ShantanuT01/DACTYL-Pretraining huggingface dataset:
The solution of the classical Coupon Collector's Problem is based on the assumptions that all stickers are independently and uniformly distributed. We can prove statistically as well as analytically that in particular the assumption of independence is not fulfilled in the field. To achieve this result we have systematically modeled and analyzed the production process of the stickers by combinatorial methods. We have also derived measures for the deviation from independence. Thus the results for the classical Coupon Collector's Problem are not valid for practical applications anymore, but we can show that they constitute an a upper bound. This means in particular that in practice the deviation is advantageous for the collectors, not a fraud of the vendors.   ------   Im klassischen Sammelbilderproblem wird angenommen, dass die Sammelbilder gleichverteilt und unabh\"angig sind, d. h. alle Bilder kommen gleich h\"aufig vor und werden zuf\"allig auf die P\"ackchen verteilt. Wir k\"onnen sowohl statistisch als auch analytisch nachweisen, dass insbesondere die Annahme der Zuf\"alligkeit bzw. Unabh\"angigkeit in der Praxis nicht erf\"ullt ist. Dazu haben wir den Herstellungsprozess der Sammelbilder systematisch mit kombinatorischen Methoden untersucht. Wir k\"onnen auch Ma{\ss}e daf\"ur angeben, wie gro{\ss} die Abweichung vom Zufall ist. Damit sind die Ergebnisse des klassischen Sammelbildermodells in der Praxis nicht mehr g\"ultig, aber wir k\"onnen zeigen, dass sie eine Absch\"atzung nach oben darstellen. Dies bedeutet insbesondere, dass kein Betrug durch den Hersteller vorliegt, sondern vielmehr eine Bevorteilung der Sammler.
An electron moving in a magnetically ordered background feels an effective magnetic field that can be both stronger and more rapidly varying than typical externally applied fields. One consequence is that insulating magnetic materials in three dimensions can have topologically nontrivial properties of the effective band structure. For the simplest case of two bands, these "Hopf insulators" are characterized by a topological invariant as in quantum Hall states and Z_2 topological insulators, but instead of a Chern number or parity, the underlying invariant is the Hopf invariant that classifies maps from the 3-sphere to the 2-sphere. This paper gives an efficient algorithm to compute whether a given magnetic band structure has nontrivial Hopf invariant, a double-exchange-like tight-binding model that realizes the nontrivial case, and a numerical study of the surface states of this model.
Lyman-alpha (Lya) photons that escape the interstellar medium of star-forming galaxies may be resonantly scattered by neutral hydrogen atoms in the circumgalactic and intergalactic media, thereby increasing the angular extent of the galaxy's Lya emission. We present predictions of this extended, low surface brightness Lya emission based on radiative transfer modeling in a cosmological reionization simulation. The extended emission can be detected from stacked narrowband images of Lya emitters (LAEs) or of Lyman break galaxies (LBGs). Its average surface brightness profile has a central cusp, then flattens to an approximate plateau beginning at an inner characteristic scale below ~0.2 Mpc (comoving), then steepens again beyond an outer characteristic scale of ~1 Mpc. The inner scale marks the transition from scattered light of the central source to emission from clustered sources, while the outer scale marks the spatial extent of scattered emission from these clustered sources. Both scales tend to increase with halo mass, UV luminosity, and observed Lya luminosity. The extended emission predicted by our simulation is already within reach of deep narrowband photometry using large ground-based telescopes. Such observations would test radiative transfer models of emission from LAEs and LBGs, and they would open a new window on the circumgalactic environment of high-redshift star-forming galaxies.
(Abridged) We present an analysis of X-ray observations made of the Galactic supernova remnants (SNRs) HB21 (G89.0+4.7) and CTB 1 (G116.9+0.2), two well-known mixed-morphology (MM) SNRs. We find a marked contrast between the X-ray properties of these SNRs: for HB21, the extracted ASCA spectra of the northwest and southeast regions of the X-ray emitting plasma can be fit with a single thermal model with marginally enhanced silicon and sulfur abundances. For both of these regions, the derived column density and temperature are N_H~0.3x10^22 cm^-2 and kT~0.7 keV, respectively. No significant spatial differences in temperature or elemental abundances between the two regions are detected and the X-ray-emitting plasma in both regions is close to ionization equilibrium. Our Chandra spectral analysis of CTB 1 reveals that this source is likely an oxygen-rich SNR with enhanced abundances of oxygen and neon. The extracted ASCA spectra for the southwestern and northeastern regions of CTB 1 cannot be fit with a single thermal component. Based on our fits to these spectra, we derive a column density N_H~0.6x10^22 cm^-2 and a temperature for the soft thermal component of kT_soft~0.28 keV. The hard emission from the southwest may be modeled with either a thermal component (kT_hard~3 keV) or by a power law component (Gamma~2-3) while the hard emission from the northeast may be modeled with a power law component (Gamma~1.4). We have also extracted ASCA GIS spectra of the discrete X-ray source 1WGA J0001.4+6229 which is seen in projection toward CTB 1. These spectra are best fit using a power-law model with a photon index Gamma=2.2^{+0.5}_{-1.2} which is typical for featureless power-law continua produced by rotation-powered pulsars. This source may be a neutron star associated with CTB 1.
