A random state vector (a vector of complex elements with unit length) can be generated in the following way : (i) Generate a vector such that both the real and the imaginary parts of the vector elements are random numbers that have a normal distribution with a zero mean and unit variance. (ii) Normalize the vector.
QUBIT4MATLAB V3.0: A program package for quantum information science and quantum optics for MATLAB
This is a symmetric square matrix τij of complex numbers, whose rank equals the genus of the surface and whose imaginary part is strictly positive.
Topological Strings and Quantum Curves
Let ℜP : K0 → R and ℑP : K0 → R be deﬁned by ℜP (v) := Re(cid:0)SP (v)(cid:1) and ℑP (v) := Im(cid:0)SP (v)(cid:1), respectively, where SP : K → carr(P ) ⊂ C is the associated simplicial isomorphism, and Re and Im denote real and imaginary parts of a complex number, respectively.
Random walks of circle packings
It is convenient to deﬁne local averages of diagonal and off-diagonal elements of Oαβ , O(z) = Dσ2 Xα Oαα δ(z − λα )E , Oαβ δ(z1 − λα ) δ(z2 − λβ )E . O(z1 , z2 ) = Dσ2 Xα6=β Here, z = x + iy is a complex number with real and imaginary parts x and y and δ(z) denotes a delta-function in both coordinates.
Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
The user-deﬁned functions VALUE, REALVALUE and IMAGVALUE, which return the value of a taylor ob ject as as a complex number, and the real and imaginary part of this value, respectively, as a real number are also provided.
TaylUR 3, a multivariate arbitrary-order automatic differentiation package for Fortran 95
The amplitude δf can be a complex number, the imaginary part of which represents the relative phase shift between different quantities.
A Godunov Method for Multidimensional Radiation Magnetohydrodynamics based on a variable Eddington tensor
The real part of ω , R(ω ), describes a drift of the pattern along the azimuth which only exists for nonaxisymmetric ﬂows. L is a complex quantity, both its real part and its imaginary part must vanish for the critical Reynolds number (Fig. 2).
Nonaxisymmetric patterns in the linear theory of MHD Taylor-Couette instability
For axisymmetric ﬂows (m = 0) the real part of ω , i.e. R(ω ), is zero for stationary patterns of ﬂow and ﬁeld and it is nonzero for oscillating solutions, which are called overstability. L is a complex quantity, both its real part and its imaginary part must vanish for the critical Reynolds number.
The linear MHD Taylor-Couette instability for liquid sodium
The real and the imaginary part of a complex-valued vector x are denoted by xI and xQ , respectively, |x| denotes the absolute value of a complex number x and |S | denotes the cardinality of the set S .
A Low ML-decoding Complexity, Full-diversity, Full-rate MIMO Precoder
SL A special linear group. τ A complex number with positive imaginary part, or the number of orbits of cusps. T r The trace of something.
Reflection groups of Lorentzian lattices