Miracle-dz/UniFAR-FTUs huggingface dataset:
Bistable reaction equations with doubly nonlinear diffusion

Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions \begin{document}$ 0\leq u(x, t)\leq 1 $\end{document} of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r. t. the Fisher-KPP framework recently studied in [ 5 ]. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries. Finally, as a complement of [ 5 ], we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").
Geometric Foundation of Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit

A geometric foundation thermo-statistics is presented with the only axiomatic assumption of Boltzmann’s principle S ( E, N, V ) = k ln W . This relates the entropy to the geometric area e S ( E,N,V ) /k of the manifold of constant energy in the ( ﬁnite - N )-body phase space. From the principle, all thermodynamics and especially all phenomena of phase transitions and critical phenomena can unambiguously be identiﬁed for even small systems . The topology of the curvature matrix C ( E, N ) of S ( E, N ) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Phase transitions are linked to convex (upwards bending) intruders of S ( E, N ), where the canonical ensemble deﬁned by the Laplace transform to the intensive variables becomes multi-modal, non-local, (it mixes widely diﬀerent conserved quan-tities). Here the one-to-one mapping of the Legendre transform gets lost. Within Boltzmann’s principle, Statistical Mechanics becomes a geometric theory addressing the whole ensemble or the manifold of all points in phase space which are consistent with the few macroscopic conserved control parameters. This interpretation leads to a straight derivation of irreversibility and the Second Law of Thermodynamics out of the time-reversible, microscopic, mechanical dynamics. It is the whole ensemble that spreads irreversibly over the accessible phase space not
Colorful hypergraphs in Kneser hypergraphs

Using a $Z_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a hypergraph).
A CURVILINEAR LEVEL SET FORMULATION FOR HIGHLY DEFORMABLE FREE SURFACE PROBLEMS WITH APPLICATION TO SOLIDIFICATION

A curvilinear level set formulation has been developed for highly deformable free surface problems. In this new scheme, the grid lines follow the irregular domain generated by the multizone adaptive grid-generation (MAGG) scheme [1] and free surfaces are captured by level set functions among the curvilinear grids. Navier-Stokes equations are discretized and solved based on a multiphase curvilinear finite-volume method [2], and the level set function is solved based on a finite-difference method using the second-order essentially nonoscillatory (ENO) scheme [3]. The scheme is capable of accurately and efficiently representing the deformation, oscillation, merger, and separation of free surfaces. The effectiveness and robustness of the algorithm are demonstrated by using it for problems involving merger of bubbles, mold filling, and the spreading and solidification of molten droplets on a cold substrate where both free surface and solidification interfaces move and the mass of the liquid phase is continuous.
Mesh Independence of an Accelerated Block Coordinate Descent Method for Sparse Optimal Control Problems

An accelerated block coordinate descent (ABCD) method in Hilbert space is analyzed to solve the sparse optimal control problem via its dual. The finite element approximation of this method is investigated and convergence results are presents. Based on the second order growth condition of the dual objective function, we show that iteration sequence of dual variables has the iteration complexity of $O(1/k)$. Moreover, we also prove iteration complexity for the primal problem. Two types of mesh-independence for ABCD method are proved, which asserts that asymptotically the infinite dimensional ABCD method and finite dimensional discretizations have the same convergence property, and the iterations of ABCD method remain nearly constant as the discretization is refined.
Fitting regression models with response‐biased samples

This paper extends the work in Lawless, Kalbfleisch, & Wild ( 1999 ) on fitting regression models with response‐biased samples, that is, samples where some or all the covariates are missing for some units and the probability that this happens depends in part on the value of the reponse of that unit. In general, the resulting likelihood depends on the distribution of the covariates but we are only interested in methods that do not involve modelling this distribution. We look at a variety of methods based on estimating equations, at the relationship of these methods to semi‐parametric efficient methods in cases where such methods exist, and show ways of obtaining efficiency gains that can sometimes be dramatic. The Canadian Journal of Statistics 39: 519–536; 2011 © 2011 Statistical Society of Canada
Strongly Polynomial 2-Approximations of Discrete Wasserstein Barycenters

Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, which arise in a wide range of fields. In many applications, data is given as a set of probability measures with finite support. The discrete barycenters in this setting exhibit favorable properties: All barycenters have finite support, and there always is one with a provably sparse support. Further, each barycenter allows a non-mass splitting optimal transport to each of the marginals. It is open whether the computation of a discrete barycenter is possible in polynomial time. The best known exact algorithms are based on linear programming, but the sizes of these programs scale exponentially. In this paper, we prove that there is a strongly polynomial, tight 2-approximation, based on restricting the possible support of an approximate barycenter to the union of supports of the measures. The resulting measure is sparse, but an optimal transport will generally split mass. We then exhibit a strongly polynomial algorithm to improve this measure to another one, for which there exists a non-mass splitting transport of lower cost. Finally, we present an iterative scheme that alternates between these two algorithms. It terminates with a 2-approximation that has a sparse support and an associated non-mass splitting optimal transport. We conclude with some practical computations.
Closeness to spheres of hypersurfaces with normal curvature bounded below

