In this paper we consider the finite element approximation of Maxwell's problem and analyse the prescription of essential boundary conditions in a weak sense using Nitsche's method. To avoid indefiniteness of the problem, the original equations are augmented with the gradient of a scalar field that allows one to impose the zero divergence of the magnetic induction, even if the exact solution for this scalar field is zero. Two finite element approximations are considered, namely, one in which the approximation spaces are assumed to satisfy the appropriate inf-sup condition that render the standard Galerkin method stable, and another augmented and stabilised one that permits the use of finite element interpolations of arbitrary order. Stability and convergence results are provided for the two finite element formulations considered.
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We investigate the effect of the well-known Mycielski construction on the Shannon capacity of graphs and on one of its most prominent upper bounds, the (complementary) Lov\'asz theta number. We prove that if the Shannon capacity of a graph, the distinguishability graph of a noisy channel, is attained by some finite power, then its Mycielskian has strictly larger Shannon capacity than the graph itself. For the complementary Lov\'asz theta function we show that its value on the Mycielskian of a graph is completely determined by its value on the original graph, a phenomenon similar to the one discovered for the fractional chromatic number by Larsen, Propp and Ullman. We also consider the possibility of generalizing our results on the Sperner capacity of directed graphs and on the generalized Mycielsky construction. Possible connections with what Zuiddam calls the asymptotic spectrum of graphs are discussed as well.
Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps implies that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constant in the complex case.
In this paper we present a low-rank method for conforming multipatch discretizations of compressible linear elasticity problems using Isogeometric Analysis. The proposed technique is a non-trivial extension of [M. Montardini, G. Sangalli, and M. Tani. A low-rank isogeometric solver based on Tucker tensors. Comput. Methods Appl. Mech. Engrg., page 116472, 2023.] to multipatch geometries. We tackle the model problem using an overlapping Schwarz method, where the subdomains can be defined as unions of neighbouring patches. Then on each subdomain we approximate the blocks of the linear system matrix and of the right-hand side vector using Tucker matrices and Tucker vectors, respectively. We use the Truncated Preconditioned Conjugate Gradient as a linear solver, coupled with a suited preconditioner. The numerical experiments show the advantages of this approach in terms of memory storage. Moreover, the number of iterations is robust with respect to the relevant parameters.
In this paper we consider the numerical solution of fractional terminal value problems (FDE-TVPs). In particular, the proposed procedure uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems (FDE-IVPs), able to produce spectrally accurate solutions of FDE problems. Some numerical tests are reported to make evidence of its effectiveness.
In this paper we introduce a proposal to provide students in labs with an alternative to the traditional visible range spectrophotometers, whose acquisition and maintenance entails high costs, based on smartphones. Our solution faced two aspects. On the one hand, the software for the smartphone, able to perform the typical functionalities of the traditional spectrophotometers. On the other hand, the portable peripheral support needed to capture the images to be analyzed in the smartphone. The promising results allow this solution to be applied in Bring Your Own Devices (BYOD) contexts.
In this paper, we propose a general unified tracking-servoing approach for controlling the shape of elastic deformable objects using robotic arms. Our approach works by forming a lattice around the object, binding the object to the lattice, and tracking and servoing the lattice instead of the object. This makes our approach have full 3D control over deformable objects of any general form (linear, thin-shell, volumetric). Furthermore, it decouples the runtime complexity of the approach from the objects' geometric complexity. Our approach is based on the As-Rigid-As-Possible (ARAP) deformation model. It requires no mechanical parameter of the object to be known and can drive the object toward desired shapes through large deformations. The inputs to our approach are the point cloud of the object's surface in its rest shape and the point cloud captured by a 3D camera in each frame. Overall, our approach is more broadly applicable than existing approaches. We validate the efficiency of our approach through numerous experiments with deformable objects of various shapes and materials (paper, rubber, plastic, foam). Experiment videos are available on the project website: //sites.google.com/view/tracking-servoing-approach.
In this paper, we propose a generic approach to perform global sensitivity analysis (GSA) for compartmental models based on continuous-time Markov chains (CTMC). This approach enables a complete GSA for epidemic models, in which not only the effects of uncertain parameters such as epidemic parameters (transmission rate, mean sojourn duration in compartments) are quantified, but also those of intrinsic randomness and interactions between the two. The main step in our approach is to build a deterministic representation of the underlying continuous-time Markov chain by controlling the latent variables modeling intrinsic randomness. Then, model output can be written as a deterministic function of both uncertain parameters and controlled latent variables, so that it becomes possible to compute standard variance-based sensitivity indices, e.g. the so-called Sobol' indices. However, different simulation algorithms lead to different representations. We exhibit in this work three different representations for CTMC stochastic compartmental models and discuss the results obtained by implementing and comparing GSAs based on each of these representations on a SARS-CoV-2 epidemic model.
Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal $a$-$b$ separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P = NP.
In this paper, we present a rigorous analysis of root-exponential convergence of Hermite approximations, including projection and interpolation methods, for functions that are analytic in an infinite strip containing the real axis and satisfy certain restrictions on the asymptotic behavior at infinity within this strip. Asymptotically sharp error bounds in the weighted and maximum norms are derived. The key ingredients of our analysis are some remarkable contour integral representations for the Hermite coefficients and the remainder of Hermite spectral interpolations. Further extensions to Gauss--Hermite quadrature, Hermite spectral differentiations, generalized Hermite spectral approximations and the scaling factor of Hermite approximation are also discussed. Numerical experiments confirm our theoretical results.
This paper introduces the sequential composition of answer set programs. On the semantic side, we show that the immediate consequence operator of a program can be represented via composition, which allows us to compute the least model semantics of Horn programs without any explicit reference to operators. As a result, we can characterize answer sets algebraically, which provides an algebraic characterization of strong and uniform equivalence. In a broader sense, this paper is a first step towards an algebra of answer set programs and in the future we plan to lift the methods of this paper to wider classes of programs, most importantly to higher-order and disjunctive programs and extensions thereof.
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