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Several subadditivity results and conjectures are given for matrices (or operators), block-matrices, concave functions and norms.
In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of $d \times d$ matrices, if their traces are known. As a next step $4 \times 4$…
The attitude space has been parameterized in various ways for practical purposes. Different representations gain preferences over others based on their intuitive understanding, ease of implementation, formulaic simplicity, and physical as…
There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a…
In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient…
Random tensor models can be used as combinatorial devices to generate Euclidean dynamical triangulations. A physical continuum limit of dynamical triangulations requires a suitable generalization of the double-scaling limit of random…
We describe the images of multilinear polynomials of degree up to four on the upper triangular matrix algebra.
The problem of extracting a well conditioned submatrix from any rectangular matrix (with normalized columns) has been studied for some time in functional and harmonic analysis; see…
On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for…
We adopt an operator-theoretic perspective to analyze a class of nonlinear fixed-point iterations and discrete-time dynamical systems. Specifically, we study the Krasnoselskij iteration - at the heart of countless algorithmic schemes and…
Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process an equivalent to Witt's…
We compute to all loop orders correlation function of four heavy BPS operators in $\mathcal{N}$= 4 SYM with special polarisations considered recently by Frank Coronado. Our main result is an expression for the octagon form factor as…
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…
We find an algorithmic procedure that enables to compute and to describe the structure of the isotropy subgroups of the group of complex orthogonal matrices with respect to the action of similarity on complex symmetric matrices. A key step…
Recently, a new and powerful separability criterion was introduced in [O. Rudolph, quant-ph/0202121] and [Chen {\it et al.}, quant-ph/0205017]. Composing the main idea behind the above criterion and the necessary and sufficient condition in…
In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of {\em triangles}, for a matroid to have either $U_{2,4}$ or $F_7$ or $M(K_5)$ as a minor.
Contraction theory is a mathematical framework for studying the convergence, robustness, and modularity properties of dynamical systems and algorithms. In this opinion paper, we provide five main opinions on the virtues of contraction…
In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking…
We obtain several sharp lower and upper bounds for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. As applications of these bounds we deduce a chain of new bounds for the classical…
We derive upper and lower bounds on the determinant of an exponential matrix. They can be transformed into corresponding bounds for the determinant of a univariate Gaussian matrix.