Miracle-dz/UniFAR-FTUs huggingface dataset:
Specular reflection on the surface of a sphere: compass and ruler constructions

We provide an explicit geometric algorithm involving only ruler and compass constructions in order to specify the specular reflection point on the surface of a reflecting sphere of radius $r$ given two focal points $A$ and $B$ lying outside of it. By numerically implementing the algorithm we compute the point in question for a number of cases. We conclude by discussing how the first iteration of the algorithm constitutes a first order approximation to the real solution by providing a closed expression for it as well as the error involved in doing so, as a function of the distances of the two focal points from the origin and the angle formed between them.
Narratives About the Macroeconomy

We provide evidence on narratives about the macroeconomy—the stories people tell to explain macroeconomic phenomena—in the context of a historic surge in inflation. We measure economic narratives in open-ended survey responses and represent them as Directed Acyclic Graphs. We apply this approach in surveys with more than 8,000 U.S. households and 100 academic experts. We document three main findings. First, compared to experts, households’ narratives are coarser, focus less on the demand side, and are more likely to feature politically loaded explanations. Second, households’ narratives strongly shape their inflation expectations, which we demonstrate with descriptive survey data and a series of experiments. Third, an experiment varying news consumption shows that the media is an important source of narratives. Our findings demonstrate the relevance of narratives for understanding macroeconomic expectation formation. (JEL: D83, D84, E31, E52, E71)
Conforming mixed triangular prism and nonconforming mixed tetrahedral elements for the linear elasticity problem

We propose two families of mixed finite elements for solving the classical Hellinger-Reissner mixed problem of the linear elasticity equations in three dimensions. First, a family of conforming mixed triangular prism elements is constructed by product of elements on triangular meshes and elements in one dimension. The well-posedness is established for all elements with $k\geq1$, which are of $k+1$ order convergence for both the stress and displacement. Besides, a family of reduced stress spaces is proposed by dropping the degrees of polynomial functions associated with faces. As a result, the lowest order conforming mixed triangular prism element has 93 plus 33 degrees of freedom on each element. Second, we construct a new family of nonconforming mixed tetrahedral elements. The shape function spaces of our stress spaces are different from those of the elements in literature.
Asymptotic modeling of the JKR adhesion contact for a thin elastic layer

The Johnson--Kendall--Roberts model of frictionless adhesive contact is extended to the case of a thin transversely isotropic elastic layer bonded to a rigid base. The leading-order asymptotic models are obtained for both compressible and incompressible elastic layers. The boundary conditions for the contact pressure approximation on the contour of the contact area have been derived from the boundary-layer solutions, which satisfy the condition that the stress intensity factor of the contact pressure density should have the same value all round the contact area boundary. In the incompressible case, a perturbation solution is obtained for a slightly perturbed circular contact area.
Tectonics is a Hologram

A hologram is an image in which each area contains almost all the information about the entire system. It is a metaphor commonly used for complex systems in which the whole is bigger than the sum of the parts because of self-organization. And also the whole is smaller than the sum of the parts, since the collective organization limits the behavior of dynamic features. The tectonic evolution of the Earth is an emergent behavior of the lithospheremantle system, a witness of a program defined at the scale of rocks. Modeling the physics behind tectonics at a global scale became a reachable goal entering the 21st century. Geodynamicists developed numerical models of solid-sate convection with yielding, and reproduced some fundamentals of planetary tectonics. In the past 15 years, several groups in the world have used these models to investigate how continents drift, seafloor spreads and ∗Corresponding author. Email address: nicolas.coltice@ens.fr (Nicolas Coltice ) Published as Dynamics of Plate Tectonics and Mantle Convection November 4, 2022 plates evolve. These emergent characteristics tell that the whole is bigger than the sum of the parts. Slabs, plumes, ridges, plates are interdependent and constrain each other. The whole is smaller than the sum of the parts. In this context, searching for causality relationships between tectonic features seems vain. In this chapter, I consider this point of view to describe how convection models with yielding have changed and still change our views on how tectonics shape the Earth. I finally propose an outlook about this window that remains half-opened.
Euler observers in geometrodynamics

