ShantanuT01/DACTYL-Pretraining huggingface dataset:
In 2010, Bousquet-M\'elou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and $\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has long been known to be the $n$-th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-M\'elou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder.   We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension 3 given by Rabinovitch to provide a structural description of the semiorders of dimension at most 2. From this description, we are able to determine the ordinary generating for the semiorders of dimension at most 2.
Az\'{e}ma associated with an honest time L the supermartingale $Z_{t}^{L}=\mathbb{P}[L>t|\mathcal{F}_{t}]$ and established some of its important properties. This supermartingale plays a central role in the general theory of stochastic processes and in particular in the theory of progressive enlargements of filtrations. In this paper, we shall give an additive characterization for these supermartingales, which in turn will naturally provide many examples of enlargements of filtrations. In particular, we use this characterization to establish some path decomposition results, closely related to or reminiscent of Williams' path decomposition results.
One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, \ t\geq0, \ x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ of the solution $u(t_n)$ to an equilibrium state of the equation for a large set of nonnegative initial data in $L^1$. These results are new in the literature on nonlinear Fokker-Planck equations arising in the mean field theory and are also relevant to the theory of stochastic differential equations. As a matter of fact, by the above convergence result, it follows that the solution to the McKean-Vlasov stochastic differential equation corresponding to (1), which is a nonlinear distorted Brownian motion, has this equilibrium state as its unique invariant measure. Keywords: Fokker-Planck equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10.
The flat, homogeneous, and isotropic universe with a massless scalar field is a paradigmatic model in Loop Quantum Cosmology. In spite of the prominent role that the model has played in the development of this branch of physics, there still remain some aspects of its quantization which deserve a more detailed discussion. These aspects include the kinematical resolution of the cosmological singularity, the precise relation between the solutions of the densitized and non-densitized versions of the quantum Hamiltonian constraint, the possibility of identifying superselection sectors which are as simple as possible, and a clear comprehension of the Wheeler-DeWitt (WDW) limit associated with the theory in those sectors. We propose an alternative operator to represent the Hamiltonian constraint which is specially suitable to deal with these issues in a satisfactory way. In particular, with our constraint operator, the singularity decouples in the kinematical Hilbert space and can be removed already at this level. Thanks to this fact, we can densitize the quantum Hamiltonian constraint in a rigorous manner. Besides, together with the physical observables, this constraint superselects simple sectors for the universe volume, with a support contained in a single semiaxis of the real line and for which the basic functions that encode the information about the geometry possess optimal physical properties. Namely, they provide a no-boundary description around the cosmological singularity and admit a well-defined WDW limit in terms of standing waves. Both properties explain the presence of a generic quantum bounce replacing the singularity at a fundamental level, in contrast with previous studies where the bounce was proved in concrete regimes and focusing on states with a marked semiclassical behavior.
We consider a system of PDEs of Monge-Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain $\Omega\subset\mathbb{R}^n$), whose construction is based on an asymmetric Minkowski distance from the boundary of $\Omega$, was already established in [G. Crasta and A. Malusa, The distance function from the boundary in a Minkowski space, to appear in Trans. Amer. Math. Soc.]. In this paper we prove that this solution is essentially unique. A fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form, which is of some interest by itself.
We use the recently developed Kinetic Field Theory (KFT) for cosmic structure formation to show how non-linear power spectra for cosmic density fluctuations can be calculated in a mean-field approximation to the particle interactions. Our main result is a simple, closed and analytic, approximate expression for this power spectrum. This expression has two parameters characterising non-linear structure growth which can be calibrated within KFT itself. Using this self-calibration, the non-linear power spectrum agrees with results obtained from numerical simulations to within typically $\lesssim10\,\%$ up to wave numbers $k\lesssim10\,h\,\mathrm{Mpc}^{-1}$ at redshift $z = 0$. Adjusting the two parameters to optimise agreement with numerical simulations, the relative difference to numerical results shrinks to typically $\lesssim 5\,\%$. As part of the derivation of our mean-field approximation, we show that the effective interaction potential between dark-matter particles relative to Zel'dovich trajectories is sourced by non-linear cosmic density fluctuations only, and is approximately of Yukawa rather than Newtonian shape.
In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $\lambda_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $Re(\lambda_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involving the generalized Euler-Stieltjes constants of the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$ unconditionally contains all automorphic $L$-functions attached to irreducible cuspidal unitary representations of $GL_N(\mathbb{Q})$, hence, as a corollary we also derive relations equivalent to the GRH for automorphic $L$-functions.
