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For a graph $G$, let Conf$(G,n)$ denote the classical configuration space of $n$ labelled points in $G$. Abrams introduced a cubical complex, denoted here by DConf$(G,n)$, sitting inside Conf$(G,n)$ as a strong deformation retract provided $G$ is suitably subdivided. Using discrete Morse Theory techniques, we extend Abrams' result to the realm of configurations having no $k$-fold collisions., Comment: 23 pages, 4 figures

We formulate a simple model for the bounded derived category of gentle algebras in terms of marked ribbon graphs and their walks, in order to analyze indecomposable objects, Auslander-Reiten triangles and homological bilinear forms, and to provide some relevant derived invariants in a graph theoretic setting. Among others, we exhibit the non-negativity and Dynkin type of the homological quadratic form of a gentle algebra, describe its roots as the classes of indecomposable perfect complexes in the Grothendieck group, and express its Coxeter polynomial in terms of the Avella-Alaminos Geiss invariant. We also derive some consequences for Brauer graph algebras.

Let $M^{(k)}_{d}(n)$ be the manifold of $n$-tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ having non-$k$-equal coordinates. We show that, for $d\geq2$, $M^{(3)}_{d}(n)$ is rationally formal if and only if $n\leq 6$. This stands in sharp contrast with the fact that all classical configuration spaces $M^{(2)}_d(n)=\text{Conf}(\hspace{.2mm}\mathbb{R}^d,n)$ are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non formality of $M^{(3)}_{d}(n)$ for $n>6$ is established via detection of non-trivial triple Massey products assessed through Poincar\'e duality., Comment: 21 pages

For a positive integer $n$ and a finite simplicial complex $K$, we describe an algorithmic procedure constructing a maximal discrete gradient field $W(K,n)$ on Abrams' discretized configuration space $\text{DConf}(K,n)$. Computer experimentation shows that the field is generically optimal. We study the field $W(K,n)$ for $n=2$ and $K=\Delta^{m,d}$, the $d$-dimensional skeleton of the $m$-dimensional simplex. In particular, we prove that $\text{DConf}(\Delta^{m,d},2)$ is $(\min\{d,m-1\}-1)$-connected, has torsion-free homology and admits a minimal cell structure. We compute the Betti numbers of $\text{DConf}(\Delta^{m,d},2)$ and, for certain values of $d$, we prove that $\text{DConf}(\Delta^{m,d},2)$ breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres., Comment: 24 pages, 2 figures

Bidirected graphs are multigraphs where every edge has an independent direction at each end. In the paper, with an arbitrary bidirected graph we associate a non-negative integral quadratic form (called the incidence form of the graph), and determine all forms that appear in this way in two main results: first, among non-negative connected unit forms, precisely those of Dynkin type $\mathbb{A}$ or $\mathbb{D}$ are incidence forms; second, we give simple conditions on the coefficients of a non-negative connected non-unitary form to be an incidence form. We say that those non-unitary forms have Dynkin type $\mathbb{C}$, and justify such nomenclature by generalizing known classifications and properties of non-negative quadratic forms of Dynkin types $\mathbb{A}$ and $\mathbb{D}$ to the introduced type~$\mathbb{C}$. We also show that the graphical framework of an incidence form is an useful tool to visualize its arithmetical properties, to prove new facts and to perform efficient computations for integral quadratic forms and related problems in number theory, algebra and graph theory. For instance, in a third main result we relate the walks of a bidirected graph with the $0,1,2$-roots of the associated incidence form (and to the classical root systems in the positive case). Moreover, we prove the universality property for a large class of integral quadratic forms, provide computational methods to find solutions or to characterize the finiteness of the sets of solutions of various related Diophantine equations, show a variant of Whitney's theorem on line graphs using switching classes, and apply our techniques to give a conceptual and constructive proof of the non-negativity (and possible Dynkin types) of the Euler quadratic forms of a class of finite-dimensional gentle algebras.

