A topological approach to inferring the intrinsic dimension of convex sensing data

We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasi-convex functions, under natural assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We construct a filtration of Dowker complexes, associated to measurements by quasi-convex functions. Topological features of these complexes are then used to infer the intrinsic dimension. We prove convergence theorems that guarantee obtaining the correct intrinsic dimension in the limit of large data, under natural assumptions. We also illustrate the usability of this method in simulations.

A topological approach for emerging D-Branes and its implications for gravity

In this paper, we introduce a new geometric/topological approach to the emerging braneworld scenario in the context of D-branes using partially negative-dimensional product (PNDP) manifolds. The working hypothesis is based on the fact that the orientability of PNDP manifolds can be arbitrary, and starting from this, we propose that gravitational interaction can derive naturally from the non-orientability. According to this hypothesis, we show that topological defects can emerge from non-orientability and they can be identified as gravitational interaction at macroscopic level. In other words, the orientability of fundamental PNDPs can be related to the appearance of curvature at low-energy scales.

Multiple Solutions for Some Classes Integro-Dynamic Equations on Time Scales

In this chapter we study a class of second-order integro-dynamic equations on time scales. A new topological approach is applied to prove the existence of at least two non-negative solutions. The arguments are based upon a recent theoretical result.

A computer program for electromechanical system operational diagnostics based on the topological approach

The paper presents a method, a mathematical model, and a computer program for the operational diagnostics of an electromechanical system (EMS). During EMS operation, service aging changes the properties of the parametric matrices of the windings and, as a consequence, the characteristics of the EMS vector space. Periodic testing of the vector space offers relevant and reliable data on the current health of the EMS, its changes during operation, and the risk of loss of function. The object of the study is an asynchronous electric motor (AEM). It is urgent to automate the process of assessing the current health of an AEM and to organize the storage of information on its states at different stages of its life cycle. To solve the problem, software (SW) for accumulation of information on AEM operation and for evaluation of its basic performance metrics has been developed in the Python programming language. The SW is based on the topological approach to diagnostics, which implies the analysis of the current responses of motor rotor windings to phase voltage pulses. The SW enables one to determine the rate of the service aging of an item, the probability of its survival and residual life, to obtain access to the history of previous diagnostics, and to visualize the in-service history of the above-mentioned performance metrics. The developed SW can be used to increase the AEM operation efficiency and to plan engineering or repair work; it can also be used as an information source for re- engineering and modification of existing AEMs. The described SW can be extended to perform operational diagnostics based on the topological approach of devices of various types. Also, this SW can be considered as a separate information component of the digital twin of a complex EMS, which will allow us to study the main indicators of its reliability, fault tolerance and operational efficiency at all stages of the life cycle.