Miracle-dz/UniFAR-FTUs huggingface dataset:
Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation: $$ -\varepsilon^2\Delta v+V(x)v = \frac{1}{\varepsilon^\alpha}(I_\alpha*F(v))f(v) \quad \hbox{in}\ \mathbb{R}^N, $$ where $N\geq 3$, $\alpha\in (0,N)$, $I_\alpha(x)={A_\alpha\over |x|^{N-\alpha}}$ is the Riesz potential, $F\in C^1(\mathbb{R},\mathbb{R})$, $F'(s) = f(s)$ and $\varepsilon>0$ is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as $\varepsilon\to 0$, to a local minima of $V(x)$ under general conditions on $F(s)$. Our result is new also for $f(s)=|s|^{p-2}s$ and applicable for $p\in (\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2})$. Especially, we can give the existence result for locally sublinear case $p\in (\frac{N+\alpha}{N}, 2)$, which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least $\hbox{cupl}(K)+1$ solutions concentrating around $K$ as $\varepsilon\to 0$, where $K\subset \Omega$ is the set of minima of $V(x)$ in a bounded potential well $\Omega$, that is, $m_0 \equiv \inf_{x\in \Omega} V(x) < \inf_{x\in \partial\Omega}V(x)$ and $K=\{x\in\Omega;\, V(x)=m_0\}$.
Le lemme fondamental pondéré pour le groupe métaplectique

Abstract Dans cet article, on énonce une variante du lemme fondamental pondéré d’Arthur pour le groupe métaplectique de Weil, qui sera un ingrédient indispensable de la stabilisation de la formule des traces. Pour un corps de caractéristique résiduelle suffisamment grande, on en donne une démonstration à l’aide de la méthode de descente, qui est conditionnelle : on admet le lemme fondamental pondéré non standard sur les algèbres de Lie. Vu les travaux de Chaudouard et Laumon, on s’attend à ce que cette condition soit ultérieurement vérifiée.
Regular colored graphs of positive degree

Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of regular colored graphs of fixed positive degree and perform their exact and asymptotic enumeration. In particular we show that the generating function of the family of graphs of fixed degree is an algebraic series with a positive radius of convergence, independant of the degree. We describe the singular behavior of this series near its dominant singularity, and use the results to establish the double scaling limit of colored tensor models.
Generated coherent systems and a conjecture of D. C. Butler

Let [Formula: see text] be a general generated coherent system of type [Formula: see text] on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of [Formula: see text] to the semistability of the kernel of the evaluation map [Formula: see text]. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler’s Conjecture in some cases. The strongest results are obtained for type [Formula: see text], which is the first previously unknown case.
On the geometry of random lemniscates

We investigate the geometry of a random rational lemniscate Γ , the level set {|r(z)|=1} on the Riemann sphere C^=C∪{∞} of the modulus of a random rational function r . We assign a probability distribution to the space of rational functions r=p/q of degree n by sampling p and q independently from the complex Kostlan ensemble of random polynomials of degree n .
A universal U(1)-RCC invariant of links and rationality conjecture

We define a graph algebra version of the stationary phase integration over the coadjoint orbits in the Reshetikhin formula for the colored Jones-HOMFLY polynomial. As a result, we obtain a `universal' U(1)-RCC invariant of links in rational homology spheres, which determines the U(1)-RCC invariants based on simple Lie algebras. We formulate a rationality conjecture about the structure of this invariant.
On the global regularity for the dissipative surface quasi-geostrophic equation

In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. Global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques such as integration by parts no longer apply, to overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic type Gronwall inequality, which may be of independent interest and potential applications.
Generation of 24 T at 4.2 K using a layer-wound GdBCO insert coil with Nb3Sn and Nb–Ti external magnetic field coils

