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We present solutions to the Einstein-Klein Gordon system representing boson stars in the slow rotation approximation. By considering slow rotation we are able to reduce the number of equations yielding a system of ordinary differential equations that is conveniently solved numerically without the need of expensive computational resources. We find sequences of solutions and describe some of their physical properties such as, total mass, angular momentum and compactness. We also consider the dynamics of particles (geodesics) in the resulting spacetime. A detailed comparison with fully rotating boson stars (non-linear treatment) is performed by showing the region of validity of the slow-rotation approximation., Comment: 21 pages, 12 figures

Quantum machine learning has gained considerable attention as quantum technology advances, presenting a promising approach for efficiently learning complex data patterns. Despite this promise, most contemporary quantum methods require significant resources for variational parameter optimization and face issues with vanishing gradients, leading to experiments that are either limited in scale or lack potential for quantum advantage. To address this, we develop a general-purpose, gradient-free, and scalable quantum reservoir learning algorithm that harnesses the quantum dynamics of neutral-atom analog quantum computers to process data. We experimentally implement the algorithm, achieving competitive performance across various categories of machine learning tasks, including binary and multi-class classification, as well as timeseries prediction. Effective and improving learning is observed with increasing system sizes of up to 108 qubits, demonstrating the largest quantum machine learning experiment to date. We further observe comparative quantum kernel advantage in learning tasks by constructing synthetic datasets based on the geometric differences between generated quantum and classical data kernels. Our findings demonstrate the potential of utilizing classically intractable quantum correlations for effective machine learning. We expect these results to stimulate further extensions to different quantum hardware and machine learning paradigms, including early fault-tolerant hardware and generative machine learning tasks., Comment: 10 + 14 pages, 4 + 7 figures

Orthogonal arrays play a fundamental role in many applications. However, constructing orthogonal arrays with the required parameters for an application usually is extremely difficult and, sometimes, even impossible. Hence there is an increasing need for a relaxation of orthogonal arrays to allow a wider flexibility. The latter has lead to various types of arrays under the name of ``nearly-orthogonal arrays'', and less often ``almost orthogonal arrays''. The aim of this paper is twofold. On the one hand, we review all the existing relaxations, comparing and discussing them in depth. On the other hand, we explore how to find almost orthogonal arrays three ways: using integer programming, local search meta-heuristics and algebraic methods. We compare all our search results with the ones existing in the literature, and we show that they are competitive, improving some of the existing arrays for many non-orthogonality measures. All our found almost orthogonal arrays are available at a public repository., Comment: 59 pages

The industrial manufacturing of chemicals consumes a significant amount of energy and raw materials. In principle, the development of new catalysts could greatly improve the efficiency of chemical production. However, the discovery of viable catalysts can be exceedingly challenging because it is difficult to know the efficacy of a candidate without experimentally synthesizing and characterizing it. This study explores the feasibility of using fault-tolerant quantum computers to accelerate the discovery of homogeneous catalysts for nitrogen fixation, an industrially important chemical process. It introduces a set of ground-state energy estimation problems representative of calculations needed for the discovery of homogeneous catalysts and analyzes them on three dimensions: economic utility, classical hardness, and quantum resource requirements. For the highest utility problem considered, two steps of a catalytic cycle for the generation of cyanate anion from dinitrogen, the economic utility of running these computations is estimated to be $200,000, and the required runtime for double-factorized phase estimation on a fault-tolerant superconducting device is estimated under conservative assumptions to be 139,000 QPU-hours. The computational cost of an equivalent DMRG calculation is estimated to be about 400,000 CPU-hours. These results suggest that, with continued development, it will be feasible for fault-tolerant quantum computers to accelerate the discovery of homogeneous catalysts., Comment: 27 pages, 11 tables, 8 figures plus appendix

