The generating functional of correlation functions for theXXZ spin chain is considered in the thermodynamic limit. We derive a system of integro-difference equations that prescribe this functional. On the basis of this system we establish the operator-valued Riemann–Hilbert problem for correlation functions of the XXZ spin chain.

A weighted Hilbert space approach to the study of zero-energy states of supersymmetric matrix models is introduced. Applied to a related but technically simpler model, it is shown that the spectrum of the corresponding weighted Hamiltonian simplifies to become purely discrete for sufficient weights. This follows from a bound for the number of negative eigenvalues of an associated matrix-valued ...

Let X1, X2, . . . , Xn be independent random variables and Sk = Pk i=1 Xi. We show that for any constants ak, P( max 1≤k≤n ||Sk| − ak| > 11t) ≤ 30 max 1≤k≤n P(||Sk| − ak| > t). We also discuss similar inequalities for sums of Hilbert and Banach space valued random vectors.

The in-process changes of weld nugget growth during the Resistance Spot Welding were investigated based on the resistance of input electrical impedance. To compute the time varying resistance of input electrical impedance, the welding voltage and current signals are measured simultaneously and then converted into complex-valued signals by using Hilbert transform. Comparing with the dynamic cont...

Principal component analysis (PCA) has been generalized to complex principal component analysis (CPCA), which has been widely applied to complex-valued data, two-dimensional vector fields, and complexified real data through the Hilbert transform. Nonlinear PCA (NLPCA) can also be performed using auto-associative feed-forward neural network (NN) models, which allows the extraction of nonlinear f...

This article focuses on a recent concept of covariation for processes taking values in a separable Banach space B and a corresponding quadratic variation. The latter is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace χ of the dual of the projective tensor product of B with itself. We also introduce the notion of a convolution type ...

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.

In this chapter we study cones in the real Hilbert spaces of Hermitian matrices and real valued trigonometric polynomials. Based on an approach using such cones and their duals, we establish various extension results for positive semidefinite matrices and nonnegative trigonometric polynomials. In addition, we show the connection with semidefinite programming and include some numerical experiments.

Set-valued mappings play an important role in a number of optimization problems. We examine several of those problems in these notes. We discuss iterative algorithms for solving these problems and prove convergence. We use the finite-dimensional space RJ through-out, although the results also hold for weak convergence in infinite-dimensional Hilbert space.