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G Q Hu - One of the best experts on this subject based on the ideXlab platform.
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classification of finite dimensional estimation algebras of Maximal Rank with arbitrary state space dimension and mitter conjecture
International Journal of Control, 2005Co-Authors: G Q HuAbstract:In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of non-linear filtering problems. In the early eighties, Brockett proposed to classify finite dimensional estimation algebras and Mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one. Despite the massive effort in understanding the finite dimensional estimation algebras, the 20 year old problem of Brockett and Mitter conjecture remains open. In this paper, we give a classification of finite dimensional estimation algebras of Maximal Rank and solve the Mitter conjecture affirmatively for finite dimensional estimation algebras of Maximal Rank. In particular, for an estimation algebra E of Maximal Rank, we give a necessary and sufficient conditions for E to be finite dimensional in terms of the drift fi (x) and observation hj (x). As an important corollary, we show that the numb...
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Classification of finite-dimensional estimation algebras of Maximal Rank with arbitrary state–space dimension and mitter conjecture
International Journal of Control, 2005Co-Authors: G Q HuAbstract:In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of non-linear filtering problems. In the early eighties, Brockett proposed to classify finite dimensional estimation algebras and Mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one. Despite the massive effort in understanding the finite dimensional estimation algebras, the 20 year old problem of Brockett and Mitter conjecture remains open. In this paper, we give a classification of finite dimensional estimation algebras of Maximal Rank and solve the Mitter conjecture affirmatively for finite dimensional estimation algebras of Maximal Rank. In particular, for an estimation algebra E of Maximal Rank, we give a necessary and sufficient conditions for E to be finite dimensional in terms of the drift fi (x) and observation hj (x). As an important corollary, we show that the numb...
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Recent results on classification of finite dimensional Maximal Rank estimation algebras with state space dimension up to 5
Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), 1998Co-Authors: G Q HuAbstract:The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed independently by Brockett (1981) and Mitter (1979). Brockett proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow one to construct all possible finite dimensional recursive filters from the Lie algebraic point of view. In this paper we classify all Maximal Rank finite dimensional estimation algebras with state space dimension five.
E. Ballico - One of the best experts on this subject based on the ideXlab platform.
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REMARKS ON THE Maximal Rank CONJECTURE
International journal of pure and applied mathematics, 2020Co-Authors: E. BallicoAbstract:Here we point out a key lemma in an old paper which should be improved to get a weak form of the Maximal Rank Conjecture.
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ON THE POSTULATION OF GENERAL PROJECTIVE CURVES: THE Maximal Rank CONJECTURE FOR CURVES IN P 5
2020Co-Authors: E. BallicoAbstract:Fix integers g; d such that g � 0 and 6d � 5d + 30. Let CP 5 be a general degree d embedding of a general curve of genus g. Here we prove that C has Maximal Rank, i.e. the Maximal Rank conjecture for curves in P 5 is true.
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THE Maximal Rank CONJECTURE FOR LINEARLY NORMAL CURVES CP r with h 1 (C,OC(1)) = 1
2020Co-Authors: E. BallicoAbstract:Let C ⊂ P r be a general linearly normal curve with prescribed genus and h 1 (C,OC(1)) = 1. Here we prove that C has Maximal Rank, i.e. that for all integers t the restriction map H 0 (P r ,OPr(t)) → H 0 (C,OC(t)) is either
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Maximal Rank of space curves in the range a
European Journal of Mathematics, 2018Co-Authors: E. Ballico, Philippe Ellia, Claudio FontanariAbstract:We prove the following statement, which has been conjectured since 1985: There exists a constant K such that for all natural numbers d, g with \(g\leqslant Kd^{3/2}\) there exists an irreducible component of the Hilbert scheme of \(\mathbb {P}^3\) whose general element is a smooth, connected curve of degree d and genus g of Maximal Rank.
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on the Maximal Rank of a general union of a multiple linear space and a general rational curve
Boletin De La Sociedad Matematica Mexicana, 2016Co-Authors: E. BallicoAbstract:We prove in several cases that a general union of a multiple linear space and a general rational curve with assigned degree has Maximal Rank, i.e., its postulation is the expected one. In this case, its Hilbert function is uniquely determined by its Hilbert polynomial.