We study supersymmetry conditions for the heterotic pure superstring preserving and N=1 supersymmetry in four dimensions directly from the curved superspace defined by the Berkovits-Howe constraints.
This paper concerns the hypersurface Bohm-Dirac model, i.e., the version of Bohmian mechanics in a relativistic space-time proposed by D\"urr et al. [1], which assumes a preferred foliation of space-time into spacelike hypersurfaces (called the time foliation) as given. We show that the leaves of the time foliation do not have to be smooth manifolds but can be allowed to have kinks. More precisely, we show that, also for leaves with kinks, the trajectories are still well defined and the appropriate $|\psi|^2$ distribution is still equivariant, so that the theory is still empirically equivalent to standard quantum mechanics. This result applies to the case where the time foliation is determined by the previously proposed law $dn=0$, since such a foliation generically has kinks.
We present a brief summary of results from our FUSE program designed to study OVI absorption in the disk of the Milky Way. As a full analysis of our data has now been published, we focus on the improvements that FUSE afforded us compared to Copernicus data published thirty years ago. We discuss FUSE's limitations in studying OVI absorption from nearby galaxies using background QSOs, but present FUSE spectra of two probes which indicate the absence of OVI (but the presence of Lyman-beta) absorption 8 and 63 kpc from a foreground galaxy. Finally, we discuss the need for a more sensitive UV spectrograph to map out the physical conditions of baryons around galaxies.
We discuss the algebraic structure of the spin chains related to high energy scattering in QCD. We study the sl(2) Yangian symmetry and possible generalizations to nonzero spin and anisotropy parameter.
B$_5$Se this work we investigate the Li adsorption properties of a hybrid 2D material, namely monolayer B5Se with first principles calculations. 2 dimensional B$_5$Se was found to have a distorted hexagonal structure with five B atoms and one Se atom at the vertices of each hexagon. The density functional theory (DFT) calculations are performed with the generalized gradient approximation (GGA) and Perdew-Burke-Ernzerhoff (PBE) exchange correlation functional. The results were inclusive of van der Waals corrections with Grimmes DFT-D2 scheme. Electrode performance metrics, as the most preferred adsorption sites and adsorption energies, open circuit anode potentials, charge density differences and specific capacities for varying adatom coverage were evaluated with the DFT calculations. The adatom diffusion barriers were evaluated with a nudged elastic band (NEB) method. The ab-initio calculations predict a maximum theoretical specific capacity of 1486.87 mAhg$^{-1}$ for Li adsorption on 2D B$_5$Se, which is over four times that of conventional Li ion battery anode materials. This coupled with an open circuit anode potential of 0.291-0.179V for different degrees of Li coverage, and a small Li diffusion barrier of 0.15eV, metallic nature of the sheet under pure and lithiated conditions and good charge density variations, make monolayer B$_5$Se a potent anode material for Li-ion battery applications.
In this paper, we prove that Linearization Stability of Einstein Field Equations is a Generic Property in the sense that within the class $\mathcal{V}$ of space-times which admit a compact Cauchy hypersurface of constant mean curvature, the subclass $\mathcal{V}_K$ of space-times which are linearization stable forms an open and dense subset of $\mathcal{V}$ under $C^\infty$ - topology. The work is based upon the work of Ebin $[1]$, Fischer, Marsden and Moncrief $[2,3]$ and Beig, Chrusciel and Schoen $[4]$.
We prove that any prime $p$ satisfying $\phi(p-1)\leq (p-1)/4$ contains two consecutive quadratic non-residues modulo $p$ neither of which is a primitive root modulo $p$.
We study little string theory on R^1 x S^5, defined by a theory which lives on type IIA N NS5 branes wrapped on S^5, using its supergravity dual. In particular we study semiclassical rotating closed strings in this background. We also consider Penrose limit of this background that leads to a plane wave on which string theory is exactly solvable.
We have recently shown that for the special case of a bipartite system with binary inputs and outputs there exist equalities in local theories which are violated by quantum theory. The amount of white noise tolerated by these equalities are twice that of inequalities. In this paper we will first introduce an inequality in bipartite qutrits systems which, if non-maximally entangled state is used instead of maximally entangled state, is violated more strongly by quantum theory. Hence reproducing the results obtained in the literature. We will then prove that our equalities in this case are violated by quantum theory too, and they tolerate much more white noise than inequalities.