In addition to a plethora of emergent phenomena, the spatial topology of optical vortices enables an array of applications spanning communications to quantum photonics. Nonlinear optics is essential in this context, providing access to an infinitely large set of quantum states associated with the orbital angular momentum of light. Nevertheless, the realization of such processes have failed to keep pace with the ever-growing need to shrink the fundamental length-scale of photonic technologies to the nanometer regime6. Here, we push the boundaries of vortex nonlinear optics to the ultimate limits of material dimensionality. By exploiting second and third-order frequency-mixing processes in semiconducting monolayers, we demonstrate the independent manipulation of the wavelength, orbital angular momentum, and spatial distribution of vortex light-fields. Due to the atomically-thin nature of the host quantum material, this control spans a broad spectral bandwidth in a highly-integrable platform, unconstrained by the traditional limits of bulk nonlinear optical materials. Our work heralds a new avenue for ultra-compact and scalable hybrid nanotechnologies empowered by twisted nonlinear light-matter interactions in van der Waals quantum nanomaterials.

Blockchain technology has become a trusted method for establishing secure and transparent transactions through a distributed, encrypted network. The operation of blockchain is governed by consensus algorithms, among which Proof of Stake (PoS) is popular yet has its drawbacks, notably the potential for centralising power in nodes with larger stakes or higher rewards. Fuzzychain, our proposed solution, introduces the use of fuzzy sets to define stake semantics, promoting decentralised and distributed processing control. This system selects validators based on their degree of membership to the stake fuzzy sets rather than just the size of their stakes. As a pioneer proposal in applying fuzzy sets to blockchain, Fuzzychain aims to rectify PoS's limitations. Our results indicate that Fuzzychain not only matches PoS in functionality but also ensures a fairer distribution of stakes among validators, leading to more inclusive validator selection and a better-distributed network., Comment: 16 pages, 13 figures, 2 tables, this article was submitted to a JCR journal

We revisit the concepts of acyclic orderings and number of acyclic orderings of acyclic digraphs in terms of dispositions and counters for arbitrary multidigraphs. We prove that when we add a sequence of nested directed paths to a directed graph there is a unique polynomial such that the generatrix function of the family of counters is the product of the polynomial and the exponential function. We give an application, by considering a kind of digraphs arranged in rows introduced by the authors in a previous paper, called dispositional digraphs, in the particular case in which the digraph has two rows, to obtain new families of linear differential equations of small order whose coefficients are polynomials of small degree which admit polynomial solutions. In particular, we obtain a new differential equation associated to Catalan numbers, and the corresponding associated polynomials, which are solution of this differential equation; we term them Catalan differencial equation and Catalan polynomials, respectively. We prove that the Catalan polynomials obtained when we connect the directed path to the second vertex of the lower row of the digraph are orthogonal polynomials for an appropriate weight function. We characterize the digraphs that maximize the counter of connected dispositional digraphs and we find a new differential equation associated to these digraphs. We introduce also dispositions and counters in any multidigraph with non-strict inequalities in the dispositions, and we find new differential equations associated to some of them.

We provide a complete study of existence and uniqueness (uniqueness up to multiples in the case $\mathbf{d} = \mathbf{0}$) of non-negative and non-trivial solutions $\mathbf{x}$ for the linear system $(\mathbf{I}- \mathbf{A})\mathbf{x} = \mathbf{d}$ with $\mathbf{A} \geq \mathbf{0}, \mathbf{d} \geq 0$ (which, in particular, applies to Leontief's Input Output Model). This study is done in terms of the block triangular form of the matrix $\mathbf{A}$ and is related to a directed graph associated to $\mathbf{A}$. In particular, this study of existence and uniqueness up to multiples of solutions provides a framework that allows us to rigorously perform a sensitivity analysis of the normalized solutions of the linear system $(\mathbf{I} - \mathbf{A})\mathbf{x} = \mathbf{d}$.

The classical notion of twisted product is studied in the context of partial actions, in particular, we show that the globalization of a partial action is a twisted product. In addition, we establish conditions for the metrizability of twisted products, and some homotopy and categorical properties are proved. Furthermore, sufficient conditions for the enveloping space to be an equivariant absolute neighborhood extensor are also studied.

