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Magnetic nanoparticles play a crucial role in different fields such as biomedicine or information and quantum technologies. These applications require nanoparticles with a single, well-defined energy minimum, free of metastable states, and characterized by narrow switching field distributions. Here, we demonstrate that high-transition-temperature nanoSQUIDs can be successfully applied to the characterization of individual nanodiscs hosting magnetic vortices. We present measurements performed under varying temperature and external magnetic field, revealing signatures of ubiquitous, multiple metastable configurations. We also demonstrate that metastability can be reduced by introducing an intended asymmetry. NanoSQUID measurements can be applied to optimize the fabrication of on-demand spin-texture states, such as degenerated vortices or particles with fixed circulation and deterministic and narrow switching probabilities.

We present an extension of the well-known micromagnetic package Mumax3 to simulate magnon-polaritons in realistic magnetic materials and nanostructures. Mumax3-cQED leverages the full GPU-accelerated capabilities of Mumax3 to model standard spin-spin interactions and the coupling of magnetic moments to external space- and time-dependent magnetic fields, with the additional unique feature of including the coupling to a cavity. We validate the code against results obtained from the Dicke model in both the paramagnetic and the superradiant phases. We show that hybrid magnon-light states can be calculated, as well as the non-equilibrium dynamics and their approach to equilibrium. In addition, we demonstrate the potential of Mumax3-cQED to reproduce experimental results and design magnon-cavity experiments, including three-dimensional and coplanar waveguide resonators. The code is fully available and will be useful for designing experiments involving microscopic saturated ferromagnets as well as systems featuring spin textures such as domain walls, vortices, or skyrmions., Comment: 34 pages, 6 figures

We consider cycle decompositions of even, $2an$-dimensional hypercubes $Q_{2an},$ where $a \geq 3$ is odd and $n \geq 1.$ Prior work done by Axenovich, Offner, and Tompkins focused on obtaining the existence of cycle decompositions for even-dimensional hypercubes using long cycles of a given form, leaving out cycles of shorter lengths and, in fact, cycles of even longer lengths than those obtained there, such as $C_{7 \cdot 2^{11}}$ in the case of $Q_{14}.$ In this paper, we provide two novel methods for explicitly constructing cycle decompositions of virtually all possible cycle lengths, using cycles of a given form, on Cartesian products of cycles up to those known by the work of Axenovich, Offner, and Tompkins. In particular, we show that we can explicitly obtain cycle decompositions of even dimensional hypercubes $Q_{2an}$ for all lengths mentioned above while on the same Cartesian product of cycles. With this, the current understanding of cycle decompositions of even dimensional hypercubes is furthered constructively and is featured with some interesting consequences for when $a$ is a positive, even integer. Additionally, progress is made towards obtaining cycle decompositions using the longest admissible cycle lengths with the incorporation of a more explicit starting point from which such decompositions of $Q_{2an}$ can be studied further., Comment: 146 pages, 38 figures, submitted to Discrete Mathematics journal

Performing nanoscale scanning electron paramagnetic resonance (EPR) requires three essential ingredients. First, a static magnetic field together to field gradients to Zeeman split the electronic energy levels with spatial resolution. Second, a radiofrequency (rf) magnetic field capable of inducing spin transitions. Finally, a sensitive detection method to quantify the energy absorbed by spins. This is usually achieved by combining externally applied magnetic fields with inductive coils or cavities, fluorescent defects or scanning probes. Here, we theoretically propose the realization of a EPR scanning sensor merging all three characteristics into a single device: the vortex core stabilized in ferromagnetic thin-film discs. On one hand, the vortex ground state generates a significant static magnetic field and field gradients. On the other hand, the precessional motion of the vortex core around its equilibrium position produces a circularly polarized oscillating magnetic field, which is enough to produce spin transitions. Finally, the spin-magnon coupling broadens the vortex gyrotropic frequency, suggesting a direct measure of the presence of unpaired electrons. Moreover, the vortex core can be displaced by simply using external magnetic fields of a few mT, enabling EPR scanning microscopy with large spatial resolution. Our numerical simulations show that, by using low damping magnets, it is theoretically possible to detect single spins located on the disc's surface. Vortex nanocavities could also attain strong coupling to individual spin molecular qubits, with potential applications to mediate qubit-qubit interactions or to implement qubit readout protocols.

