Aspects of a Phase Transition in High-Dimensional Random Geometry

A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization with a ban on short-selling, the storage capacity of the perceptron, the solvability of a set of linear equations with random coefficients, and competition for resources in an ecological system. These examples shed light on various aspects of the underlying geometric phase transition, create links between problems belonging to seemingly distant fields, and offer the possibility for further ramifications.

Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.

$$ T\overline{T} $$ Deformation of stress-tensor correlators from random geometry

We study stress-tensor correlators in the $$ T\overline{T} $$ T T ¯ -deformed conformal field theories in two dimensions. Using the random geometry approach to the $$ T\overline{T} $$ T T ¯ deformation, we develop a geometrical method to compute stress-tensor correlators. More specifically, we derive the $$ T\overline{T} $$ T T ¯ deformation to the Polyakov-Liouville conformal anomaly action and calculate three and four-point correlators to the first-order in the $$ T\overline{T} $$ T T ¯ deformation from the deformed Polyakov-Liouville action. The results are checked against the standard conformal perturbation theory computation and we further check consistency with the $$ T\overline{T} $$ T T ¯ -deformed operator product expansions of the stress tensor. A salient feature of the $$ T\overline{T} $$ T T ¯ -deformed stress-tensor correlators is a logarithmic correction that is absent in two and three-point functions but starts appearing in a four-point function.

Pengaruh Permainan Acak Geometri Terhadap Perkembangan Kecerdasan Logika-Matematika Anak

Based on the results of observations in the field on the development of mathematical logical intelligence in students has not developed optimally so it is necessary to apply the geometry random game media. This study aims to determine the effect of random geometry on the development of mathematical-logic intelligence in children at Nasyithatun Nisa Kindergarten, Teluk Kiambang, Tempuling District, Indragiri Hilir Regency. The sample used in this study was 13 students consisting of one class. The data collection techniques used are test, observation and documentation. The data analysis technique used the Prerequisite test and Hypothesis test using the SPSS 17. The research hypothesis was that the activity of using geometric random games had an influence on the development of mathematical-logic intelligence in group B children at Nasyithatun Nisa Kindergarten. This can be seen from the results of data analysis on the comparison of the pretest and posttest classes obtained by tcount = 50,229 and Sig. (2-tailed) = 0,000. because of Sig. (2-tailed) = 0,000 <0,05, it can be concluded that there is a significant effect after using geometric random play in learning. So that means Ho is rejected and Ha is accepted, which means that in this study there is the influence of random geometry games before and after treatment. The effect of random geometry on the development of logic-mathematical intelligence in children at Nasyithatun Nisa Kindergarten was 82,005%.