For a Riemannian manifold and a compact domain bounded by a hypersurface with normal curvature bounded below, estimates are obtained in terms of the distance from to for the angle between the geodesic line joining a fixed interior point in to a point on and the outward normal to the surface. Estimates for the width of a spherical shell containing such a hypersurface are also presented. Bibliography: 9 titles.
Iterative identification and restoration of images

The blur identification problem is formulated as a constrained maximum-likelihood problem. The constraints directly incorporate a priori known relations between the blur (and image model) coefficients, such as symmetry properties, into the identification procedure. The resulting nonlinear minimization problem is solved iteratively, yielding a very general identification algorithm. An example of blur identification using synthetic data is given. <<ETX>>
A Survey of the Statistical Theory of Shape

This is a review of the current state of the "theory of shape" introduced by the author in 1977. It starts with a definition of "shape" for a set of k points in m dimensions. The first task is to identify the shape spaces in which such objects naturally live, and then to examine the probability structures induced on such a shape space by corresponding structures in Rm. Against this theoretical background one formulates and solves statistical problems concerned with shape characteristics of empirical sets of points. Some applications (briefly sketched here) are to archeology, astronomy, geography and physical chemistry. We also outline more recent work on "size-and-shape," on shapes of sets of points in riemannian spaces, and on shape-theoretic aspects of random Delaunay tessellations.
Elliptic Sklyanin integrable systems for arbitrary reductive groups

We present the analogue, for an arbitrary complex reductive group G, of the elliptic integrable systems of Sklyanin. The Sklyanin integrable systems were originally constructed on symplectic leaves, of a quadratic Poisson structure, on a loop group of type A. The phase space, of our integrable systems, is a group-like analogue of the Hitchin system over an elliptic curve E. We consider the moduli space of pairs (P,f), where P is a principal G-bundle on E, and f is a meromorphic section of the adjoint group bundle. We show: 1) The moduli space admits an algebraic Poisson structure. It is related to the poisson structures on loop groupoids, constructed by Etingof and Varchenko, using Felder's elliptic solutions of the Classical Dynamical Yang-Baxter Equation. 2) The symplectic leaves are finite dimensional. A symplectic leaf is determined by labeling, finitely many points of E, each by a dominant co-character of the maximal torus of G. 3) Each leaf is an algebraically completely integrable system.
Finite field restriction estimates based on Kakeya maximal operator estimates

In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$ dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which $-1$ \emph{is} a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein-Tomas exponent in this setting.
Constructing positive maps from block matrices

Positive maps are useful for detecting entanglement in quantum information theory. Any entangled state can be detected by some positive map. In this paper, the relation between positive block matrices and completely positive trace-preserving maps is characterized. Consequently, a new method for constructing decomposable maps from positive block matrices is derived. In addition, a method for constructing positive but not completely positive maps from Hermitian block matrices is also obtained.
Estimation and Inference in Two-Step Econometric Models

A commonly used procedure in a wide class of impirical applications is to impute unobserved regressors, such as expectations, from an auxiliary econometric model. This two-step (T-S) procedure fails to account for he fact that imputed regessors are measured with sampling error, so hypothesis tests based on the estimated covariance matrix of the second-step estimator are biased, even in large samples. We present a simple yet general method of calculating asymptotically correct standard errors in T-S models. The proceedure may be applied even when joint estimation methods, such as full information maximum likelihood, are inappropriate or computationally infeasible. We present two examples from recent empirical literature in which these corrections have a major impact on hypothesis testing.
Integrating speaker-nonspecific timbre transformation to an HNM-based speech synthesis scheme

In this article, a speech signal synthesis scheme based on harmonic-plus-noise model (HNM) is extended to provide the function of speaker-nonspecific timbre transformation. To transform synthetic speech's timbre, we have developed a formant-based frequency mapping method called piece-wise linear frequency mapping (PLFM). Additionally, a commonly adopted method is frequency axis scaling (FAS). Both methods have been integrated into the HNM-based signal synthesis scheme, and a real-time speech synthesis system has been implemented according to this scheme. Using the speech files synthesized, perception tests are conducted. The results show that the proposed scheme can indeed transform the source timbre of a female adult into the timbre of a male adult, boy, or girl. In addition, the method PLFM is shown to be better than FAS in obtaining more masculine timbre.
Atmospheric Variations: Vacillations and Index Cycles