High-temperature superconducting (HTS) magnets are believed to be a practical option in the development of high field nuclear magnetic resonance (NMR) systems. The development of a 600 MHz NMR system that uses an HTS magnet and a probe with an HTS radio frequency coil is underway. The HTS NMR magnet is expected to reduce the volume occupied by the magnet and to encourage users to install higher field NMR systems. The tolerance to high tensile stress is expected for HTS conductors in order to reduce the magnet in volume. A layer-wound Gd–Ba–Cu–O (GdBCO) insert coil was fabricated in order to investigate its properties under a high electromagnetic force in a high magnetic field. The GdBCO insert coil was successfully operated at a current of up to 321 A and an electromagnetic force BJR of 408 MPa in an external magnetic field generated by Nb3Sn and Nb–Ti low-temperature superconducting coils. The GdBCO insert coil also managed to generate a magnetic field of 6.8 T at the center of the coil in an external magnetic field of 17.2 T. The superconducting magnet consisting of GdBCO, Nb3Sn and Nb–Ti coils successfully generated a magnetic field of 24.0 T at 4.2 K, which represents a new record for a superconducting magnet.
D-groups and the Dixmier-Moeglin equivalence

A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for $D$-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if $R$ is a commutative affine Hopf algebra over a field of characteristic zero, and $A$ is an Ore extension to which the Hopf algebra structure extends, then $A$ satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such $A$ are Hopf Ore extensions
Greatest lower bounds on Ricci curvature for Fano T ‐manifolds of complexity one

In this short note we determine the greatest lower bounds on Ricci curvature for all Fano T ‐manifolds of complexity one, generalizing the result of Chi Li. Our method of proof is based on the work of Datar and Székelyhidi, using the description of complexity one special test configurations given by Ilten and Süß.
Forced Convective Heat Transfer from Unconfined Isothermal and Isoflux Elliptic Cylinders

This article presents the numerical study of laminar forced convective heat transfer from elliptic cylinders of various axis ratios (AR = 0.1, 0.4, 0.6, 0.8, and 1.0), angles of attack (AOA = 30°, 45°, 60°, and 90°), and Reynolds numbers (Re = 50, 100, 150, and 200). Simulations are carried out for both isothermal and isoflux wall boundary conditions. A detailed study of flow field reveals distinct instantaneous and time-averaged flow patterns behind the elliptic cylinder. The effect of flow patterns on isotherms and, thus, on heat transfer, is analyzed in detail. Local and surface averaged Nusselt number (Nu and Nu avg ) is computed and their variation due to change in AR, AOA, and Re is studied. It is observed that increasing AR and Re increases Nu avg monotonically, while increasing AOA decreases Nu avg . Finally, correlations are proposed for Nu avg with respect to AR, AOA, and Re with minimum rms error.
Transformation & Uncertainty. Some Thoughts on Quantum Probability Theory, Quantum Statistics, and Natural Bundles

This PHD thesis is concerned with uncertainty relations in quantum probability theory, state estimation in quantum stochastics, and natural bundles in differential geometry. After some comments on the nature and necessity of decoherence in open systems and its absence in closed ones, we prove sharp, state-independent inequalities reflecting the Heisenberg principle, the necessity of decoherence and the impossibility of perfect joint measurement. These bounds are used to judge how far a particular measurement is removed from the optimal one. We do this for a qubit interacting with the quantized EM field, continually probed using homodyne detection. We calculate to which extent this joint measurement is optimal. We then propose a two-step strategy to determine the (possibly mixed) state of n identically prepared qubits, and prove that it is asymptotically optimal in a local minimax sense, using `Quantum Local Asymptotic Normality' for qubits. We propose a physical implementation of QLAN, based on interaction with the quantized EM field. In differential geometry, a bundle is called `natural' if diffeomorphisms of the base lift to automorphisms of the bundle. We slightly extend this notion to `infinitesimal naturality', requiring only vector fields (infinitesimal diffeomorphisms) to lift. We classify these bundles. Physical fields that transform under (infinitesimal) space-time transformations must be described in terms of (infinitesimally) natural bundles. All spin structures are infinitesimally natural. Our framework thus encompasses Fermionic fields, for which the notion of a natural bundle is too restrictive. Interestingly, generalized spin structures (e.g. spin-c structures) are not always infinitesimally natural. We classify the ones that are. Depending on the gauge group at hand, this can significantly reduce the number of allowed space-time topologies.
First Passage Percolation on Inhomogeneous Random Graphs

In this paper we investigate first passage percolation on an inhomogeneous random graph model introduced by Bollobás et al. (2007). Each vertex in the graph has a type from a type space, and edge probabilities are independent, but depend on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal-weight path, properly normalized, follows a central limit theorem. We handle the cases where the average number of neighbors λ̃ n of a vertex tends to a finite λ̃ in full generality and consider λ̃ = ∞ under mild assumptions. This paper is a generalization of the paper of Bhamidi et al. (2011), where first passage percolation is explored on the Erdős-Rényi graphs.
Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice

In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE$_{6}$. This is an important conjecture in mathematical physics and probability. The case of critical site percolation on the hexagonal lattice was established in the seminal work of Smirnov via proving Cardy's formula. Our proof uses a series of transformations and conditioning to construct a pair of paths: the $+\partial$CBP and the $-\partial$CBP. The convergence in the site percolation case on the hexagonal lattice allows us to obtain certain estimates on the scaling limit of the $+\partial$CBP and the $-\partial$CBP. By considering a path which is the concatenation of $+\partial$CBPs and $-\partial$CBPs in an alternating manner, we can prove the convergence in the case of bond percolation on the square lattice.
Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases

The refractive-index change in a collisionless gas is evaluated from the Stark shifts of the rotational energy levels that arise from the polarizability anisotropy. In the limit of an extremely-short-duration excitation, a multilevel coherent effect results in delayed refractive-index bursts. Both stationary and transient responses of this birefringence to an optical field are considered for symmetric-top molecules, with particular emphasis on the special case of linear molecules.
On laminar groups, Tits alternatives and convergence group actions on 𝑆2

Abstract Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups, so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.
Unboundedness of solutions to a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 1 and to a critical one in higher dimensions

Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and chemoattractant. The proof is achieved by contradiction. It uses a virial type inequality together with a boundedness from below of the Liapunov functional associated to the system. Moreover, the same method is applied also in higher dimensions and an unboundedness result is shown in the case of critical quasilinear fully parabolic Keller-Segel system for mass large enough in dimensions n\geq 3. Key words: chemotaxis, infinite-time blowup.
Reynolds number and fan/inlet coupling effects on subsonic transportinlet distortion

The paper shows that the common practice of using scale model inlet distortion to evaluate high angle-of- attack engine/inlet compatibility on subsonic commercial aircraft is conservative, especially when the inlet lower lip is separated. The angle of attack at separation is shown to he Reynolds number sensitive; the higher Reynolds number full-scale configuration separates at a higher angle. The magnitude of distortion with a separated inlet is reduced by the favorable coupling effect of the fan and inlet. HE engine inlet of a commercial transport is designed to provide optimum cruise drag and pressure recovery without any sacrifice of operational safety. The aircraft must operate trouble-free over a range of flight conditions (takeoff, climb, cruise, etc.), including emergency avoidance maneuvers. The inlet must allow satisfactory engine perfor- mance throughout these operational requirements; and during the development and certification process, tests are conducted to determine the acceptability of the design. Experience has shown that the most adverse case of total pressure distortion, which can cause engine surge or relatively high fan stresses, occurs when the inlet entrance conditions are severe enough to cause flow separation. During the development of the engine, tests are conducted which evaluate the tolerance of the engine to severe inlet distortion. The distortion selected for the test is based on wind tunnel model tests designed to investigate the aircraft operating envelope for normal and emergency operation and is simulated in engine ground tests through the use of screens (see Fig. 1) or a cross-wind generator (see Fig. 2). It has been known for some time that at a given condition, the model tests can produce higher distortions than are normally experienced in flight, resulting in conservative engine evaluations. This report will identify the magnitude of these differences and show that the major reasons for these differences can be at- tributed to scale effects and engine/inlet coupling.
Seasonal correction factors for indoor radon measurements in the United Kingdom

Indoor radon concentrations are subject to seasonal variation and, in order to reflect annual averages, measurements made over periods other than twelve months need to be adjusted accordingly. A series of measurements made in the radon affected area of southwest England as part of an epidemiological study have been used to estimate seasonal correction factors. These are found to be substantial, and result in a change in the estimated annual average of up to 35% for measurements taken over a six month period, and 56% for measurements taken over a three month period. The results agree closely with previously published correction factors based on measurements made throughout the UK in an earlier time period. This confirms the stability of such correction factors over time, and the applicability of general UK correction factors to measurements of indoor radon in the radon affected area of southwest England.
On a weighted Trudinger-Moser inequality in $\mathbb{R}^N$

