Derivatives and Fréchet differentials

Zorn, Max

doi:10.1090/s0002-9904-1946-08524-9

Cited by 24 publications

(10 citation statements)

References 3 publications

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“…We recall some standard definitions and facts about analytic functions between Banach spaces. For background, see [35], [36], [37]. For the reader's convenience, we include only the level of generality that will be used in this paper and give sketches of the proofs.…”

Section: Analytic Functions Between Banach Spacesmentioning

confidence: 99%

Operator-valued chordal Loewner chains and non-commutative probability

Jekel

2020

Journal of Functional Analysis

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We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a C * -algebra A. We define an A-valued chordal Loewner chain as a subordination chain of analytic self-maps of the A-valued upper halfplane, such that each F t is the reciprocal Cauchy transform of an A-valued law µ t , such that the mean and variance of µ t are continuous functions of t.We relate A-valued Loewner chains to processes with A-valued free or monotone independent independent increments just as was done in the scalar case by Bauer [1] and Scheißinger [2].We show that the Loewner equation, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains F t and vector fieldsBased on the Loewner equation, we derive a combinatorial expression for the moments of µ t in terms of ν t . We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws µ t . Finally, we prove a version of the monotone central limit theorem which describes the behavior of F t as t → +∞ when ν t has uniformly bounded support.Remark 2.11. Note that by the previous lemma and some basic results on L ∞ Boch , there is an isometric inclusion ι : L ∞ Boch ([0, T ], X ) → L(L 1 [0, T ], X ) given by, so in the sequel we will regard L ∞ Boch ([0, T ], X ) as a subspace of L(L 1 [0, T ], X ).If we had a bounded function R : [0, T ] × [0, T ] → X denoted R(s, t), then could define the diagonal restriction R(t, t). We claim that under appropriate hypotheses, this operation still makes sense when R(s, ·) is an element of L(L 1 [0, T ], X ) rather than a bounded function [0, T ] → X . For this to be rigorous, we must view R as a map [0, T ] → L(L 1 [0, T ], X ). Lemma 2.12 (Diagonal restriction). There exists a unique linear map diag : L ∞ Boch ([0, T ], L(L 1 [0, T ], X )) → L(L 1 [0, T ], X ) such that

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Section: Analytic Functions Between Banach Spacesmentioning

confidence: 99%

Operator-valued chordal Loewner chains and non-commutative probability

Jekel

2020

Journal of Functional Analysis

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“…Furthermore, the gradient of this cost function, with respect to the parameters μ and β , can be expressed as a Gâteaux, or directional, derivative (e.g. Zorn, 1946), defined by d J ≡ d μ J + d β J , with…”

Section: Theorymentioning

confidence: 99%

Initialization of ice-sheet forecasts viewed as an inverse Robin problem

2010

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As simulations of 21st-century climate start to include components with longer timescales, such as ice sheets, the initial conditions for those components will become critical to the forecast. This paper describes an algorithm for specifying the initial state of an ice-sheet model, given spatially continuous observations of the surface elevation, the velocity at the surface and the thickness of the ice. The algorithm can be viewed as an inverse procedure to solve for the viscosity or the basal drag coefficient. It applies to incompressible Stokes flow over an impenetrable boundary, and is based upon techniques used in electric impedance tomography; in particular, the minimization of a type of cost function proposed by Kohn and Vogelius. The algorithm can be implemented numerically using only the forward solution of the Stokes equations, with no need to develop a separate adjoint model. The only requirement placed upon the numerical Stokes solver is that boundary conditions of Dirichlet, Neumann and Robin types can be implemented. As an illustrative example, the algorithm is applied to shear flow down an impenetrable inclined plane. A fully three-dimensional test case using a commercially available solver for the Stokes equations is also presented.

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“…One of the main ingredients in the proof is the Earle-Hamilton theorem, which is a fixed-point theorem for holomorphic functions between Banach spaces. Definition 4.2 (See [48,49,50]). Let X and Y be Banach spaces and let Ω be an open subset of X .…”

Section: Tree Convolutionsmentioning

confidence: 99%

Tree convolution for probability distributions with unbounded support

Davis¹,

Jekel²,

Wang³

2021

Preprint

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We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in [27], which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree T of the N -regular tree (with vertices labeled by alternating strings), we define the convolution ⊞ T (µ 1 , . . . , µ N ) for arbitrary probability measures µ 1 , . . . , µ N on R using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from [27]. We prove a general limit theorem for iterated T -free convolution similar to Bercovici and Pata's results in the free case [12], and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.

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Derivatives and Fréchet differentials

Cited by 24 publications

References 3 publications

Operator-valued chordal Loewner chains and non-commutative probability

Operator-valued chordal Loewner chains and non-commutative probability

Initialization of ice-sheet forecasts viewed as an inverse Robin problem

Tree convolution for probability distributions with unbounded support

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