Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems

Manibo CNC. Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems. Bielefeld: Universität Bielefeld; 2019.In this work, we consider primitive inflation rules as generators of aperiodic tilings, and subsequently, of aperiodic point sets (which are toy models for quasicrystals) deemed adequate for diffraction analysis. We harvest the combinatorial-geometric properties of these systems to obtain renormalisation relations for the pair correlation functions, which carry over to measures that generate the diffraction measure. This yields a measure-valued renormalisation satisfied by each of the components of the diffraction. Using tools from the theory of Lyapunov exponents, we provide a sufficient criterion to rule out the presence of absolutely continuous components in the diffraction and a necessary condition to have a non-trivial absolutely continuous part. Moreover, we provide a computable bound which one can use to use invoke this criterion. We show that this holds for large classes of systems, and, as a sanity check, show that the necessary criterion for existence is satisfied by systems which are a priori known to have absolutely continuous diffraction. Furthermore, we present the recovery of known singularity results and point out connections to number-theoretic quantities which naturally arise from these objects, such as logarithmic Mahler measures

Spectral theory of spin substitutions

We introduce substitutions in Zm which have non-rectangular domains based on an endomorphism Q of Zm and a set D of coset representatives of Zm/QZm, which we call digit substitutions. Using a finite abelian ‘spin’ group we define ‘spin digit substitutions’ and their subshifts (Σ, Zm). Conditions under which the subshift is measure-theoretically isomorphic to a group extension of an m-dimensional odometer are given, inducing a complete decomposition of the function space L2 (Σ, µ). This enables the use of group characters in Ĝ to derive substitutive factors and analyze the spectra of specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous, and singular continuous spectral measures, together with some bounds on their spectral multiplicity

Monochromatic arithmetic progressions in automatic sequences with group structure

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence {dn} of differences along which the maximum length A (dn) of a monochromatic arithmetic progression (with fixed difference dn) grows at least polynomially in dn. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution

A study on the applicability of a practical experience requirement for the licensure of Certified Public Accountants in the Philippines

Several studies conducted by international bodies conclude that the licensure standards for Certified Public Accountants (CPA) in the Philippines are not at par with international standards and practices. In particular, the lack of a practical experience requirement has been pointed out as a major weakness. The International Federation of Accountants (IFAC), through its International Education Guidelines (IEG), includes practical experience as a requirement prior to licensure and entry into the profession. Its guidelines stipulate that all candidates should undergo a period of work experience whose length and coverage permits prospective accountants to demonstrate knowledge, skills and values sufficient for performing with professional competence . The international body recommends a period of no less than three years under a supervised environment in the scope of work of accountancy

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

Baake M, Gähler F, Manibo N. Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction. Communications in Mathematical Physics. 2019;370(2):591-635.The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. We also discuss some consequences for the spectral measures of such systems. While we develop the theory first for the classic setting in one dimension, we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations

Zaremba, Salem and the fractal nature of ghost distributions

http://dx.doi.org/10.1017/S000497271200033

Spectral properties of substitutions on compact alphabets

We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non-constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry

Transparent HTTPS web filtering system with no end user configuration - Prozilla

Prozilla is a network application which is capable of filtering network traffic without additional installation of certificates/configuration on the end user system. One distinct technology in network security is the web filtering system, a network application responsible for screening and filtering the traffic at the end users based on their network policy. HTTPS connections secures the connection between client and the web server by sharing keys between the client and the web server which provides security against tampering or eavesdropping - and because of these shared keys, systems are unable to filter the requests without additional configuration on the end user\u27s system. This study aims to develop an in-line HTTPS web filtering application which does not require any configuration on the client\u27s system. The system Prozilla will be located between a switch and a router in which the switch is connected to the client\u27s system and the router being connected to the Internet. The system can prohibit access to the end user\u27s requests by the use of [a] content based filtering by website categorization. When the end user requests for a restricted website, the system terminates the connection of the request and provides notifications by returning the system\u27s block page