Abstract Zonal indices, momentum flux, and mean and perturbation kinetic energies of the Southern Hemisphere upper troposphere, derived from the EOLE constant density balloon data set, are found to possess strong 18–23 day variations which are statistically significant to a posteriori levels. Cross-spectra and phase relationships suggest that the variation appears as a barotropic interaction between the middle latitude mean westerlies and ultra large-scale perturbations. Long-period and high-amplitude variations in energy conversion between eddy and mean kinetic energies are found which average out in the long term mean to be very small and positive, consistent with previously measured average hemispheric energetics. Furthermore, the variation is shown to possess an ultra-long wave preference. Examination of two independent data sources, time series of total ozone and stratospheric thickness data from the Selective Chopper Radiometer, suggests that the periodicity is a property of general hemispheric moti.
Perfect Sampling for Gibbs Processes with a focus on Hard-sphere Models

We consider the problem of generating perfect samples from a Gibbs point process, a spatial process that is absolutely continuous with respect to a Poisson point process. Examples include area-interaction processes, Strauss processes, hard-sphere model and the Ising model. Traditionally, this problem is addressed using {\it coupling from the past} (CFTP) based methodologies. In this paper, we focus on {\it acceptance-rejection} based methods. Our key contribution is a novel {\it importance sampling} based acceptance-rejection methodology for generating perfect samples from Gibbs point processes, with a specific focus on a simpler setting of hard-sphere model (defined on a unit cube) that we analyze in an asymptotic regime where the number of spheres generated increases to infinity while the sphere radius decreases to zero at varying rates. We compare analytically and numerically, the computational effort required by the proposed method with naive acceptance-rejection based methods as well as with popular dominated CFTP based algorithms. Our analysis of the hard-sphere models relies upon identifying the large deviations decay rates of no overlap probability of spheres when their centers are distributed as a homogeneous Poisson point process, these results may also be of independent interest.
Quantization of Hitchin's equations for Higgs Bundles I

We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit towards the solution of the Hitchin equation. We relate the existence of balanced metrics to the Gieseker stability of the Higgs bundle.
A new embedded variable stepsize, variable order family of low computational complexity

Variable Stepsize Variable Order (VSVO) methods are the methods of choice to efficiently solve a wide range of ODEs with minimal work and assured accuracy. However, VSVO methods have limited impact in timestepping methods in complex applications due to their computational complexity and the difficulty to implement them in legacy code. We introduce a family of implicit, embedded, VSVO methods that require only one BDF solve at each time step followed by adding linear combinations of the solution at previous time levels. In particular, we construct implicit and linearly implicit VSVO methods of orders two, three and four with the same computational complexity as variable stepsize BDF3. The choice of changing the order of the method is simple and does not require additional solves of linear or nonlinear systems.
Representations of $\mathbb{Z}_{2}$-orbifold of the parafermion vertex operator algebra $K(sl_2,k)$

In this paper, the irreducible modules for the $\mathbb{Z}_{2}$-orbifold vertex operator subalgebra of the parafermion vertex operator algebra associated to the irreducible highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ of level $k$ are classified and constructed.
The continuum limit of the Kuramoto model on sparse random graphs

In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal. , 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs. There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems on sparse graphs and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices. Finally, a Galerkin scheme is developed to show convergence of the averaged model to the continuum limit. The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erdős-R{\' e}nyi, small-world, and power law graphs.
A Direct Algorithm for Convective Adjustment of the Vertical Temperature Profile for an Arbitrary Critical Lapse Rate

Abstract An efficient direct algorithm of convective adjustment for an arbitrary critical value of the vertical temperature lapse rate γ is proposed. The algorithm provides an exact and unique solution of a standard convective adjustment problem for models with temperature specified either on nonuniformly spaced levels or for layers of different thicknesses in pressure, sigma, or other vertical coordinate related to pressure. The algorithm may be recommended for use either directly in atmospheric models not explicitly including a hydrologic cycle with prescribed γ, or as a part of more complicated parameterizations of moist convection, where γ may be calculated depending on relative humidity.
Testing for nonlinear deterministic components when the order of integration is unknown

We consider testing for the presence of nonlinearities in the deterministic component of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear deterministic components in a series.
The domination equivalence classes of paths

A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. %The domination number $G$, denoted $\gamma (G)$, is the cardinality of the smallest dominating set of $G$. The domination polynomial is defined by $D(G,x) = \sum d(G,i)x^i$ where $d(G,i)$ is the number of dominating sets in $G$ with cardinality $i$. Two graphs $G$ and $H$ are considered $\mathcal{D}$-equivalent if $D(G,x)=D(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\mathcal{D}$-equivalent to $G$. Extending previous results, we determine the equivalence classes of all paths.
Perturbation analysis of Markov modulated fluid models

ABSTRACT We consider perturbations of positive recurrent Markov modulated fluid models. In addition to the infinitesimal generator of the phases, we also perturb the rate matrix, and analyze the effect of those perturbations on the matrix of first return probabilities to the initial level. Our main contribution is the construction of a substitute for the matrix of first return probabilities, which enables us to analyze the effect of the perturbation under consideration.
Equivariant maps between representation spheres

Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal $G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V) \to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup $H\subseteq G$. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when $G$ is a torus.