Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This monograph deals with mathematical constructions that are foundational in such an important area of data mining as pattern recognition. By using combinatorial and graph theoretic techniques, a closer look is taken at infeasible systems of linear inequalities, whose generalized solutions act as building blocks of geometric decision rules for pattern recognition.Infeasible systems of linear inequalities prove to be a key object in pattern recognition problems described in geometric terms thanks to the committee method. Such infeasible systems of inequalities represent an important special subclass of infeasible systems of constraints with a monotonicity property - systems whose multi-indices of feasible subsystems form abstract simplicial complexes (independence systems), which are fundamental objects of combinatorial topology.The methods of data mining and machine learning discussed in this monograph form the foundation of technologies like big data and deep learning, which play a growing role in many areas of human-technology interaction and help to find solutions, better solutions and excellent solutions. Contents:PrefacePattern recognition, infeasible systems of linear inequalities, and graphsInfeasible monotone systems of constraintsComplexes, (hyper)graphs, and inequality systemsPolytopes, positive bases, and inequality systemsMonotone Boolean functions, complexes, graphs, and inequality systemsInequality systems, committees, (hyper)graphs, and alternative coversBibliographyList of notationIndex

Inequalities (Mathematics) --- Graph theory. --- Graph theory --- Graphs, Theory of --- Theory of graphs --- Extremal problems --- Combinatorial analysis --- Topology --- Processes, Infinite

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Graph theory --- Théorie des graphes --- Periodicals. --- Périodiques --- Graph theory. --- Graphs, Theory of --- Theory of graphs --- Extremal problems --- colourings --- hereditary properties --- independence and dominating structure --- local properties --- product of graphs --- graph algorithms --- Combinatorial analysis --- Topology

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

combinatorics --- graph theory --- discrete mathematics --- Combinatorial analysis --- Graph theory --- Graphs, Theory of --- Theory of graphs --- Topology --- Extremal problems --- Graph theory. --- Combinatorial analysis. --- Combinatorics --- Algebra --- Mathematical analysis

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Computer science --- Computer science. --- Informatics --- computer science --- algorithms --- combinatorics --- graphs and networks --- optimization --- Science

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems.

quantum graphs --- ground states --- open sets converging to metric graphs --- norm convergence of operators --- NLD --- scaling limit --- standing waves --- bound states --- networks --- localized nonlinearity --- nonlinear Schrödinger equation --- metric graphs --- convergence of spectra --- sine-Gordon equation --- NLS --- star graph --- point interactions --- Laplacians --- nonrelativistic limit --- nonlinear wave equations --- quantum graph --- soliton --- nonlinear shallow water equations --- Kre?n formula --- breather --- non-linear Schrödinger equation --- Schrödinger equation --- nodal structure