ShantanuT01/DACTYL-Pretraining huggingface dataset:
Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ are prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.
Left-Right symmetric theories solve the strong CP problem and explain the small Higgs quartic coupling at high energy scales via the Higgs Parity mechanism, which forces the Higgs quartic coupling to vanish at the Left-Right symmetry breaking scale. They also predict three right-handed neutrinos; one may be stable and provide dark matter, and another may decay and explain the baryon asymmetry of the universe through leptogenesis. For the dark matter abundance to arise from freeze-out, the required range of the Left-Right symmetry breaking scale is $10^{10}$-$10^{13}$ GeV, in remarkable agreement with the energy scale at which the Higgs quartic coupling vanishes. The allowed parameter space can be probed by the warmness of dark matter, precise measurements of the top quark mass and QCD coupling constant by future colliders and lattice computations, and measurement of the neutrino mass hierarchy.
In this work, we exploit Strichartz average approach \cite{Strichartz2001} to define the Laplacian on Sierpi\'{n}ski gasket, in the construction of the finite volume method. The approach present sum similarities with the finite difference approach in terms of stability and convergence.
For each $\Delta>0$, we prove that there exists some $C=C(\Delta)$ for which the binomial random graph $G(n,C\log n/n)$ almost surely contains a copy of every tree with $n$ vertices and maximum degree at most $\Delta$. In doing so, we confirm a conjecture by Kahn.
The Laser Interferometer Space Antenna (LISA) is expected to detect gravitational radiation from a large number of compact binary systems. We present a method by which these signals can be identified and have their parameters estimated. Our approach uses Bayesian inference, specifically the application of a Markov chain Monte Carlo method. The simulation study that we present here considers a large number of sinusoidal signals in noise, and our method estimates the number of periodic signals present in the data, the parameters for these signals and the noise level. The method is significantly better than classical spectral techniques at performing these tasks and does not use stopping criteria for estimating the number of signals present.
The concept of spectrum for a class of non-linear wave equations is studied. Instead of looking for stability, the key to the spectral structure is found in the instability phenomena (bifurcations). This aspect is best seen in the `classical model' of the non-linear wave mechanics. The solitons (macro-localizations) are a part of the non-linear spectral problem; their bifurcations reflect the dynamical symmetry breaking. The computer simulations suggest that the bifurcations of the asymptotic behaviour occur also for the general, non-stationary states. A~phenomenon of the soliton splitting is observed.
Phases of elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix can be obtained using decays of $B$ mesons to $\pi^+ \pi^-$, $\pi^\pm K^\mp$, and $\pi^+ K^0$ or $\pi^- \overline{K}^0$. For $B^0~{\rm or}~ \overline{B}^0 \to \pi^+ \pi^-$, one identifies the flavor of the neutral $B$ meson at time of production and studies the time-dependence of the decay rate. The other processes are self-tagging and only their rates need be measured. By assuming flavor SU(3) symmetry and first-order SU(3) breaking, one can separately determine the phases $\gamma \equiv {\rm Arg}~V_{ub}^*$ and $\alpha = \pi - \beta - \gamma$, where $\beta \equiv {\rm Arg}~V_{td}^*$. Special cases include the vanishing of strong interaction phase differences between amplitudes, the possibility of recovering partial information when $\pi^+ \pi^-$ and $\pi^\pm K^\mp$ decays cannot be distinguished from one another, and the use of a correlation between $\gamma$ and $\alpha$ in the region of allowed parameters.
