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The Brasselet–Schürmann–Yokura Conjecture on L-Classes of Projective Rational Homology Manifolds
No full textIn 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson -class and the Hirzebruch homology class for a compact complex algebraic variety that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest
Intertwining operators beyond the Stark Effect
No full textThe main mathematical manifestation of the Stark effect in quantum mechanics is the shift and the formation of clusters of eigenvalues when a spherical Hamiltonian is perturbed by lower order terms. Understanding this mechanism turned out to be fundamental in the description of the large-time asymptotics of the associated Schrödinger groups and can be responsible for the lack of dispersion in Fanelli, Felli, Fontelos and Primo [Comm. Math. Phys., 324(2013), 1033-1067; 337(2015), 1515-1533]. Recently, Miao, Su, and Zheng introduced in [Tran. Amer. Math. Soc., 376(2023), 1739--1797] a family of spectrally projected intertwining operators, reminiscent of the Kato's wave operators, in the case of constant perturbations on the sphere (inverse-square potential), and also proved their boundedness in Lp. Our aim is to establish a general framework in which some suitable intertwining operators can be defined also for non constant spherical perturbations in space dimensions 2 and higher. In addition, we investigate the mapping properties between Lp-spaces of these operators. In 2D, we prove a complete result, for the Schrödinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and Lp-bounds of Bochner--Riesz means. In higher dimensions, apart from recovering the example of inverse-square potential, we can conjecture a complete result in presence of some symmetries (zonal potentials), and open some interesting spectral problems concerning the asymptotics of eigenfunctions
Error bounds for Physics Informed Neural Networks in Nonlinear Schrödinger equations placed on unbounded domains
No full textWe consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a new PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided a reasonable integration scheme is available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well
A revisited branch-and-cut algorithm for large-scale orienteering problems
Get PDFThe orienteering problem is a route optimization problem which consists of finding a simple cycle that maximizes the total collected profit subject to a maximum distance limitation. In the last few decades, the occurrence of this problem in real-life applications has boosted the development of many heuristic algorithms to solve it. However, during the same period, not much research has been devoted to the field of exact algorithms for the orienteering problem. The aim of this work is to develop an exact method which is able to obtain the optimum in a wider set of instances than with previous methods, or to improve the lower and upper bounds in its disability. We propose a revisited version of the branch-and-cut algorithm for the orienteering problem which includes new contributions in the separation algorithms of inequalities stemming from the cycle problem, in the separation loop, in the variables pricing, and in the calculation of the lower and upper bounds of the problem. Our proposal is compared to three state-of-the-art algorithms on 258 benchmark instances with up to 7397 nodes. The computational experiments show the relevance of the designed components where 18 new optima, 76 new best-known solutions and 85 new upper-bound values were obtained
Linear combinations of i.i.d. strictly stable variables with random coefficients and their application to anomalous diffusion processes
No full textWe show that linear combinations of independent and identically distributed strictly stable
variables with positive random coefficients is equal in distribution to a function of these random
coefficients times a random variable from the same stable distribution. Furthermore, this result
is used to show that a random linear combination of independent standard Wiener processes has
the distribution of a function of these random coefficients times one standard Wiener process.
In opposition to the central limit theorem, this result does not require a large number of terms
but it holds with two or more terms. This has implications to simplify stochastic differential
equations with a finite number of noises with random coefficients that can be used in modeling
anomalous diffusion
Symmetry in a multi-strain epidemiological model with distributed delay as a general cross-protection period and disease enhancement factor
Get PDFImportant biological features of viral infectious diseases caused by multiple agents with interacting strain dynamics continue to pose challenges for mathematical modelling development. Motivated by dengue fever epidemiology, we study a system of Integro-Differential Equations (IDE) considering strain structure of pathogens. Knowing that complex dynamics observed in dengue models are driven by the combination of two biological features, the temporary cross-immunity (TCI) and disease enhancement via the antibody-dependent enhancement process (ADE), our IDE system incorporates the TCI with a general time delay term, and the ADE effect by a constant factor to differentiate the susceptibility of individuals experiencing a primary or a secondary infection. Aiming at analysing the effect of the symmetry on dengue serotypes in the IDE framework, a detailed qualitative analysis of the model is performed and the instability of the coexistence steady state is shown using the perturbation theory approach. Numerical simulations identify the bifurcation structures and confirm the stability analysis. Results for the symmetric and asymmetric models are discussed
Reducing spatial discretization error on coarse CFD simulations using an openFOAM-embedded deep learning framework
Get PDFWe propose a method for reducing the spatial discretization error of coarse computational fluid dynamics (CFD) problems by enhancing the quality of low-resolution simulations using deep learning. We feed the model with fine-grid data after projecting it to the coarse-grid discretization. We substitute the default differencing scheme for the convection term by a feed-forward neural network that interpolates velocities from cell centers to face values to produce velocities that approximate the down-sampled fine-grid data well. The deep learning framework incorporates the open-source CFD code OpenFOAM, resulting in an end-to-end differentiable model. We automatically differentiate the CFD physics using a discrete adjoint code version. We present a fast communication method between TensorFlow (Python) and OpenFOAM (c++) that accelerates the training process. We applied the model to the flow past a square cylinder problem, reducing the error from 120% to 25% in the velocity for simulations inside the training distribution compared to the traditional solver using an x8 coarser mesh. For simulations outside the training distribution, the error reduction in the velocities was about 50%. The training is affordable in terms of time and data samples since the architecture exploits the local features of the physics.PID2023-146678OB-I00
PRE2020-09309
ON PAULI PAIRS AND FOURIER UNIQUENESS PROBLEMS
Get PDFWe investigate the concept of Pauli pairs and a discrete counterpart to it. In partic- ular, we make substantial progress on the question of when a discrete Pauli pair is automatically a classical Pauli pair.