The approximately symmetric form of the PMNS matrix suggests that there could exist a hidden symmetry which makes the PMNS matrix different from the CKM matrix for quarks. In literature, all the proposed fully symmetric textures exhibit an explicit $\mu-\tau$ symmetry in addition to other symmetries which may be different for various textures. Observing obvious deviations of the practical PMNS matrix elements from those in the symmetric textures, there must be a mechanism to distort the symmetry. It might be due to existence of light sterile neutrinos. In this work, we study the case of the Tribimaximal texture and propose that its apparent symmetry disappears due to existence of a sterile neutrino. We observe that introducing just one sterile neutrino is still not sufficient to recover the data, thus a slight $\mu-\tau$ symmetry breaking is also needed. By considering those factors, we obtain the PMNS matrix elements which are consistent with data within the experimental tolerance.
We study the theory of noncommutative U(N) Yang-Mills field interacting with scalar and spinor fields in the fundamental and the adjoint representations. We include in the action both the terms describing interaction between the gauge and the matter fields and the terms which describe interaction among the matter fields only. Some of these interaction terms have not been considered previously in the context of noncommutative field theory. We find all counterterms for the theory to be finite in the one-loop approximation. It is shown that these counterterms allow to absorb all the divergencies by renormalization of the fields and the coupling constants, so the theory turns out to be multiplicatively renormalizable. In case of 1PI gauge field functions the result may easily be generalized on an arbitrary number of the matter fields. To generalize the results for the other 1PI functions it is necessary for the matter coupling constants to be adapted in the proper way. In some simple cases this generalization for a part of these 1PI functions is considered.
The ground states of Bose-Einstein condensates of spin-2 bosons are classified into three distinct (ferromagnetic, ^^ ^^ antiferromagnetic", and cyclic) phases depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while low-lying excitation spectra are investigated in the hermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied agnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet ^^ ^^ pairs" undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson ^^ ^^ trios" condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one single-particle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.
We have derived the global mass functions of a sample of 35 Galactic globular clusters by comparing deep Hubble Space Telescope photometry with suitable multimass dynamical models. For a subset of 29 clusters with available radial velocity information we were also able to determine dynamical parameters, mass-to-light ratios and the mass fraction of dark remnants. The derived global mass functions are well described by single power-laws in the mass range $0.2 < m/M_\odot < 0.8$ with mass function slopes $\alpha>-1$. Less evolved clusters show deviations from a single-power law, indicating that the original shape of their mass distribution was not a power-law. We find a tight anticorrelation between the present-day mass function slopes and the half-mass relaxation times, which can be understood if clusters started from the same universal IMF and internal dynamical evolution is the main driver in shaping the present-day mass functions. Alternatively, IMF differences correlated with the present-day half-mass relaxation time are needed to explain the observed correlation. The large range of mass function slopes seen for our clusters implies that most globular clusters are dynamically highly evolved, a fact that seems difficult to reconcile with standard estimates for the dynamical evolution of clusters. The mass function slopes also correlate with the dark remnant fractions indicating a preferential retention of massive remnants in clusters subject to high mass-loss rates.
In this paper, we obtain new drift-type conditions for nonexplosion and explosion for nonhomogeneous Markov pure jump processes in Borel state spaces. The conditions are sharp; e.g., the one for nonexplosion is necessary if the state space is in addition locally compact and the $Q$-function satisfies weak Feller-type and local boundedness conditions. We comment on the relations of our conditions with the existing ones in the literature, and demonstrate some possible applications.
In Molecular Dynamics (MD) simulations, interactions between water molecules and graphitic surfaces are often modeled as a simple Lennard-Jones potential between oxygen and carbon atoms. A possible method for tuning this parameter consists of simulating a water nanodroplet on a flat graphitic surface, measuring the equilibrium contact angle, extrapolating it to the limit of a macroscopic droplet and finally matching this quantity to experimental results. Considering recent evidence demonstrating that the contact angle of water on a graphitic plane is much higher than what was previously reported, we estimate the oxygen-carbon interaction for the recent SPC/Fwwater model. Results indicate a value of about 0.2 kJ/mol, much lower than previous estimations. We then perform simulations of cylindrical water filaments on graphitic surfaces, in order to compare and correlate contact angles resulting from these two different systems. Results suggest that modified Young's equation does not describe the relation between contact angle and drop size in the case of extremely small systems and that contributions different from the one deriving from contact line tension should be taken into account.