We present Keck spectroscopic observations of z>6 Lyman-break galaxy (LBG) candidates in the Subaru Deep Field (SDF). The candidates were selected as i'-dropout objects down to z'=27 AB magnitudes from an ultra-deep SDF z'-band image. With the Keck spectroscopy we identified 19 LBGs with prominent Ly_alpha emission lines at 6< z < 6.4. The median value of the Ly_alpha rest-frame equivalent widths (EWs) is ~50 A, with four EWs >100 A. This well-defined spectroscopic sample spans a UV-continuum luminosity range of -21.8< M_{UV}<-19.5 (0.6~5 L*_{UV}) and a Ly_alpha luminosity range of (0.3~3) x 10^{43} erg s^{-1} (0.3~3 L*_ {Ly_alpha}). We derive the UV and Ly_alpha luminosity functions (LFs) from our sample at <z>~6.2 after we correct for sample incompleteness. We find that our measurement of the UV LF is consistent with the results of previous studies based on photometric LBG samples at 5<z<7. Our Ly_alpha LF is also generally in agreement with the results of Ly_alpha-emitter surveys at z~5.7 and 6.6. This study shows that deep spectroscopic observations of LBGs can provide unique constraints on both the UV and Ly_alpha LFs at z>6.
Bell state measurements, in which two quantum bits are projected onto a maximally entangled state, are an essential component of quantum information science. We propose and experimentally demonstrate the projection of two quantum systems with three states (qutrits) onto a generalized maximally entangled state. Each qutrit is represented by the polarization of a pair of indistinguishable photons - a biphoton. The projection is a joint measurement on both biphotons using standard linear optics elements. This demonstration enables the realization of quantum information protocols with qutrits, such as teleportation and entanglement swapping.
Recent results demonstrating the chaotic behavior of geodesics in non-homogeneous vacuum pp-wave solutions are generalized. Here we concentrate on motion in non-homogeneous sandwich pp-waves and show that chaos smears as the duration of these gravitational waves is reduced. As the number of radial bounces of any geodesic decreases, the outcome channels to infinity become fuzzy, and thus the fractal structure of the initial conditions characterizing chaos is cut at lower and lower levels. In the limit of impulsive waves, the motion is fully non-chaotic. This is proved by presenting the geodesics in a simple explicit form which permits a physical interpretation, and demonstrates the focusing effect. It is shown that a circle of test particles is deformed by the impulse into a family of closed hypotrochoidal curves in the transversal plane. These are deformed in the longitudinal direction in such a way that a specific closed caustic surface is formed.
This paper aims at studying the Iwasawa $\lambda$-invariant of the $p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer groups in $p$-power degree extensions over non-cyclotomic $\mathbb{Z}_p$-extensions of a number field. We prove a generalization of Kida's formula in such a case. Unlike the cyclotomic $\mathbb{Z}_p$-extension, where all primes are finitely decomposed; in the $\mathbb{Z}_p$-extensions we consider, primes may be infinitely decomposed. In the second part of the paper, we study the relationship for Iwasawa invariants with respect to congruences, obtaining refinements of the results of R. Greenberg-V. Vatsal and K. Kidwell. As an application, we provide an algorithm for constructing elliptic curves with large anticyclotomic $\lambda$-invariant. Our results are illustrated by explicit computation.
Bootstrap percolation in (random) graphs is a contagion dynamics among a set of vertices with certain threshold levels. The process is started by a set of initially infected vertices, and an initially uninfected vertex with threshold $k$ gets infected as soon as the number of its infected neighbors exceeds $k$. This process has been studied extensively in so called \textit{rank one} models. These models can generate random graphs with heavy-tailed degree sequences but they are not capable of clustering. In this paper, we treat a class of random graphs of unbounded rank which allow for extensive clustering. Our main result determines the final fraction of infected vertices as the fixed point of a non-linear operator defined on a suitable function space. We propose an algorithm that facilitates neural networks to calculate this fixed point efficiently. We further derive criteria based on the Fr\'echet derivative of the operator that allows one to determine whether small infections spread through the entire graph or rather stay local.
