Miracle-dz/UniFAR-FTUs huggingface dataset:
Surrogate accelerated Bayesian inversion for the determination of the thermal diffusivity of a material

Determination of the thermal properties of a material is an important task in many scientific and engineering applications. How a material behaves when subjected to high or fluctuating temperatures can be critical to the safety and longevity of a system’s essential components. The laser flash experiment is a well-established technique for indirectly measuring the thermal diffusivity, and hence the thermal conductivity, of a material. In previous works, optimization schemes have been used to find estimates of the thermal conductivity and other quantities of interest which best fit a given model to experimental data. Adopting a Bayesian approach allows for prior beliefs about uncertain model inputs to be conditioned on experimental data to determine a posterior distribution, but probing this distribution using sampling techniques such as Markov chain Monte Carlo methods can be incredibly computationally intensive. This difficulty is especially true for forward models consisting of time-dependent partial differential equations. We pose the problem of determining the thermal conductivity of a material via the laser flash experiment as a Bayesian inverse problem in which the laser intensity is also treated as uncertain. We introduce a parametric surrogate model that takes the form of a stochastic Galerkin finite element approximation, also known as a generalized polynomial chaos expansion, and show how it can be used to sample efficiently from the approximate posterior distribution. This approach gives access not only to the sought-after estimate of the thermal conductivity but also important information about its relationship to the laser intensity, and information for uncertainty quantification. Moreover, this approach leads to significant speed up over traditional methods by orders of magnitude. We also investigate the effects of the spatial profile of the laser on the estimated posterior distribution for the thermal conductivity.
Optimal guaranteed cost control of discrete-time linear systems subject to structured uncertainties

ABSTRACT In this paper, we propose state- and static output-feedback generalised guaranteed cost control (GCC) approaches for discrete-time linear systems subject to norm-bounded structured parametric uncertainties. This method enables the convex synthesis for a more general class of systems, where uncertainties are uncorrelated block diagonal, and no feed-through uncertainty is multiplicative with control input ones. It also provides necessary and sufficient conditions for state-feedback and sufficient conditions for static output-feedback. We also present a comparative study among the proposed controllers, standard Linear Quadratic Regulator, and Guaranteed Cost Controller found in the literature.
On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations

We study the convergence of iterative linear solvers for discontinuous Galerkin discretizations of systems of hyperbolic conservation laws with polygonal mesh elements compared with that of traditional triangular elements. We solve the semi-discrete system of equations by means of an implicit time discretization method, using iterative solvers such as the block Jacobi method and GMRES. We perform a von Neumann analysis to analytically study the convergence of the block Jacobi method for the two-dimensional advection equation on four classes of regular meshes: hexagonal, square, equilateral-triangular, and right-triangular. We find that hexagonal and square meshes give rise to smaller eigenvalues, and thus result in faster convergence of Jacobi's method. We perform numerical experiments with variable velocity fields, irregular, unstructured meshes, and the Euler equations of gas dynamics to confirm and extend these results. We additionally study the effect of polygonal meshes on the performance of block ILU(0) and Jacobi preconditioners for the GMRES method.
Positive topological entropy of Reeb flows on spherizations

Abstract Let M be a closed manifold whose based loop space Ω (M) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T*M which is fiberwise starshaped with respect to the origin. Choose a function H : T*M → ℝ such that Σ is a regular energy surface of H, and let ϕt be the restriction to Σ of the Hamiltonian flow of H. Theorem 1. The topological entropy of ϕt is positive. This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows. Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive. For q ∈ M abbreviate Σq = Σ ∩ Tq*M. The following corollary extends results of Morse and Gromov on the number of geodesics between two points. Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕt from Σq to Σq′ grows exponentially in time. In the lowest dimension, Theorem 1 yields the existence of many closed, orbits. Corollary 2. Let M be a closed surface different from S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.
Convergence Theorems for Gradient Descent

Here you will find a growing collection of simple proofs of the convergence of gradient and stochastic gradient descent type method on convex, strongly convex and smooth functions. Our focus is on “good proofs” that are also simple. Each section can also be consulted separately. Some noteworthy proofs here include a “good” convergence proof of SGD with momentum. Disclaimer: Theses notes are not proper review of the literature, and we welcome earlier pointers to missing bibliog-raphy, of which we are sure is missing. Further disclaimer: These notes do not compare to a good book or well prepared lecture notes. If you are new to convex optimization I highly recommend reading instead the first few chapters of the books [4] and [1].
Asymptotic Theory of a Test for the Constancy of Regression Coefficients Against the Random Walk Alternative

one component of the coefficients follows a random walk process. We discuss the limiting null behavior of the test statistic without assuming normality under two situations, where the initial value of the random walk process is known or unknown. The limiting distribution is that of a quadratic functional of Brownian motion and the characteristic function is obtained from the Fredhohm determinant associated with a certain integral equation. The limiting distribution is then computed by numerical inversion of the characteristic function.
Interpreting formulas of divisible lattice ordered abelian groups