One of the ways to encode many-body Hamiltonians on a quantum computer to obtain their eigen-energies through Quantum Phase Estimation is by means of the Trotter approximation. There were several ways proposed to assess the quality of this approximation based on estimating the norm of the difference between the exact and approximate evolution operators. Here, we would like to explore how these different error estimates are correlated with each other and whether they can be good predictors for the true Trotter approximation error in finding eigenvalues. For a set of small molecular systems we calculated the exact Trotter approximation errors of the first order Trotter formulas for the ground state electronic energies. Comparison of these errors with previously used upper bounds show almost no correlation over the systems and various Hamiltonian partitionings. On the other hand, building the Trotter approximation error estimation based on perturbation theory up to a second order in the time-step for eigenvalues provides estimates with very good correlations with the Trotter approximation errors. The developed perturbative estimates can be used for practical time-step and Hamiltonian partitioning selection protocols, which are paramount for an accurate assessment of resources needed for the estimation of energy eigenvalues under a target accuracy., Comment: 3 figures

This work concerns the analysis of electromagnetic dispersive media modelled by generalized Lorentz models. More precisely, this paper is the second of two articles dedicated to the long time behaviour of solutions of Maxwell's equations in dissipative Lorentz media, via the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. In opposition to the frequency dependent Lyapunov functions approach used in [Cassier, Joly, Rosas Mart\'inez, Z. Angew. Math. Phys. 74 (2023), 115], we develop a method based on the spectral analysis of the underlying non-self-adjoint operator of the model. Although more involved, this approach is closer to physics, as it uses the dispersion relation of the model, and has the advantage to provide more precise and more optimal results, leading to distinguish the notion of weak and strong dissipation., Comment: 67 pages, 4 figures

We consider the question of how correlated the system hardness is between classical algorithms of electronic structure theory in ground state estimation and quantum algorithms. To define the system hardness for classical algorithms we employ empirical criterion based on the deviation of electronic energies produced by coupled cluster and configuration interaction methods from the exact ones along multiple bonds dissociation in a set of molecular systems. For quantum algorithms, we have selected the Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE) methods. As characteristics of the system hardness for quantum methods, we analyzed circuit depths for the state preparation, the number of quantum measurements needed for the energy expectation value, and various cost characteristics for the Hamiltonian encodings via Trotter approximation and linear combination of unitaries (LCU). Our results show that the quantum resource requirements are mostly unaffected by classical hardness, with the only exception being the state preparation part, which contributes to both VQE and QPE algorithm costs. However, there are clear indications that constructing the initial state with a significant overlap with the true ground state (>10%) is easier than obtaining the state with an energy expectation value within chemical precision. These results support optimism regarding the identification of a molecular system where a quantum algorithm excels over its classical counterpart, as quantum methods can maintain efficiency in classically challenging systems., Comment: 11 pages

We provide a different proof of the equivariant version of the Borsuk-Whitehead-Hanner Theorem in the category of proper G-spaces which are metrizable by a G-invariant metric.

Given the lack of an absolute time parameter in general relativistic systems, quantum cosmology often describes the expansion of the universe in terms of relational changes between two degrees of freedom, such as matter and geometry. However, if clock degrees of freedom (self-)interact non-trivially, they in general have turning points where their momenta vanish. At and beyond a turning point, the evolution of other degrees of freedom is no longer described directly by changes of the clock parameter because it stops and then turns back, while time is moving forward. Previous attempts to describe quantum evolution relative to a clock with turning points have failed and led to frozen evolution in which degrees of freedom remain constant while the clock parameter, interpreted directly as a substitute for monotonic time, is being pushed beyond its turning point. Here, a new method previously used in oscillator systems is applied to a tractable cosmological model, given by an isotropic universe with spatial curvature and scalar matter. The re-collapsing scale factor presents an example of a clock with a single turning point. The method succeeds in defining unitary and freeze-free evolution by unwinding the turning point of the clock, introducing an effective monotonic time parameter that is related to but not identical with the non-monotonic clock degree of freedom. Characteristic new quantum features are found around the turning point, based on analytical and numerical calculations., Comment: 24 pages, 4 figures