We consider the effective evolution of a phenomenological model from FLRW supersymmetric quantum cosmology with a scalar field. The scalar field acts as a clock and inflaton. We examine a family of simple superpotentials that produce an inflation whose virtual effect on inhomogeneous fluctuations shows very good agreement with PLANCK observational evidence for the tensor-to-scalar ratio and the scalar spectral index., Comment: 14 pages, 6 figures. arXiv admin note: text overlap with arXiv:2104.12914

Cavity photons and ferromagnetic spins excitations can exchange information coherently in hybrid architectures, at speeds set by their mutual coupling strength. Speed enhancement is usually achieved by optimizing the geometry of the electromagnetic cavity. Here we show that the geometry of the ferromagnet plays also an important role, by setting the fundamental frequency of the magnonic resonator. Using focused ion beam patterning, we vary the aspect ratio of different Permalloy samples reaching operation frequencies above 10 GHz while working at low external magnetic fields. Additionally, we perform broad band ferromagnetic resonance measurements and cavity experiments that demonstrate that the magnon-photon coupling strength can be estimated using either open transmission lines or resonant cavities, yielding very good agreement. Finally, we describe a simple theoretical framework based on electromagnetic and micromagnetic simulations that successfully accounts for the experimental results. This approach can be used to design hybrid quantum systems exploiting whatsoever magnetostatic mode excited in ferromagnets of arbitrary size and shape and to tune their operation conditions.

The aim of this paper is to obtain new inequalities for a large family of generalizations of the Wiener Index and to characterize the set of extremal graphs with respect to them. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs., Comment: 11 pages, 1 figure

We study two homogeneous supersymmetric extensions for the $f(R)$ modified gravity model of Starobinsky with the FLRW metric. The actions are defined in terms of a superfield $\mathcal{R}$ that contains the FLRW scalar curvature. One model has N=1 local supersymmetry, and its bosonic sector is the Starobinsky action; the other action has N=2, its bosonic sector contains, in additional to Starobinsky, a massive scalar field without self-interaction. As expected, the bosonic sectors of these models are consistent with cosmic inflation, as we show by solving numerically the classical dynamics. Inflation is driven by the $R^2$ term during the large curvature regime. In the N=2 case, the additional scalar field remains in a low energy state during inflation. Further, by means of an additional superfield, we write equivalent tensor-scalar-like actions from which we can give the Hamiltonian formulation., Comment: 17 pages, 2 figures

In this paper we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of Geometric Group Theory. Using these new tools, we are able to give two new characterizations for a group to be virtually free. Furthermore, we prove that the Baumslag-Solitar group $BS(1,n)$ is $k$-chordal for some $k$ if and only if $|n|<3$ and we give an application of generalized chordality to the study of the word problem.

We consider inflationary scenarios for FRLW supersymetric models with a scalar field, whose Wheeler-DeWitt equation has analytic solutions. Following previous work of the authors, we analyse the setting of the scalar field as clock, leading to an effective time dependent wave function, and a time dependent mean value of the scale factor. We study several superpotentials, which lead to an evolution that corresponds to consistent inflationary scenarios., Comment: 15 pages, 12 figures

The aim of this paper is to obtain new inequalities for a large family of topological indices restricted to unicyclic graphs and to characterize the set of extremal unicyclic graphs with respect to them. This family includes variable first Zagreb, variable sum exdeg, multiplicative second Zagreb and Narumi-Katayama indices. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs, fixing or not the maximum degree or the number of pendant vertices., Comment: 15 pages