Effectively, if one of the functions has space and frequency Gaussian decay, and one has that
?
p
|f| “ |g| and |f| “ |gp| on two sets which accumulate like suitable small multiples of
n at infinity, then |f| ” |g| and |f| “ |gp|. Furthermore, we show that if one drops either the assumption that one of the functions has space-frequency decay or that the discrete sets accumulate at a high
rate, then the desired property no longer holds.
Our techniques are inspired by and directly connected to several recent results in the realm
of Fourier uniqueness problems [42, 45, 48], and our results may be seen as a nonlinear gener- alization of those. As a consequence of said techniques, we are able to prove a sharp discrete version of Hardy’s uncertainty principle
Gaussian Mixture autoencoder for uncertainty-aware damage identification in a Floating Offshore Wind Turbine
Get PDFThis work proposes an uncertainty-aware approach to the inverse problem of damage identification in a Floating Offshore Wind Turbine (FOWT). We design an autoencoder architecture, where the latent space represents the features of the target damaged condition. The inverse operator (encoder) is a Deep Neural Network that maps the measurable response to the parameters (means, variances, and weights) of a multivariate Gaussian Mixture model. The Gaussian Mixture model provides a convenient distributional description that is flexible enough to accommodate complex solution spaces. The decoder receives samples from the Gaussian Mixture and maps the damaged condition (states) to the system’s measurable response. In such a problem, and depending on the quantities being observed (sensor positioning), it is possible that multiple damaged states may correspond to similar measurement records. In this context, the main contribution of this work lies in developing a method to quantify the uncertainty within the context of a possibly ill-posed damage identification problem. We employ the Gaussian Mixture to express the multimodal solution space and explain the uncertainty in the damaged condition estimates. We design and validate the methodology using synthetic data from a FOWT in the commonly adopted OpenFAST software and consider two damage types frequently occurring in mooring lines: biofouling and anchor displacement. The method allows for estimating the damaged state while capturing the uncertainty in the estimations and the multimodality of the solution under the availability of a limited number of response measurements.PID2023-146678OB-I0
Modelling swelling effects in real espresso extraction using a 1-dimensional coarse-grained model
Get PDFSwelling of coffee particle is difficult to measure and control, but it could have significant effects on espresso extraction. In this article, we incorporate a particle-level swelling model to a one dimensional bed-level extraction model to investigate the effects of swelling on extraction. Realistic geometric parameters and brewing parameters are used. A small degree of swelling is assumed to be present (≈3.6% in size). Our simulation results show that swelling only slightly affects the yield and the strength if the flow rate is fixed. However, when the pressure drop is fixed swelling will considerably enhance the strength at both fixed brewing time and fixed brewing mass. The finer the coffee particles the more pronounced the enhancement. By tuning the partition coefficient, the one-dimensional extraction model is shown to be able to capture very well the extraction kinetics of a fixing-flow-rate espresso machine in low beverage mass regime (Mc<30g). Our results suggest that better yield and strength control can be achieved by fixing the flow rate in an espresso machine. Moreover, it is indicated that when properly incorporated with relevant physical effects, numerical extraction simulations can be used to predict the extraction kinetics of espresso machine and guide their design and production