Miracle-dz/UniFAR-FTUs huggingface dataset:
Mean-Variance Risk-Averse Optimal Control of Systems Governed by PDEs with Random Parameter Fields Using Quadratic Approximations

We present a method for optimal control of systems governed by partial differential equations (PDEs) with uncertain parameter fields. We consider an objective function that involves the mean and variance of the control objective, leading to a risk-averse optimal control problem. To make the problem tractable, we invoke a quadratic Taylor series approximation of the control objective with respect to the uncertain parameter. This enables deriving explicit expressions for the mean and variance of the control objective in terms of its gradients and Hessians with respect to the uncertain parameter. The risk-averse optimal control problem is then formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs defining these gradients and Hessians. The expressions for the mean and variance of the control objective under the quadratic approximation involve the trace of the (preconditioned) Hessian and are thus prohibitive to evaluate. To address this, we employ trace estimators that only require a modest number of Hessian-vector products. We illustrate our approach with two problems: the control of a semilinear elliptic PDE with an uncertain boundary source term, and the control of a linear elliptic PDE with an uncertain coefficient field. For the latter problem, we derive adjoint-based expressions for efficient computation of the gradient of the risk-averse objective with respect to the controls. Our method ensures that the cost of computing the risk-averse objective and its gradient with respect to the control, measured in the number of PDE solves, is independent of the (discretized) parameter and control dimensions, and depends only on the number of random vectors employed in the trace estimation. Finally, we present a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with uncertain permeability field.
Representing numbers as the sum of squares and powers in the ring $\mathbb{Z}_n$

We examine the representation of numbers as the sum of two squares in $\mathbb{Z}_n$ for a general positive integer $n$. Using this information we make some comments about the density of positive integers which can be represented as the sum of two squares and powers of $2$ in $\mathbb{N}$.
Common Belief and the Theory of Games with Perfect Information

Abstract The statement "it is common belief that player i is Bayesian rational" (where Bayesian rational means an expected utility maximizer) is defined for two-person games with perfect information. It is shown that in most such games it is not possible for a theory so postulate the Bayesian rationality of all players and to be common belief. This bears directly upon the salience of standard solution concepts such as subgame perfect, and sequential equilibria as well as upon the extensive form rationalizability theories of D. Bernheim ( Econometrica 52 , 1984, 1009–1028) and D. Pearce ( Econometrica 52 , 1984, 1029–1050) which rely heavily on the common belief of Bayesian rationality. Journal of Economic Literature Classification Number: C72.
On the box dimension of graph of harmonic functions on the Sierpi\'nski gasket

In this paper, we have obtained bounds for the box dimension of graph of harmonic function on the Sierpi\'nski gasket. Also we get upper and lower bounds for the box dimension of graph of functions that belongs to $\text{dom}(\mathcal{E}),$ that is, all finite energy functionals on the Sierpi\'nski gasket. Further, we show the existence of fractal functions in the function space $\text{dom}(\mathcal{E})$ with the help of fractal interpolation functions. Moreover, we provide bounds for the box dimension of some functions that belong to the family of continuous functions and arise as fractal interpolation functions.
An Accretive Operator Approach to Ergodic Problems for Zero-Sum Games

Mean payoff stochastic games can be studied by means of a nonlinear spectral problem involving the Shapley operator: the ergodic equation. A solution consists in a scalar, called the ergodic constant, and a vector, called bias. The existence of such a pair entails that the mean payoff per time unit is equal to the ergodic constant for any initial state, and the bias gives stationary strategies. By exploiting two fundamental properties of Shapley operators, monotonicity and additive homogeneity, we give a necessary and sufficient condition for the solvability of the ergodic equation for all the Shapley operators obtained by perturbation of the transition payments of a given stochastic game with finite state space. If the latter condition is satisfied, we establish that the bias is unique (up to an additive constant) for a generic perturbation of the transition payments. To show these results, we use the theory of accretive operators, and prove in particular some surjectivity condition.
The Circular Law for random regular digraphs

Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral distribution of $A_n$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_n$.
Chemical Abundances and Kinematics in Globular Clusters and Local Group Dwarf Galaxies and Their Implications for Formation Theories of the Galactic Halo

