Structural and topological phase transitions on the German Stock Exchange

We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two transitions of the topology of a complex network representing FSE were found. First transition is from a hierarchical scale-free MST representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the market's state for the period containing the crash. Subsequently, a transition is observed from this transient, (meta)stable state of the crash, to a hierarchical scale-free MST decorated by several star-like trees after the worldwide financial crash. The phase transitions observed are analogous to the ones we obtained earlier for the Warsaw Stock Exchange and more pronounced than those found by Onnela-Chakraborti-Kaski-Kert\'esz for S&P 500 index in the vicinity of Black Monday (October 19, 1987) and also in the vicinity of January 1, 1998. Our results provide an empirical foundation for the future theory of dynamical, structural and topological phase transitions on financial markets

On Van, R and S entropies of graphenylene

Applications in the disciplines of chemistry, pharmaceuticals, communication, physics, and aeronautics all heavily rely on graph theory. To examine the properties of chemical compounds, the molecules are modelled as a graph. A few physical characteristics of the substance, including its boiling point, enthalpy, pi-electron energy, and molecular weight, are related to its geometric shape. Through the resolution of one of the interdisciplinary problems characterizing the structures of benzenoid hydrocarbons and graphenylene, the essay seeks to ascertain the practical applicability of graph theory. The topological index, which displays the correlation of chemical structures using numerous physical, chemical, and biological processes, is an invariant of a molecular graph connected with the chemical structure. Shannon's concept of entropy served as the basis for the graph entropies with topological indices, which are now used to measure the structural information of chemical graphs. Using various graph entropy metrics, the theory of graphs can be used to establish the link between particular chemical structural features. This study uses the appropriate R, S, Van topological indices to introduce some unique degree-based entropy descriptors. Additionally, the graphenylene structure's entropy measurements indicated above were computed

On Van, R and S entropies of graphenylene

Applications in the disciplines of chemistry, pharmaceuticals, communication, physics, and aeronautics all heavily rely on graph theory. To examine the properties of chemical compounds, the molecules are modelled as a graph. A few physical characteristics of the substance, including its boiling point, enthalpy, pi-electron energy, and molecular weight, are related to its geometric shape. Through the resolution of one of the interdisciplinary problems characterizing the structures of benzenoid hydrocarbons and graphenylene, the essay seeks to ascertain the practical applicability of graph theory. The topological index, which displays the correlation of chemical structures using numerous physical, chemical, and biological processes, is an invariant of a molecular graph connected with the chemical structure. Shannon's concept of entropy served as the basis for the graph entropies with topological indices, which are now used to measure the structural information of chemical graphs. Using various graph entropy metrics, the theory of graphs can be used to establish the link between particular chemical structural features. This study uses the appropriate R, S, Van topological indices to introduce some unique degree-based entropy descriptors. Additionally, the graphenylene structure's entropy measurements indicated above were computed

On Van, R and S entropies of graphenylene

Applications in the disciplines of chemistry, pharmaceuticals, communication, physics, and aeronautics all heavily rely on graph theory. To examine the properties of chemical compounds, the molecules are modelled as a graph. A few physical characteristics of the substance, including its boiling point, enthalpy, pi-electron energy, and molecular weight, are related to its geometric shape. Through the resolution of one of the interdisciplinary problems characterizing the structures of benzenoid hydrocarbons and graphenylene, the essay seeks to ascertain the practical applicability of graph theory. The topological index, which displays the correlation of chemical structures using numerous physical, chemical, and biological processes, is an invariant of a molecular graph connected with the chemical structure. Shannon's concept of entropy served as the basis for the graph entropies with topological indices, which are now used to measure the structural information of chemical graphs. Using various graph entropy metrics, the theory of graphs can be used to establish the link between particular chemical structural features. This study uses the appropriate R, S, Van topological indices to introduce some unique degree-based entropy descriptors. Additionally, the graphenylene structure's entropy measurements indicated above were computed

Integrity Constraints Revisited: From Exact to Approximate Implication

Integrity constraints such as functional dependencies (FD), and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Finally, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Our results recover, and sometimes extend, several previously known results about the implication problem: implication of MVDs can be checked by considering only 2-tuple relations, and the implication of differential constraints for frequent item sets can be checked by considering only databases containing a single transaction