Indices of fixed point classes play a central role in Nielsen theory. Jiang-Wang-Zhang proved that for selfmaps graphs and surfaces, the index any class has an upper bound called its characteristic. In this paper, we study difference between characteristic graphs. First, free groups, extend notion attracting words at infinity automorphisms into injective endomorphisms. Then, by using relative t...

This paper introduces a new real-time concept of reconfigurable P-timed Petri nets. Our goal is to provide a formal model to build and modify a net on the fly. In the first part of the article, the original P-timed extensions are summarized. Then we define an endomorphism that alters the original Petri net in real time; for instance one can change the number of tokens or the net structure. The ...

Introduction 1 1. Invariant Subspaces 3 2. Eigenvectors, Eigenvalues and Eigenspaces 11 3. Cyclic Spaces 14 4. Prime and Primary Vectors 15 5. The Cyclic Decomposition Theorem 20 6. Rational and Jordan Canonical Forms 22 7. Similarity 23 8. The Cayley-Hamilton Polynomial (Or: Up With Determinants?) 24 9. Extending The Ground Field 25 9.1. Some Invariances Under Base Extension 25 9.2. Semisimpli...

For discrete groups G, we introduce equivariant Nielsen invariants. They are equivariant analogs of the Nielsen number and give lower bounds for the number of fixed point orbits in the G-homotopy class of an equivariant endomorphism f : X → X. Under mild hypotheses, these lower bounds are sharp. We use the equivariant Nielsen invariants to show that a G-equivariant endomorphism f is G-homotopic...

Let V be a vector space over some field k, and let α : V V be a linear map (an ‘endomorphism of V ’). Given any polynomial p with coefficients in k, there is an endomorphism p(α) of V , and we say that p is an annihilating polynomial for α if p(α) = 0. Our first major goal is to see that for any α, the annihilating polynomials can easily be classified: they’re precisely the multiples of a certa...

We consider digital expansions in lattices with endomorphisms acting as base. We focus on the w-non-adjacent form (w-NAF), where each block of w consecutive digits contains at most one non-zero digit. We prove that for su ciently large w and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a w-NAF. If the eigenvalues of the endomorphism...

Let T be a tilting object in a triangulated category which is equivalent to the bounded derived category of a finite-dimensional hereditary algebra. The text investigages the strong global dimension, in the sense of Ringel, of the opposite algebra A of the endomorphism algebra of T . This invariant is expressed in terms of the lengths of the sequences T0, . . . , Tl of tilting objects such that...

Let R be an associative ring with non-zero identity. For a Serre subcategory C of the category R-mod of left R-modules, we consider the class AC of all modules that do not belong to C, but all of their proper submodules belong to C. Alongside of basic properties of such associated classes of modules, we will prove that every uniform module of AC has a local endomorphism ring. Moreover, if R is ...