A Quantum Geometric Model of Color Similarity Judgements

Since Tversky (1977) argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been offered as particularly difficult challenges for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky (1977), asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize to different kinds of stimuli, especially perceptual stimuli for which there are no apparent differences in extent of knowledge. To investigate similarity judgments between perceptual stimuli, across three experiments we collected data where individuals would rate the similarity of a pair of temporally separated color patches. We identified several violations of symmetry in the empirical results, which the conventional multidimensional scaling model cannot readily capture. Pothos et al. (2013) proposed a quantum geometric model of similarity to account for Tversky’s (1977) findings. In the present work, we developed this model to a form that can be fit to similarity judgments. We fit several variants of quantum and multidimensional scaling models to the behavioral data and concluded in favor of the quantum approach. Without further modifications of the model, the quantum model additionally predicted violations of the triangle inequality that we observed in the same data. Overall, by offering a different form of geometric representation, the quantum geometric model of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity.

Positive Polynomials and Boundary Interpolation with Finite Blaschke Products

The famous Jones–Ruscheweyh theorem states that n distinct points on the unit circle can be mapped to n arbitrary points on the unit circle by a Blaschke product of degree at most $$n-1$$ n - 1 . In this paper, we provide a new proof using real algebraic techniques. First, the interpolation conditions are rewritten into complex equations. These complex equations are transformed into a system of polynomial equations with real coefficients. This step leads to a “geometric representation” of Blaschke products. Then another set of transformations is applied to reveal some structure of the equations. Finally, the following two fundamental tools are used: a Positivstellensatz by Prestel and Delzell describing positive polynomials on compact semialgebraic sets using Archimedean module of length N. The other tool is a representation of positive polynomials in a specific form due to Berr and Wörmann. This, combined with a careful calculation of leading terms of occurring polynomials finishes the proof.

Compiler for physical and geometric construction of sets

This article is devoted to the geometric representation of sets. The paper presents an implementation that allows using operations on sets to build any images. For this algorithm, the equations of first-and second-order lines on the plane and first-and second-order surfaces in space are used. Restrictions are introduced by setting segments on the coordinate axes, indicating whether lines or surfaces belong to these segments. Inequalities are used to create the shaded area of the drawings. In the course of research on this topic, a software tool was developed that allows you to model drawings using the entered formulas, build images based on a set of operations on given sets of points, which allows you to have an analytical description of the drawings. This software tool can be used both in the educational process to test the correctness of the assimilation of the material, and to create flat geometric shapes and three-dimensional bodies for the purpose of engineering the optimal forms of structures and various parts. This application allows the user, performing operations with mathematical expressions and inequalities, to get a graphical display of the result with a mathematical description of any part of the figure.

From the BTZ black hole to JT gravity: geometrizing the island

We study the evaporation of two-dimensional black holes in JT gravity from a three-dimensional point of view. A partial dimensional reduction of AdS3 in Poincaré coordinates leads to an extremal 2D black hole in JT gravity coupled to a ‘bath’: the holographic dual of the remainder of the 3D spacetime. Partially reducing the BTZ black hole gives us the finite temperature version. We compute the entropy of the radiation using geodesics in the three-dimensional spacetime. We then focus on the finite temperature case and describe the dynamics by introducing time-dependence into the parameter controlling the reduction. The energy of the black hole decreases linearly as we slowly move the dividing line between black hole and bath. Through a re-scaling of the BTZ parameters we map this to the more canonical picture of exponential evaporation. Finally, studying the entropy of the radiation over time leads to a geometric representation of the Page curve. The appearance of the island region is explained in a natural and intuitive fashion.

Semantics in High-Dimensional Space

Geometric models are used for modelling meaning in various semantic-space models. They are seductive in their simplicity and their imaginative qualities, and for that reason, their metaphorical power risks leading our intuitions astray: human intuition works well in a three-dimensional world but is overwhelmed by higher dimensionalities. This note is intended to warn about some practical pitfalls of using high-dimensional geometric representation as a knowledge representation and a memory model—challenges that can be met by informed design of the representation and its application.

Geometric Interpretation and General Classification of Three-Dimensional Polarization States through the Intrinsic Stokes Parameters

In contrast with what happens for two-dimensional polarization states, defined as those whose electric field fluctuates in a fixed plane, which can readily be represented by means of the Poincaré sphere, the complete description of general three-dimensional polarization states involves nine measurable parameters, called the generalized Stokes parameters, so that the generalized Poincaré object takes the complicated form of an eight-dimensional quadric hypersurface. In this work, the geometric representation of general polarization states, described by means of a simple polarization object constituted by the combination of an ellipsoid and a vector, is interpreted in terms of the intrinsic Stokes parameters, which allows for a complete and systematic classification of polarization states in terms of meaningful rotationally invariant descriptors.

Modelling purposeful processes based on the geometric representation of their trajectories

The possibilities of studying processes based on the geometric representation of their trajectories in a multidimensional space of factors and using projections of trajectories on planes suitable for analysis are discussed. A connection is established between such constructions and the spiral and snail of development. Based on the proposed projection, a development snail is defined as a special representation format of the purposeful process, expanding the possibilities of analysis of modelling results.