We investigate non-equilibrium transport in the absence of spin-flip energy relaxation in a few-electron quantum dot artificial atom. Novel non-equilibrium tunneling processes involving high-spin states which cannot be excited from the ground state because of spin-blockade, and other processes involving more than two charge states are observed. These processes cannot be explained by orthodox Coulomb blockade theory. The absence of effective spin relaxation induces considerable fluctuation of the spin, charge, and total energy of the quantum dot. Although these features are revealed clearly by pulse excitation measurements, they are also observed in conventional dc current characteristics of quantum dots.
Two-dimensional semi-Dirac fermions are quasiparticles that disperse linearly in one direction and quadratically in the other. We investigate instabilities of semi-Dirac fermions towards charge, spin-density wave and superconducting orders, driven by short-range interactions. We analyze the critical behavior of the Yukawa theories for the different order parameters using Wilson momentum shell RG. We generalize to a large number $N_f$ of fermion flavors to achieve analytic control in 2+1 dimensions and calculate critical exponents at one-loop order, systematically including $1/N_f$ corrections. The latter depend on the specific form of the bosonic infrared propagator in 2+1 dimensions, which needs to be included to regularize divergencies. The $1/N_f$ corrections are surprisingly small, suggesting that the expansion is well controlled in the physical dimension. The order-parameter correlations inherit the electronic anisotropy of the semi-Dirac fermions, leading to correlation lengths that diverge along the spatial directions with distinct exponents, even at the mean-field level. We conjecture that the proximity to the critical point may stabilize novel modulated order phases.
We present new deep images of the Coma Cluster from the ESA Herschel Space Observatory at wavelengths of 70, 100 and 160 microns, covering an area of 1.75 x 1.0 square degrees encompassing the core and southwest infall region. Our data display an excess of sources at flux densities above 100 mJy compared to blank-field surveys, as expected. We use extensive optical spectroscopy of this region to identify cluster members and hence produce cluster luminosity functions in all three photometric bands. We compare our results to the local field galaxy luminosity function, and the luminosity functions from the Herschel Virgo Cluster Survey (HeViCS). We find consistency between the shapes of the Coma and field galaxy luminosity functions at all three wavelengths, however we do not find the same level of agreement with that of the Virgo Cluster.
In this contribution we summarize recent results on the transport properties of strongly correlated dilute Fermi gases. We discuss the hydrodynamic equations in the normal phase and present new results on the structure of second order terms in the gradient expansion. We also discuss predictions from kinetic theory, and show how these predictions can be tested using experimental data on elliptic flow. We summarize current constraints on the shear viscosity to entropy density ratio $\eta/s$.
The leading-order nucleon-nucleon (NN) potential derived from chiral perturbation theory consists of one-pion exchange plus a short-distance contact interaction. We show that in the 1S0 and 3S1-3D1 channels renormalization of the Lippmann-Schwinger equation for this potential can be achieved by performing one subtraction. This subtraction requires as its only input knowledge of the NN scattering lengths. This procedure leads to a set of integral equations for the partial-wave NN t-matrix which give cutoff-independent results for the corresponding NN phase shifts. This reformulation of the NN scattering equation offers practical advantages, because only observable quantities appear in the integral equation. The scattering equation may then be analytically continued to negative energies, where information on bound-state energies and wave functions can be extracted.
(abridged) We present an investigation of the clustering of i'AB<24.5 galaxies in the redshift interval 0.2<z<1.2. Using 100,000 precise photometric redshifts in the four ultra-deep fields of the Canada-France Legacy Survey, we construct a set of volume-limited galaxy catalogues. We study the dependence of the amplitude and slope of the galaxy correlation function on absolute B-band rest-frame luminosity, redshift and best-fitting spectral type. We find: 1. The comoving correlation length for all galaxies decreases steadily from z~0.3 to z~1. 2. At all redshifts and luminosities, galaxies with redder rest-frame colours have clustering amplitudes between two and three times higher than bluer ones. 3. For bright red and blue galaxies, the clustering amplitude is invariant with redshift. 4. At z~0.5, less luminous galaxies have higher clustering amplitudes of around 6 h-1 Mpc. 5. The relative bias between galaxies with red and blue rest-frame colours increases gradually towards fainter absolute magnitudes. One of the principal implications of these results is that although the full galaxy population traces the underlying dark matter distribution quite well (and is therefore quite weakly biased), redder, older galaxies have clustering lengths which are almost invariant with redshift, and by z~1 are quite strongly biased.
