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In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of nonlocal elliptic systems: \begin{equation*} \left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u = \lambda|u|^{p - 2}u+ \frac{\alpha}{\alpha+\beta}\theta |u|^{\alpha - 2}u|v|^{\beta}, \;\;\; \mbox{in}\;\;\; \mathbb{R}^N, (-\Delta)^sv +V_2(x)v= \lambda|v|^{q - 2}v+ \frac{\beta}{\alpha+\beta}\theta |u|^{\alpha}|v|^{\beta-2}v, \;\;\; \mbox{in}\;\;\; \mathbb{R}^N, (u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N). \end{array}\right. \end{equation*} Here we mention that $\alpha > 1, \beta > 1, 1 \leq p \leq q < 2 < \alpha + \beta < 2^*_s$, $\theta > 0, \lambda > 0, N > 2s$, and $s \in (0,1)$. Notice also that continuous potentials $V_1, V_2: \mathbb{R}^N \to \mathbb{R}$ satisfy some extra assumptions. Furthermore, we find the largest positive number $\lambda^* > 0$ such that our main problem admits at least two positive solutions for each $ \lambda \in (0, \lambda^*)$. This can be done by using the nonlinear Rayleigh quotient together with the Nehari method. The main feature here is to minimize the energy functional in Nehari manifold which allows us to prove our main results without any restriction on size of parameter $\theta > 0$., Comment: In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of nonlocal elliptic systems

In the present work, we consider existence and multiplicity of positive solutions for nonlocal elliptic problems driven by the Stein-Weiss problem with concave-convex nonlinearities defined in the whole space $\mathbb{R}^N$. More precisely, we consider the following nonlocal elliptic problem: \begin{equation*} - \Delta u + V(x)u = \lambda a(x) |u|^{q-2} u + \displaystyle \int \limits_{\mathbb{R}^N}\frac{b(y)\vert u(y) \vert^p dy}{\vert x\vert^\alpha\vert x-y\vert^\mu \vert y\vert^\alpha} b(x)\vert u\vert^{p-2}u, \,\, \hbox{in}\ \mathbb{R}^N, \,\, u\in H^1(\mathbb{R}^N), \end{equation*} where $\lambda >0, \alpha \in (0,N), N\geq3, 0<\mu 0$ in $\mathbb{R}^N$ and $b\in{L}^{t}(\mathbb{R}^N), b>0$ in $\mathbb{R}^N$ for some specific $r, t > 1$. We assume also that $1\leq q<2$ and $2_{\alpha,\mu} < p<2_{\alpha,\mu}^*$ where $2_{\alpha ,\mu}=(2N-2\alpha-\mu)/N$ and $2_{\alpha,\mu}^*= (2N-2\alpha-\mu)/(N-2)$. Our main contribution is to find the largest $\lambda^* > 0$ in such way that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda^*)$. In order to do that we apply the nonlinear Rayleigh quotient together with the Nehari method. Moreover, we prove a Brezis-Lieb type Lemma and a regularity result taking into account our setting due to the potentials $a, b : \mathbb{R}^N \to \mathbb{R}$., Comment: In the present work, we consider existence and multiplicity of positive solutions for nonlocal elliptic problems driven by the Stein-Weiss problem with concave-convex nonlinearities defined in the whole space $\mathbb{R}^N$

In this work we study how to extend a partial action of a Hopf Algebra $A$ on an algebra $R$ to a partial action of a Hopf-Ore extension of $A$ on $R$. As consequence, we characterize all partial actions of rank one Hopf algebras (in particular, generalized Taft algebras and Radford algebras), under suitable conditions., Comment: 27 pages. The second and third author were partially supported by FAPERGS (Brazil), projects n. 23/2551-0000803-3 and 23/2551-0000913-7, respectively

It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems $$ \left\{ \begin{array}{lr} -\Delta_\Phi u = \lambda a(x) |u|^{q-2}u + |u|^{p-2}u, & x\in\Omega, u = 0, & x \in \partial \Omega, \end{array} \right. $$ where $\Omega \subset \mathbb{R}^N, N \geq 2,$ is a smooth bounded domain, $1 < q < \ell \leq m < p < \ell^*$ and $\Phi: \mathbb{R} \to \mathbb{R}$ is suitable $N$-function. The main feature here is to show whether the Nehari method can be applied to find the largest positive number $\lambda^* > 0$ in such way that our main problem admits at least two distinct solutions for each $\lambda \in (0, \lambda^*)$. Furthermore, using some fine estimates and some extra assumptions on $\Phi$, we prove the existence of at least two positive solutions for $\lambda = \lambda^*$ and $\lambda \in (\lambda^*, \overline{\lambda})$ where $\overline{\lambda} > \lambda^*$.

