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Antisymmetric Matrix
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Cristinel Mardare – One of the best experts on this subject based on the ideXlab platform.

A New Approach to the Fundamental Theorem of Surface Theory
Archive for Rational Mechanics and Analysis, 2008CoAuthors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positivedefinite symmetric matrices ( a _ αβ ) of order two and a field of symmetric matrices ( b _ αβ ) of order two together satisfy the Gauss and CodazziMainardi 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 threedimensional 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.

A new approach to the fundamental theorem of surface theory
Archive for Rational Mechanics and Analysis, 2008CoAuthors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positivedefinite symmetric matrices (a_{αβ}) of order two and a field of symmetric matrices (b_{αβ}) of order two together satisfy the Gauss and CodazziMainardi 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 threedimensional 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.

Differential Geometry New compatibility conditions for the fundamental theorem of surface theory
, 2007CoAuthors: Philippe G. Ciarlet, Liliana Gratie, Cristinel MardareAbstract:The fundamental theorem of surface theory classically asserts that, if a field of positivedefinite 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 simplyconnected 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
Erwin Frey – One of the best experts on this subject based on the ideXlab platform.

Topologically robust zerosum games and Pfaffian orientation: How network topology determines the longtime dynamics of the Antisymmetric LotkaVolterra equation
Physical Review E, 2018CoAuthors: Philipp Geiger, Johannes Knebel, Erwin FreyAbstract:To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the Antisymmetric LotkaVolterra equation (ALVE). The ALVE is the replicator equation of zerosum 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 zerosum games, such as the rockpaperscissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zerosum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the longtime dynamics of the ALVE to the algebra of Antisymmetric matrices, we identify simple graphtheoretical 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 nonHamiltonian networks. In graphtheoretical terms, we extend the concept of a Pfaffian orientation from evensized to oddsized networks. Our results show that the topology of interaction networks alone can determine the longtime 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.

σ₅equivariant syzygies for the Del Pezzo surface of degree 5
Rendiconti del Circolo Matematico di Palermo Series 2, 2020CoAuthors: 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 arithmeticallyGorenstein 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. 
$${\mathfrak {S}}_5$$S5equivariant syzygies for the Del Pezzo surface of degree 5
Rendiconti del Circolo Matematico di Palermo Series 2, 2020CoAuthors: 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 arithmeticallyGorenstein 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.

$${\mathfrak {S}}_5$$S5equivariant syzygies for the Del Pezzo surface of degree 5
Rendiconti del Circolo Matematico di Palermo Series 2, 2020CoAuthors: 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 arithmeticallyGorenstein 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.