Euler observers are a fundamental tool for the study of spacetime evolution. Cauchy surfaces are evolved through the use of hypersurface orthogonal fields and their relationship to coordinate observers, that enable the use of already developed algorithms. In geometrodynamics new tetrad vectors have been introduced with outstanding simplifying properties. We are going to use these already introduced tetrad vectors in the case where we consider a curved four dimensional Lorentzian spacetime with the presence of electromagnetic fields. These Einstein-Maxwell geometries will provide the new tetrad that we are going to use in order to develop an algorithm to produce Cauchy evolution with additional simplifying properties.
Quantum Tomography twenty years later

We show a sample of some relevant developments in classical and quantum tomography that have taken place over the last twenty years. We will present a general conceptual framework that provides a simple unifying mathematical picture for them and, as an effective use of it, three subjects have been chosen that offer a wide panorama of the scope of classical and quantum tomography: tomography along lines and submanifolds, coherent state tomography and tomography in the abstract algebraic setting of quantum systems.
Heat flow in anharmonic crystals with internal and external stochastic baths: a convergent polymer expansion for a model with discrete time and long range interparticle interaction

We investigate a chain of oscillators with anharmonic on-site potentials, with long range interparticle interactions, and coupled both to external and internal stochastic thermal reservoirs of Ornstein–Uhlenbeck type. We develop an integral representation, a` la Feynman–Kac, for the correlations and the heat current. We assume the approximation of discrete times in the integral formalism (together with a simplification in a subdominant part of the harmonic interaction) and develop a suitable polymer expansion for the model. In the regime of strong anharmonicity, strong harmonic pinning, and for the interparticle interaction with integrable polynomial decay, we prove the convergence of the polymer expansion uniformly in volume (number of sites and time). We also show that the two-point correlation decays in space such as the interparticle interaction. The existence of a convergent polymer expansion is of practical interest: it establishes a rigorous support for a perturbative analysis of the heat flow problem and for the computation of the thermal conductivity in related anharmonic crystals, including those with inhomogeneous potentials and long range interparticle interactions. To show the usefulness and trustworthiness of our approach, we compute the thermal conductivity of a specific anharmonic chain, and make a comparison with related numerical results presented in the literature.
Top degree part in $b$-conjecture for unicellular bipartite maps

A BSTRACT . Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ ( x , y , z ; 1 , 1 + β ) with an additional parameter β that may be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has the property that for β ∈ { 0 , 1 } , it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefﬁcients of ψ are polynomials in β with positive integer coefﬁcients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of β is an integer-valued statistic that in some sense “measures the non-orientability” of the corresponding bipartite map. We show that except for two special values of β = 0 , 1 for which the combinatorial interpretation of the coefﬁcients of ψ is known, there exists a third special value β = − 1 for which the coefﬁcients of ψ indexed by two partitions µ, ν , and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black/white vertex degrees given by µ / ν , respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefﬁcients of ψ . As a consequence, we introduce a collection of integer-valued statistics of maps ( η ) such that the top-degree of the multivariate generating series of rooted bipartite maps with only one face (called unicellular ) with respect to η gives the top-degree of the appropriate coefﬁcients of ψ . Finally, we show that the b -conjecture holds true for all rooted, unicellular bipartite maps of genus at most 2 .
Volume of the space of qubit channels and some new results about the distribution of the quantum Dobrushin coefficient

The simplest building blocks for quantum computations are the qbit-qbit quantum channels. In this paper we analyse the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of general and unital qubit channels over real and complex state spaces with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. Moreover an algorithm is presented to generate uniformly distributed channels with respect to the Lebesgue measure, which enables further studies. With this algorithm the distribution of trace-distance contraction coefficient (Dobrushin) is investigated numerically by Monte-Carlo simulations, which leads to some conjectures and points out the strange behaviour of the real state space.
Semiclassical dynamics of Dirac and Weyl particles in rotating coordinates