It is established ground states and multiplicity of solutions for a nonlocal Schr\"{o}dinger equation $(-\Delta )^s u + V(x) u = \lambda a(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $00,$ under general conditions over the measurable functions $a,$ $b$, $V$ and $f.$ The nonlinearity $f$ is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights $a$ and $b$ are not necessarily bounded and the potential $V$ can change sign. We obtained a sharp $\lambda^*> 0$ which guarantees the existence of at least two nontrivial solutions for each $\lambda \in (0, \lambda^*)$. Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered.

We study the spectral properties and eigenvector statistics of the Laplacian on highly-connected networks with random coupling strengths and a gamma distribution of rescaled degrees. The spectral density, the distribution of the local density of states, the singularity spectrum and the multifractal exponents of this model exhibit a rich behaviour as a function of the first two moments of the coupling strengths and the variance of the rescaled degrees. In the case of random coupling strengths, the spectral density diverges within the bulk of the spectrum when degree fluctuations are strong enough. The emergence of this singular behaviour marks a transition from non-ergodic delocalized states to localized eigenvectors that exhibit pronounced multifractal scaling. For constant coupling strengths, the bulk of the spectrum is characterized by a regular spectral density. In this case, the corresponding eigenvectors display localization properties reminiscent of the critical point of the Anderson localization transition on random graphs., Comment: 34 pages, 9 figures

Unmagnetized cold plasma modes are investigated in the context of the chiral Maxwell-Carroll-Field-Jackiw (MCJF) electrodynamics, where the axion chiral factor acts retrieving some typical properties of magnetized plasmas. The Maxwell equations are rewritten for a cold, uniform, and collisionless fluid plasma model, allowing us to determine the dispersion relation, new refractive indices and propagating modes. We find four distinct refractive indices modified by the purely timelike CFJ background that plays the magnetic conductivity chiral parameter role, associated with right-circularly polarized [RCP] and left-circularly polarized [LCP] waves. For each refractive index, the propagation and absorption zones are determined and illustrated for some specific parameter values. Modified RCP and LCP helicons are found in the low-frequency regime. The optical behavior is investigated, revealing that the chiral factor induces birefringence, measured in terms of the rotatory power (RP). The dichroism coefficient is carried out for the absorbing zones. The negative refraction zones may enhance the involved rotatory power, yielding RP sign reversion, a feature of rotating plasmas and MCFJ chiral plasmas., Comment: 14 pages, 15 figures

In this work, we investigate the reflection properties on the interface between an ordinary dielectric medium and a dielectric supporting a magnetic current (equivalent to a dielectric governed by axion electrodynamics). Considering the usual Maxwell equations and constitutive relations, we derive the general Fresnel coefficients for reflection for an incident wave with $s$ and $p$ polarization, assuming an isotropic magnetic current on a dielectric substrate. We determine all total internal reflection and critical angles (Brewster angles) conditions, which are given by strict relations between all relevant electromagnetic quantities of the system and the frequency. For $s$- and $p$-polarized incident waves, total internal reflection can occur under certain conditions on the constitutive parameters (for each propagating mode) at specific frequency windows and certain incidence angle intervals. All possible conditions to define critical angles for null reflection are determined. This scenario allows polarization changes by reflection. Considering a $p$-polarized incident wave, the frequency and the Brewster angle, allowing null reflection for both propagating modes, are determined. The Goos-H\"anchen shift and the complex Kerr rotation are also evaluated. The Kerr ellipticity angle presents a frequency-dependent behavior, also reported in Weyl semimetals, having maximum values $\eta_{Kerr}=\pm \pi/4$ for specific values of frequency, which may work as a signature of this axion chiral dielectric., Comment: 15 pages, 16 figures, 5 tables

Multi-controlled gates are fundamental components in the design of quantum algorithms, where efficient decompositions of these operators can enhance algorithm performance. The best asymptotic decomposition of an n-controlled X gate with one borrowed ancilla into single qubit and CNOT gates produces circuits with degree 3 polylogarithmic depth and employs a divide-and-conquer strategy. In this paper, we reduce the number of recursive calls in the divide-and-conquer algorithm and decrease the depth of n-controlled X gate decomposition to a degree of 2.799 polylogarithmic depth. With this optimized decomposition, we also reduce the depth of n-controlled SU(2) gates and approximate n-controlled U(2) gates. Decompositions described in this work achieve the lowest asymptotic depth reported in the literature. We also perform an optimization in the base of the recursive approach. Starting at 52 control qubits, the proposed n-controlled X gate with one borrowed ancilla has the shortest circuit depth in the literature. One can reproduce all the results with the freely available open-source code provided in a public repository., Comment: 6 pages, 8 figures