We prove some existence, uniqueness and non-existence results of stochastic strong solutions for a class of stochastic transport equations with a $q$-integrable (in time), bounded and $\alpha$-H\"{o}lder continuous (in space) drift coefficient. More precisely, we show that for a Sobolev differentiable initial condition, there exists a unique stochastic strong solution when $\alpha>2/q$, while for $\alpha+1<2/q$ with spatial dimension higher than one, we can choose proper initial data and drift coefficients so that there is no stochastic strong solutions.
Borrowing techniques from cosmology, I compute the power spectrum of quantum fluctuations in (2+1)-dimensional causal dynamical triangulations, a promising discrete path integral approach to quantum gravity. The results agree with those of canonical quantization to a high degree of precision, providing strong evidence for the equivalence of the two approaches and for the validity of the discrete method.
The Perey-effect in two-body channels of $(d,p)$ reactions has been known for a long time. It arises when the nonlocal two-body deuteron-target and/or proton-target problem is approximated by a local one, manifesting itself in a reduction of the scattering channel wave functions in the nuclear interior. However, the $(d,p)$ reaction mechanism requires explicit accounting for three-body dynamics involving the target and the neutron and proton in the deuteron. Treating nonlocality of the nucleon-target interactions within a three-body context requires significant effort and demands going beyond the widely-used adiabatic approximation, which can be done using a continuum-discretized coupled-channel (CDCC) method. However, the inclusion of nonlocal interactions into the CDCC description of $(d,p)$ reactions has not been developed yet. Here, we point out that, similarly to the two-body nonlocal case, nonlocality in a three-body channel can be accounted for by introducing the Perey factors. We explain this procedure and present the first CDCC calculations to our knowledge including the Perey-effect.
Non-supersymmetric black holes carrying both electric and magnetic charge with respect to a single Kaluza-Klein gauge field have much in common with supersymmetric black holes. Angular momentum conservation and other general physics principles underlies some of their basic features. Kaluza-Klein black holes are interpreted in string theory as bound states of D6-branes and D0-branes. The microscopic theory reproduces the full nonlinear mass formula of the extremal black holes.
We have combined perturbative unitarity and renormalisation group equation arguments in order to find a dynamical way to constrain the space of the gauge couplings ($g'_1$, $\widetilde{g}$) of the so-called "Minimal $Z'$ Models". We have analysed the role of the gauge couplings evolution in the perturbative stability of the two-to-two body scattering amplitudes of the vector and scalar sectors of these models and we have shown that perturbative unitarity imposes an upper bound that is generally stronger than the triviality constraint. We have also demonstrated how this method quantitatively refines the usual triviality bound in the case of benchmark scenarios such as the $U(1)_\chi$, the $U(1)_R$ or the "pure" $U(1)_{B-L}$ extension of the Standard Model. Finally, a description of the underlying model structure in Feynman gauge is provided.
The velocity field in the lower solar atmosphere undergoes strong interactions with magnetic fields. Many authors have pointed out that power is reduced by a factor between two and three within magnetic regions, depending on frequency, depth, the radius and the magnetic strength of the flux tube. Many mechanisms have been proposed to explain the observations. In this work, SDO dopplergrams and magnetograms of 12 bipolar active regions ($\beta$ARs) at a 45 second cadence, are used to investigate the relation between velocity fluctuations and magnetic fields. We show that there is an asymmetry within $\beta$ARs, with the velocity oscillation amplitude being more suppressed in the leading polarities compared to the trailing polarities. Also, the strongest magnetic fields do not completely suppress the five-minute oscillation amplitude, neither in the spot innermost umbrae.
We present near--infrared (NIR) imaging of FBQS J164442.5+261913, one of the few $\gamma$--ray emitting Narrow Line Seyfert 1 ($\gamma-$NLSy1) galaxies detected at high significance level by $Fermi$--LAT. This study is the first morphological analysis performed of this source and the third performed of this class of objects. Conducting a detailed two--dimensional modeling of its surface brightness distribution and analysing its $J-K_s$ colour gradients, we find that FBQS J164442.5+261913 is statistically most likely hosted by a barred lenticular galaxy (SB0). We find evidence that the bulge in the host galaxy of FBQS J164442.5+261913 is not classical but pseudo, against the paradigm of powerful relativistic jets exclusively launched by giant ellipticals. Our analysis, also reveal the presence of a ring with diameter equalling the bar length ($r_{bar} = 8.13\ \textrm{kpc}\pm 0.25$), whose origin might be a combination of bar--driven gas rearrangement and minor mergers, as revealed by the apparent merger remnant in the $J$--band image. In general, our results suggest that the prominent bar in the host galaxy of FBQS J164442.5+261913 has mostly contributed to its overall morphology driving a strong secular evolution, which plays a crucial role in the onset of the nuclear activity and the growth of the massive bulge. Minor mergers, in conjunction, are likely to provide the necessary fresh supply of gas to the central regions of the host galaxy.