We review Galactic halo formation theories and supporting evidence, in particular, kinematics and detailed chemical abundances of stars in some relevant globular clusters as well as Local Group dwarf galaxies. Outer halo red HB clusters tend to have large eccentricities and inhabit the area of the Lee diagram populated by dwarf spheroidal stars, favoring an extragalactic origin. Old globular clusters show the full range of eccentricities, while younger ones seem to have preferentially high eccentricities, again hinting at their extragalactic origin. However, the three outer halo second parameter clusters with well‐determined orbits indicate they come from three independent systems. We compare detailed abundances of a variety of elements between the halo and all dwarf galaxies studied to date, including both dwarf spheroidals and irregulars. The salient feature is that halo abundances are essentially unique. In particular, the general α vs. [Fe/H] pattern of 12 of the 13 galaxies studied are similar to each other and very different from the Milky Way. Sgr appears to be the only possible exception. At the metal‐poor end the extragalactic sample is only slightly deficient compared to the halo but begins to diverge by [Fe/H] ∼ −2 and the difference is particularly striking for stars with [Fe/H] ∼ −1. Only Sgr, the most massive dSph, has some stars similar in α‐abundance to Galactic stars at intermediate metallicities, even the most extreme low‐α subset most likely to have been accreted. It appears very unlikely that a significant fraction of the metal‐rich halo could have come from disrupted dSphs of low mass. However, at least some of the metal‐poor halo may have come from typical dSphs, and a portion of the intermediate metallicity and metal‐rich halo may have come from very massive systems like Sgr. This argues against the standard hierarchical galaxy formation scenario and the Searle‐Zinn paradigm for the formation of the Galactic halo via accretion of “fragments” composed of stars like those we see in typical present‐day dSphs. The chemical differences between the dwarfs and the halo are due to a combination of a low star formation efficiency and a high galactic wind efficiency in the former. AGB stars are also more important in the chemical evolution of the dwarfs. The formation problem may be solved if the majority of halo stars formed within a few, very massive satellites accreted very early. However, any such satellites must either be accreted much earlier than postulated, before the onset of SNe Ia, or star formation must be prevented to occur in them until only shortly before they are accreted. The intrinsic scatter in many elements, particularly the α‐elements, indicates that the halo was also mixed on a surprisingly short timescale, a further problem for hierarchical formation theories.
Experiments on laminar film flow along a periodic wall

Experimental results are reported on the structure of gravity-driven film flow along an inclined periodic wall with rectangular corrugations. A fluorescence imaging method is used to capture the evolution of film height in space and time with accuracy of a few microns. The steady flow is found to exhibit a statically deformed free surface, as predicted by previous asymptotic and numerical studies. Though usually unstable, its characteristics determine much of the subsequent non-stationary dynamics. Travelling disturbances are observed to evolve into solitary multi-peaked humps, and pronounced differences from the respective phenomena along a flat wall are noted. Finally, a remarkable stabilization of the flow at high Reynolds numbers is documented, which proceeds through the development of a three-dimensional flow structure and leads to a temporary decrease in film thickness and recession of solitary waves.
Counting points of schemes over finite rings and counting representations of arithmetic lattices

If $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(\Gamma)$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(\Gamma)=O(n^C)$ for every such $\Gamma$. We also prove similar results for lattices in positive characteristic.
The Resurgence of Instantons in String Theory

Nonperturbative effects in string theory are usually associated to D-branes. In many cases it can be explicitly shown that D-brane instantons control the large-order behavior of string perturbation theory, leading to the well-known (2g)! growth of the genus expansion. This paper presents a detailed treatment of nonperturbative solutions in string theory, and their relation to the large-order behavior of perturbation theory, making use of transseries and resurgent analysis. These are powerful techniques addressing general nonperturbative contributions within non-linear systems, which are developed at length herein as they apply to string theory. The cases of topological strings, the Painleve I equation describing 2d quantum gravity, and the quartic matrix model, are explicitly addressed. These results generalize to minimal strings and general matrix models. It is shown that, in order to completely understand string theory at a fully nonperturbative level, new sectors are required beyond the standard D-brane sector.
Default Bayesian Priors for Regression Models with First‐Order Autoregressive Residuals

Abstract. The objective of this paper is to develop default priors when the parameter of interest is the autocorrelation coefficient in normal regression models with first‐order autoregressive residuals. Jeffreys’ prior as well as reference priors are found. These priors are compared in the light of how accurately the coverage probabilities of Bayesian credible intervals match the corresponding frequentist coverage probabilities. It is found that the reference priors have a definite edge over Jeffreys’ prior in this respect. Also, the credible intervals based on these reference priors seem superior to similar intervals based on certain divergence measures.
Stochastic homogenisation of high-contrast media

