1=0 in ω, where A_1 and A_2 are Antisymmetric Matrix fields of order three that are functions of the fields (a_{αβ}) and (b_{αβ}), the field (a_{αβ}) appearing in particular through its square root. The unknown immersion θ : ω → R^3 is found in the present approach in function spaces ‘with little regularity’, viz., W^{2,p}_loc(ω;R^3), p > 2.
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New compatibility conditions for the fundamental theorem of surface theory
Comptes Rendus Mathematique, 2007Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:Abstract The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices ( a α β ) of order two and a field of symmetric matrices ( b α β ) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ : ω → R 3 such that these fields are the first and second fundamental forms of the surface θ ( ω ) and this surface is unique up to proper isometries in R 3 . In this Note, we identify new compatibility conditions, expressed again in terms of the functions a α β and b α β , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂ 1 A 2 − ∂ 2 A 1 + A 1 A 2 − A 2 A 1 = 0 in ω , where A 1 and A 2 are Antisymmetric Matrix fields of order three that are functions of the fields ( a α β ) and ( b α β ) , the field ( a α β ) appearing in particular through its square root. The unknown immersion θ : ω → R 3 is found in the present approach in function spaces ‘with little regularity’, viz., W loc 2 , p ( ω ; R 3 ) , p > 2 . To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).
Erwin Frey - One of the best experts on this subject based on the ideXlab platform.
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Topologically robust zero-sum games and Pfaffian orientation: How network topology determines the long-time dynamics of the Antisymmetric Lotka-Volterra equation
Physical Review E, 2018Co-Authors: Philipp Geiger, Johannes Knebel, Erwin FreyAbstract:To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the Antisymmetric Lotka-Volterra equation (ALVE). The ALVE is the replicator equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an Antisymmetric Matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of Antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio phi = 1.6180 ..., cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.
Fabrizio Catanese - One of the best experts on this subject based on the ideXlab platform.
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σ₅-equivariant syzygies for the Del Pezzo surface of degree 5
Rendiconti del Circolo Matematico di Palermo Series 2, 2020Co-Authors: Ingrid Bauer, Fabrizio CataneseAbstract:The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$. We give canonical explicit $${\mathfrak {S}}_5$$-invariant Pfaffian equations through a 6$$\times $$6 Antisymmetric Matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$. Finally, we give $${\mathfrak {S}}_5$$-invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.
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$${\mathfrak {S}}_5$$S5-equivariant syzygies for the Del Pezzo surface of degree 5
Rendiconti del Circolo Matematico di Palermo Series 2, 2020Co-Authors: Ingrid Bauer, Fabrizio CataneseAbstract:The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$ P 5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$ S 5 . We give canonical explicit $${\mathfrak {S}}_5$$ S 5 -invariant Pfaffian equations through a 6 $$\times $$ × 6 Antisymmetric Matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$ S 5 . Finally, we give $${\mathfrak {S}}_5$$ S 5 -invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$ ( P 1 ) 5 , and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.
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$${\mathfrak {S}}_5$$S5-equivariant syzygies for the Del Pezzo surface of degree 5
Rendiconti del Circolo Matematico di Palermo Series 2, 2020Co-Authors: Ingrid Bauer, Fabrizio CataneseAbstract:The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$ P 5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$ S 5 . We give canonical explicit $${\mathfrak {S}}_5$$ S 5 -invariant Pfaffian equations through a 6 $$\times $$ × 6 Antisymmetric Matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$ S 5 . Finally, we give $${\mathfrak {S}}_5$$ S 5 -invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$ ( P 1 ) 5 , and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.
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$\mathfrak S_5$-equivariant syzygies for the Del Pezzo Surface of Degree 5
arXiv: Algebraic Geometry, 2018Co-Authors: Ingrid Bauer, Fabrizio CataneseAbstract:The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4 general points, embedded in $\mathbb{P}^5$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum-Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a $6 \times 6$ Antisymmetric Matrix. We give concrete geometric descriptions of the irreducible representations of $\mathfrak S_5$. Finally, we give $\mathfrak S_5$-invariant equations for the embedding of $Y$ inside $(\mathbb{P}^1)^5$, and show that they have the same Hilbert resolution as for the Del Pezzo of degree $4$.
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$\mathfrak S_5$-symmetric equations for the Del Pezzo Surface of Degree 5
arXiv: Algebraic Geometry, 2018Co-Authors: Ingrid Bauer, Fabrizio CataneseAbstract:The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4 general points, embedded in $\mathbb{P}^5$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum-Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a $6 \times 6$ Antisymmetric Matrix. We give concrete geometric descriptions of the irreducible representations of $\mathfrak S_5$. Finally, we give $\mathfrak S_5$-invariant equations for the embedding of $Y$ inside $(\mathbb{P}^1)^5$, and show that they have the same Hilbert resolution as for the Del Pezzo of degree $4$.