We explicitly compute the homology groups with coefficients in a field of characteristic zero of cocyclic subgroups or even Artin groups of FC-type. We also give some partial results in the case when the coefficients are taken in a field of prime characteristic., Comment: 18 pages, 2 figures. To appear in AGT

We develop a version of the Bass-Serre theory for Lie algebras (over a field $k$) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to $\mathbb{N}$-graded Lie algebras: for example, we show that one relator $\mathbb{N}$-graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras., Comment: 27 pages

In this paper we study Sigma invariants of even Artin groups of FC-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong homological $n$-link condition for a graph $\Gamma$ and prove that it gives a sufficient condition for a character $\chi:A_\Gamma\to \mathbb{Z}$ to satisfy $[\chi]\in\Sigma^n(A_\Gamma,\mathbb{Z})$. This implies that the kernel $A^\chi_\Gamma=\ker \chi$ is of type $FP_n$. The homotopy counterpart is also proved. Partial results on the converse are discussed. We also provide a general formula for the free part of $H_n(A^\chi_\Gamma;\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $\chi$. This gives a characterization of when $H_n(A^\chi_\Gamma;\mathbb{F})$ is a finite dimensional vector space over $\mathbb{F}$. In the last version we correct a problem in the proof of Lemma 4.3 and also a remark at the end of subsection 3.3., Comment: 19 pages

We theoretically study quantum size effects in the magnetic response of a spherical metallic nanoparticle (e.g. gold). Using the Jellium model in spherical coordinates, we compute the induced magnetic moment and the magnetic susceptibility for a nanoparticle in the presence of a static external magnetic field. Below a critical magnetic field the magnetic response is diamagnetic, whereas above such field the magnetization is characterized by sharp, step-like increases of several tenths of Bohr magnetons, associated with the Zeeman crossing of energy levels above and below the Fermi sea. We quantify the robustness of these regimes against thermal excitations and finite linewidth of the electronic levels. Finally, we propose two methods for experimental detection of the quantum size effects based on the coupling to superconducting quantum interference devices., Comment: 5+6 pages, 3+3 figures

In this paper, we deal with a simple geometric problem: Is it possible to partition a rectangle into $k$ non-congruent rectangles of equal area? This problem is motivated by the so-called `Mondrian art problem' that asks a similar question for dissections with rectangles of integer sides. Here, we generalize the Mondrian problem by allowing rectangles of real sides. In this case, we show that the minimum value of $k$ for a rectangle to have a `perfect Mondrian partition' (that is, with non-congruent equal-area rectangles) is seven. Moreover, we prove that such a partition is unique (up to symmetries) and that there exist exactly two proper perfect Mondrian partitions for $k=8$. Finally, we also prove that any square has a perfect Mondrian decomposition for $k\ge 7$.

We combine top-down and bottom-up nanolithography to optimize the coupling of small molecular spin ensembles to $1.4$ GHz on-chip superconducting resonators. Nanoscopic constrictions, fabricated with a focused ion beam at the central transmission line, locally concentrate the microwave magnetic field. Drops of free-radical molecules have been deposited from solution onto the circuits. For the smallest ones, the molecules were delivered at the relevant circuit areas by means of an atomic force microscope. The number of spins $N_{\rm eff}$ effectively coupled to each device was accurately determined combining Scanning Electron and Atomic Force Microscopies. The collective spin-photon coupling constant has been determined for samples with $N_{\rm eff}$ ranging between $2 \times 10^{6}$ and $10^{12}$ spins, and for temperatures down to $44$ mK. The results show the well-known collective enhancement of the coupling proportional to the square root of $N_{\rm eff}$. The average coupling of individual spins is enhanced by more than four orders of magnitude (from $4$ mHz up to above $180$ Hz) when the transmission line width is reduced from $400$ microns down to $42$ nm, and reaches maximum values near $1$ kHz for molecules located on the smallest nanoconstrictions. This result opens promising avenues for the realization of magnetic spectroscopy experiments at the nanoscale and for the development of hybrid quantum computation architectures based on molecular spin qubits., Comment: 31 pages, 6 figures, supporting information in ancillary PDF file