The semiclassical kinetic theory of Dirac particles in the presence of external electromagnetic fields and global rotation is established. To provide the Hamiltonian formulation of Dirac particles, a symplectic two-form which is a matrix in spin indices is proposed. The particle number and current densities for the Dirac particles are acquired in the helicity basis. Following a similar procedure, the semiclassical kinetic theory of the Weyl particles is accomplished. It is shown that the phase-space dynamics of the Weyl and Dirac particles is directly linked. The anomalous chiral effects due to the external electromagnetic fields and angular velocity of the frame are calculated.
Skill learning approach based on impedance control for spine surgical training simulators with haptic playback

In this article, a robust adaptive impedance control based on the modified function approximation technique, and augmented with a novel integral nonsingular terminal sliding mode control, is proposed for a surgical haptic system. The haptic device is maneuvered by surgeons, whereby the mode of action, motion path, and haptic dynamics are rendered as unknowns. The inclusion of integral nonsingular terminal sliding mode control leverages the capability of ensuring consistent tracking of system’s trajectories performance, fast transient response, finite-time convergence, improved robustness, and reduced chattering. However, the modified function approximation technique strategy allows to estimate model’s dynamics regardless of the availability of dynamic uncertainties’ lower and upper bounds. In addition, an estimate of the intended motion mode and path is integrated in the evolved adaptive impedance control, via radial basis function neural network, permitting the haptic arm to track the target impedance model. Finally, using the haptic arm, controlled experimental cases and comparative study were done, after which the proposed surgical simulator adopted the results for real-time validation.
A Model Structure On The Category Of Small Acyclic Categories

In this paper, we show that the Thomason model structure restricts to a Quillen equivalent cofibrantly generated model structure on the category of acyclic categories, whose generating cofibrations are the same as those generating the Thomason model structure. To understand the Thomason model structure, we need to have a closer look at the (barycentric) subdivision endofunctor on the category of simplicial sets. This functor has a well known right adjoint, called Kan's Ex functor. Taking the subdivision twice and then the fundamental category yields a left adjoint of an adjunction between the category of simplicial sets and the category of small categories, whose right adjoint is given by applying the Ex functor twice on the nerve of a category. This adjunction lifts the cofibrantly generated Quillen model structure on simplicial sets to a cofibrantly generated model structure on the category of small categories, the Thomason model structure. The generating sets are given by the image of the generating sets of the Quillen model structure on simplicial sets under the aforementioned adjunction. We furthermore show that the category of acyclic categories is proper and combinatorial with respect to said model structure. That is weak equivalences behave nicely with respect to pushouts along fibrations and cofibrations, and cofibrations satisfy certain smallness conditions which allow us to work with sets instead of proper classes.
On the estimation of the Mori-Zwanzig memory integral

We develop rigorous estimates and provably convergent approximations for the memory integral in the Mori-Zwanzig (MZ) formulation. The new theory is built upon rigorous mathematical foundations and is presented for both state-space and probability density function space formulations of the MZ equation. In particular, we derive errors bounds and sufficient convergence conditions for short-memory approximations, the $t$-model, and hierarchical (finite-memory) approximations. In addition, we derive computable upper bounds for the MZ memory integral, which allow us to estimate (a priori) the contribution of the MZ memory to the dynamics. Numerical examples demonstrating convergence of the proposed algorithms are presented for linear and nonlinear dynamical systems evolving from random initial states.
Conceptual Proofs of L log L Criteria

The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution.
Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle

Inside a fixed bounded domain $\Omega$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of the qualitative properties of the maximizers, moving toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with convex holes. In these situations, we prove symmetry and, in some cases non symmetry results, identifying the solution. We choose to work with the outer Minkowski content as the "good" notion of perimeter. Therefore, we are led to prove some new properties for it as its lower semicontinuity with respect to the Hausdorff convergence and the fact that the outer Minkowski content is equal to the Hausdorff lower semicontinuous envelope of the classical perimeter.
How Access to Resources Affects Complementor Innovation in Platform Ecosystems