Philippe G. Ciarlet - One of the best experts on this subject based on the ideXlab platform.
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A New Approach to the Fundamental Theorem of Surface Theory
Archive for Rational Mechanics and Analysis, 2008Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices ( a _ αβ ) of order two and a field of symmetric matrices ( b _ αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of $${\mathbb{R}}^{2}$$ , then there exists an immersion $${\bf \theta}:\omega \to {\mathbb{R}}^{3}$$ such that these fields are the first and second fundamental forms of the surface $${\bf \theta}(\omega)$$ , and this surface is unique up to proper isometries in $${\mathbb{R}}^3$$ . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a _ αβ and b _ αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the Matrix equation $$\partial{\bf A}_2-\partial_2{\bf A}_1+{\bf A}_1{\bf A}_2-{\bf A}_2{\bf A}_1={\bf 0}\,{\rm in}\,\omega,$$ where A _1 and A _2 are Antisymmetric Matrix fields of order three that are functions of the fields ( a _ αβ ) and ( b _ αβ ), the field ( a _ αβ ) appearing in particular through the square root U of the Matrix field $${\bf C} = \left(\begin{array}{lll} a_{11} & a_{12} & 0\\ a_{21} & a_{22} & 0\\ 0 & 0 & 1\end{array}\right).$$ The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization $${\bf \nabla}{\bf \Theta}={\bf RU}$$ of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension $${\bf \Theta}$$ of the unknown immersion $${\bf \theta}$$ . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental theorem of surface theory set out by S. M ardare [20–22], the unknown immersion $${\bf \theta}: \omega \to {\mathbb{R}}^3$$ is found in the present approach to exist in function spaces “with little regularity”, such as $$W^{2,p}_{\rm loc}(\omega;{\mathbb{R}}^3)$$ , p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells where rotation fields are introduced as bona fide unknowns.
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A new approach to the fundamental theorem of surface theory
Archive for Rational Mechanics and Analysis, 2008Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a_{αβ}) of order two and a field of symmetric matrices (b_{αβ}) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of R^2, then there exists an immersion θ : ω → R^3 such that these fields are the first and second fundamental forms of the surface θ(ω), and this surface is unique up to proper isometries in R^3. The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a_{αβ} and b_{αβ}, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the Matrix equation ∂_1A_2−∂_2A_1+A_1A_2−A_2A_1=0 inω, where A_1 and A_2 are Antisymmetric Matrix fields of order three that are functions of the fields (a_{αβ}) and (b_{αβ}), the field (a_{αβ}) appearing in particular through the square root U of the Matrix field a_{11} a_{12} 0 C = a_{21} a_{22} 0 0 0 1 The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization ∇Θ = RU of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension Θ of the unknown immersion θ. In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental theorem of surface theory set out by S. Mardare [2007], the unknown immersion θ : ω → R^3 is found in the present approach in function spaces “with little regularity”, such as W^{2,p}_loc(ω;R^3), p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells where rotation fields are introduced as bona fide unknowns.
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Differential Geometry New compatibility conditions for the fundamental theorem of surface theory
2007Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (aαβ ) of order two and a field of symmetric matrices (bαβ ) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ : ω → R 3 such that these fields are the first and second fundamental forms of the surface θ (ω) and this surface is unique up to proper isometries in R 3 . In this Note, we identify new compatibility conditions, expressed again in terms of the functions aαβ and bαβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂1A2 − ∂2A1 + A1A2 − A2A1 = 0 in ω, where A1 and A2 are Antisymmetric Matrix fields of order three that are functions of the fields (aαβ ) and (bαβ ) ,t he fi eld(aαβ ) appearing in particular through its square root. The unknown immersion θ : ω → R 3 is found in the present approach in function
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New compatibility conditions for the fundamental theorem of surface theory
Comptes Rendus Mathématique, 2007Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a_{αβ}) of order two and a field of symmetric matrices (b_{αβ}) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R^2, then there exists an immersion θ : ω → R^3 such that these fields are the first and second fundamental forms of the surface θ(ω) and this surface is unique up to proper isometries in R^3. In this Note, we identify new compatibility conditions, expressed again in terms of the functions a_{αβ} and b_{αβ}, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂_1A_2−∂_2A_1+A_1A_2−A_2A_1=0 in ω, where A_1 and A_2 are Antisymmetric Matrix fields of order three that are functions of the fields (a_{αβ}) and (b_{αβ}), the field (a_{αβ}) appearing in particular through its square root. The unknown immersion θ : ω → R^3 is found in the present approach in function spaces ‘with little regularity’, viz., W^{2,p}_loc(ω;R^3), p > 2.