ABSTRACT Using a suitable stochastic version of the compactness argument of [Zhikov VV. On an extension of the method of two-scale convergence and its applications. Sb Math. 2000;191(7–8):973–1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled ‘macroscopic’ and ‘microscopic’ components of the field, giving rise to an analogue of the ‘Zhikov function’, which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.
GLOBAL ASYMPTOTICS OF THE HAHN POLYNOMIALS

In this paper, we study the asymptotics of the Hahn polynomials Qn(x; α, β, N) as the degree n grows to infinity, when the parameters α and β are fixed and the ratio of n/N = c is a constant in the interval (0, 1). Uniform asymptotic formulas in terms of Airy functions and elementary functions are obtained for z in three overlapping regions, which together cover the whole complex plane. Our method is based on a modified version of the Riemann–Hilbert approach introduced by Deift and Zhou.
Identifying Red Variables in the Northern Sky Variability Survey

We present a catalog of 8678 slowly varying stars with near-infrared colors corresponding to the evolved asymptotic giant branch population. Objects were selected from the Northern Sky Variability Survey (NSVS) covering the entire sky above declination δ = -38° in a single unfiltered photometric band corresponding to a V-band magnitude range of ∼8–15.5 mag. After quality cuts, the number of measurements for a typical star is approximately 150, but it ranges up to ∼1000 for high-declination stars. We show that the use of support vector machines, a modern machine-learning algorithm, can reliably distinguish Mira variables from other types of red variables, namely, semiregular and irregular. We also identify a region of parameter space that is dominated by carbon stars. Our classification is based on period, amplitude, and three independent colors possible with photometry from the NSVS and the Two Micron All Sky Survey. The overall classification accuracy is ∼90% despite the relatively short survey baseline of 1 yr and limited set of features. There are 6474 stars in our sample without identifications in the General Catalogue of Variable Stars, which, as such, are most likely new discoveries. Period-amplitude and period-color diagrams of both our previously known and newly identified Mira stars are in good agreement with published studies based on smaller samples.
Geodesics and geodesic deviation for impulsive gravitational waves

The geometry of impulsive pp-waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric. The geodesic equation involves formally ill-defined products of distributions due to the nonlinearity of the equations and the presence of the Dirac δ-distribution in the space–time metric. Thus, strictly speaking, it cannot be treated within Schwartz’s linear theory of distributions. To cope with this difficulty we proceed by first regularizing the δ-singularity, then solving the regularized equation within classical smooth functions and, finally, obtaining a distributional limit as solution to the original problem. Furthermore, it is shown that this limit is independent of the regularization without requiring any additional condition, thereby confirming earlier results in a mathematically rigorous fashion. We also treat the Jacobi equation which, despite being linear in the deviation vector field, involves even more delicate singular expressions, like the “square” of the Dirac δ-distribution. Again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization-independent distributional limit. Hence it is concluded that the geometry of impulsive pp-waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill-defined products.
Algebraic Properties of Product of Graphs

Let G and H be two simple graphs and let G*H denotes the graph theoretical product of G by H. In this article, we provide graded Betti numbers, Castelnuovo–Mumford regularity, projective dimension, h-vector and Hilbert series of G*H in terms of that information of G and H. More generally, we present explicit formulae to compute graded Betti numbers, h-vector, and Hilbert series of disjoint union of complexes. We will also prove that the family of graphs for which the regularity of each edge ring equals the maximum number of pairwise 3-disjoint edges, is closed under product of graphs.
The Boundary of the Gelfand-Tsetlin Graph: New Proof of Borodin-Olshanski's Formula, and its q-analogue

In the recent paper [arXiv:1109.1412], Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand-Tsetlin graph can be identified with finite indecomposable (= extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly infinite totally nonnegative sequences. A principal ingredient of Borodin-Olshanski's proof is a new explicit determinantal formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand-Tsetlin schemes of trapezoidal shape). We present a simpler and more direct derivation of that formula using the Cauchy-Binet summation involving the inverse Vandermonde matrix. We also obtain a q-generalization of that formula, namely, a new explicit determinantal formula for arbitrary q-specializations of skew Schur polynomials. Its particular case is related to the q-Gelfand-Tsetlin graph and q-Toeplitz matrices introduced and studied by Gorin [arXiv:1011.1769].
On transition region convection cells in simulations of p‐mode propagation into the solar atmosphere

