arXiv subjects

Amnon Rosenmann

Publications and source records attributed to Amnon Rosenmann.

16 recordsLinked to original sources

arXiv2026

On the chromatic number and equilateral dimension of $\mathbb{R}^n$ with the tropical norm

We study the tropical chromatic number of $\mathbb{R}^n$, $\chi_{\mathrm{tr}}(\mathbb{R}^n)$, the minimal number of colors needed to color $\mathbb{R}^n$, so that no two points at tropical unit distance have the same color. It is the tropical analogue of the well-known Hadwiger-Nelson problem in $\mathbb{R}^2$. We have $\displaystyle \binom{n+1}{\lfloor (n+1)/2 \rfloor} \leq \chi_{\mathrm{tr}}(\mathbb{R}^n) \leq 2^n$ for every $n$, where the lower bound comes from Sperner's antichain bound on a maximal equilateral set, as shown by Swanepoel. It is conjectured that $\chi_{\mathrm{tr}}(\mathbb{R}^n) = 2^n$, which is known to be the case for the measurable chromatic number. By constructing a graph with 62 vertices and 577 edges we demonstrate that $\chi_{\mathrm{tr}}(\mathbb{R}^3)=8$. We also construct a graph in $\mathbb{Z}^4$ with 37 vertices and 386 edges that is 11-colorable but not 10-colorable, which is above Sperner's lower bound of 10.

math.CO↗

arXiv2026

Tropical measures, anisotropic isoperimetric inequality and honeycomb

We introduce a tropical spherical measure on $\mathbb{R}^n$ that is based on the tropical metric and is an analogue of spherical Hausdorff measure. This measure is translation invariant but, unlike Lebesgue measure, is not invariant under rotations or reflections. It agrees with Lebesgue measure on $n$-dimensional (but not on $k$-dimensional, $k<n$) measurable subsets of $\mathbb{R}^n$, and on rectifiable curves it recovers tropical length. In dimension $2$ we prove a sharp tropical isoperimetric inequality, with equality precisely for tropical disks, and deduce a tropical honeycomb theorem. We also introduce a tropical analogue of Minkowski content and show that the tropical ball is the associated Wulff shape. This yields an anisotropic type of the tropical isoperimetric problem and consequently a tropical honeycomb theorem in $\mathbb{R}^n$. Finally, we describe the tropical dual norm and dual ball, compare the tropical spherical and Minkowski surface measures, and prove that they agree in the plane and on polytopes in $\mathbb{R}^n$ whose facets are parallel to facets of the tropical ball or its dual.

math.MG↗

arXiv2026

Tropical balls, geodesics and honeycomb

In these notes we describe the geometry of tropical balls in $\mathbb{R}^n$ equipped with the tropical metric. After defining the tropical length of rectifiable curves (and not just piecewise linear curves), we characterize compact tropically geodesic sets in $\mathbb{R}^n$. Next, we describe the tropical unit ball as a zonotope, via its tropical generating set, as a union of $n+1$ tropical unit hypercubes, and as the tropical geodesic hull of the tropical unit vectors. Finally, we show that translates of the tropical unit ball whose centers lie in a sublattice of $\mathbb{Z}^n$ form a facet-to-facet honeycomb tiling of $\mathbb{R}^n$. We note that a great part of the material presented here is either known or implied from known results.

math.CO↗

arXiv2025

On matrices in finite free position

We study pairs $(A,B)$ of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial $χ_{A+B}(x)$ (resp. $χ_{AB}(x)$) equals the additive finite free convolution $χ_{A}(x) \boxplus χ_{B}(x)$ (resp. the multiplicative finite free convolution $χ_{A}(x) \boxtimes χ_{B}(x)$), which equals the expected characteristic polynomial $\mathbb{E}_U [ χ_{A+U^* BU}(x) ]$ (resp. $\mathbb{E}_U [ χ_{AU^* BU}(x) ]$) over the set of unitary matrices $U$. We examine the lattice of (non-irreducible) affine algebraic sets of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with constant diagonal, and the scalar matrices vs. the set of all square matrices.