We show that a large class of divisible abelian $\ell$-groups (lattice ordered groups) of continuous functions is interpretable (in a certain sense) in the lattice of the zero sets of these functions. This has various applications to the model theory of these $\ell$-groups, including decidability results.
Distinguished curves and integrability in Riemannian, conformal, and projective geometry

We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is a type of moving incidence relation. The characterisation is used to provide a very general theory and construction of quantities that are necessarily conserved along the curves. The usual role of Killing tensors and conformal Killing tensors is recovered, but the construction shows that a significantly larger class of equation solutions can also yield curve first integrals. In particular any normal solution to an equation from the class of first BGG equations can potentially yield such a conserved quantity. For some equations the condition of normality is not required. For nowhere-null curves in pseudo-Riemannian and conformal geometry additional results are available. We provide a fundamental tractor-valued invariant of such curves and this quantity is parallel if and only if the curve is an unparametrised conformal circle.
Some Results on the $b$-Colouring Parameters of Graphs

A vertex colouring of a given graph $G$ can be considered as a random experiment. A discrete random variable $X$, corresponding to this random experiment, can be defined as the colour of a randomly chosen vertex of $G$ and a probability mass function for this random variable can be defined accordingly. In this paper, we study the concepts of mean and variance corresponding to the $b$-colouring of $G$ and hence determine the values of these parameters for a number of standard graphs.
Achieving Rental Harmony with a Secretive Roommate

Abstract Given the subjective preferences of n roommates in an n-bedroom apartment, one can use Sperner’s lemma to find a division of the rent such that each roommate prefers a distinct room. At the resulting rent division, no roommate has a strictly stronger preference for a different room. We give a new elementary proof that the subjective preferences of only n – 1 of the roommates actually suffice to achieve this envy-free rent division. Our proof, in particular, yields an algorithm to find such a division of rent. The techniques also give generalizations of Sperner’s lemma including a new proof of a conjecture of Meunier.
The projective parabolic geometry of Riemannian, Kähler and quaternion-Kähler metrics

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.
An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type

The Legendre-based ultraspherical spectral method for ordinary differential equations (Olver, S. & Townsend, A. (2013) A fast and well-conditioned spectral method. SIAM Rev., 55, 462–489. ) is combined with a formula for the convolution of two Legendre series (Hale, N. & Townsend, A. (2014a) An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput. , 36, A1207–A1220. ) to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth, then the method is spectrally accurate and the resulting almost-banded linear systems can be solved with linear complexity.
Dispersion enhancement in atom-cavity and coupled cavity systems

We present an entirely linear all-optical method of dispersion enhancement using coupled cavities that leads to a substantial increase in system transmission in comparison with atom-cavity systems. This is achieved by tuning the system to an anomalous dispersion condition by under-coupling at least one of the cavities to the other. The intracavity anomalous dispersion is then associated with a dip in reflection (and in turn with a peak in transmission) rather than with an absorption resonance as in the case of the atomic vapor. We find that in contrast with the atom-cavity system where mode reshaping always contributes to the mode pushing, in coupled cavity systems reshaping of the mode profile can either contribute to or oppose the mode pushing, and even reverse it under appropriate conditions leading to a reduced scale factor in transmission. We demonstrate a method for further optimizing the transmission of both atom-cavity and coupled-cavity systems, but show that this leads to a more rectangular mode profile and a reduction in the scale factor bandwidth. We also derive the cavity scale factor in reflection for both atom-cavity and coupled cavity systems and show that in reflection the reshaping of the mode profile can either contribute to or oppose the mode pushing, but cannot reverse it.
The Merrifield-Simmons conjecture also holds for parity graphs

The Merrifield-Simmons conjectures states a relation between the distance of vertices in a simple graph $G$ and the number of independent sets, denoted as $\sigma(G)$, in vertex-deleted subgraphs. Namely, that the sign of the term $\sigma(G_{-u}) \cdot \sigma(G_{-v}) - \sigma(G) \cdot \sigma(G_{-u-v})$ only depends on the parity of the distance of $u$ and $v$ in $G$. We prove this statement in the case of parity graphs and give some evidence that this result may not be further generalized to other classes of graphs.
Casimir and mean-field interactions between membrane inclusions subject to external torques

Inclusions embedded in fluctuating fluid membranes have been shown to experience membrane mediated forces decaying as 1/R4. Modeling the inclusions as local constraints on the membrane curvature tensor, we show that the presence of external torques (mechanical or field-induced) strongly enhances and increases the range of the interactions. Repulsive mean-field contributions and attractive fluctuation (Casimir) contributions of range 1/R2 or longer are found, which may combine to yield equilibrium distances.
The Racah algebra as a commutant and Howe duality

The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the -sphere. It is further identified as the commutant of the subalgebra of in oscillator representations of the universal algebra of the latter. How this observation relates to the Racah problem and the superintegrable model on the -sphere is discussed on the basis of the Howe duality associated to the pair .
Ground states of nonlinear fractional Schrödinger equation involving critical growth