We consider Brox's model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t;m_(log t) + x)=t; x \in R), where m_(log t) is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case which same questions have been solved recently by N. Gantert, Y. Peres and Z. Shi.
We present an investigation of the ultraviolet and X-ray spectra of the Seyfert 1.5 galaxy Markarian 817. The ultraviolet analysis includes two recent observations taken with the Cosmic Origins Spectrograph in August and December 2009, as well as archival spectra from the International Ultraviolet Explorer and the Hubble Space Telescope. Twelve Ly-alpha absorption features are detected in the 1997 GHRS and 2009 COS spectra - of these, four are associated with high-velocity clouds in the interstellar medium, four are at low-significance, and the remaining four are intrinsic features, which vary between the GHRS and COS observations. The strongest intrinsic absorber in the 1997 spectrum has a systemic velocity of ~ -4250 km/s. The corresponding feature in the COS data is five times weaker than the GHRS absorber. The three additional weak (equivalent width from 13-54 mA) intrinsic Ly-alpha absorbers are at systemic velocities of -4100 km/s, -3550 km/s, and -2600 km/s. However, intrinsic absorption troughs from highly ionized C IV and N V, are not detected in the COS observations. No ionized absorption signatures are detected in the ~ 14 ks XMM-Newton EPIC spectra. The factor of five change in the intrinsic Ly$\alpha$ absorber is most likely due to bulk motions in the absorber, since there is no drastic change in the UV luminosity of the source from the GHRS to the COS observations. In a study of variability of Mrk 817, we find that the X-ray luminosity varies by a factor of ~40 over 20 years, while the UV continuum/emission lines vary by at most a factor of ~2.3 over 30 years. The variability of the X-ray luminosity is strongly correlated with the X-ray power-law index, but no correlation is found with the simultaneous optical/UV photometry.
The systematics of strangeness enhancement is calculated using the HIJING and VENUS models and compared to recent data on $\,pp\,$, $\,pA\,$ and $\,AA\,$ collisions at CERN/SPS energies ($200A\,\, GeV\,$). The HIJING model is used to perform a {\em linear} extrapolation from $pp$ to $AA$. VENUS is used to estimate the effects of final state cascading and possible non-conventional production mechanisms. This comparison shows that the large enhancement of strangeness observed in $S+Au$ collisions, interpreted previously as possible evidence for quark-gluon plasma formation, has its origins in non-equilibrium dynamics of few nucleon systems. % Strangeness enhancement %is therefore traced back to the change in the production dynamics %from $pp$ to minimum bias $pS$ and central $SS$ collisions. A factor of two enhancement of $\Lambda^{0}$ at mid-rapidity is indicated by recent $pS$ data, where on the average {\em one} projectile nucleon interacts with only {\em two} target nucleons. There appears to be another factor of two enhancement in the light ion reaction $SS$ relative to $pS$, when on the average only two projectile nucleons interact with two target ones.
We prove global existence of solutions to multiple speed, Dirichlet-wave equations with quadratic nonlinearities satisfying the null condition in the exterior of compact obstacles. This extends the result of our previous paper by allowing general higher order terms. In the currect setting, these terms are much more difficult to handle than for the free wave equation, and we do so using an analog of a pointwise estimate due to Kubota and Yokoyama.
We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrodinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability.
