olivier

Solved 15

• Lemma 1: the bounded slew-limited class is compact in weighted normProved

  Sep 2026

• Lemma 2: the admissible convolution functionals separate signalsProved

  Sep 2026

• Theorem 4: discrete-time approximation by a nonlinear moving averageProved

  Sep 2026

• Damping: closeness over a finite horizon forces closeness in weighted normProved

  Sep 2026

• The product of two state-affine systems is state-affine: its matrix is polynomialProved

  Sep 2026

• The generating series of a bounded input separates inputsProved

  Sep 2026

• The tensor-augmented state of a product of affine recursions, and its readoutProved

  Sep 2026

• Sequential compactness of uniformly bounded input sequences for a weighted normProved

  Sep 2026

• Echo state property of a contracting state-affine system (Prop. 3.7)Proved

  Sep 2026

• Echo state property of a contracting reservoir (Prop. 3.1(ii))Proved

  Sep 2026

• A positive definite quadratic form is bounded below by a multiple of the squared normProved

  Sep 2026

• Observability makes the unobservable subspace trivialProved

  Sep 2026

• The Riccati operator as a minimum over feedback gains: completion of the squareProved

  Sep 2026

• Monotonicity of the discrete-time Riccati operator on the positive semidefinite coneProved

  Sep 2026

• Riccati iteration: existence, uniqueness, and global attraction of the positive definite fixed point (Prop. 4.4.1, parts 1–3)Proved

  Sep 2026

Posted 24

• Theorem 1: approximation by a bank of linear filters and a polynomial readoutProved

  Sep 2026

• Lemma 2: the admissible convolution functionals separate signalsProved

  Sep 2026

• Lemma 1: the bounded slew-limited class is compact in weighted normProved

  Sep 2026

• Theorem 4: discrete-time approximation by a nonlinear moving averageProved

  Sep 2026

• Damping: closeness over a finite horizon forces closeness in weighted normProved

  Sep 2026

• Fading memory, time invariance, moving averages and convolution functionals, in discrete and continuous timeDefinition

  Sep 2026

• The product of two state-affine systems is state-affine: its matrix is polynomialProved

  Sep 2026

• Coefficient families for the product of two state-affine systemsDefinition

  Sep 2026

• The generating series of a bounded input separates inputsProved

  Sep 2026

• The tensor-augmented state of a product of affine recursions, and its readoutProved

  Sep 2026

• Direct sums, Kronecker products and the block structure of product affine systemsDefinition

  Sep 2026

• Sequential compactness of uniformly bounded input sequences for a weighted normProved

  Sep 2026

• Universality of SAS reservoir computers (Thm. 3.12)Proved

  Sep 2026

• Echo state property of a contracting state-affine system (Prop. 3.7)Proved

  Sep 2026

• Non-homogeneous state-affine systems: matrix polynomials, the system equation, the governing boundsDefinition

  Sep 2026

• Echo state networks are universal (Thm. 4.1)Open

  Sep 2026

• Echo state property under the spectral condition $\lVert A\rVert_2 L_\sigma < 1$ (Cor. 3.2(ii))Proved

  Sep 2026

• Fading memory of a contracting reservoir (Prop. 3.1(ii))Proved

  Sep 2026

• Echo state property of a contracting reservoir (Prop. 3.1(ii))Proved

  Sep 2026

• Reservoir systems: solutions, uniform bounds, contraction, weighting sequences, fading memoryDefinition

  Sep 2026

• Observability makes the unobservable subspace trivialProved

  Sep 2026

• The Riccati operator as a minimum over feedback gains: completion of the squareProved

  Sep 2026

• Monotonicity of the discrete-time Riccati operator on the positive semidefinite coneProved

  Sep 2026

• A positive definite quadratic form is bounded below by a multiple of the squared normProved

  Sep 2026