In this paper, we are concerned with the ground state solutions of nonlinear fractional Schr\"odinger equation involving critical growth. Without Ambrosetti-Rabinowitz condition and monotonicity condition on the nonlinearity, we get the existence of ground state solutions of such equation when the potential $V(x)$ is not a constant and not radial.
Topology On BCK-Modules

In this paper, we introduce the notion of a BCK-topological module in a natural way and establish that every decreasing sequence of submodules on a BCK-module M over bounded commutative BCK-algebra X is indeed a BCK- topological module. We have defined the notion of compatible and strict BCK- module homomorphisms, and establish that a strict BCK-module homomorphism is an open as well as a continuous mapping. Also, we establish the necessary and sufficient condition for a compatible mapping to be strict.
Sensitivity of Methods of Variance Component Estimation to Culling Type of Selection

Abstract Two pseudoexpectation methods of variance component estimation were examined for selection bias from culling and were compared with Henderson's simple method and restricted maximum likelihood. Milk yield data were simulated for 100 herds, 50 sires, and approximately 2000 cows per replicate. Heritability was . 25, and repeatability was . 50. Each cow had two records. Twenty replicates were made. Two data sets were created; one was unselected and included both records of the cow, and the second excluded second records of cows with first records below the herd mean. Sire, cow, and error variances were estimated according to a mixed model that included fixed herd, parity, and random sire, cow, and error effects. Estimates of sire, cow, and error variances and heritability and repeatability were unbiased and were similar for all methods for unselected data. For selected data, sire and cow variances were overestimated and error variances were underestimated with Henderson's simple method and the pseudoexpectation methods. Accordingly, heritabilities and repeatabilities were overestimated. Biases were greatest for Henderson's simple method and differences between the two pseudoexpectation methods were small. With restricted maximum likelihood, sire variances were unbiased but error variances were slightly overestimated and cow variances underestimated. Use of restricted maximum likelihood is preferred with selected data.
The Cohen-Macaulay property in derived commutative algebra

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementing results of J{\o}rgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natural examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allowing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but we also discuss extensions of it to non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in topology.
The second realization of the ICRF with VLBI

Abstract The ICRF derived from VLBI observations of extragalactic radio sources up to 1995.6 and effective since 1998.0 was a radical change from the FK5 stellar/equinox celestial system. Since then the number of geodetic/astrometric VLBI observations has tripled and the number of radio sources with astrometrically useful data has quadrupled. These data along with advances in modeling and estimation will be used to generate the next ICRF realization in the microwave band. Analysis of source position time series and source structure evolution will be used to select better “defining” sources. Working groups have been established by the IAU, IERS and IVS with the goal of presenting the second realization of the ICRF at the IAU General Assembly in 2009.
Enhancing joint reconstruction and segmentation with non-convex Bregman iteration

All imaging modalities such as computed tomography, emission tomography and magnetic resonance imaging require a reconstruction approach to produce an image. A common image processing task for applications that utilise those modalities is image segmentation, typically performed posterior to the reconstruction. Recently, the idea of tackling both problems jointly has been proposed. We explore a new approach that combines reconstruction and segmentation in a unified framework. We derive a variational model that consists of a total variation regularised reconstruction from undersampled measurements and a Chan–Vese-based segmentation. We extend the variational regularisation scheme to a Bregman iteration framework to improve the reconstruction and therefore the segmentation. We develop a novel alternating minimisation scheme that solves the non-convex optimisation problem with provable convergence guarantees. Our results for synthetic and real data show that both reconstruction and segmentation are improved compared to the classical sequential approach.
Perturbed gradient flow trees and $A_\infty$-algebra structures on Morse cochain complexes

We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse functions. The purpose of this article is to provide a coherent and detailed treatment of Abouzaid's approach including a discussion of all relevant analytic notions and results.
A stable quantum Darmois-Skitovich theorem

The Darmois-Skitovich theorem is a simple characterization of the normal distribution in terms of the independence of linear forms. We present here a non-commutative version of this theorem in the context of Gaussian bosonic states and show that this theorem is stable under small errors in its underlying conditions. An explicit estimate of the stability constants which depend on the physical parameters of the problem is given.
A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions

In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Ling et al. 2016 Physica D 327, 13–29 (doi:10.1016/j.physd.2016.03.012); Feng et al. 2016 Phys. Rev. E 93, 052227 (doi:10.1103/PhysRevE.93.052227)).
Measures of full dimension on self-affine sets

Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we show that for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set itself.
A transitive HORPO for curried systems

The higher-order recursive path ordering is one of the oldest, but still very eﬀective, methods to prove termination of higher-order TRSs. A limitation of this ordering is that it is not transitive (and its transitive closure is not computable). We will present a transitive variation of HORPO. Unlike previous HORPO deﬁnitions, this method can be used directly on terms in curried notation. 2012 ACM Subject Classiﬁcation Theory of computation → Equational logic and rewriting