math.RA↗

arXiv2023

Dependence over subgroups of free groups

Given a finitely generated subgroup $H$ of a free group $F$, we present an algorithm which computes $g_1,\ldots,g_m\in F$, such that the set of elements $g\in F$, for which there exists a non-trivial $H$-equation having $g$ as a solution, is, precisely, the disjoint union of the double cosets $H\sqcup Hg_1H\sqcup \cdots \sqcup Hg_mH$. Moreover, we present an algorithm which, given a finitely generated subgroup $H\leqslant F$ and an element $g\in F$, computes a finite set of elements of $H * \langle x \rangle$ that generate (as a normal subgroup) the ``ideal" $I_H(g) \unlhd H * \langle x \rangle$ of all ``polynomials" $w(x)$, such that $w(g)=1$. The algorithms, as well as the proofs, are based on the graph-theory techniques introduced by Stallings and on the more classical combinatorial techniques of Nielsen transformations. The key notion here is that of dependence of an element $g\in F$ on a subgroup $H$. We also study the corresponding notions of dependence sequence and dependence closure of a subgroup.

math.GR↗

arXiv2021

Computing the sequence of $k$-cardinality assignments

The $k$-cardinality assignment problem asks for finding a maximal (minimal) weight of a matching of cardinality $k$ in a weighted bipartite graph $K_{n,n}$, $k \leq n$. The algorithm of Gassner and Klinz from 2010 for the parametric assignment problem computes in time $O(n^3)$ the set of $k$-cardinality assignments for those integers $k \leq n$ which refer to "essential" terms of a corresponding maxpolynomial. We show here that one can extend this algorithm and compute in a second stage the other "semi-essential" terms in time $O(n^2)$, which results in a time complexity of $O(n^3)$ for the whole sequence of $k=1,...,n$-cardinality assignments. The more there are assignments left to be computed at the second stage the faster the two-stage algorithm runs. In general, however, there is no benefit for this two-stage algorithm on the existing algorithms, e.g. the simpler network flow algorithm based on the successive shortest path algorithm which also computes all the $k$-cardinality assignments in time $O(n^3)$.

math.OC↗

arXiv2020

Circular automata synchronize with high probability

In this paper we prove that a uniformly distributed random circular automaton $\mathcal{A}_n$ of order $n$ synchronizes with high probability (whp). More precisely, we prove that $$ \mathbb{P}\left[\mathcal{A}_n \text{ synchronizes}\right] = 1- O\left(\frac{1}{n}\right). $$ The main idea of the proof is to translate the synchronization problem into properties of a random matrix; these properties are then handled with tools of the probabilistic method. Additionally, we provide an upper bound for the probability of synchronization of circular automata in terms of chromatic polynomials of circulant graphs.

math.CO↗

arXiv2019

On the Distance between Timed Automata

The problem of inclusion of the language accepted by timed automaton $A$ (e.g., the implementation) in the language accepted by $B$ (e.g., the specification) is, in general, undecidable in the class of non-deterministic timed automata. In order to tackle this disturbing problem we show how to effectively construct deterministic timed automata $A_d$ and $B_d$ that are discretizations (digitizations) of the non-deterministic timed automata $A$ and $B$ and differ from the original automata by at most $\frac{1}{6}$ time units on each occurrence of an event. Language inclusion in the discretized timed automata is decidable and it is also decidable when instead of $\mathfrak{L}(B)$ we consider $\overline{\mathfrak{L}(B)}$, the closure of ${\mathfrak{L}(B)}$ in the Euclidean topology: if $\mathfrak{L}(A_d) \nsubseteq \mathfrak{L}(B_d)$ then $\mathfrak{L}(A) \nsubseteq \mathfrak{L}(B)$ and if $\mathfrak{L}(A_d) \subseteq \mathfrak{L}(B_d)$ then $\mathfrak{L}(A) \subseteq \overline{\mathfrak{L}(B)}$. Moreover, if $\mathfrak{L}(A_d) \nsubseteq \mathfrak{L}(B_d)$ we would like to know how far away is $\mathfrak{L}(A_d)$ from being included in $\mathfrak{L}(B_d)$. For that matter we define the distance between the languages of timed automata as the limit on how far away a timed trace of one timed automaton can be from the closest timed trace of the other timed automaton. We then show how one can decide under some restriction whether the distance between two timed automata is finite or infinite.

