To infer transfer operators from data is usually take as a classical problem that hinges on the Ulam method. The usual description is in terms of projection onto basis functions that are characteristic functions supported over a fine grid of rectangles, that we have previously called the Ulam-Galerkin method when taken in terms of finite time. Here, we present that the same problem can be understood by statistical density estimation formalism. In these terms, the usual Ulam-Galerkin approach is a density estimation by the histogram method. This perspective allows us other methods such as spectral density estimation, Kernel density estimation, etc. However, this is not the only popular method of density estimation, and we point out inherent efficiencies available by the popular kernel density estimation method, and this general phrasing of the problem allows for analysis of bias and variance, toward a discussion of the mean square error for example

Learning high dimensional nonlinear dynamical systems multivariate time series data, without knowing the governing equation is a trending topic in the field of dynamical systems, fluid dynamics, aerodynamics, neuroscience, financial trading markets, multimedia, etc. In the spirit of random projection theory and especially the Johnson-Lindenstrauss theorem, we will re-interpret a major tool known as Dynamic Mode Decomposition (DMD) and its extensions. Simple and computationally efficient random projections improve aspects of the DMD, the Extended DMD (EDMD), and also kernelized DMD approximations of the Koopman operator. The Johnson-Lindenstrauss Lemma provides a theoretical framework for random feature selection by random projection matrices.

Oceanic flow around the Strait of Gibraltar comprises dynamic sub-mesoscale features arising due to topographic and tidal forcing, instabilities, and strongly nonlinear hydraulic processes, all governed by nonlinear equations of fluid motion. The purpose of this study is to isolate dominant features from 3D MIT general circulation model simulations and to investigate their physics. To this end, we use the Dynamic Mode Decomposition (DMD) that decomposes the sequence of simulation snapshots into a sum of Koopman modes: spatial profiles with well-defined exponential growth/decay rates and oscillation frequencies. To identify known features, we correlate identified DMD modes with the tidal forcing and demonstrate that DMD is able to non-parametrically detect the prominent waves known to occur in the western Mediterranean. Additionally, the analysis reveals previously undocumented Kelvin waves and demonstrates that meandering motions in the Atlantic Jet entering the Mediterranean Sea are associated with the diurnal tidal forcing. The DMD thus recovers the results obtained by classical harmonic analysis of tidal constituents, and also highlights features that have eluded attention so far, suggesting that DMD could be a useful part of an oceanographer's toolbox.

We applied Dynamic Mode Decomposition (DMD) to identify flow constituents in a three-dimensional simulation of tidal flows through the Strait of Gibraltar over a six-day period. DMD managed to isolate well-known tidal patterns, such as the Western Alboran Gyre, the twelve-hour lunar tide cycle (M2), and the tidal bore discharging Atlantic waters into the Mediterranean. Additionally, the DMD modes were able to clarify the structure of a recurrent flow pattern occurring as the water passes the Camarinal Sill just to the Atlantic side of Gibraltar.

DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. For a given graph $G$, the self-assembled structure represents a thickened graph $F(G)$ containing $G$ as a deformation retract. We show that for every postman tour $\tau$ in a connected $3$-valent (multi)graph $G$ there exists a thickened graph $F(G)$ with a reporter strand $\sigma$ which contains $\tau$. In cases where the postman tour $\tau$ and the reporter strand $\sigma$ are non-identical, DNA self-assembly may not be able to replicate the tour without including an extra cycle. Further reduction of the reporter strands to isolate the postman tour contained within $\sigma$ would yield all non-maximal optimal postman tours of G.

By using iterative three-degree perturbations, any graph can be represented as a self-assembled DNA graph structure of three armed junction molecules. In representing these graph structures as deformation retracts of closed compact DNA manifolds, a single strand can be identified which traverses each edge of the graph structure at least once. We show various applications of the property to traditional graph theory problems, and consider weighting algorithms and their applications to DNA computing.

As per business entities, customer satisfaction is the growing fact towards profit maximization and competitiveness. So, it tends to increase researcher’s interest in the topic of Service Quality. It always depends on the behavior of the stakeholders of the business. Especially, as it is being used to measure the level of satisfaction of the customers. Evaluating this satisfactory level is basically finding a Mathematical model for the behavior of the customers. Here, we try to quantify the level of service quality via mathematical modeling approach.
When selecting a suitable tool, the vagueness of the customers’ opinions has to be considered. The logical tools that people can rely on are generally considered the outcome of a bivalent logic, but the problems posed by real-life situations and human thought processes and approaches to problemsolving are by no means bivalent. So, fuzzy sets have to be used for the representation of human opinion. Neural network can be used to train the human behavioral patterns and it can be used to find the relationship between the respondent’s view of the service quality criterion and overall ranking of the service. Therefore, fuzzy model & Hybrid model (i.e. combine fuzzy logic with neural network) are used in this research. A case study is done to find out the validity of the proposed model.

It is important to identify recession before a country becomes a recession state. So development of forecast models is an important subject area. Leading economic indicators change before the economy changes and can be used to predict what the economy will be like in the future. So, recession can be forecast using leading economic indicators and neural networks can detect relationships among the selected leading indicators and the recession signals. In recent years, neural networks have been increasingly used for a wide variety of applications. Due to the structure of the neural network, a reasonably large number of indicators can be used. In this project, seven leading indicators of USA were used. The result shows that this model could predict the recession with 90 percent of confidence.