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In this manuscript, the decoherence dynamics and spectral response of an optomechanical system, with linear and quadratic couplings, is addressed. The decoherence considered arises from pure dephasing, described by the Milburn stochastic evolution of the Schr\"odinger equation. In the first part of the manuscript, it is shown how the decoherence rate influences the evolution of the number of phonons, and the quadrature of the mechanical resonator. In the second part of the manuscript, an attempt to look at the spectral response of the mechanical part of the system is given using non-stationary spectroscopy. It is emphasized the response of the resonator in its equilibrium position when a single photon excitation in the cavity field is prepared. Coherent states are also considered in the cavity field and the mechanical resonator. Results and discussion comparing the inclusions of the linear, quadratic, and linear-quadratic couplings are given., Comment: 13 pages, 6 figures

In this work, we make use of Lie algebraic methods to obtain the time evolution operator for an optomechanical system with linear and quadratic couplings between the field and the mechanical oscillator. Firstly, we consider the case of a non-driven system and find its exact time evolution operator, secondly we consider the case of a forced system whose time evolution operator is obtained in an approximate form. We confront our analytical results with a numerical simulation and find a good agreement between them.

We classify all projective surfaces with only T-singularities, $K^2=2p_g-4$, and ample canonical class. In this way, we identify all surfaces, smoothable or not, with only T-singularities in the Koll\'ar--Shepherd-Barron--Alexeev (KSBA) moduli space of Horikawa surfaces. We also prove that they are not smoothable when $p_g \geq 10$, except for the Lee-Park (Fintushel-Stern) examples, which we show to have only one deformation type unless $p_g=6$ (in which case they have two). This demonstrates that the challenging Horikawa problem cannot be addressed through complex T-degenerations, and we propose new questions regarding diffeomorphism types based on our classification. Furthermore, the techniques developed in this paper enable us to classify all KSBA surfaces with only T-singularities and $K^2=2p_g-3$, e.g. quintic surfaces and I-surfaces.

The Jaynes-Cummings quantum optics model allows us to understand the dialogue between light and matter at its most fundamental level, which is crucial for advancements in quantum science and technology. Several generalizations of the model have long been proposed, emphasizing their dynamic behavior but paying less attention to their spectroscopy. Here, we obtain analytical expressions of the time-dependent spectral response of a nonlinear Jaynes-Cummings model based on deformed field operators. We show that the long-time response of the resulting nonlinear cavity field resembles the one experimentally obtained in the strong-dispersive regime of circuit quantum electrodynamics. The spectrum is intrinsically asymmetric with the nonlinear coupling, a signature of the impossibility of getting resonant conditions for finite field excitations., Comment: 14 pages, 7 figures

Within the expansive domain of optical sciences, achieving the precise characterization of light beams stands as a fundamental pursuit, pivotal for various applications, including telecommunications and imaging technologies. This study introduces an innovative methodology aimed at reconstructing the Wigner distribution function of optical signals; a crucial tool in comprehending the time-frequency behavior exhibited by these signals. The proposed approach integrates two robust mathematical tools: the discrete realization of fractional Fourier transform, and the propagator of the quantum harmonic oscillator in waveguide arrays. This integration offers a direct and efficient method for characterizing optical signals by reconstructing their Wigner distribution function in the scope of integrated optics. We provide evidence of how having knowledge of the signal propagation amid the phase-space reconstruction, has desirable advantages in respect to only knowing the signal state., Comment: 10 pages, 7 figures

This paper has been written to illustrate the power of techniques from tropical geometry and mirror symmetry for studying the KSBA moduli space of surfaces on or near the Noether line. We focus on the moduli space of octic double planes ($K^2 = 2$, $p_g = 3$) and use methods from tropical and toric geometry to classify the strata corresponding to normal KSBA-stable surfaces, focusing on the non-Gorenstein case., Comment: 46 pages, 19 figures; v2 results strengthened, exposition streamlined, title made more specific

In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $\operatorname{V}$ for the wave equation as a minimization problem in $L^2(\Sigma)$, where $\Sigma := \partial \Omega \times (0,T)$ is the lateral boundary of the space-time domain $Q := \Omega \times (0,T)$. For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and, moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.

We show an example of an embedded copy of 5RP^2 in the four-sphere which is topologically standard but smoothly knotted, i.e. smoothly not isotopic to the standard embedding., Comment: 18 pages, 3 figures. Mistake in the construction from previous version has been fixed, two new authors have been added

It is known that all degenerations of the complex projective plane into a surface with only quotient singularities are controlled by the positive integer solutions $(a,b,c)$ of the Markov equation $$x^2+y^2+z^2=3xyz.$$ It turns out that these degenerations are all connected through finite sequences of other simpler degenerations by means of birational geometry. In this paper, we explicitly describe these birational sequences and show how they are bridged among all Markov solutions. For a given Markov triple $(a,b,c)$, the number of birational modifications depends on the number of branches that it needs to cross in the Markov tree to reach the Fibonacci branch. We show that each of these branches corresponds exactly to a Mori train of the flipping universal family of a particular cyclic quotient singularity defined by $(a,b,c)$. As a byproduct, we obtain new numerical/combinatorial data for each Markov number, and new connections with the Markov conjecture (Frobenius Uniqueness Conjecture), which rely on Hirzebruch-Jung continued fractions of Wahl singularities.