A $k$-improper edge coloring of a graph $G$ is a mapping $\alpha:E(G)\longrightarrow \mathbb{N}$ such that at most $k$ edges of $G$ with a common endpoint have the same color. An improper edge coloring of a graph $G$ is called an improper interval edge coloring if the colors of the edges incident to each vertex of $G$ form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph $G$ defined as the smallest $k$ such that $G$ has a $k$-improper interval edge coloring; we denote the smallest such $k$ by $\mu_{\mathrm{int}}(G)$. We prove upper bounds on $\mu_{\mathrm{int}}(G)$ for general graphs $G$ and for particular families such as bipartite, complete multipartite and outerplanar graphs; we also determine $\mu_{\mathrm{int}}(G)$ exactly for $G$ belonging to some particular classes of graphs. Furthermore, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer $k$, there exists a graph $G$ with $\mu_{\mathrm{int}}(G) =k$. Finally, for graphs with at least two vertices we prove a new upper bound on the number of colors used in an improper interval edge coloring.
We analyze the geometric phase for an open quantum system when computed by resorting to a stochastic unravelling of the reduced density matrix (quantum jump approach or stochastic Schrodienger equations). We show that the resulting phase strongly depends on the type of unravelling used for the calculations: as such, this phase is 'not' a geometric object since it depends on non-physical parameters which are not related to the path followed by the density matrix during the evolution of the system.
The Einstein equations for an isotropic and homogeneous Friedmann--Robertson--Walker Universe in the presence of a quintessence scalar field are shown to be described in a unified way, formally identical to the dynamics of a relativistic particle moving on a two--dimensional spacetime. The correct Lagrangian for the system is presented and used to construct a spinor quantum cosmology theory using Breit's prescription. The theory is supersymmetric when written in Majorana representation. The spinor components interact through a potential that unifies spacetime curvature and quintessence. An exact supersymmetric solution for $k=0$ case is exhibited. This quantum cosmology model may be interpreted as a theory of interacting universes.
Given two or more non-commuting observables, it is generally not possible to simultaneously assign precise values to each. This quantum mechanical uncertainty principle is widely understood to be encapsulated by some form of uncertainty relation, expressing a trade-off between the standard deviations or other measures of uncertainty of two (or more) observables, resulting from their non-commutativity. Typically, such relations are coarse, and miss important features. It was not until very recently that a broader perspective on quantum uncertainty was envisaged and explored, one that utilises the notion of an uncertainty region. Here we review this new approach, illustrating it with pairs or triples of observables in the case of qubit and qutrit systems. We recall some of the shortcomings of traditional uncertainty relations, and highlight their inability to identify the full uncertainty region. These shortcomings suggest a precautionary note that, we surmise, ought to accompany the presentation of the uncertainty principle in introductory quantum mechanics courses.
Luttinger's theorem for Fermi liquids equates the volume enclosed by the Fermi surface in momentum space to the electron filling, independent of the strength and nature of interactions. Motivated by recent momentum balance arguments that establish this result in a non-perturbative fashion [M. Oshikawa, Phys. Rev. Lett. {\bf 84}, 3370 (2000)], we present extensions of this momentum balance argument to exotic systems which exhibit quantum number fractionalization focussing on $Z_2$ fractionalized insulators, superfluids and Fermi liquids. These lead to nontrivial relations between the particle filling and some intrinsic property of these quantum phases, and hence may be regarded as natural extensions of Luttinger's theorem. We find that there is an important distinction between fractionalized states arising naturally from half filling versus those arising from integer filling. We also note how these results can be useful for identifying fractionalized states in numerical experiments.
A subset $R\subseteq V(G)$ of a graph $G$ is a general position set if any triple set $R_0$ of $R$ is non-geodesic in $G$, that is, no vertex of $R_0$ lies on any geodesic between the other two vertices of $R_0$ in $G$. Let $\mathcal{R}$ be the set of general position sets of a graph $G$. The general position number of a graph $G$, denoted by $gp(G)$, is defined as $gp(G)=\max\{|R|:R\in\mathcal{R}\}$. In this paper, we determine the bounds on the gp-numbers for any maximal outerplane graph and characterize the corresponding extremal graphs.
