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Complement-reducible graphs (or cographs) are the graphs formed from the single-vertex graph by the operations of complement and disjoint union. By combining the Johnson-Newman theorem on generalized cospectrality with the standard tools in…
We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph…
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism…
We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the…
A geometrical formulation for adjoint-symmetries as 1-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution…
We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m \leq n-2$, or a complete…
Abstract separation systems are a new unifying framework in which separations of graph, matroids and other combinatorial structures can be expressed and studied. We characterize the abstract separation systems that have representations as…
In this paper, ideas of open ball, closed ball, compact set are introduced and some related basic properties are studied. Some topological properties and some other well known results of metric spaces including Cantor intersection theorem…
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types…
We establish new pair correlation results for certain generic homogenous diagonal forms evaluated on the integers. Methods are analytic leading to explicit quantitative statements.
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of…
We introduce a new spectral method for image segmentation that incorporates long range relationships for global appearance modeling. The approach combines two different graphs, one is a sparse graph that captures spatial relationships…
Dilated Convolutions have been shown to be highly useful for the task of image segmentation. By introducing gaps into convolutional filters, they enable the use of larger receptive fields without increasing the original kernel size. Even…
There are several notions of duality between lines and points. In this note, it is shown that all these can be studied in a unified way. Most interesting properties are independent of specific choices. It is also shown that either dual…
We extend results of our previous papers, on ordinary multiple points of curves, and on the computation of their conductor, to ordinary multiple subvarieties of codimension one.
Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to…
We introduce a special subset of the graph of a homogeneous coupled cell network, called a projection block, and show that the network obtained from identifying this block to a single point can be used to understand the generic bifurcations…
In this paper we give a characterization of the possible homology groups that can occur for compact simply connected cohomogeneity one manifolds in dimensions seven and lower.
We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix. This construction can be used to form pairs of cospectral…