We aim to refine the sample of isolated early-type galaxies in the AMIGA catalogue via high resolution imaging. Here we report the result from a pilot study investigating two candidates, KIG 685 and KIG 895, in K-band with the laser guide star and wavefront sensing facility ARGOS} at LBT. Observations, obtained during commissioning time, achieved a PSF of ~0.25". We present the data reduction and the PSF analysis from the best closed loop exposures to investigate the galaxies' morphological structure, including their nuclear region. We used PROFILER for the decomposition of the azimuthal 1D light distribution and GALFIT for the 2D analysis, accounting for ARGOS's PSF. KIG 685 was found to be a S0 galaxy and has been modeled with two Sersic components representing a pseudo-bulge (n_{1D}=2.87+-0.21, n_{2D}=2.29+-0.10) and a disk (n_{1D}=0.95+-0.16, n_{2D}=0.78+-0.10). Nearly symmetric ring/shell-like structures emerge after subtracting the GALFIT model from the image. KIG 895 shows a clear irregular arm-like structure, in which the northern outer arm is reminiscent of a tail. The galaxy body is a disk, best fitted by a single Sersic profile (n_{1D}=1.22+-0.1; n_{2D}=1.32+-0.12), i.e. KIG 895 is a bulge-less very late-type spiral. ARGOS high resolution images clearly revealed interaction signatures in KIG 895. We suggest that the ring/shell like residuals in KIG 685, a "bona fide" early-type galaxy, point towards a past accretion event.
We show, by means of a perturbative weakly nonlinear analysis, that the axisymmetric magneto-rotational instability (MRI) in a magnetic Taylor-Couette (mTC) flow in a thin-gap gives rise, for very small magnetic Prandtl numbers (P_m), to a real Ginzburg-Landau equation for the disturbance amplitude. The saturation amplitude A_s is found to scale in this regime as P^\delta, with 1/2<\delta<2/3 (depending on the boundary conditions adopted). The asymptotic results are shown to comply with numerical calculations performed by using a spectral code. They suggest that the transport due to the nonlinearly developed MRI may be vanishingly small for P_m << 1.
Theories of stellar spiral arms in disk galaxies can be grouped into two classes based on the longevity of a spiral arm. Although the quasi-stationary density wave theory supposes that spirals are rigidly-rotating, long-lived patterns, the dynamic spiral theory predicts that spirals are differentially-rotating, transient, recurrent patterns. In order to distinguish between the two spiral models from observations, we performed hydrodynamic simulations with steady and dynamic spiral models. Hydrodynamics simulations in steady spiral models demonstrated that the dust lane locations relative to the stellar spiral arms (hereafter, arm-gas offsets) depend on radius, regardless of the strength and pitch angle of the spiral and the model of the inter-stellar medium (ISM). In contrast, we found that the dynamic spiral models show no systematic radial dependence of the arm-gas offsets. The arm-gas offset radial profile method, together with the other test methods, will help us to distinguish between the two spiral models in observed spiral galaxies.
We give a complete description of the relationship between the vanishing cycles of a complex of sheaves along a function $f$ and Thom's $a_f$ condition.
In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0,T]. The Laplace transform of A(T) follows as a consequence. The main proof involves a double Laplace transform of A(T) and is based on excursion theory and local time for Brownian motion.
An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$ with $k\le 2^{n/4300}$, the number of Gallai colorings of $K_n$ that use at most $k$ given colors is $(\binom{k}{2}+o_n(1))\,2^{\binom{n}{2}}$. Our result is asymptotically best possible and implies that, for those $k$, almost all Gallai $k$-colorings use only two colors. However, this is not true for $k \ge \Omega (2^{2n})$.
The structure of leading nonperturbative corrections to the static Coulomb potential in QCD at small distances is analyzed. We argue in favor of the correction linearly dependent on distance and remark that lattice measurements of static potential for charges in higher representations can distinguish between different phenomenological models used to describe it. Related problems of validity of Dirac quantization condition for running charges in abelian theory and significance of the quantity <A^2> are briefly discussed.