This paper examines the way that transition region surface waves, generated in 2-D numerical simulations of the nonmagnetic solar atmosphere when various synthetic photospheric drivers are applied, drive the granulation of the transition region/lower coronal region into convection cells. It is shown that these cells are generated by both synthetic point drivers and synthetic horizontally coherent p-mode drivers. These cells cause the conversion of driven signals in vertical velocity into coronal signals predominantly in horizontal velocity, which if carried over to a case with a magnetic field included could cause mode conversion. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Are two modes of thermohaline circulation stable

The standard version of the coupled ocean–atmosphere model developed at the GeophysicalFluid Dynamics Laboratory (GFDL) of NOAA has at least two stable equilibria. One has arealistic and active thermohaline circulation (THC) with sinking regions in the northern NorthAtlantic Ocean. The other has a reverse THC with extremely weak upwelling in the NorthAtlantic and sinking in the Circumpolar Ocean of the Southern Hemisphere. Although themodel has the seasonal variation of insolation, the structure of these two stable equilibria arevery similar to those of a previous GFDL model without the seasonal variation. It is notedthat the inactive mode of the reverse THC mentioned above is not a stable equilibrium foranother version of the same coupled model which has a large coefficient of vertical subgridscale diffusion. Although the reverse THC cell was produced in the Atlantic Ocean by a massivedischarge of freshwater, it began to transform back to the original direct THC as soon as thefreshwater discharge was terminated. It appears that there is a critical value of diffusivity, abovewhich two stable equilibria do not exist. Based upon paleoceanographic evidence, we suggestthat the stable state of the reverse THC mentioned above did not prevail during the cold periodsof Younger Dryas event which occurred during the last deglacial period. Instead, it is likelythat the THC weakened temporarily, but reintensified before it reached the state of the reverseTHC with no deep water formation in the North Atlantic Ocean. DOI: 10.1034/j.1600-0870.1999. t01-3-00005.x
Falsication Of The Atmospheric CO2 Greenhouse Eects Within The Frame Of Physics

The atmospheric greenhouse effect, an idea that many authors trace back to the traditional works of Fourier (1824), Tyndall (1861), and Arrhenius (1896), and which is still supported in global climatology, essentially describes a fictitious mechanism, in which a planetary atmosphere acts as a heat pump driven by an environment that is radiatively interacting with but radiatively equilibrated to the atmospheric system. According to the second law of thermodynamics, such a planetary machine can never exist. Nevertheless, in almost all texts of global climatology and in a widespread secondary literature, it is taken for granted that such a mechanism is real and stands on a firm scientific foundation. In this paper, the popular conjecture is analyzed and the underlying physical principles are clarified. By showing that (a) there are no common physical laws between the warming phenomenon in glass houses and the fictitious atmospheric greenhouse effects, (b) there are no calculations to determine an average surface temperature of a planet, (c) the frequently mentioned difference of 33° is a meaningless number calculated wrongly, (d) the formulas of cavity radiation are used inappropriately, (e) the assumption of a radiative balance is unphysical, (f) thermal conductivity and friction must not be set to zero, the atmospheric greenhouse conjecture is falsified.
On Landis’ conjecture in the plane for some equations with sign-changing potentials

In this article, we investigate the quantitative unique continuation properties of real-valued solutions to elliptic equations in the plane. Under a general set of assumptions on the operator, we establish quantitative forms of Landis' conjecture. Of note, we prove a version of Landis' conjecture for solutions to $-\Delta u + V u = 0$, where $V$ is a bounded function whose negative part exhibits polynomial decay at infinity. The main mechanism behind the proofs is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing result, we present a new idea for constructing positive multipliers and use it reduce the equation to a Beltrami system. The resulting first-order equation is analyzed using the similarity principle and the Hadamard three-quasi-circle theorem.
Spatial discretization error in Kalman filtering for discrete-time infinite dimensional systems

We derive a reduced-order state estimator for discrete-time infinite dimensional linear systems with finite dimensional Gaussian input and output noise. This state estimator is the optimal one-step estimate that takes values in a fixed finite dimensional subspace of the system's state space --- consider, for example, a Finite Element space. We then derive a Riccati difference equation for the error covariance and use sensitivity analysis to obtain a bound for the error of the state estimate due to the state space discretization.