Characterizing matrices with XX-simple image eigenspace in max-min semiring

A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector $x$ belonging to the interval $X=\{x\colon \underline{x}\leq x\leq\overline{x}\}$ is the unique solution of the system $A\otimes y=x$ in $X$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.Comment: 23 page

On the max-algebraic core of a nonnegative matrix

The max-algebraic core of a nonnegative matrix is the intersection of column spans of all max-algebraic matrix powers. Here we investigate the action of a matrix on its core. Being closely related to ultimate periodicity of matrix powers, this study leads us to new modifications and geometric characterizations of robust, orbit periodic and weakly stable matrices.Comment: 27 page

Tolerance problems for generalized eigenvectors of interval fuzzy matrices

summary:Fuzzy algebra is a special type of algebraic structure in which classical addition and multiplication are replaced by maximum and minimum (denoted $\oplus$ and $\otimes$, respectively). The eigenproblem is the search for a vector $x$ (an eigenvector) and a constant $\lambda$ (an eigenvalue) such that $A\otimes x=\lambda\otimes x$, where $A$ is a given matrix. This paper investigates a generalization of the eigenproblem in fuzzy algebra. We solve the equation $A\otimes x = \lambda\otimes B\otimes x$ with given matrices $A,B$ and unknown constant $\lambda$ and vector $x$. Generalized eigenvectors have interesting and useful properties in the various computational tasks with inexact (interval) matrix and vector inputs. This paper studies the properties of generalized interval eigenvectors of interval matrices. Three types of generalized interval eigenvectors: strongly tolerable generalized eigenvectors, tolerable generalized eigenvectors and weakly tolerable generalized eigenvectors are proposed and polynomial procedures for testing the obtained equivalent conditions are presented

Controllable and tolerable generalized eigenvectors of interval max-plus matrices

summary:By max-plus algebra we mean the set of reals $\mathbb{R}$ equipped with the operations $a\oplus b=\max\{a,b\}$ and $a\otimes b= a+b$ for $a,b\in \mathbb{R}.$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B\in\mathbb{R}(m,n)$ if $A\otimes x=\lambda\otimes B\otimes x$ for some $\lambda\in \mathbb{R}$. The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the controllable, the tolerable and the strong generalized eigenproblem in max-plus algebra are presented. As a consequence of the obtained results, efficient algorithms for checking equivalent conditions are introduced

The numerical approach to quantum field theory in a non-commutative space

Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is presented, and their implications are reviewed. In addition, we also discuss how related numerical techniques have been recently applied in computer simulations of dimensionally reduced supersymmetric theories.Comment: 15 pages, 6 figures, invited talk presented at the Humboldt Kolleg "Open Problems in Theoretical Physics: the Issue of Quantum Space-Time", to appear in the proceedings of the Corfu Summer Institute 2015 "School and Workshops on Elementary Particle Physics and Gravity" (Corfu, Greece, 1-27 September 2015

Dealing with non-metric dissimilarities in fuzzy central clustering algorithms

Clustering is the problem of grouping objects on the basis of a similarity measure among them. Relational clustering methods can be employed when a feature-based representation of the objects is not available, and their description is given in terms of pairwise (dis)similarities. This paper focuses on the relational duals of fuzzy central clustering algorithms, and their application in situations when patterns are represented by means of non-metric pairwise dissimilarities. Symmetrization and shift operations have been proposed to transform the dissimilarities among patterns from non-metric to metric. In this paper, we analyze how four popular fuzzy central clustering algorithms are affected by such transformations. The main contributions include the lack of invariance to shift operations, as well as the invariance to symmetrization. Moreover, we highlight the connections between relational duals of central clustering algorithms and central clustering algorithms in kernel-induced spaces. One among the presented algorithms has never been proposed for non-metric relational clustering, and turns out to be very robust to shift operations. (C) 2008 Elsevier Inc. All rights reserved

On design of robust fault detection filter in finite-frequency domain with regional pole assignment

This brief is concerned with the fault detection (FD) filter design problem for an uncertain linear discrete-time system in the finite-frequency domain with regional pole assignment. An optimized FD filter is designed such that: 1) the FD dynamics is quadratically D-stable; 2) the effect from the exogenous disturbance on the residual is attenuated with respect to a minimized H∞-norm; and 3) the sensitivity of the residual to the fault is enhanced by means of a maximized H--norm. With the aid of the generalized Kalman-Yakubovich-Popov lemma, the mixed H--/H∞ performance and the D-stability requirement are guaranteed by solving a convex optimization problem. An iterative algorithm for designing the desired FD filter is proposed by evaluating the threshold on the generated residual function. A simulation result is exploited to illustrate the effectiveness of the proposed design technique.This work was supported in part by the Deanship of Scientific Research (DSR) at King Abdulaziz University in Saudi Arabia under Grant 16-135- 35-HiCi, the National Natural Science Foundation of China under Grants 61134009 and 61203139, the Royal Society of the U.K., and the Alexander von Humboldt Foundation of Germany

Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry

Recent vigorous investigations of topological order have not only discovered new topological states of matter but also shed new light to "already known" topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking but most typically manifest topological order known as hidden string order on 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy super-geometry in the construction of supersymmetric versions of VBS (SVBS) states, and give a pedagogical introduction of SVBS models and their properties [arXiv:0809.4885, 1105.3529, 1210.0299]. As concrete examples, we present detail analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e. UOSp(N|2) and UOSp(N|4) SVBS states whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with superconducting property that interpolate various VBS states depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate regardless of the parity of bulk (super)spins. Stability of topological phase with supersymmetry is discussed with emphasis on its relation to particular edge (super)spin states.Comment: Review article, 1+104 pages, 37 figures, published versio

Flow-based reputation: more than just ranking

The last years have seen a growing interest in collaborative systems like electronic marketplaces and P2P file sharing systems where people are intended to interact with other people. Those systems, however, are subject to security and operational risks because of their open and distributed nature. Reputation systems provide a mechanism to reduce such risks by building trust relationships among entities and identifying malicious entities. A popular reputation model is the so called flow-based model. Most existing reputation systems based on such a model provide only a ranking, without absolute reputation values; this makes it difficult to determine whether entities are actually trustworthy or untrustworthy. In addition, those systems ignore a significant part of the available information; as a consequence, reputation values may not be accurate. In this paper, we present a flow-based reputation metric that gives absolute values instead of merely a ranking. Our metric makes use of all the available information. We study, both analytically and numerically, the properties of the proposed metric and the effect of attacks on reputation values