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New compatibility conditions for the fundamental theorem of surface theory
Comptes Rendus Mathematique, 2007Co-Authors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:Abstract The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices ( a α β ) of order two and a field of symmetric matrices ( b α β ) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ : ω → R 3 such that these fields are the first and second fundamental forms of the surface θ ( ω ) and this surface is unique up to proper isometries in R 3 . In this Note, we identify new compatibility conditions, expressed again in terms of the functions a α β and b α β , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂ 1 A 2 − ∂ 2 A 1 + A 1 A 2 − A 2 A 1 = 0 in ω , where A 1 and A 2 are Antisymmetric Matrix fields of order three that are functions of the fields ( a α β ) and ( b α β ) , the field ( a α β ) appearing in particular through its square root. The unknown immersion θ : ω → R 3 is found in the present approach in function spaces ‘with little regularity’, viz., W loc 2 , p ( ω ; R 3 ) , p > 2 . To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).
Ying Tang - One of the best experts on this subject based on the ideXlab platform.
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Fast orthogonal recurrent neural networks employing a novel parametrisation for orthogonal matrices
Signal Processing, 2019Co-Authors: Ying TangAbstract:Abstract Training Recurrent Neural Networks (RNNs) is challenging due to the vanishing/exploding gradient problem. Recent progress suggests to solve this problem by constraining the recurrent transition Matrix to be unitary/orthogonal during training, but all of which are either limited-capacity, or involve time-consuming operators, e.g., evaluation for the derivation of lengthy Matrix chain multiplication, the Matrix exponential, or the singular value decomposition. This paper addresses this problem based on the exponentials of sparse Antisymmetric matrices with one or more nonzero columns and an equal number of nonzero rows from a geometric view. An analytical expression is presented to simplify the computation of the sparse Antisymmetric Matrix exponential, which is actually a novel formula for parameterizing orthogonal matrices. The algorithms of this paper are fast, tunable, and full-capacity, where the target variable is updated by optimizing a Matrix multiplier, instead of using the explicit gradient descent. Experiments demonstrate the superior performance of our proposed algorithms.
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AICI - A novel neural network approach for computing eigen-pairs of real Antisymmetric matrices
Artificial Intelligence and Computational Intelligence, 2012Co-Authors: Hang Tan, Xianhe Huang, Huachun Tan, Ying TangAbstract:In the present paper, we focus on the problem how to compute all eigen-pairs of any real Antisymmetric Matrix by the conventional neural network approach without modification the original structure of the neural network. Given any n-dimensional real Antisymmetric Matrix, our proposed method is based on a n-dimensional ODEs and the preprocessing become comparatively easy. The contributions of this paper are mainly come from two aspects, on the one hand, we constructed the eigen-pairs relationship between those of symmetric Matrix and anti-symmetric Matrix; on the other hand, we presented a simple method to compute all eigen-pairs of any Antisymmetric Matrix. Simulations verify the computational capability of the proposed method.
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A Simple and Accurate ICA Algorithm for Separating Mixtures of Up to Four Independent Components
Acta Automatica Sinica, 2011Co-Authors: Ying TangAbstract:Abstract This paper introduces an algorithm for independent component analysis (ICA) using explicit closed forms of two-, three-and four-dimensional Antisymmetric Matrix exponentials, based on which both the search direction and Matrix exponentials can be directly computed in each iteration without any approximation. In addition, two errors have been corrected for the representation of four-dimensional Antisymmetric Matrix exponentials that were established in other works. Simulations show that the algorithm converges fast and can achieve better performance than the well-known Extended InfoMax and FastICA algorithms for mixtures of up to four independent components.
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Computing eigenvectors and corresponding eigenvalues with largest or smallest modulus of real Antisymmetric Matrix based on neural network with less scale
2010 The 2nd International Conference on Computer and Automation Engineering (ICCAE), 2010Co-Authors: Ying Tang, Jianping LiAbstract:In this paper, we extend the neural network based approaches, which can asymptotically compute the largest or smallest eigenvalues and the corresponding eigenvectors of real symmetric Matrix, to the real Antisymmetric Matrix case. Given any n-by-n real Antisymmetric Matrix, unlike the previous neural network based methods that were summarized by some ordinary differential equations (ODEs) with 2n dimension, our proposed method can be represented by some n dimensional ODEs, which can much reduce the scale of networks and achieve higher computing performance. Simulations verify the computational capability of our proposed method.