Astronomy and astrophysics are regarded as highly motivating topics for students in primary and secondary schools, and they have been a recurrent and effective resource to inspire passion about science. In fact, during the last years we have witnessed a boost of facilities providing small robotic telescopes for teachers and students to remotely undertake their own observing projects. A step forward is presented here, where we describe the experience of secondary school students attending professional observations of Venus at NASA's Infrared Telescope Facility (IRTF) and, in a second observing run, conducting the observations by themselves. In addition to quickly mastering the basic operation of the control software for the SpeX instrument, the students successfully performed different types of data acquisition, including drift scan imaging., Comment: 4 pages, 1 figure

The higher topological complexity of a space $X$, $\text{TC}_r(X)$, $r=2,3,\ldots$, and the topological complexity of a map $f$, $\text{TC}(f)$, have been introduced by Rudyak and Pave\v{s}i\'{c}, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map~$f$, $\text{TC}_{r,s}(f)$, for $1\leq s\leq r\geq2$, which simultaneously extends Rudyak's and Pave\v{s}i\'{c}'s notions. Our unified concept is relevant in the $r$-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in $s$ prescribed stages of the motion task. We study the homotopy invariance and the behavior of $\text{TC}_{r,s}$ under products and compositions of maps, as well as the dependence of $\text{TC}_{r,s}$ on $r$ and $s$. We draw general estimates for $\text{TC}_{r,s}(f\colon X\to Y)$ in terms of categorical invariants associated to $X$, $Y$ and $f$. In particular, we describe within one the value of $\text{TC}_{r,s}$ in the case of the non-trivial double covering over real projective spaces, as well as for their complex counterparts., Comment: 27 pages. Improved presentation

For finite connected graphs $\Gamma$ and $G$, with $\Gamma$ admitting a free involution $\tau$, we characterize the based homotopy classes $\alpha\in[\Gamma,G]$ for which the Borsuk-Ulam property holds in the sense of Gon\c{c}alves, Guaschi and Casteluber-Laass, i.e., the homotopy classes $\alpha$ so that each of its representatives $f\in\alpha$ satisfies $f(x) = f(\tau\cdot x)$ for some $x\in\Gamma$. This is attained through a graph-braid-group perspective aided by the use of discrete Morse theory., Comment: 11 pages, 2 figures

We introduce an algorithm that constructs a discrete gradient field on any simplicial complex. We show that, in all situations, the gradient field is maximal possible and, in a number of cases, optimal. We make a thorough analysis of the resulting gradient field in the case of Munkres' discrete model for $\text{Conf}(K_m,2)$, the configuration space of ordered pairs of non-colliding particles on the complete graph $K_m$ on $m$ vertices. Together with the use of Forman's discrete Morse theory, this allows us to describe in full the cohomology $R$-algebra $H^*(\text{Conf}(K_m,2);R)$ for any commutative unital ring $R$. As an application we prove that, although $\text{Conf}(K_m,2)$ is outside the "stable" regime, all its topological complexities are maximal possible when $m\geq4$., Comment: 29 pages, 18 figures

We compute the topological complexity of a polyhedral product $\mathcal{Z}$ defined in terms of an LS-logarithmic family of locally compact connected CW topological groups. The answer is given by a combinatorial formula that involves the LS category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for $\mathcal{Z}$. The proof methodology uses a Fadell-Husseini viewpoint for the monoidal topological complexity (MTC) of a space, which, under mild conditions, recovers Iwase-Sakai's original definition. In the Fadell-Husseini context, the stasis condition -- MTC's raison d'\^etre -- can be encoded at the covering level. Our Fadell-Husseini-inspired definition provides an alternative to the MTC variant given by Dranishnikov, as well as to the ones provided by Garc\'ia-Calcines, Carrasquel-Vera and Vandembroucq in terms of relative category., Comment: 15 pages

The authors of the study conduct a legal analysis of the concept of energy security. Energy is vital for sustainable development, and sustainability is not only at the heart of development, but also economic, environmental, social and military policies. To ensure the sustainability of the policy, 'security' seems to be a mandatory goal to achieve. The article critically assesses the change in the energy paradigm., Comment: 7 pages. Key words: energy security, energy concept, energy paradigm. JEL codes: E-00; E-6

For a tree $T$ and a positive integer $n$, let $B_nT$ denote the $n$-strand braid group on $T$. We use discrete Morse theory techniques to show that the cohomology ring $H^*(B_nT)$ is encoded by an explicit abstract simplicial complex $K_nT$ that measures $n$-local interactions among essential vertices of $T$. We show that, in many cases (for instance when $T$ is a binary tree), $H^*(B_nT)$ is the exterior face ring determined by $K_nT$., Comment: The paper has been reorganized so to spell out the explicit description of the cohomology ring of arbitrary tree braid groups. Furthermore, the exterior face ring structure is now proved for a large family of trees (which includes binary trees). Final version, submitted for publication. 29 pages, 11 figures