A quantum anomalous Hall (QAH) insulator breaks reciprocity by combining magnetic polarization and spin-orbit coupling to generate a unidirectional transmission of signals in the absence of an external magnetic field. Such behavior makes QAH materials a good platform for the innovation of circulator technologies. However, it remains elusive as to how the wavelength of the chiral edge plasmon relates to its frequency and how the plasmon wave packet is excited in the time domain in a QAH insulator. Here, we investigate the edge magnetoplasmon (EMP) resonances in Cr-(Bi,Sb)$_2$Te$_3$ by frequency and time domain measurements. From disk shaped samples with various dimensions, we obtain the dispersion relation of EMPs and extract the drift velocity of the chiral edge state. From the time-resolved transport measurements, we identify the velocity of the plasmon wave packet and observe a transition from the edge to bulk transport at an elevated temperature. We show that the frequency and time domain measurements are well modeled by loss from the microwave induced dissipative channels in the bulk area. Our results demonstrate that the EMP decay rate can be significantly reduced by applying a low microwave power and fabricating devices of larger diameter $\ge100~\mu$m. In a $R=125~\mu$m sample, a non-reciprocity of 20 dB has been realized at 1.3 GHz, shining light on using QAH insulators to develop on-chip non-reciprocal devices.

This study introduces the P5 model - a foundational method that utilizes reinforcement learning (RL) to augment control, effectiveness, and scalability in molecular dynamics simulations (MD). Our innovative strategy optimizes the sampling of target polymer chain conformations, marking an efficiency improvement of over 37.1%. The RL-induced control policies function as an inductive bias, modulating Brownian forces to steer the system towards the preferred state, thereby expanding the exploration of the configuration space beyond what traditional MD allows. This broadened exploration generates a more varied set of conformations and targets specific properties, a feature pivotal for progress in polymer development, drug discovery, and material design. Our technique offers significant advantages when investigating new systems with limited prior knowledge, opening up new methodologies for tackling complex simulation problems with generative techniques., Comment: ICML 2023 Workshop on Structured Probabilistic Inference (SPIGM) and Generative Modeling, of the International Conference of Machine Learning (ICML)

The levitation of a macroscopic object within a superconducting resonator provides a unique and novel platform to study optomechanics, quantum information, and gravitational wave detection. Existing mirror-method and single-loop models for calculating magnet levitation are insufficient for predicting the position and motion of the levitated magnet. If the cavity-magnet interaction is modeled using a large number of smaller surface current loops, one can quantitatively model the dynamics of the levitation of the magnet within the cavity. The magnet's most-likely position and orientation can be predicted for non-trivial cavity geometries and cavity orientations. Knowing the potential energy landscape within the cavity configuration also provides a means to estimate the resonant mechanical frequencies at which the levitated magnet vibrates, and enables tailoring the cavity design for specific outcomes., Comment: 10 pages, 10 figures, submitting to PRA

Research and education in machine learning needs diverse, representative, and open datasets that contain sufficient samples to handle the necessary training, validation, and testing tasks. Currently, the Recommender Systems area includes a large number of subfields in which accuracy and beyond accuracy quality measures are continuously improved. To feed this research variety, it is necessary and convenient to reinforce the existing datasets with synthetic ones. This paper proposes a Generative Adversarial Network (GAN)-based method to generate collaborative filtering datasets in a parameterized way, by selecting their preferred number of users, items, samples, and stochastic variability. This parameterization cannot be made using regular GANs. Our GAN model is fed with dense, short, and continuous embedding representations of items and users, instead of sparse, large, and discrete vectors, to make an accurate and quick learning, compared to the traditional approach based on large and sparse input vectors. The proposed architecture includes a DeepMF model to extract the dense user and item embeddings, as well as a clustering process to convert from the dense GAN generated samples to the discrete and sparse ones, necessary to create each required synthetic dataset. The results of three different source datasets show adequate distributions and expected quality values and evolutions on the generated datasets compared to the source ones. Synthetic datasets and source codes are available to researchers.

