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53
Stability Results For Fractional Differential Equations With Applications To Control Processing
 In Computational Engineering in Systems Applications
, 1996
"... In this paper, stability results of main concern for control theory are given for finitedimensional linear fractional differential systems. For fractional differential systems in statespace form, both internal and external stabilities are investigated. For fractional differential systems in polyno ..."
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Cited by 92 (3 self)
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In this paper, stability results of main concern for control theory are given for finitedimensional linear fractional differential systems. For fractional differential systems in statespace form, both internal and external stabilities are investigated. For fractional differential systems in polynomial representation, external stability is thoroughly examined. Our main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector jarg(oe)j ff=2, thus generalizing in a stupendous way the wellknown results for the integer case ff = 1. 1. INTRODUCTION Fractional differential systems have proved to be useful in control processing for the last two decades (see [21, 22]). The notion of fractional derivative dates back two centuries; some references that have now become classical were written two decades ago (see [20, 25]). Several authors published reference books on the subject very recently: see [26] for a thorough mathem...
On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach
, 2008
"... In this contribution, the capacityachieving input covariance matrices for coherent blockfading correlated MIMO Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are avai ..."
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Cited by 43 (19 self)
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In this contribution, the capacityachieving input covariance matrices for coherent blockfading correlated MIMO Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed by numerical techniques. As the corresponding optimization algorithms are not very attractive, an approximation of the average mutual information is evaluated in this paper in the asymptotic regime where the number of transmit and receive antennas converge to + ∞ at the same rate. New results related to the accuracy of the corresponding large system approximation are provided. An attractive optimization algorithm of this approximation is proposed and we establish that it yields an effective way to compute the capacity achieving covariance matrix for the average mutual information. Finally, numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information, while being much more computationally attractive.
Extension of the structure theorem of Borchers and its application to halfsided modular inclusions" manuscript, preliminary version
, 1995
"... Abstract. A result of H.W. Wiesbrock is extended from the case of a common cyclic and separating vector for the halfsided modular inclusion N ⊂ M of von Neumann algebras to the case of a common faithful normal semifinite weight and at the same time a gap in Wiesbrock’s proof is filled in. 1. ..."
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Cited by 20 (0 self)
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Abstract. A result of H.W. Wiesbrock is extended from the case of a common cyclic and separating vector for the halfsided modular inclusion N ⊂ M of von Neumann algebras to the case of a common faithful normal semifinite weight and at the same time a gap in Wiesbrock’s proof is filled in. 1.
D.: Free diffusions and matrix models with strictly convex interaction
 Geom. Funct. Anal
"... We study solutions to the free stochastic differential equation dXt = dSt − 1 2DV (Xt)dt, where V is a locally convex polynomial potential in m noncommuting variables. We show that for selfadjoint V, the law µV of a stationary solution is the limit law of a random matrix model, in which an mtuple ..."
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Cited by 12 (1 self)
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We study solutions to the free stochastic differential equation dXt = dSt − 1 2DV (Xt)dt, where V is a locally convex polynomial potential in m noncommuting variables. We show that for selfadjoint V, the law µV of a stationary solution is the limit law of a random matrix model, in which an mtuple of selfadjoint matrices are chosen according to the law exp(−NTr(V (A1,..., Am)))dA1 · · · dAm. We show that if V = Vβ depends on complex parameters β1,..., βk, then the law µV is analytic in β at least for those β for which Vβ is locally convex. In particular, this gives information on the region of convergence of the generating function for planar maps. We show that the solution dXt has nice convergence properties with respect to the operator norm. This allows us to derive several properties of C ∗ and W ∗ algebras generated by an mtuple with law µV. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II1 factor. We show that the microstates free entropy χ(τV) is finite. A corollary of these results is the fact that the support of the law of any selfadjoint polynomial in X1,...,Xn under the law µV is connected, vastly generalizing the case of a single random matrix. 1
Symmetry of minimizers for some nonlocal variational problems
 J. Funct. Anal
"... We present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems. This approach which generalizes the Reflection method is based on the obtention of some integral identities. We study the identities that lead to symmetry results, the functionals that c ..."
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Cited by 11 (1 self)
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We present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems. This approach which generalizes the Reflection method is based on the obtention of some integral identities. We study the identities that lead to symmetry results, the functionals that can be considered and the function spaces that can be used. Then we use our method to prove the symmetry of minimizers for a class of variational problems involving the fractional powers of Laplacian, for the generalized Choquard functional and for the standing waves of the DaveyStewartson equation.
Monodromy Groups of Parameterized Linear Differential Equations with Regular Singular Points, arXiv:1106.2664v1 [math.CA] 14
, 2011
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Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 RAPPORT DE RECHERCHE 911, INRIA, OCTOBRE
, 1988
"... We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implem ..."
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Cited by 9 (7 self)
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We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number.
On the selfdisplacement of deformable bodies in a potential fluid flow
 in "Math. Models Methods Appl. Sci
"... Abstract. Understanding fishlike locomotion as a result of internal shape changes may result in improved underwater propulsion mechanism. In this article, we study a coupled system of partial differential equations and ordinary differential equations which models the motion of selfpropelled deform ..."
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Cited by 7 (3 self)
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Abstract. Understanding fishlike locomotion as a result of internal shape changes may result in improved underwater propulsion mechanism. In this article, we study a coupled system of partial differential equations and ordinary differential equations which models the motion of selfpropelled deformable bodies (called swimmers) in an potential fluid flow. The deformations being prescribed, we apply the least action principle of Lagrangian mechanics to determine the equations of the inferred motion. We prove that the swimmers degrees of freedom solve a second order system of nonlinear ordinary differential equations. Under suitable smoothness assumptions on the fluid’s domain boundary and on the given deformations, we prove the existence and regularity of the bodies rigid motions, up to a collision between two swimmers or between a swimmer with the boundary of the fluid. Then we compute explicitly the EulerLagrange equations in terms of the geometric data of the bodies and of the value of the fluid’s harmonic potential on the boundary of the fluid. 1.
PROJECTIVE ISOMONODROMY AND GALOIS GROUPS
"... (Communicated by) Abstract. In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy evolving deformation of linear differential equations, based on the example of the DarbouxHalphen equation. We give an algebraic condition for a paramaterized linear ..."
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Cited by 6 (1 self)
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(Communicated by) Abstract. In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy evolving deformation of linear differential equations, based on the example of the DarbouxHalphen equation. We give an algebraic condition for a paramaterized linear differential equation to be projectively isomonodromic, in terms of the derived group of its parameterized PicardVessiot group. 1.