Platform owners must ensure that the ecosystems around their platforms remain as innovative as possible to meet market needs and keep pace with competing platform ecosystems. To this end, platform owners either attract new complementors with innovative complements or foster innovation among existing complementors. This study takes the perspective of complementors to understand the conditions under which access to the platform owner’s resources contributes to the complementor’s innovativeness. Based on survey data from 179 complementors of different software ecosystems, our findings support the supposition that access to the platform owner's valuable technical and commercial capital drives the complementor’s product and process innovativeness. When it comes to access to social capital (e.g., reputation effects, quality signaling), benefits for innovation (limited to production innovation) only accrue under the condition that the platform owner invests in partner-specific information sharing. Our findings contribute to a better understanding of the role of platform design and partnership management practices by the platform owner in shaping complementor innovation.
A New Family of Elliptic Curves with Unbounded Rank

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the Mordell--Weil group $E_d(K)$ is unbounded as $d$ varies. The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order of vanishing of its $L$-function at the central point. We provide an explicit expression for the $L$-function of $E_d$, and use it to study this order of vanishing in terms of $d$.
The interaction of atoms with LiF(001) revisited

Pairwise additive potentials for multielectronic atoms interacting with a LiF(001) surface are revisited by including an improved description of the electron density associated with the different lattice sites, as well as non-local electron density contributions. Within this model, the electron distribution around each ionic site of the crystal is described by means of an onion approach that accounts for the influence of the Madelung potential. From such densities, binary interatomic potentials are then derived by using well-known non-local functionals for the kinetic, exchange and correlation terms. Rumpling and long-range contributions due to projectile polarization and van der Waals forces are also included in an analogous fashion. We apply this pairwise additive approximation to evaluate the interaction potential between closed-shell He, Ne, Ar, Kr, and Xe and open-shell N, S, and Cl atoms and the LiF surface, analyzing the relative importance of the different contributions. The performance of the proposed potentials is assessed by contrasting angular positions of rainbow and supernumerary rainbow maxima produced by fast grazing incidence with available experimental data. The good agreement found for normal energies in the eVrange represents a meaningful evidence of the quality of the present description.
Simple error bounds for the multivariate Laplace approximation

In a recent paper Inglot and Majerski introduced new proof of the Laplace approximation for multiple integrals and obtained simple and accurate bounds for the error term. In contrast to most previous attempts to the problem their result is rigorous. However, it involves a constricting condition, which actually was conjectured superfluous. In this paper a suitable modification of the proof that removes the undesirable assumption is provided. An illustrative application to a classical and previously inaccessible problem is presented.
Convergence to the Stochastic Burgers Equation from a degenerate microscopic dynamics

In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [Gon\c{c}alves, Jara 2014]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.
Robustness of DC Power Networks under Weight Control

We study, possibly distributed, robust weight control policies for DC power networks that change link susceptances, or weights in response to balanced disturbances to the supply-demand vector. The margin of robustness for a given control policy is defined as the radius of the largest $l_1$ ball in the space of balanced disturbances under which the link flows can be asymptotically contained within their specified limits. For centralized control policies, the control design as well as margin of robustness are obtained from solution to an non-convex weight control problem. We establish relationship between feasible sets for DC power flow and associated network flow, which is used to establish an upper bound on the margin of robustness in terms of the min cut capacity. This bound is proven to be tight if the network is tree-like, or if the lower bound of the operation range of weight control is zero. An explicit expression for the flow-weight Jacobian is derived and is used to devise a projected sub-gradient algorithm to solve the relaxed weight control problem. An exact multi-level programming approach to solve the weight control problem for reducible networks, based on recursive application of equivalent bilevel formulation for relevant class of non-convex network optimization problems, is also proposed. The lower level problem in each recursion corresponds to replacing a sub-network by a (virtual) link with equivalent weight and capacities. The equivalent capacity function for tree-reducible networks is shown to possess a strong quasi-concavity property, facilitating easy solution to the weight control problem. Robustness analysis for natural decentralized control policies that decrease weights on overloaded links, and increase weights on underloaded links with increasing flows is provided for parallel networks. Illustrative simulation results for a benchmark IEEE network are included.