We provide a fairly complete discussion of electronic properties of nanochains modelling the simplest quantum nanowires, within the recently proposed approach combining Exact Diagonalization in the Fock space with an Ab Initio calculations (EDABI method). In particular, the microscopic parameters of the second-quantized Hamiltonian are determined and the evolution of the system properties is traced in a systematic manner as a function of the interatomic distance (the lattice parameter, R). Both the many-particle ground state and the dynamical correlation functions are discussed within a single scheme. The principal physical results are: (i) the evolution of the electron momentum distribution and its analysis in terms of the Tomonaga-Luttinger scaling, (ii) the appearance of mixed metallic and insulating features partial localization) for the half-filled band case, (iii) the appearence of a universal renormalized dispersion relation of electron energy, which incorporates both the band-structure and the Hubbard-splitting features in the presence of electron interactions, and (iv) the transformation from a highly-conducting nanometalic state to the charge-ordered nanoinsulator in the quarter-filled case. The analysis is performed using an adjustable Gaussian 1s-like basis set composing the Wannier functions, as well as includes the long-range Coulomb interaction.
We study the presence of entangled states of nucleon pairs from nuclear decays and in reactions with exotic nuclei, e.g., 11Li, or 6He. It is shown that the fingerprints of entangled states in these subsystems are visible in correlation measurements and can be accessed with present experimental techniques. This shows that not only atomic and optical systems, but also nuclear systems serve as important tools to obtain dichotomic outcomes for tests of the Einstein-Podolsky-Rosen paradox.
It is shown that, when f'' is non-vanishing, metric f(R) gravity is completely equivalent to a scalar-tensor theory (with zero Brans-Dicke parameter) with respect to perturbations of de Sitter space, contrary to previous expectations. Moreover, the stability conditions of de Sitter space with respect to homogeneous and inhomogeneous perturbations coincide in most scalar-tensor theories, as is the case in metric f(R) gravity.
Let $F$ be a finite field of order $q$ and characteristic $p$. Let $\mathbb{Z}_F=F[t]$, $\mathbb{Q}_F=F(t)$, $\mathbb{R}_F=F((1/t))$ equipped with the discrete valuation for which $1/t$ is a uniformizer, and let $\mathbb{T}_F=\mathbb{R}_F/\mathbb{Z}_F$ which has the structure of a compact abelian group. Let $d$ be a positive integer and let $A$ be a $d\times d$-matrix with entries in $\mathbb{Z}_F$ and non-zero determinant. The multiplication-by-$A$ map is a surjective endomorphism on $\mathbb{T}_F^d$. First, we compute the entropy of this endomorphism; the result and arguments are analogous to those for the classical case $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Second and most importantly, we resolve the algebraicity problem for the Artin-Mazur zeta function of all such endomorphisms. As a consequence of our main result, we provide a complete characterization and an explicit formula related to the entropy when the zeta function is algebraic.
A global definition of time-asymmetry is presented. Schulman's two arrows of time model is criticized.
We show that the solutions of SU(2) Yang-Mills-dilation and Einstein-Yang-Mills-dilaton theories described in a sequence of papers by Kleihaus and Kunz are not regular in the gauge field part.
Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.
This paper presents OpenREALM, a real-time mapping framework for Unmanned Aerial Vehicles (UAVs). A camera attached to the onboard computer of a moving UAV is utilized to acquire high resolution image mosaics of a targeted area of interest. Different modes of operation allow OpenREALM to perform simple stitching assuming an approximate plane ground, or to fully recover complex 3D surface information to extract both elevation maps and geometrically corrected orthophotos. Additionally, the global position of the UAV is used to georeference the data. In all modes incremental progress of the resulting map can be viewed live by an operator on the ground. Obtained, up-to-date surface information will be a push forward to a variety of UAV applications. For the benefit of the community, source code is public at https://github.com/laxnpander/OpenREALM.
We prove a Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations, which improves a classical result by Crandall and Rabinowitz in the case where basic spaces are Hilbert spaces. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular, our theorem can be applied to semilinear equations in unbounded domains of $\mathbb{R}^n$.
The magnetic circular dichroism and the Faraday rotation are the fundamental phenomena of great practical importance arising from the breaking of the time reversal symmetry by a magnetic field. In most materials the strength and the sign of these effects can be only controlled by the field value and its orientation. Furthermore, the terahertz range is lacking materials having the ability to affect the polarisation state of the light in a non-reciprocal manner. Here we demonstrate, using broadband terahertz magneto-electro-optical spectroscopy, that in graphene both the magnetic circular dichroism and the Faraday rotation can be modulated in intensity, tuned in frequency and, importantly, inverted using only electrostatic doping at a fixed magnetic field. In addition, we observe strong magneto-plasmonic resonances in a patterned array of graphene antidots, which potentially allows exploiting these magneto-optical phenomena in a broad THz range.