Diophantine conditions and real or complex Brjuno functions

The continued fraction expansion of the real number x=a_0+x_0, a_0\in {\ZZ}, is given by 0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in {\NN}, for $n\geq 0.$ The Brjuno function is then $B(x)=\sum_{n=0}^{\infty}x_0x_1... x_{n-1}\ln(x_n^{-1}),$ and the number $x$ satisfies the Brjuno diophantine condition whenever $B(x)$ is bounded. Invariant circles under a complex rotation persist when the map is analytically perturbed, if and only if the rotation number satisfies the Brjuno condition, and the same holds for invariant circles in the semi-standard and standard maps cases. In this lecture, we will review some properties of the Brjuno function, and give some generalisations related to familiar diophantine conditions. The Brjuno function is highly singular and takes value $+\infty$ on a dense set including rationals. We present a regularisation leading to a complex function holomorphic in the upper half plane. Its imaginary part tends to the Brjuno function on the real axis, the real part remaining bounded, and we also indicate its transformation under the modular group.Comment: latex jura.tex, 6 files, 19 pages Proceedings on `Noise, Oscillators and Algebraic Randomness' La Chapelle des Bois, France 1999-04-05 1999-04-10 April 5-10, 1999 [SPhT-T99/116

Study of leptonic CP violation

The "complementary" Ansatz, Tr$(M_\nu)=0$, where $M_\nu$ is the prediagonal neutrino mass matrix, seems a plausible approximation for capturing in a self-contained way some of the content of Grand Unification. We study its consequences in the form of relations between the neutrino masses and CP violation phases.Comment: 8 pages, 1 figure, talk at MRST200

Investigating the Light Scalar Mesons

We first briefly review a treatment of the scalars in meson meson scattering based on a non-linear chiral Lagrangian, with unitarity implemented by a "local" modification of the scalar propagators. It is shown that the main results are confirmed by a treatment in the SU(3) linear sigma model in which unitarity is implemented "globally". Some remarks are made on the speculative subject of the scalars' quark structure.Comment: 9 pages,3 figures,talk at hadron2001, Protvin

The effect of political connections on firms' auditor choice decisions and audit opinions: evidence from Egypt

Purpose – This paper aims to examine how different types of corporate political connections (PCs) affect auditor choice decisions (and, therefore, audit quality) and audit opinions following the 2013 Egyptian uprising. Design/methodology/approach – This paper utilizes a unique hand-collected dataset on the type of PCs of Egyptian listed companies from 2014 to 2019. Several analyses are employed to test the hypotheses, including logit regression, probit regression and generalized linear mixed models (GLMM). A number of additional analyses are conducted to ensure the robustness of the results, including the instrumental variables (IVs) probit models and propensity score matching (PSM). Findings – The results show that firms’ choice of auditor and audit opinion is heavily influenced by firms’ PCs. Companies with PCs through boards of directors and major shareholders hire Big 4 audit firms to enhance corporate legitimacy; however, government-linked companies usually retain non-Big 4 audit firms to avoid increased transparency and to conceal improper activities, including tunneling and rent-seeking. Further, the results indicate that companies with PCs through boards of directors or major shareholders are more likely to receive favorable audit opinions, whereas government-owned businesses are less likely to receive such opinions. Research limitations/implications – This study provides additional evidence to policymakers that binding regulations and guidelines are necessary to oversee politically connected firms (PCFs) and to enhance governance and investor protection. Originality/value – This study provides the first empirical evidence on how corporate PCs influence the choice of auditors and the opinions of audit firms in Egypt. This paper also sheds light on the impact of different types of corporate PCs on the choice of auditors and audit opinions

On learning and visualizing lexicographic preference trees

Preferences are very important in research fields such as decision making, recommendersystemsandmarketing. The focus of this thesis is on preferences over combinatorial domains, which are domains of objects configured with categorical attributes. For example, the domain of cars includes car objects that are constructed withvaluesforattributes, such as ‘make’, ‘year’, ‘model’, ‘color’, ‘body type’ and ‘transmission’.Different values can instantiate an attribute. For instance, values for attribute ‘make’canbeHonda, Toyota, Tesla or BMW, and attribute ‘transmission’ can haveautomaticormanual. To this end,thisthesis studiesproblemsonpreference visualization and learning for lexicographic preference trees, graphical preference models that often are compact over complex domains of objects built of categorical attributes. Visualizing preferences is essential to provide users with insights into the process of decision making, while learning preferences from data is practically important, as it is ineffective to elicit preference models directly from users. The results obtained from this thesis are two parts: 1) for preference visualization, aweb- basedsystem is created that visualizes various types of lexicographic preference tree models learned by a greedy learning algorithm; 2) for preference learning, a genetic algorithm is designed and implemented, called GA, that learns a restricted type of lexicographic preference tree, called unconditional importance and unconditional preference tree, or UIUP trees for short. Experiments show that GA achieves higher accuracy compared to the greedy algorithm at the cost of more computational time. Moreover, a Dynamic Programming Algorithm (DPA) was devised and implemented that computes an optimal UIUP tree model in the sense that it satisfies as many examples as possible in the dataset. This novel exact algorithm (DPA), was used to evaluate the quality of models computed by GA, and it was found to reduce the factorial time complexity of the brute force algorithm to exponential. The major contribution to the field of machine learning and data mining in this thesis would be the novel learning algorithm (DPA) which is an exact algorithm. DPA learns and finds the best UIUP tree model in the huge search space which classifies accurately the most number of examples in the training dataset; such model is referred to as the optimal model in this thesis. Finally, using datasets produced from randomly generated UIUP trees, this thesis presents experimental results on the performances (e.g., accuracy and computational time) of GA compared to the existent greedy algorithm and DPA