We consider homological finiteness properties $FP_n$ of certain $\mathbb{N}$-graded Lie algebras. After proving some general results, see Theorem A, Corollary B and Corollary C, we concentrate on a family that can be considered as the Lie algebra version of the generalized Bestvina-Brady groups associated to a graph $\Gamma$. We prove that the homological finiteness properties of these Lie algebras can be determined in terms of the graph in the same way as in the group case. In the last version we have corrected some missprints, in particular the statement of Theorem D (from $n-1$-acyclicity to $n-1 - |w|$-acyclicity)., Comment: 22 pages

The primary cell surface receptor for SARS-CoV-2 is the angiotensin-converting enzyme 2 (ACE2). Recently it has been noticed that the viral Spike protein has an RGD motif, suggesting that cell surface integrins may be co-receptors. We examined the sequences of ACE2 and integrins with the Eukaryotic Linear Motif resource, ELM, and were presented with candidate short linear motifs (SLiMs) in their short, unstructured, cytosolic tails with potential roles in endocytosis, membrane dynamics, autophagy, cytoskeleton and cell signalling. These SLiM candidates are highly conserved in vertebrates. They suggest potential interactions with the AP2 mu2 subunit as well as I-BAR, LC3, PDZ, PTB and SH2 domains found in signalling and regulatory proteins present in epithelial lung cells. Several motifs overlap in the tail sequences, suggesting that they may act as molecular switches, often involving tyrosine phosphorylation status. Candidate LIR motifs are present in the tails of ACE2 and integrin beta3, suggesting that these proteins can directly recruit autophagy components. We also noticed that the extracellular part of ACE2 has a conserved MIDAS structural motif, which are commonly used by beta integrins for ligand binding, potentially supporting the proposal that integrins and ACE2 share common ligands. The findings presented here identify several molecular links and testable hypotheses that might help uncover the mechanisms of SARS-CoV-2 attachment, entry and replication, and strengthen the possibility that it might be possible to develop host-directed therapies to dampen the efficiency of viral entry and hamper disease progression. The strong sequence conservation means that these putative SLiMs are good candidates: Nevertheless, SLiMs must always be validated by experimentation before they can be stated to be functional., Comment: 38 pages, 7 figures, 2 tables. Corresponding authors are Luc\'ia B. Chemes, Toby J. Gibson

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition and, therefore, it can be removed as hypothesis., Comment: arXiv admin note: substantial text overlap with arXiv:1605.04394, arXiv:1806.04619; text overlap with arXiv:1501.02288 by other authors

Given a metric space $(X,d)$, a set $S\subseteq X$ is called a $k$-\emph{metric generator} for $X$ if any pair of different points of $X$ is distinguished by at least $k$ elements of $S$. A $k$-\emph{metric basis} is a $k$-metric generator of the minimum cardinality in $X$. We prove that ultrametric spaces do not have finite $k$-metric bases for $k>2$. We also characterize when the metric and 2-metric bases of an ultrametric space are finite and, when they are finite, we characterize them. Finally, we prove that an ultrametric space can be easily recovered knowing only the metric basis and the coordinates of the points in it., Comment: 10 pages

Coherent exchange between photons and different matter excitations (like qubits, acoustic surface waves or spins) allows for the entanglement of light and matter and provides a toolbox for performing fundamental quantum physics. On top of that, coherent exchange is a basic ingredient in the majority of quantum information processors. In this work, we develop the theory for coupling between magnetic textures (vortices and skyrmions) stabilized in ferromagnetic nanodiscs and photons generated in a circuit. In particular, we show how to perform broadband spectroscopy of the magnetic textures by sending photons through a transmission line and recording the transmission. We also discuss the possibility of reaching the strong coupling regime between these texture excitations and a single photon residing in a cavity.

Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the largest integer such that there exists a $k$-metric generator for $G$. This paper studies some bounds on the number $k$ for which a graph is $k$-metric dimensional., Comment: 11 pages, 3 figures

Strong light-matter coupling is a necessary condition for exchanging information in quantum information protocols. It is used to couple different qubits (matter) via a quantum bus (photons) or to communicate different type of excitations, e.g. transducing between light and phonons or magnons. An unexplored, so far, interface is the coupling between light and topologically protected particle like excitations as magnetic domain walls, skyrmions or vortices. Here, we show theoretically that a single photon living in a superconducting cavity can be coupled strongly to the gyrotropic mode of a magnetic vortex in a nanodisc. We combine numerical and analytical calculations for a superconducting coplanar waveguide resonator and different realizations of the nanodisc (materials and sizes). We show that, for enhancing the coupling, constrictions fabricated in the resonator are beneficial, allowing to reach the strong coupling in CoFe discs of radius $200-400$ nm having resonance frequencies of few GHz. The strong coupling regime permits to exchange coherently a single photon and quanta of vortex excitations. Thus, our calculations show that the device proposed here serves as a transducer between photons and gyrating vortices, opening the way to complement superconducting qubits with topologically protected spin-excitations like vortices or skyrmions. We finish by discussing potential applications in quantum data processing based on the exploitation of the vortex as a short-wavelength magnon emitter., Comment: published version

A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric inequality in a Riemann surface by using a graph related to it, even if the surface has injectivity radius zero (this graph is inspired in Kanai's graph, but it is different from it). We also present an application relating Gromov boundary and isoperimetric inequality., Comment: 18 pages

The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the first variable Zagreb index $M_1^\alpha$, and to characterize the set of extremal graphs with respect to them. Our main results provide lower bounds on this family of topological indices involving just the minimum and the maximum degree of the graph. These inequalities are new even for the first Zagreb, the inverse and the forgotten indices., Comment: 17 pages

In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. In this paper we study the relation between vertex separator sets, some chordality properties which are natural generalizations of being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons., Comment: 16 pages, 3 figures

We consider the Lagrangian formulation with duplicated variables of dissipative mechanical systems. The application of Noether theorem leads to physical observable quantities which are not conserved, like energy and angular momentum, and conserved quantities like the Hamiltonian, that generate symmetry transformations and do not correspond to observables. We show that there are simple relations among the equations satisfied by these two types of quantities. In the case of the damped harmonic oscillator, from the quantities obtained by Noether theorem follows the algebra of Feshbach and Tikochinsky. Further, if we consider the whole dynamics, the degrees of freedom separate into a physical and an unphysical sector. We analyze several cases, with linear and nonlinear dissipative forces; the physical consistency of the solutions is ensured observing that the unphysical sector has always the trivial solution., Comment: 14 pages

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this subgroup, based on the number and degree of a certain type of vertices, which we call deep. We then use combinatorial methods to analyze the average value of this Betti number, in terms of the size of the defining tree.

We prove that even Artin groups of FC type are poly-free and residually finite.

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them. Our main results provide lower bounds $GA_1(G)$ involving just the minimum and the maximum degree of the graph $G$., Comment: 12 pages

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them., Comment: 13 pages

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X)=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}.$ Some previous works characterize the hyperbolic product graphs (for the Cartesian, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product $G_1\times G_2$ is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain formulae or good bounds for the hyperbolicity constant of the direct product of some important graphs., Comment: 20 pages

We show Lie algebra versions of some results on homological finiteness properties of subdirect products of groups, including a version of the 1-2-3 Theorem.