cs.FL↗

arXiv2019

The Timestamp of Timed Automata

Given a member A of the class of non-deterministic timed automata with silent transitions (eNTA), we effectively compute its timestamp: the set of all pairs (time value, action) of all observable timed traces of A, a generalization of the reachability problem. We show that the timestamp is eventually periodic and that one can compute a simple deterministic timed automaton with the same timestamp as that of A. As a consequence, we have a partial method, not bounded by time or number of steps, for the general language non-inclusion problem for eNTA. We also show that the language of A is periodic with respect to suffixes.

cs.FL↗

arXiv2018

Polynomial convolutions in max-plus algebra

Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava and then Marcus studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability. We explore analogues of these types of convolutions in the setting of max-plus algebra. In this setting the max-permanent replaces the determinant, the maximum is the analogue of the expected value and real-rootedness is replaced by full canonical form. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots.

math.RA↗

arXiv2015

A Multiple-Valued Logic Approach to the Design and Verification of Hardware Circuits

We present a novel approach, which is based on multiple-valued logic (MVL), to the verification and analysis of digital hardware designs, which extends the common ternary or quaternary approaches for simulations. The simulations which are performed in the more informative MVL setting reveal details which are either invisible or harder to detect through binary or ternary simulations. In equivalence verification, detecting different behavior under MVL simulations may lead to the discovery of a genuine binary nonequivalence or to a qualitative gap between two designs. The value of a variable in a simulation may hold information about its degree of truth and its "place of birth" and "date of birth." Applications include equivalence verification, initialization, assertions generation and verification, partial control on the flow of data by prioritizing and block-oriented simulations. Much of the paper is devoted to theoretical aspects behind the MVL approach, including the reason for choosing a specific algebra for computations, and the introduction of the verification complexity of a Boolean expression. Two basic algorithms are presented.

cs.LO↗

arXiv2015

Bounded Determinization of Timed Automata with Silent Transitions

Deterministic timed automata are strictly less expressive than their non-deterministic counterparts, which are again less expressive than those with silent transitions. As a consequence, timed automata are in general non-determinizable. This is unfortunate since deterministic automata play a major role in model-based testing, observability and implementability. However, by bounding the length of the traces in the automaton, effective determinization becomes possible. We propose a novel procedure for bounded determinization of timed automata. The procedure unfolds the automata to bounded trees, removes all silent transitions and determinizes via disjunction of guards. The proposed algorithms are optimized to the bounded setting and thus are more efficient and can handle a larger class of timed automata than the general algorithms. The approach is implemented in a prototype tool and evaluated on several examples. To our best knowledge, this is the first implementation of this type of procedure for timed automata.

cs.FL↗

arXiv2014

On the intersection of subgroups in free groups: echelon subgroups are inert

A subgroup $H$ of a free group $F$ is called inert in $F$ if for every $G < F$ the rank of the intersection of $H$ with $G$ is no grater than the rank of $G$. In this paper we expand the known families of inert subgroups. We show that the inertia property holds for 1-generator endomorphisms. Equivalently, echelon subgroups in free groups are inert. An echelon subgroup is defined through a set of generators that are in echelon form with respect to some ordered basis of the free group, and may be seen as a generalization of a free factor. For example, the fixed subgroups of automorphisms of finitely generated free groups are echelon subgroups. The proofs follow mostly a graph-theoretic or combinatorial approach.