We discuss how classical electromagnetic techniques are useful to describe optical effects in conventional and chiral dielectric systems endowed with optical activity. Starting from the Maxwell equations and constitutive relations of the medium, we obtain the wave equation (for the electric field) that yields the refractive indices and the corresponding propagating modes that define the electromagnetic wave polarization. This procedure is employed for bi-isotropic and bi-anisotropic dielectrics, allowing us to determine birefringence, a manifestation of the space or time inversion breaking, typical of the chiral media. Such a method can also be applied for axion electrodynamics and a dielectric supporting an isotropic chiral magnetic current, similar to the Chiral Magnetic Effect (CME). For a diagonal magnetic conductivity, two distinct refractive indices are found, associated with left circularly polarized (LCP) and right circularly polarized (RCP) waves, and yielding circular birefringence (evaluated in terms of the optical rotatory power). Additionally, the scenario of a bi-isotropic medium endowed with isotropic magnetic current is examined, where a rotatory power marked by sign reversal may work as an optical signature of this particular system., Comment: 15 pages, 4 figures

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. However, most findings were obtained using a case-by-case approach, and it is often unclear what are consequences of the specificities of the ambient manifold and what could be generic. In this work, we propose a theoretical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices; we prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux vector field of the directrix. We characterize rectifying surfaces of constant angle; in other words, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, then the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction., Comment: 27 pages. Keywords: Helix, Slant helix, Surface of constant angle, Rectifying surface

In this work, we study the propagation and absorption of plasma waves in the context of the Maxwell-Carroll-Field-Jackiw (MCFJ) electrodynamics with a purely spacelike background playing the role of the anomalous Hall conductivity, concerning the anomalous Hall current. Such a current is also found in an axion field which increases linearly with a space coordinate. The Maxwell equations are rewritten for a cold, uniform, and collisionless fluid plasma model, allowing us to determine the new refractive indices and propagating modes. The analysis begins for propagation along the magnetic axis, examined in the cases of chiral vectors parallel and orthogonal to the magnetic field. Two distinct refractive indices (associated with right-handed circularly polarized [RCP] and left-handed circularly polarized [LCP] waves) are obtained and the associated propagation and absorption zones are determined. The low-frequency regime is discussed and we obtain RCP and LCP helicons. We scrutinize optical effects, such as birefringence and dichroism, and observe rotatory power sign reversion, a property of chiral MCFJ plasmas. We also examine the case of transversal propagation in the direction orthogonal to the magnetic field, providing much more involved results., Comment: 21 pages, 18 figures, 2 tables

The dispersive propagation and absorption in the domain of a Drude-Lorentz dielectric modified by a magnetic current are classically investigated, taking as the starting point the Drude electric permittivity, in connection with the chiral magnetic effect. Using standard electromagnetic methods, the dispersion relations and refractive indices are evaluated and discussed, as well as the group and the energy velocities. We consider three configurations for the magnetic conductivity tensor: isotropic, antisymmetric, and symmetric. In all these cases, we carry out the group velocity and the energy velocity, for investigating the properties of signal propagation. We observe that the isotropic magnetic conductivity enhances the signal propagation at low and intermediary frequencies, while the symmetric and antisymmetric conductivity severely constrains the wave propagation in these frequency windows., Comment: 16 pages, two columns, 13 figures

We provide a method for compiling approximate multi-controlled single qubit gates into quantum circuits without ancilla qubits. The total number of elementary gates to decompose an n-qubit multi-controlled gate is proportional to 32n, and the previous best approximate approach without auxiliary qubits requires 32nk elementary operations, where k is a function that depends on the error threshold. The proposed decomposition depends on an optimization technique that minimizes the CNOT gate count for multi-target and multi-controlled CNOT and SU(2) gates. Computational experiments show the reduction in the number of CNOT gates to apply multi-controlled U(2) gates. As multi-controlled single-qubit gates serve as fundamental components of quantum algorithms, the proposed decomposition offers a comprehensive solution that can significantly decrease the count of elementary operations employed in quantum computing applications.