We obtain a time-evolution operator for a forced optomechanical quantum system using Lie algebraic methods when the normalized coupling between the electromagnetic field and a mechanical oscillator, $G/\omega_m$, is not negligible compared to one. Due to the forcing term, the interaction picture Hamiltonian contains the number operator in the exponents, and in order to deal with it, we approximate these exponentials by their average values taken between initial coherent states. Our approximation is justified when we compare our results with the numerical solution of the number of photons, phonons, Mandel parameter, and the Wigner function, showing an excellent agreement., Comment: 25 pages, 11 figures. Peer-reviewed version

We show some geometric properties of Enriques surfaces via $\mathbb Q$-Gorenstein smoothings of Coble surfaces. In particular, we explicitly identify the Enriques lattice of the general fiber with the Coble-Mukai lattice. At the end, we discuss applications to Gorenstein $\mathbb Q$-Homology projective planes with trivial canonical class., Comment: Submitted to the proceedings of the Moduli and Hodge Theory 2021 conference at the Institute of the Mathematical Sciences of the Americas

We effectively bound T-singularities on non-rational projective surfaces with an arbitrary amount of T-singularities and ample canonical class. This fully generalizes the previous work for the case of one singularity, and illustrates the vast increase in combinatorial complexity as the number of singularities grows. We find that certain combinatorial configurations lead to relatively high bounds. We classify all such configurations, and show that their non-existence gives a strong and optimal bound. As an application, we work out in detail the case of two singularities.

We study the decaying dynamics in the mirror-field interaction by means of the intrinsic decoherence scheme. Factorization of the mirror-field Hamiltonian with the use of displacement operators, allows us to calculate the explicit solution to Milburn's equation for arbitrary initial conditions. We show expectation values, correlations, and Husimi functions for the solutions obtained., Comment: 16 pages, 6 figures

We propose a preconditioner for the Helmholtz exterior problems on multi-screens. For this, we combine quotient-space BEM and operator preconditioning. For a class of multi-screens (which we dub \emph{type A} multi-screens), we show that this approach leads to block diagonal Calder\'on preconditioners and results in a spectral condition number that grows only logarithmically with $h$, just as in the case of simple screens. Since the resulting scheme contains many more DoFs than strictly required, we also present strategies to remove almost all redundancy without significant loss of effectiveness of the preconditioner. We verify these findings by providing representative numerical results. Further numerical experiments suggest that these results can be extended beyond type A multi-screens and that the numerical method introduced here can be applied to essentially all multi-screens encountered by the practitioner, leading to a significantly reduced simulation cost.

We construct the first exotic $\mathbb C \mathbb P^2 \# 4 \overline{\mathbb C \mathbb P^2}$ by means of rational blowdowns. Similarly, we construct the first exotic $3\mathbb C \mathbb P^2 \# b^- \overline{\mathbb C \mathbb P^2}$ for $b^-=9,8,7$ . All of them are minimal and symplectic, as they are produced from projective surfaces $W$ with Wahl singularities, and $K_W$ big and nef. In more generality, we elaborate on the problem of finding exotic $(2\chi(\mathcal{O}_W)-1) \mathbb C \mathbb P^2 \# (10\chi(\mathcal{O}_W)-K^2_W-1) \overline{\mathbb C \mathbb P^2}$ from these Koll\'ar--Shepherd-Barron--Alexeev surfaces $W$, obtaining explicit geometric obstructions.

Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su($3$) algebra of Cartesian three-dimensional oscillator multiplet states into so($3$) angular momentum submultiplets, we implement unitary rotations of three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information., Comment: 19 pages, 5 figures, 3 tables

It is well known that a $2$-dimensional cyclic quotient singularity $\overline{W}$ has the same singularity category as a finite dimensional associative algebra $\overline{R}$ introduced by Kalck and Karmazyn. We study the deformations of the algebra $\overline{R}$ induced by the deformations of the surface $\overline{W}$ to a smooth surface. We show that they are Morita--equivalent to path algebras $\hat{R}$ of acyclic quivers for general smoothings within each irreducible component of the versal deformation space of $\overline{W}$ (as described by Koll\'ar and Shepherd-Barron). Furthermore, $\hat{R}$ is semi-simple if and only if the smoothing is $\mathbb{Q}$-Gorenstein (one direction is due to Kawamata). We provide many applications. For example, we describe strong exceptional collections of length $10$ on all Dolgachev surfaces and classify admissible embeddings of derived categories of quivers into derived categories of rational surfaces.