We demonstrate the operation of low-noise nano superconducting quantum interference devices (SQUIDs) based on the high critical field and high critical temperature superconductor YBa$_2$Cu$_3$O$_7$ (YBCO) as ultra-sensitive magnetometers for single magnetic nanoparticles (MNPs). The nanoSQUIDs exploit the Josephson behavior of YBCO grain boundaries and have been patterned by focused ion beam milling. This allows to precisely define the lateral dimensions of the SQUIDs so as to achieve large magnetic coupling between the nanoloop and individual MNPs. By means of focused electron beam induced deposition, cobalt MNPs with typical size of several tens of nm have been grown directly on the surface of the sensors with nanometric spatial resolution. Remarkably, the nanoSQUIDs are operative over extremely broad ranges of applied magnetic field (-1 T $< \mu_0 H <$ 1 T) and temperature (0.3 K $< T<$ 80 K). All these features together have allowed us to perform magnetization measurements under different ambient conditions and to detect the magnetization reversal of individual Co MNPs with magnetic moments (1 - 30) $\times 10^6\,\mu_{\rm B}$. Depending on the dimensions and shape of the particles we have distinguished between two different magnetic states yielding different reversal mechanisms. The magnetization reversal is thermally activated over an energy barrier, which has been quantified for the (quasi) single-domain particles. Our measurements serve to show not only the high sensitivity achievable with YBCO nanoSQUIDs, but also demonstrate that these sensors are exceptional magnetometers for the investigation of the properties of individual nanomagnets.

Weanalyzethecomputationalcomplexityofanalgorithmtosolve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most exponential. For a subfamily of groups we prove that the conjugacy search problem is polynomial. We also show that for a different subfamily the conjugacy search problem reduces to the discrete logarithm problem.

We consider generalisations of Thompson's group $V$, denoted by $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type $\mathrm{F}_\infty$. Under more general conditions we show that the groups $V_r(\Sigma)$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure., Comment: 24 pages, no figures

Superconducting Quantum Interference Devices (SQUIDs) are one of the most popular devices in superconducting electronics. They combine the Josephson effect with the quantization of magnetic flux in superconductors. This gives rise to one of the most beautiful manifestations of macroscopic quantum coherence in the solid state. In addition, SQUIDs are extremely sensitive sensors allowing to transduce magnetic flux into measurable electric signals. As a consequence, any physical observable that can be converted into magnetic flux, e.g., current, magnetization, magnetic field or position, becomes easily accessible to SQUID sensors. In the late 1980's it became clear that downsizing the dimensions of SQUIDs to the nanometric scale would encompass an enormous increase of their sensitivity to localized tiny magnetic signals. Indeed, nanoSQUIDs opened the way to the investigation of, e.g., individual magnetic nanoparticles or surface magnetic states with unprecedented sensitivities. The purpose of this review is to present a detailed survey of microscopic and nanoscopic SQUID sensors. We will start by discussing the principle of operation of SQUIDs, placing the emphasis on their application as ultrasensitive detectors for small localized magnetic signals. We will continue by reviewing a number of existing devices based on different kinds of Josephson junctions and materials, focusing on their advantages and drawbacks. The last sections are left for applications of nanoSQUIDs in the fields of scanning SQUID microscopy and magnetic particle characterization, putting special stress on the investigation of individual magnetic nanoparticles.

The physics learning is very important to techniques teaching and the engineers upbringing. However, there is evidence that science classes from elementary school through to university are generally failing to provide most students with an understanding of physics We have explored the use of a novel aspect of information technology: interactive simulations to learning of energys concepts. This way, students can construct their understanding through semi-guided exploration. Research has shown this process to be an effective and engaging way to learn., Comment: 12 pages, in Spanish, 3 figures, XVI Congreso Universitario de Innovacion Educativa en las Ensenanzas Tecnicas, Cadiz, Espana, 23 a 26 de Septiembre de 2008

Stackability for finitely presented groups consists of a dynamical system that iteratively moves paths into a maximal tree in the Cayley graph. Combining with formal language theoretic restrictions yields auto- or algorithmic stackability, which implies solvability of the word problem. In this paper we give two new characterizations of the stackable property for groups, and use these to show that every HNN extension of a stackable group is stackable. We apply this to exhibit a wide range of Dehn functions that are admitted by stackable and autostackable groups, as well as an example of a stackable group with unsolvable word problem. We use similar methods to show that there exist finitely presented metabelian groups that are non-constructible but admit an autostackable structure., Comment: 34 pages