Recently Seiberg and Witten have proposed that noncommutative gauge theories realized as effective theories on D-brane are equivalent to some ordinary gauge theories. This proposal has been proved, however, only for the Dirac-Born-Infeld action in the approximation of neglecting all derivative terms. In this paper we explicitly construct general forms of the 2n-derivative terms which satisfy this equivalence under their assumption in the approximation of neglecting (2 n+2)-derivative terms. We also prove that the D-brane action computed in the superstring theory is consistent with the equivalence neglecting the fourth and higher order derivative terms.
We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for "energy solutions" of the stochastic Burgers equation of [GP18a] to a wider class of equations.
This article identifies the process of crystallization of thin high-$\kappa$ dielectric films and an optimal range of annealing temperature in the field of high-$\kappa$ dielectric-based metal-oxide-semiconductor (MOS) technology for its improved electrical performances. Differential Scanning Calorimetry (DSC) technique is employed to understand the thermal behaviour of thin high-$\kappa$ dielectric films of HfO$_2$, deposited by rf sputtering, on Si. The exothermic trends of the DSC signal and Grazing Incidence X-ray diffraction (GIXRD) data indicate an amorphous to crystalline transition in the high-$\kappa$ film at higher temperature. The enthalpy-temperature variation shows a glass temperature (T$_g$) at $\sim$ 590 $^o$C beyond which an amorphous to m-HfO$_2$ crystalline transition takes place. Further, the Hf-Silicate formation, observed in DSC measurement and corroborated by Fourier transformed Infrared Spectroscopy (FT-IR) studies, indicates that the process of formation of Hf-Silicate begins at $\sim$ 717 $^oC$. High-frequency (HF) capacitance-voltage $(C-V)$ and current density - voltage $(J-V)$ characteristics establish that the crystallization of the film is not the root cause of degradation of the electrical properties of the high-$\kappa$-based MOS devices, rather the device degrades due to formation of interfacial Hf-Silicate.
Massive hypermultiplets admit degenerate discrete vacua only if they form nonlinear sigma models or have gauge interaction. We discuss BPS domain walls in these theories. This talk is based on the original papers hep-th/0307274 and hep-th/0211103.
We show how to construct families of stationary hydrodynamical configurations that reproduce the observed vertical gradient of the rotation velocity of the extra-planar gas in spiral galaxies. We then present a simple model for the lagging halo of the spiral galaxy NGC 891, which is in agreement with the HI observations. Our method is based on well known properties of baroclinic solutions, and it is an elementary application of a much more general and flexible method.
We introduce an extension of the M/M/1 queueing process with a spatial structure and excluded- volume effect. The rule of particle hopping is the same as for the totally asymmetric simple exclusion process (TASEP). A stationary-state solution is constructed in a slightly arranged matrix product form of the open TASEP. We obtain the critical line that separates the parameter space depending on whether the model has the stationary state. We calculate the average length of the model and the number of particles and show the monotonicity of the probability of the length in the stationary state. We also consider a generalization of the model with backward hopping of particles allowed and an alternate joined system of the M/M/1 queueing process and the open TASEP.
Motivated by the phenomenologies of dynamic roughening of strings in random media and magnetohydrodynamics, we examine the universal properties of driven diffusive system with coupled fields. We demonstrate that cross-correlations between the fields lead to amplitude-ratios and scaling exponents varying continuosly with the strength of these cross-correlations. The implications of these results for experimentally relevant systems are discussed.
Let $\Omega(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $\omega(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $\omega(N)$ and $\Omega(N)$ when $N$ is an odd perfect number. We improve on these inequalities. In particular, we show that if $3 \not| N$, then $\Omega \geq \frac{8}{3}\omega(N)-\frac{7}{3}$ and if $3 |N$ then $\Omega(N) \geq \frac{21}{8}\omega(N)-\frac{39}{8}.$
Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level $N$. As an application, we consider a Waring-type problem with sums of mixed powers.