We describe parametrised motion planning algorithms for systems controlling objects represented by points that move without collisions in an even dimensional Euclidean space and in the presence of up to three obstacles with \emph{a priori} unknown positions. Our algorithms are optimal in the sense that the parametrised local planners have minimal posible size., Comment: 16 pages. Final version. To appear in Morfismos

We introduce the effectual topological complexity (ETC) of a $G$-space $X$. This is a $G$-equivariant homotopy invariant sitting in between the effective topological complexity of the pair $(X,G)$ and the (regular) topological complexity of the orbit space $X/G$. We study ETC for spheres and surfaces with antipodal involution, obtaining a full computation in the case of the torus. This allows us to prove the vanishing of twice the non-trivial obstruction responsible for the fact that the topological complexity of the Klein bottle is 4. In addition, this gives a counterexample to the possibility -- suggested in Pave\v{s}i\'c's work on the topological complexity of a map -- that ETC of $(X,G)$ would agree with Farber's $TC(X)$ whenever the projection map $X\to X/G$ is finitely sheeted. We conjecture that ETC of spheres with antipodal action recasts the Hopf invariant one problem, and describe (conjecturally optimal) effectual motion planners., Comment: 19 pages

For a graph $\Gamma$, let $K(H_{\Gamma},1)$ denote the Eilenberg-Mac Lane space associated to the right-angled Artin (RAA) group $H_{\Gamma}$ defined by $\Gamma$. We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma},1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal{O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_{\Gamma},1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups., Comment: The paper is now written in a more concise and succinct way. In particular, our main results are highlighted and contextualized in an introductory section. 25 pages, 4 figures

Two integral quadratic unit forms are called strongly Gram congruent if their upper triangular Gram matrices are Z-congruent. The paper gives a combinatorial strong Gram invariant for those unit forms that are non-negative of Dynkin type A, within the framework introduced in [Fundamenta Informaticae 184(1):49-82, 2021], and uses it to determine all corresponding Coxeter polynomials and (reduced) Coxeter numbers., Comment: Integral quadratic form, Gram congruence, Dynkin type, Coxeter polynomial, edge-bipartite graph, quiver, incidence matrix, signed line graph

In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among positive unit forms of Dynkin type A. The concept of inverse of a quiver is also introduced, and is used to obtain and analyze the Coxeter matrix of non-negative unit forms of Dynkin type A. Connected principal unit forms of such type are also classified., Comment: Integral quadratic form, Gram congruence, quiver, Dynkin type, Coxeter matrix, edge-bipartite graph, signed line graph

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle $\mathcal{R}$, and outputs an explicit system of piecewise linear motion planners for $\mathcal{R}$. The algorithm is designed in such a way that the cardinality of the outputed system is probabilistically close (with parameters chosen by the user) to minimal possible. This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber's topological complexity TC. Besides its relevance toward technological applications, our work revels that, unlike other discrete approaches to TC, the SC model can recast Farber's invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Mac\'ias-Virg\'os and Mosquera-Lois' notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra., Comment: 21 pages. New author added. The algorithm is now set in the realm of homotopic distance, thus encompassing other sectional category invariants

We compute the Lusternik-Schnirelmann category and the topological complexity of no $k$-equal manifolds $M^{(k)}_d(n)$ for certain values of $d$, $k$ and $n$. This includes instances where $M^{(k)}_d(n)$ is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring $H^*(M^{(k)}_d(n))$ as described by Dobrinskaya and Turchin. A fine tuning comes from the use of obstruction theory techniques.

Wiltshire-Gordon has introduced a homotopy model for ordered configuration spaces on a given simplicial complex. That author asserts that, after a suitable subdivision, his model also works for unordered configuration spaces. We supply details justifying Wiltshire-Gordon's assertion and, more importantly, uncover the equivariant properties of his more-general simplicial-difference model for the complement of a subcomplex inside a larger complex. This is achieved by proving an equivariant version of the Nerve Lemma. In addition, in the case of configuration spaces, we show that a slight variation of the model has better properties: it is regular and sits inside the configuration space as a strong and equivariant deformation retract. Our variant for the configuration-space model comes from a comparison, in the equivariant setting, between Wiltshire's simplicial difference and a well known model for the complement of a full subcomplex on a simplicial complex., Comment: Version 2 of this manuscript extends the result in version 1 by developing the equivariant properties of the simplicial difference. A needed equivariant version of the Nerve Lemma is proved. 19 pages