In this work we present a matrix generalization of the Euler identity about exponential representation of a complex number. The concept of matrix exponential is used in a fundamental way. We define a notion of matrix imaginary unit which generalizes the usual complex imaginary unit. The Euler-like identity so obtained is compatible with the classical one. Also, we derive some exponential repres...

Introduction: He and Yablonskiy introduced a quantitative BOLD method (qBOLD) for estimation of the oxygenation extraction fraction (OEF) and the deoxygenated blood volume (DBV) in brain tissue [1]. In qBOLD, the signal course of a spin echo scan is compared to tissue model curves. The model incorporates assumptions about the blood capillaries, parenchyma, interstitial/cerebrospinal fluid, and ...

Abstract. In this paper we propose a variant of the Generalized Matrix Learning Vector Quantization (GMLVQ) for dissimilarity learning on complex-valued data. Complex features can be encountered in various data domains, e.g. Fourier transformed mass spectrometry or image analysis data. Current approaches deal with complex inputs by ignoring the imaginary parts or concatenating real and imaginar...

We consider saddle point integrals in d variables whose phase functions are neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The proofs are via contour shifting in comp...

We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real- and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, a...

One suggestion for determining the properties of QCD at finite temperatures and densities is to carry out lattice simulations with an imaginary chemical potential whereby no sign problem arises, and to convert the results to real physical observables only afterwards. We test the practical feasibility of such an approach for a particular class of physical observables, spatial correlation lengths...

transmittance and reflectance, the imaginary part can be determined from transmittance data alone. It is possible to use Kramers-Kronig analysis to calculate the real part if the imaginary part is known over a sufficiently broad range. We show that the delta calculated from reflection and transmission data without taking into account roughness may underestimate the real part of the refractive i...

Sensitivity analysis of aerosol parameter (refractive index which consists of real and imaginary parts, size distribution which is represented by Junge parameter) estimations with measured solar direct and diffuse irradiance is made. Through experiments with the measured solar direct and diffuse irradiance, it is found that the results from the sensitivity analysis is valid and adequate. Keywor...

We consider thermal n-point Green functions in the framework of quantum field theory at finite temperature. We show how analytic continuations from imaginary to real energies relate these functions originally defined in the imaginary-time formalism to retarded and advanced real-time ones. The described method is valid to all orders of perturbation theory. It has the further advantage that it is...

These bounds result in IXkl < v'Na, k = 0,1 N — 1. With scaling by 1/N. and the requirement that the output magnitude be less than one, a = i//. However, this is not the tightest bound on the real and imaginary components of the input data. It can be shown that the magnitudes of the real and imaginary parts of the output of the complex DFT are bounded by NaSN. For N a multiple of 4. — 4cos(r/N)...