Unitary rotations of polychromatic images on finite two-dimensional pixellated screens provide invertibility, group composition, and thus conservation of information. Rotations have been applied on monochromatic image data sets, where we now examine closer the Gibbs-like oscillations that appear due to discrete "discontinuities" of the input images under unitary transformations. Extended to three-color images we examine here the display of color at the pixels where, due to the oscillations, some pixel color values may fall outside their required common numerical range [0, 1], between absence and saturation of the red, green, and blue formant color images., Comment: 11 pages, 4 figures

Under the most common experimental conditions, the adsorption of proteins to solid surfaces is an spontaneous process that leads to a rather compact layer of randomly oriented molecules. Due to the importance of this process for the development of catalytic surfaces, a number of existing computational and experimental approaches try to predict and control the orientation of such molecules. However, and despite their own advantages, these tend to be either too expensive computationally, or oversimplified, undermining their ability to predict the most appropriate experimental conditions to maximize the catalytic activity of adsorbed proteins. To address this current need, we present an efficient computational approach to model the behavior of proteins near surfaces in the presence of an external electric field, based on continuum electrostatics. Our model can not only estimate the overall affinity of the protein with the surface, but also their most likely orientation as a function of the potential applied. In this way, a rational selection of the potential can be performed to maximize the accessibility of the protein's active site to the solvent. The model relies on the Poisson-Boltzmann equation and was implemented in an extension of the code PyGBe that includes an external electric field, and renders the electrostatic component of the solvation free energy. To demonstrate the feasibility of this technique, we investigate the adsorption of trypsin onto a carbon electrode under potentiostatic conditions both numerically and experimentally. We found that even though the adsorption process is largely dominated by hydrophobic effects, the orientation of trypsin can be controlled through an external potential, influencing the position of the active sites, and resulting in an important change in the catalytic activity of the surface., Comment: 34 pages, 11 figures

By using the complete solution of the Milburn equation (beyond the Lindblad form that it is generally used) that describes intrinsic decoherence, we study the decaying dynamics of a displaced harmonic oscillator. We calculate the expectation values of position quadrature, and the number operator in initial coherent and squeezed states.

We prove the existence of $(20-2K^2)$-dimensional families of simply-connected surfaces with ample canonical class, $p_g=1$, and $1 \leq K^2 \leq 9$, and we study the relation with configurations of rational curves in K3 surfaces via $\mathbb Q$-Gorenstein smoothings. Our surfaces with $K^2=7$ and $K^2=9$ are the first surfaces known in the literature, together with the existence of a $4$-dimensional family for $K^2=8$.

Applying the Milburn equation to describe intrinsic decoherence, we study the interaction of three-coupled quantum harmonic oscillators or quantized fields. We give an explicit solution for the complete equation, i.e., beyond the usual second order approximation used to arrive to the Lindblad form. Then we calculate the expectation value of the number operator of each oscillator or mode for one of the modes given in an initial coherent, Comment: 10 pages, 2 figures

In this note, we discuss the ellipticity of the single layer boundary integral operator for the wave equation in one space dimension. This result not only generalizes the well-known ellipticity of the energetic boundary integral formulation in $L^2$, but it also turns out to be a particular case of a recent result on the inf-sup stability of boundary integral operators for the wave equation. Instead of the time derivative in the energetic formulation, we use a modified Hilbert transformation, which allows us to stay in Sobolev spaces of the same order. This results in the applicability of standard boundary element error estimates, which are confirmed by numerical results.

Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator equations with a large number of discrete variables. In this paper we show that the accuracy can be improved significantly if the equation is transformed before discretization, a process we call full operator preconditioning (FOP). It bears many similarities with traditional preconditioning for iterative methods but, crucially, transformations are applied at the operator level. We show that while condition-number improvements from traditional preconditioning generally do not improve the accuracy of the solution, FOP can. A number of topics in numerical analysis can be interpreted as implicitly employing FOP; we highlight (i) Chebyshev interpolation in polynomial approximation, and (ii) Olver-Townsend's spectral method, both of which produce solutions of dramatically improved accuracy over a naive problem formulation. In addition, we propose a FOP preconditioner based on integration for the solution of fourth-order differential equations with the finite-element method, showing the resulting linear system is well-conditioned regardless of the discretization size, and demonstrate its error-reduction capabilities on several examples. This work shows that FOP can improve accuracy beyond the standard limit for both direct and iterative methods.

We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise efficient adaptive space-time boundary element methods, which will be tackled in future work. On the other hand, the proposed approach is compatible with current time dependent boundary element method's implementations and we predict that it explains many of the behaviours observed in practice but that were not understood with the existing theory.