math.GR↗

arXiv1994

When Schrier transversals grow wild

Schreier formula for the rank of a subgroup of finite index of a finitely generated free group $F$ is generalized to an arbitrary (even infinitely generated) subgroup $H$ through the Schreier transversals of $H$ in $F$. The rank formula may also be expressed in terms of the cogrowth of $H$. We introduce the rank-growth function $rk_H(i)$ of a subgroup $H$ of a finitely generated free group $F$. $rk_H(i)$ is defined to be the rank of the subgroup of $H$ generated by elements of length less than or equal to $i$ (with respect to the generators of $F$), and it equals the rank of the fundamental group of the subgraph of the cosets graph of $H$, which consists of the paths starting at $1$ that are of length $\leq i$. When $H$ is supnormal, i.e. contains a non-trivial normal subgroup of $F$, we show that its rank-growth is equivalent to the cogrowth of $H$. A special case of this is the known result that a supnormal subgroup of $F$ is of finite index if and only if it is finitely generated. In particular, when $H$ is normal then the growth of the group $G=F/H$ is equivalent to the rank-growth of $H$. A Schreier transversal forms a spanning tree of the cosets graph of $H$, and thus its topological structure is of a contractible spanning subcomplex of a simplicial complex. The $d$-dimensional simplicial complexes that contain contractible spanning subcomplexes have the homotopy type of a bouquet of $r$ $d$-spheres. When these complexes are also $n$-regular then $r$ can be computed by generalizing the rank formula (which applies to Schreier transversals) to higher dimensions.

math.GR↗

arXiv1994

The normalized cyclomatic quotient associated with presentations of finitely generated groups

Given the Cayley graph of a finitely generated group $G$, with respect to a presentation $G^α$ with $n$ generators, the quotient of the rank of the fundamental group of subgraphs of the Cayley graph by the cardinality of the set of vertices of the subgraphs gives rise to the definition of the normalized cyclomatic quotient $Ξ(G^α)$. The asymptotic behavior of this quotient is similar to the asymptotic behavior of the quotient of the cardinality of the boundary of the subgraph by the cardinality of the subgraph. Using Følner's criterion for amenability one gets that $Ξ(G^α)$ vanishes for infinite groups if and only if they are amenable. When $G$ is finite then $Ξ(G^α)=1/|G|$, where $|G|$'> is the cardinality of $G$, and when $G$ is non-amenable then $1-n\leqΞ(G^α)\le 0$, with $Ξ(G^α)=1-n$ if and only if $G$ is free of rank $n$. Thus we see that on special cases $Ξ(G^α)$ takes the values of the Euler characteristic of $G$. Most of the paper is concerned with formulae for the value of $Ξ(G^α)$ with respect to that of subgroups and factor groups, and with respect to the decomposition of the group into direct product and free product. Some of the formulae and bounds we get for $Ξ(G^α)$ are similar to those given for the spectral radius of symmetric random walks on the graph of $G^α$, but this is not always the case. In the last section of the paper we define and touch very briefly the balanced cyclomatic quotient, which is defined on concentric balls in the graph and is related to the growth of $G$.

math.GR↗

arXiv1993

Cogrowth and essentiality in groups and algebras

The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects non-trivially every non-trivial subgroup. The main result of this paper is that every function $f:{\Bbb N}\cup \{0\}\rightarrow {\Bbb N}$ which is strictly increasing, but at most exponential, is equivalent to a cogrowth function of an essential subgroup of infinite index of the free group of rank two. This class of functions properly contains the class of growth functions of groups. The notions of growth and cogrowth of right ideals in algebras are introduced. We show that when the algebra is without zero divisors then every right ideal, whose cogrowth is less than that of the algebra, is essential.

math.GR↗