We extend the perturbation theory of Vǐsik, Ljusternik and Lidskĭı for eigenvalues of matrices, using methods of min-plus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Ou...

We first prove Kato's inequalities for the Laplacian and a Schr$\ddot{o}$dinger-type operator on smooth functions closed Riemannian manifolds. then apply result to establish some new domination spectral zeta their related kernels $n$-dimensional unit spheres using majorisation techniques. Our results are generalisations of comparison surfaces

Resource theories provide a framework for thermodynamics beyond the thermodynamic limit. For systems with definite energy, the key resource is purity. In this paper we formulate three alternative resource theories of purity in the framework of general probabilistic theories (GPTs). Each resource theory corresponds to a different choice of free operations, including i) random reversible operatio...

The concept of majorisation is explored as a tool to characterize the performance quantum Otto engine in quasi-static regime. For working substance form spin arbitrary magnitude, yields necessary and sufficient condition for operation engine, provided canonical distribution medium at hot reservoir majorised by its cold reservoir. case spin-1/2 interacting with an via isotropic Heisenberg exchan...

‎let $lambda_1,dots,lambda_n$ be positive real numbers such that‎‎$sum_{k=1}^n lambda_k=1$‎. ‎we prove that for‎‎any positive operators $a_1,a_2,cdots‎, ‎a_n$ in semifinite von‎‎neumann algebra $m$ with faithful normal trace that $t(1)

Let A,B be two positive definite matrices. The Fan-Lidskii majorisations can be subsumed as the symmetric norm rearrangement inequalities ‖A↓ +B↑‖ ‖A+B‖ ‖A↓ +B↓‖ where the up/down arrows on A,B mean the diagonal matrices with the same eigenvalues in decreasing/increasing order down to the diagonal. We refine these relations with sums of type A +UBU∗ and A +VBV∗ for two unitary matrices U,V asso...

The composite autoregressive system can be used to estimate a speech source-filter decomposition in a rigorous manner, thus having potential use in glottal inverse filtering. By introducing a suitable prior, spectral tilt can be introduced into the source component estimation to better correspond to human voice production. However, the current expectationmaximisation based composite autoregress...

Estimation of a precision matrix (i.e. inverse covariance matrix) is widely used to exploit conditional independence among continuous variables. The influence abnormal observations exacerbated in high dimensional setting as the dimensionality increases. In this work, we propose robust estimation based on an l1 regularised objective function with weighted sample matrix. robustness proposed can b...

This paper explores some connections between rank one convexity, multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant. Theorem 6.2 describes a class of non-polyconvex but multiplicative quasiconvex isotropic functions. Relevance of...