We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems, H∞ optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and matrix pencils with Hamiltonian and/or ske...

In the framework of quantum computational tensor network, which is a general framework of measurement-based quantum computation, the resource many-body state is represented in a tensor-network form (or a matrix-product form), and universal quantum computation is performed in a virtual linear space, which is called a correlation space, where tensors live. Since any unitary operation, state prepa...

Our achievements on the theoretical front will be very poor indeed if...we close our eyes to problems and can only memorize isolated conclusions or principles... A mathematical discipline is alive and well if it has many exciting open problems of different levels of difficulty. This section's goal is to show that this is the case with Ergodic Ramsey Theory. To warm up we shall start with some r...

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call...

We propose a unitary matrix model as a regularization of the IIB matrix model of Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT). The fermionic part is incorporated using the overlap formalism in order to avoid unwanted “doublers” while preserving the global gauge invariance. This regularization, unlike the one adopted by IKKT, has manifest U(1)10 symmetry, which corresponds to the ten-dimensional tra...

When considering a sequent-style proof system for quantum programs, there are certain elements of quantum mechanics that we may wish to capture, such as phase, dynamics of unitary transformations, and measurement probabilities. Traditional quantum logics which focus primarily on the abstract orthomodular lattice theory and structures of Hilbert spaces have not satisfactorily captured some of th...

We discuss matrix pencils with a double symmetry structure that arise in the Hartree– Fock model in quantum chemistry. We derive anti-triangular condensed forms from which the eigenvalues can be read off. Ideally these would be condensed forms under unitary equivalence transformations that can be turned into stable (structure preserving) numerical methods. For the pencils under consideration th...

In this letter, the ability of higher-order Hopfield networks to solve combinatorial optimization problems is assessed by means of a rigorous analysis of their properties. The stability of the continuous network is almost completely clarified: (1) hyperbolic interior equilibria, which are unfeasible, are unstable; (2) the state cannot escape from the unitary hypercube; and (3) a Lyapunov functi...

Quantum field theory of space-like particles is investigated in the framework of absolute causality scheme preserving Lorentz symmetry. It is shown that tachyons are associated with unitary orbits of Poincaré mappings induced from SO(2) little group instead of SO(2, 1) one. Therefore the corresponding elementary states are labelled by helicity. A particular case of the helicity λ = ± 1 2 is inv...

In this work we investigate a quaternion-based formulation of 3D Simultaneous Localization and Mapping with Extended Kalman Filter (EKF-SLAM) using relative pose measurements. The equations of the filter do not rely on heuristic solutions for preserving the unit norm of quaternions of rotation, nor introduce spurious measurements to force the unitary norm constraint. The proposed model is forma...