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $\displaystyle Lu:=-r^{-\theta}(r^{\alpha}\vert u'(r)\vert^{\beta}u'(r))'$, where $\theta, \beta\geq 0$ and $\alpha>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted P\'olya-Szeg{\"o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights

These lecture notes discuss several related features of the exactly solvable two-dimensional corner growth model with exponentially distributed weights. A key property of this model is the availability of a fairly explicit stationary version that possesses useful independence properties. With the help of couplings and estimates, we prove the existence of Busemann functions for this model, and the precise values of the longitudinal and transversal fluctuation exponents for the stationary corner growth model. The Busemann functions in turn furnish extremals for variational formulas that describe limiting shape functions. Early versions of these notes were used at the Research School on Random Structures in Statistical Mechanics and Mathematical Physics at CIRM in Marseille Luminy (March 2017) and developed for the Proceedings of the AMS Short Course on Random Growth Models (January 2017).
Evidence for a spinodal limit of amorphous excitations in glassy systems

What is the origin of the sharp slowdown displayed by glassy systems? Physical common sense suggests there must be a concomitant growing correlation length, but finding this length has been nontrivial. In random first-order theory, it is given by the size of amorphous excitations, which depends on a balance between their mutual interfacial energy and their configurational entropy. But how these excitations disappear when crossing over to the normal high temperature phase is unclear, chiefly due to lack of data about the surface tension. We measure the energy cost for creating amorphous excitations in a model glass-former, and discover that the surface tension vanishes at a well-defined spinodal energy, above which amorphous excitations cannot be sustained. This spinodal therefore marks the true onset of glassiness.
Study of the angular fluorescence distribution of bioaerosol particles: modelling and experiment

The results of numerical modelling of the angular fluorescence distribution of the individual particles of a bioaerosol are compared with experimental data obtained by a flowthrough-optical method when studying particles of the protein ovalbumin, as well as of Bacillus subtilis spores. It is shown that the experimental data agree on the whole with the results of the model computations.
Lattice points in rotated convex domains

If $\mathcal{B}\subset \mathbb{R}^d$ ($d\geqslant 2$) is a compact convex domain with a smooth boundary of finite type, we prove that for almost every rotation $\theta\in SO(d)$ the remainder of the lattice point problem, $P_{\theta \mathcal{B}}(t)$, is of order $O_{\theta}(t^{d-2+2/(d+1)-\zeta_d})$ with a positive number $\zeta_d$. Furthermore we extend the estimate of the above type, in the planar case, to general compact convex domains.
VARIETIES WITH ONE APPARENT DOUBLE POINT

The number of apparent double points of a smooth, irreducible projective variety X of dimension n in P 2n+1 is the number of secant lines to X passing through the general point of P 2n+1 . This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow us to treat a wider class of projective varieties.
The LIL for canonical U-statistics of order 2

Let X; Xi ;i 2 N, be independent identically distributed random variables and let h(x;y)= h(y;x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, lim supn(n log log n) 1 P 1i<jn h(Xi;Xj) <1 a.s. , holds if and only if the following three conditions are satised: h is canonical for the law of X (that is, Eh(X;y)=0 for almost all y) and there exists C<1 such that, both, E(h 2 (X1;X2)^u)C log log u for all large u and supfEh(X1;X2)f(X1)g(X2):kf(X)k21;kg(X)k21;kfk1<1;kgk1<1gC.
Mod $\ell$ representations of arithmetic fundamental groups II: A conjecture of A. J. de Jong

We study deformation rings of an n-dimensional representation $\overline{\rho}$, defined over a finite field of characteristic $\ell$, of the arithmetic fundamental group $\pi_1(X)$, where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p ($ \neq \ell$). When $\overline{\rho}$ has large image, we are able to show that the resulting rings are finite flat over $\mathbf{Z}_\ell$. The proof principally uses a Galois-theoretic lifting result of the authors in Part I of this two-part work, a lifting result for cuspidal mod $\ell$ forms of Ogilvie, Taylor–Wiles systems and the result of Lafforgue. This implies a conjecture of de Jong for representations of $\pi_1(X)$ with coefficients in power series rings over finite fields of characteristic $\ell$, that have this mod $\ell$ representation $\overline{\rho}$ as their reduction. A proof of all cases of the conjecture for $\ell>2$ follows from a result announced by Gaitsgory. The methods are different.