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We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied.
In this paper, we present a synthetic solution to a geometric open problem involving the radical axis of two strangely defined circumcircles. The solution encapsulates two generalizations, one of which uses a powerful projective result…
In this paper, we consider the problem of representing graphs by triangles whose sides touch. As a simple necessary condition, we show that pairs of vertices must have a small common neighborhood. On the positive side, we present linear…
We study characteristic polynomials of symmetric matrices with entries ${i+j\choose i}$ the binomial coefficients, over finite fields.
We study the combinatorial representation theory of the ``planar rook algebra" $P_n$. This algebra has a basis consisting of planar rook diagrams and multiplication given by diagram concatenation. For each integer $0 \le k \le n$, we…
We compute a canonical circular-arc representation for a given circular-arc (CA) graph which implies solving the isomorphism and recognition problem for this class. To accomplish this we split the class of CA graphs into uniform and…
Cycloids are particular Petri nets for modelling processes of actions or events. They belong to the fundaments of Petri's general systems theory and have very different interpretations, ranging from Einstein's relativity theory and…
In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and order. Afterwards, we combine the results from our study of…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
Motivated by the effective impact of the Pascal functional and the Wronskian matrices, we investigate several identities and differential equation for the Sheffer-Appell polynomial sequence by using matrix algebra. The matrix approach,…
The conditions determining that two triangles are congruent play a basic role in planimetry. By comparing not congruent triangles with respect to given sets of corresponding elements it is important to discover if they have any common…
A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every…
A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the…
In this article we present a remarkable concyclicity of four centroids related to the orthocenters of the triangles $ABC,$ $BPC,$ $CPA,$ and $PAB$ of a quadrangle $ABCP$. Furthermore, we establish a result about orthologic triangles…
The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from…
Using some new logarithmic formal calculus, we construct a well known vertex algebra, obtaining the Jacobi identity directly, in an essentially self-contained treatment.
Let F be a 4-regular graph with an Euler system C. We introduce a simple way to modify the interlacement matrix of C so that every circuit partition P of F has an associated modified interlacement matrix M(C,P). If C and C' are Euler…
Several recent works have identified patterns that must exist in dense subsets of either the vertices or the edges of a large hypercube. We introduce a framework, based on the concept of series-parallel graphs, that unifies and generalizes…
In this paper we develop a new geometric method to answer the log-concavity questions related to a nice class of combinatorial sequences arising from the Pascal triangle.
In this paper, we construct Pell matrices, analogous to Fibonacci matrices, to study algebraic properties of Pell numbers via linear algebra. This framework yields identities involving the trace, inverse, and determinant, as well as matrix…