The discovery of long-range magnetic ordering in atomically thin materials catapulted the van der Waals (vdW) family of compounds into an unprecedented popularity. In particular, with a current push in space exploration, it is beneficial to study how the properties of such materials evolve under proton irradiation. Owing to their robust intra-layer stability and sensitivity to external perturbations, these materials provide excellent opportunities for studying proton irradiation as a non-destructive tool for controlling their magnetic properties. Specifically, the exfoliable Cr2Si2Te6 (CST) is a ferromagnetic semiconductor with the Curie temperature (TC) of ~32 K. Here, we have investigated the magnetic properties of CST upon proton irradiation as a function of fluence (1 x 1015, 5 x 1015, 1 x 1016, 5 x 1016, and 1 x 1018 H+/cm2) by employing variable-temperature, variable-field magnetization measurements coupled with electron paramagnetic resonance (EPR) spectroscopy and detail how the magnetization, magnetic anisotropy and EPR spectral parameters vary as a function of proton fluence across the magnetic phase transition. While the TC remains constant as a function of proton fluence, we observed that the saturation magnetization and magnetic anisotropy diverge at the proton fluence of 5 x 1016 H+/cm2, which is prominent in the ferromagnetic phase, in particular. This work demonstrates that proton irradiation is a feasible method for modifying the magnetic properties and local magnetic interactions of vdWs crystals, which represents a significant step forward in the design of future spintronic and magneto-electronic applications.

Several authors have remarked the convenience of understanding the different notions of center appearing in Geometry (centroid of a set of points, incenter of a triangle, center of a conic and many others) as functions. The most general way to do so is to define centers as equivariant maps between $G$-spaces. In this paper, we prove that, under certain hypothesis, for any two $G$-spaces $\mathcal A,\mathcal X$, for every $V\in \mathcal A$ and for every point $P\in\mathcal X$ fixed by the symmetry group of $V$, there exists some equivariant map $\mathfrak Z:\mathcal A\to \mathcal X$ such that $\mathfrak Z(V)=P$. As a consequence of this fact, we prove an analogue (for non-neccessarily continuous centers) of the \emph{center conjecture for equifacetal simplices}, proposed by A. L. Edmonds.

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Frechet structure of the Riordan group described in [4]. After that we describe some linear dynamical systems in $\mathbb{K}[[x]]$ with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to the Pascal Triangle.

In this paper we revisit a result due to Franz Rellich on smoothness of solutions of parametrized linear systems. With this result as a starting point, we obtain finer smoothness results in an elementary fashion and propose an efficient adjoint algorithm for computing sensitivities of $(n-1)$-deficient systems, being $n$ the order of the system.

Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts., Comment: 13 pages, 4 figures

It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances., Comment: 33 pages, 1 figure

Single quantum emitters play a fundamental role in the development of quantum technologies such as quantum repeaters, and quantum information processing. Isolating individual molecules with stable optical emission is an essential step for these applications, specially for those molecules that present large coherence times at room temperature. Among them, vanadium-oxide phthalocyanine (VOPc) molecules stand out as promising candidates due to their large coherence times measured in ensemble. However, the optical properties of individual molecules have not yet been reported. Here we show that single VOPc molecules with stable optical properties at room temperature can be isolated. We find that the optical response of the molecule under laser illumination of different polarization agrees well with a system having pyramidal C$_{4v}$ symmetry. Furthermore, the molecule reveals a non-radiative transition rate that depends on the excitation wavelength when its lifetime is interrogated. We provide theoretical calculations that support our experimental findings and provide insight to the role of phonons and internal electronic structure of the molecule. These results demonstrate that this single paramagnetic molecule can function as a single quantum emitter while displaying optical stability under ambient conditions to have their intrinsic properties investigated.

