ShantanuT01/DACTYL-Pretraining huggingface dataset:
We generalize the Szemer\'edi-Trotter incidence theorem, to bound the number of complete \emph{flags} in higher dimensions. Specifically, for each $i=0,1,\ldots,d-1$, we are given a finite set $S_i$ of $i$-flats in $\R^d$ or in $\C^d$, and a (complete) flag is a tuple $(f_0,f_1,\ldots,f_{d-1})$, where $f_i\in S_i$ for each $i$ and $f_i\subset f_{i+1}$ for each $i=0,1,\ldots,d-2$. Our main result is an upper bound on the number of flags which is tight in the worst case.   We also study several other kinds of incidence problems, including (i) incidences between points and lines in $\R^3$ such that among the lines incident to a point, at most $O(1)$ of them can be coplanar, (ii) incidences with Legendrian lines in $\R^3$, a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in $\R^3$ (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case.   Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems.
In this paper, we study the problem of energy conservation for the solutions to the incompressible viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded Lipschitz domain $\Omega$ in $\mathbb{R}^d\,\, (d\geq 2)$. We prove that under the Shinbrot type conditions $ u \in L^{q}_{loc}\left(0, T ; L^{p}(\Omega)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } {\bf F} \in L^{r}_{loc}\left(0, T ; L^{s}(\Omega)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $, the boundary conditions $u|_{\partial\Omega}=0,\,\,{\bf F}\cdot n|_{\partial\Omega}=0$ can inhibit the boundary effect and guarantee the validity of energy equality. Next, we apply this idea to deal with the case $\Omega= \mathbb{R}^d\,\,(d=2, 3, 4)$, and showed that the energy is conserved for $u\in L_{loc}^{q}\left(0,T;L_{loc}^{p}\left(\mathbb{R}^{d}\right)\right)$ with $ \frac{2}{q}+\frac{2}{p}\leq1, p\geq 4 $ and $   {\bf F}\in L_{loc}^{r}\left(0,T;L_{loc}^{s}\left(\mathbb{R}^{d}\right)\right)\cap L^{\frac{4d+8}{d+4}}\left(0,T;L^{\frac{4d+8}{d+4}}\left(\mathbb{R}^{d}\right)\right)$ with $\frac{2}{r}+\frac{2}{s}\leq1, s\geq 4 $. This result shows that the behavior of solutions in the finite regions and the behavior at infinite play different roles in the energy conservation. Finally, we consider the problem of energy conservation for distributional solutions and show energy equality for the distributional solutions belonging to the so-called Lions class $L^4L^4$.
It is now well established that there is no lower bound for the energy dissipated during a computation. The relevance of the zero-energy limit is unclear, however, because it entails computations that are unreliable or infinitely slow, or both. In contrast the thermodynamic costs of deterministic computations that complete in finite time is less well understood. We propose a model of universal computation based on Markovian stochastic processes. While strictly deterministic computation is not possible in such systems, we show that the scaling of the energy consumption in relation to the accuracy of the computation is benign. This enables quasi-deterministic computation at modest cost in energy and completing within finite time.
We discuss a closure of commutator algebras for general functionals in terms of Nambu-Goldstone fermions and their derivative terms under nonlinear supersymmetry (NLSUSY) both in flat spacetime and in curved spacetime. We point out that variations of functionals for vector supermultiplets (uniquely) determine general LSUSY transformations for linear vector supermutiplets with general auxiliary fields in extended SUSY, where the closure of the commutator algebras for NLSUSY plays a crucial role.
A common yet potentially dangerous task is the act of crossing the street. Pedestrian accidents contribute a significant amount to the high number of annual traffic casualties, which is why it is crucial for pedestrians to use safety measures such as a crosswalk. However, people often forget to activate a crosswalk light or are unable to do so -- such as those who are visually impaired or have occupied hands. Other pedestrians are simply careless and find the crosswalk signals a hassle, which can result in an accident where a car hits them. In this paper, we consider an improvement to the crosswalk system by designing a system that can detect pedestrians and triggering the crosswalk signal automatically. We collect a dataset of images that we then use to train a convolutional neural network to distinguish between pedestrians (including bicycle riders) and various false alarms. The resulting system can capture and evaluate images in real time, and the result can be used to automatically activate systems a crosswalk light. After extensive testing of our system in real-world environments, we conclude that it is feasible as a back-up system that can compliment existing crosswalk buttons, and thereby improve the overall safety of crossing the street.
We consider a compressible flow structure interaction (FSI) PDE system which is linearized about some reference rest state. The deformable interface is under the effect of an ambient field generated by the underlying and unbounded material derivative term which further contributes to the non-dissipativity of the FSI system, with respect to the standard energy inner product. In this work we show that, on an appropriate subspace, only one dimension less than the entire finite energy space, the FSI system is wellposed, and is moreover associated with a continuous semigroup which is \emph{uniformly bounded} in time. Our approach involves establishing maximal dissipativity with respect to a special inner product which is equivalent to the standard inner product for the given finite energy space. Among other technical features, the necesssary PDE estimates require the invocation of a multiplier which is intrinsic to the given compressible FSI system.