Sam Payne - One of the best experts on this subject based on the ideXlab platform.
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on the strong Maximal Rank conjecture in genus 22 and 23
arXiv: Algebraic Geometry, 2018Co-Authors: David Jensen, Sam PayneAbstract:We develop new methods to study tropicalizations of linear series and show linear independence of sections. Using these methods, we prove two new cases of the strong Maximal Rank conjecture for linear series of degree 25 and 26 on curves of genus 22 and 23, respectively.
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combinatorial and inductive methods for the tropical Maximal Rank conjecture
Journal of Combinatorial Theory Series A, 2017Co-Authors: David Jensen, Sam PayneAbstract:Abstract We produce new combinatorial methods for approaching the tropical Maximal Rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for m = 3 , extending previous results for m = 2 .
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tropical independence ii the Maximal Rank conjecture for quadrics
Algebra & Number Theory, 2016Co-Authors: David Jensen, Sam PayneAbstract:Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the Maximal Rank conjecture for quadrics. We also prove a tropical analogue of Max Noether's theorem on quadrics containing a canonically embedded curve, and state a combinatorial conjecture about tropical independence on chains of loops that implies the Maximal Rank conjecture for algebraic curves.
C.-w. Leung - One of the best experts on this subject based on the ideXlab platform.
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Recent results on classification of finite dimensional Maximal Rank estimation algebras with state space dimension up to 5
Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), 1992Co-Authors: C.-w. LeungAbstract:R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with Maximal Rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all Maximal Rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied. >
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Recent result on classification of finite dimensional Maximal Rank estimation algebras with state space dimension 3
[1992] Proceedings of the 31st IEEE Conference on Decision and Control, 1992Co-Authors: C.-w. LeungAbstract:R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with Maximal Rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all Maximal Rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied.
Eric Larson - One of the best experts on this subject based on the ideXlab platform.
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the Maximal Rank conjecture for sections of curves
Journal of Algebra, 2020Co-Authors: Eric LarsonAbstract:Abstract Let C ⊂ P r be a general curve of genus g embedded via a general linear series of degree d. The Maximal Rank Conjecture asserts that the restriction maps H 0 ( O P r ( m ) ) → H 0 ( O C ( m ) ) are of Maximal Rank; this determines the Hilbert function of C. In this paper, we prove an analogous statement for the union of hyperplane sections of general curves. More specifically, if H ⊂ P r is a general hyperplane, and C 1 , C 2 , … , C n are general curves, we show H 0 ( O H ( m ) ) → H 0 ( O ( C 1 ∪ C 2 ∪ ⋯ ∪ C n ) ∩ H ( m ) ) is of Maximal Rank, except for some counterexamples when m = 2 . As explained in [5] , this result plays a key role in the author's proof of the Maximal Rank Conjecture [7] .
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the Maximal Rank conjecture
arXiv: Algebraic Geometry, 2018Co-Authors: Eric LarsonAbstract:Let C be a general curve of genus g, embedded in P^r via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C.
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degenerations of curves in projective space and the Maximal Rank conjecture
arXiv: Algebraic Geometry, 2018Co-Authors: Eric LarsonAbstract:In this note, we give an overview of a new technique for studying Brill--Noether curves in projective space via degeneration. In particular, we give a roadmap to the proof of the Maximal Rank Conjecture.
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the Maximal Rank conjecture for sections of curves
arXiv: Algebraic Geometry, 2012Co-Authors: Eric LarsonAbstract:Let be a general curve of genus g embedded via a general linear series of degree d in P^r. The well-known Maximal Rank Conjecture asserts that the restriction maps H^0(O_{P^r}(m)) \to H^0(O_C(m) are of Maximal Rank; if known, this conjecture would determine the Hilbert function of C. In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \to H^0(O_{(C_1 \cup C_2 \cup \cdots \cup C_n) \cap H}(m)) is of Maximal Rank, except for some counterexamples when m = 2.