We investigate rank $3$ instanton vector bundles on $\mathbb{P}^3$ of charge $n$ and its correspondence with rational curves of degree $n+3$. For $n=2$ we present a correspondence between stable rank $3$ instanton bundles and stable rank $2$ reflexive linear sheaves of Chern classes $(c_1,c_2,c_3)=(-1,3,3)$ and we use this correspondence to compute the dimension of the family of stable rank $3$ instanton bundles of charge $2$. Finally, we use the results above to prove that the moduli space of rank $3$ instanton bundles on $\mathbb{P}^3$ of charge $2$ coincides with the moduli space of rank $3$ stable locally free sheaves on $\mathbb{P}^3$ of Chern classes $(c_1,c_2,c_3)=(0,2,0)$. This moduli space is irreducible, has dimension 16 and its generic point corresponds to a \textcolor{black}{generalized} t`Hooft instanton bundle., Comment: 15 pages

We examine a dielectric medium supporting a magnetic current in connection with optical properties and energy propagation. The dispersion relations, propagating modes, and some optical effects are examined for isotropic and anisotropic magnetic conductivity tensors, with the latter ones implying nonreciprocal permittivities. The eigenvalues of the effective permittivity are carried out and associated with the optical symmetries (uniaxial, biaxial) and the subjacent crystal systems. Aspects of electromagnetic energy transport for such systems are also discussed, with the group and energy velocities being presented and carried out for all the particular cases addressed. Our results suggest that a dielectric supporting magnetic current breaks the usual equivalence between group velocity and energy velocity that holds in a nonabsorbing medium, while establishing the equivalence between them in an effective absorbing scenario, two unexpected behaviors., Comment: 23 pages, 2 columns, 10 figures, 3 tables

We analysed the nuclear region of all 56 early-type galaxies from the DIVING$^\mathrm{3D}$ Project, which is a statistically complete sample of objects that contains all 170 galaxies of the Southern Hemisphere with B < 12.0 mag and galactic latitude |b| < 15$^{\circ}$. Observations were performed with the Integral Field Unit of the Gemini Multi-Object Spectrograph. Emission lines were detected in the nucleus of 86$\pm$5% of the objects. Diagnostic diagrams were used to classify 52$\pm$7% of the objects as LINERs or Seyferts, while the other 34$\pm$6% galaxies without H$\beta$ or [O III] lines in their spectra were classified as weak emission line objects. Transition Objects are not seen in the sample, possibly because the seeing-limited data cubes of the objects allow one to isolate the nuclei of the galaxies from their circumnuclear regions, avoiding contamination from H II regions. A broad line region is seen in 29$\pm$6% of the galaxies. Of the 48 galaxies with emission-line nuclei, 41 have signs of AGNs. Some objects have also indications of shocks in their nuclei. Lenticular galaxies are more likely to have emission lines than ellipticals. Also, more luminous objects have higher [N II]/H$\alpha$ ratios, which may be associated with the mass-metalicity relation of galaxies. A direct comparison of our results with the Palomar Survey indicates that the detection rates of emission lines and also of type 1 AGNs are higher in the DIVING$^\mathrm{3D}$ objects. This is a consequence of using a more modern instrument with a better spatial resolution than the Palomar Survey observations., Comment: 16 pages (plus appendices), 6 figures, 9 tables, accepted for publication in MNRAS

In this work, we study the propagation and absorption of plasma waves in the chiral Maxwell-Carroll-Field-Jackiw (MCJF) electrodynamics. The Maxwell equations are rewritten for a cold, uniform, and collisionless fluid plasma model, allowing us to determine the new refractive indices and propagating modes. The cases of propagation parallel and orthogonal to the magnetic field are examined considering a purely timelike CFJ background that plays the role of the magnetic conductivity chiral parameter. The collective electromagnetic modes are associated with four distinct refractive indices associated with right-circularly polarized and left-circularly polarized waves. For each index, the propagation and absorption zones are illustrated for some specific parameter values. In low-frequency regime, we have obtained modified helicons with right- and left-circularly polarizations. The optical behavior is investigated by means of the rotatory power (RP) and dichroism coefficient. The existence of a negative refraction zone enhances the rotatory power. It is also observed RP sign reversal, a feature of rotating plasmas., Comment: revtex style, two columns, 19 figures, 19 pages, version published in the PRD

Artistic pieces can be studied from several perspectives, one example being their reception among readers over time. In the present work, we approach this interesting topic from the standpoint of literary works, particularly assessing the task of predicting whether a book will become a best seller. Dissimilarly from previous approaches, we focused on the full content of books and considered visualization and classification tasks. We employed visualization for the preliminary exploration of the data structure and properties, involving SemAxis and linear discriminant analyses. Then, to obtain quantitative and more objective results, we employed various classifiers. Such approaches were used along with a dataset containing (i) books published from 1895 to 1924 and consecrated as best sellers by the Publishers Weekly Bestseller Lists and (ii) literary works published in the same period but not being mentioned in that list. Our comparison of methods revealed that the best-achieved result - combining a bag-of-words representation with a logistic regression classifier - led to an average accuracy of 0.75 both for the leave-one-out and 10-fold cross-validations. Such an outcome suggests that it is unfeasible to predict the success of books with high accuracy using only the full content of the texts. Nevertheless, our findings provide insights into the factors leading to the relative success of a literary work.