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary., Comment: 24 pages

Let $V_r(\Sigma)$ be the generalised Thompson group defined as the automorphism group of a valid, bounded, and complete Cantor algebra. We show that that for every $n>0$ there is a $k>n,$ such that there exists a $k$-dimensional $n$-connected simplicial complex $K$ such that $V_r(\Sigma)$ acts on $K$ with finite stabilisers. We also determine the number of conjugacy classes of finite cyclic subgroups of a given order $m$ in Brin-Thompson groups. We apply our computations to the rationalised Farrell-Jones assembly map in algebraic $K$-theory., Comment: 12 pages There was a mistake in a previous version of this preprint. We no longer compute the rational homology of generalized Brin-Higman-Thompson groups

We present the design, realization and performance of a three-axis vector nano Superconducting QUantum Interference Device (nanoSQUID). It consists of three mutually orthogonal SQUID nanoloops that allow distinguishing the three components of the vector magnetic moment of individual nanoparticles placed at a specific position. The device is based on Nb/HfTi/Nb Josephson junctions and exhibits linewidths of $\sim 250$ nm and inner loop areas of $600 \times 90$ nm$^2$ and $500 \times 500$ nm$^2$. Operation at temperature $T=4.2$ K, under external magnetic fields up to $\sim 50$ mT is demonstrated. The experimental flux noise below $\sim 250$ ${\rm n}\Phi_0/\sqrt{\rm Hz}$ in the white noise limit and the reduced dimensions lead to a total calculated spin sensitivity of $\sim 630$ $\mu_{\rm B}/\sqrt{\rm Hz}$ and $\sim 70$ $\mu_{\rm B}/\sqrt{\rm Hz}$ for the in-plane and out-of-plane components of the vector magnetic moment, respectively. The potential of the device for studying tridimensional properties of individual nanomagnets is discussed.

The family of Directed Acyclic Graphs as well as some related graphs are analyzed with respect to extremal behavior in relation with the family of intersection graphs for families of boxes with transverse intersection., Comment: 13 pages, 4 figures

We apply Noether's theorem to show how the invariances of conservative systems are broken for nonconservative systems, in the variational formulation of Galley. This formulation considers a conservative action, extended by the inclusion of a time reversed sector and a nonconservative generalized potential. We assume that this potential is invariant under the symmetries of the initial conservative system. The breaking occurs because the time reversed sector requires inverse symmetry transformations, under which the nonconservative potential is not invariant. The resulting violation of the conservation laws is consistent with the equations of motion. We generalize this formulation for fermionic and sypersymmetric systems. In the case of a supersymmetric oscillator, the effect of damping is that the bosonic and fermionic components become different frequencies. Considering that initially the nonconservative action is invariant under supersymmetry, and that the breaking is associated to an instability, this result is reminiscent of spontaneous symmetry breaking., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1501.05018

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. In this paper we study the relation between the hyperbolicity of the graph and some chordality properties which are natural generalizations of being chordal. We find chordality properties that are weaker and stronger than being $\delta$-hyperbolic. Moreover, we obtain a characterization of being hyperbolic on terms of a chordality property on the triangles., Comment: 15 pages, 2 figures

We study the (virtual) indicability of the automorphism group $Aut(A_\Gamma)$ of the right-angled Artin group $A_\Gamma$ associated to a simplicial graph $\Gamma$. First, we identify two conditions -- denoted (B1) and (B2) -- on $\Gamma$ which together imply that $H^1(G, Z)=0$ for certain finite-index subgroups $G

We prove that if $\Gamma$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $\Gamma$-equivariantly homotopy equivalent to a proper cocompact $\Gamma$-CW complex of dimension the virtual cohomological dimension of $\Gamma$., Comment: 43 pages