Among several well-known transition metal-based compounds, the van der Waals (vdW) Fe3-xGeTe2 (FGT) magnet is a strong candidate for use in two-dimensional (2D) magnetic devices due to its strong perpendicular magnetic anisotropy, sizeable Curie temperature (TC ~ 154 K), and versatile magnetic character that is retained in the low-dimensional limit. While the TC remains far too low for practical applications, there has been a successful push toward improving it via external driving forces such as pressure, irradiation, and doping. Here we present experimental evidence of a novel room-temperature (RT) ferromagnetic phase induced by the electrochemical intercalation of common tetrabutylammonium cations (TBA+) into FGT bulk crystals. We obtained Curie temperatures as high as 350 K with chemical and physical stability of the intercalated compound. The temperature-dependent Raman measurements in combination with vdW-corrected ab initio calculations suggest that charge transfer (electron doping) upon intercalation could lead to the observation of RT ferromagnetism. This work demonstrates that molecular intercalation is a viable route in realizing high-temperature vdW magnets in an inexpensive and reliable manner.

The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $\gamma$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^{\gamma}$-algebra are presented in the case that $R$ is commutative. We study when $B$ is a partial crossed product in the following cases: $B=\operatorname{M}_n(A)$ is the ring of matrices with entries in $A$, or $B={\bf grm}{M}=\bigoplus_{l \in G}{\bf Mor}_A(M,M)_l$ is the direct sum of graded endomorphisms of left graded $A$-module $M$ with degree $l$, or $B={\bf grm}{M}$ where $M=A\otimes_{R}N$ is the induced module of a left $R$-module $N$. Finally, assuming that $R$ is semiperfect, we prove that there exists an epsilon-strongly graded subring of $A$ which is graded equivalent to a partial crossed product., Comment: 24 pages

Let $\Lambda$ be an Artin algebra and $K^b(proj(\Lambda))$ be the triangulated category of bounded co-chain complexes in $proj(\Lambda).$ It is well known that two-terms silting complexes in $K^b(proj(\Lambda))$ are described by the $\tau$-tilting theory. The aim of this paper is to give a characterization of certain $n$-term silting complexes in $K^b(proj(\Lambda))$ which are induced by $\Lambda$-modules. In order to do that, we introduce the notions of $\tau_n$-rigid, $\tau_n$-tilting and $\tau_{n,m}$-tilting $\Lambda$-modules. The latter is both a generalization of $\tau$-tilting and tilting in $mod(\Lambda).$ It is also stated and proved some variant, for $\tau_n$-tilting modules, of the well known Bazzoni's characterization for tilting modules. We give some connections between $n$-terms presilting complexes in $K^b(proj(\Lambda))$ and $\tau_n$-rigid $\Lambda$-modules. Moreover, a characterization is given to know when a $\tau_n$-tilting $\Lambda$-module is $n$-tilting. We also study more deeply the properties of the $\tau_{n,m}$-tilting $\Lambda$-modules and their connections of being $m$-tilting in some quotient algebras. We apply the developed $\tau_{n,m}$-tilting theory to the finitistic dimension of $\Lambda.$ Finally, at the end of the paper we discuss and state some open questions (conjectures) that we consider crucial for the future develop of the $\tau_{n,m}$-tilting theory.

There are several remarkable points, defined for polygons and multisets of points in the plane, called centers (such as the centroid). To make possible their study, there exists a formal definition for the concept of center in both cases. In this paper, the relation between symmetries of polygons and multisets of points in the plane and the coincidence and collinearity of their centers is studied. First, a precise statement for the problem is given. Then, it is proved that, given a polygon or a multiset of points in the plane, a given point in the plane is a center for this object if and only if it belongs to the set of points fixed by its group of symmetries.

In this paper we provide a review of the concept of center of a $n$-gon, generalizing the original idea given by C. Kimberling for triangles. We also generalize the concept of central line for $n$-gons for $n\geq 3$ and establish its basic properties.

In 1946, J. Rosenbaum proposed a family of problems asking how many power points are needed to ensure that the boundary $c$ of a given convex body is a disk. In this paper, we use Riordan matrices to show that, if this curve $c$ is analytic, then one single exterior power point is sufficient., Comment: There is a mistake in the proof of the main theorem that spoils the strategy for showing uniqueness. At this moment I am not able to ensure the vality of the main theorem