Inspired by the phenomenon of co-channel interference in communication network, a novel graph parameter, called $r$-vertex-strongly-distinguishing total coloring (abbreviate as $D(r)$-VSDTC), is proposed in this paper. Given a graph $G$, an $r$-VSDTC is an assignment of $k$ colors to $V(G)\cup E(G)$ such that any two adjacent or incident elements receive different colors and any two vertices with distance at most $r$ have distinct color-set, where the color-set of a vertex $u$ is the set of colors assigned on $u$ and its neighborhoods and incident edges. The \emph{$r$-vertex-strongly-distinguishing total chromatic number} of $G$, denoted by $\chi_{r-vsdt}(G)$, is the minimum integer $k$ for which $G$ admits a $k$-$D(r)$-VSDTC. We show that $\chi_{1-vsdt}(G)\leq 4\Delta(G)$ for every graph $G$ without isolated edges and $\chi_{1-vsdt}(G)\le k\Delta(G)+3$ for a $k$-degenerated graph $G$ without isolated edges, where $1\le k\le 3$.
We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In 'generic' cases this gives a bijection between the set Enr(X) of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank 20 we prove that the fibres of the map from Enr(X) to Br(X)[2] above the non-zero points have the same order.
We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-\Delta)^su=e^u\quad\mathrm{in}\quad \Omega\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$
A general method to easily build global and relative operators for any number n of elementary systems if they are defined for 2 is presented. It is based on properties of the morphisms valued in the tensor products of algebras of the kinematics and it allows also the generalization to any n of relations demon- strated for two. The coalgebra structures play a peculiar role in the explicit constructions. Three examples are presented concerning the Galilei, Poincare' and deformed Galilei algebras.
Phonons are known to generate a thermal Hall effect in certain insulators, including oxides with rare-earth impurities, quantum paraelectrics, multiferroic materials and cuprate Mott insulators. In each case, a special feature of the material is presumed relevant for the underlying mechanism that confers chirality to phonons in a magnetic field. The question is whether a phonon Hall effect is an unusual occurrence - linked to special characteristics such as skew scattering off rare-earth impurities, structural domains, ferroelectricity, ferromagnetism - or a much more common property of insulators than hitherto believed. To help answer this question, we have turned to a simple insulator, with none of the previously encountered special features: the cubic antiferromagnet Cu$_3$TeO$_6$. We find that it has the largest thermal Hall conductivity $\kappa_{\rm{xy}}$ of any insulator so far. We show that this record-high $\kappa_{\rm{xy}}$ signal is due to phonons and it does not require the presence of magnetic order, as it persists above the ordering temperature. We conclude that the phonon Hall effect is likely to be a fairly common property of solids.
We report the finding of an azimuthal dynamo wave of a low-order (m=1) mode in direct numerical simulations (DNS) of turbulent convection in spherical shells. Such waves are predicted by mean field dynamo theory and have been obtained previously in mean-field models. Observational results both from photometry and Doppler imaging have revealed persistent drifts of spots for several rapidly rotating stars, but, although an azimuthal dynamo wave has been proposed as a possible mechanism responsible for this behavior, it has been judged as unlikely, as practical evidence for such waves from DNS has been lacking. The large-scale magnetic field in our DNS, which is due to self-consistent dynamo action, is dominated by a retrograde m=1 mode. Its pattern speed is nearly independent of latitude and does not reflect the speed of the differential rotation at any depth. The extrema of magnetic m=1 structures coincide reasonably with the maxima of m=2 structures of the temperature. These results provide direct support for the observed drifts being due to an azimuthal dynamo wave.
The purpose of this article is a set-indexed extension of the well-known Ornstein-Uhlenbeck process. The first part is devoted to a stationary definition of the random field and ends up with the proof of a complete characterization by its $L^2$-continuity, stationarity and set-indexed Markov properties. This specific Markov transition system allows to define a general \emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial probability measure. Finally, in the multiparameter case, the SIOU process is proved to admit a natural integral representation.
In this paper, we establish automorphic properties and asymptotic behaviors of characters due to Kac-Wakimoto pertaining to $s\ell(m|n)^\wedge$ highest weight modules in the case $m=n$, extending work of the first author and Ono \cite{BOKac} and the first two authors \cite{BF} which pertains to the case $m>n$.
Domain adaptation (DA) is transfer learning which aims to leverage labeled data in a related source domain to achieve informed knowledge transfer and help the classification of unlabeled data in a target domain. In this paper, we propose a novel DA method, namely Robust Data Geometric Structure Aligned, Close yet Discriminative Domain Adaptation (RSA-CDDA), which brings closer, in a latent joint subspace, both source and target data distributions, and aligns inherent hidden source and target data geometric structures while performing discriminative DA in repulsing both interclass source and target data. The proposed method performs domain adaptation between source and target in solving a unified model, which incorporates data distribution constraints, in particular via a nonparametric distance, i.e., Maximum Mean Discrepancy (MMD), as well as constraints on inherent hidden data geometric structure segmentation and alignment between source and target, through low rank and sparse representation. RSA-CDDA achieves the search of a joint subspace in solving the proposed unified model through iterative optimization, alternating Rayleigh quotient algorithm and inexact augmented Lagrange multiplier algorithm. Extensive experiments carried out on standard DA benchmarks, i.e., 16 cross-domain image classification tasks, verify the effectiveness of the proposed method, which consistently outperforms the state-of-the-art methods.
Highly precise weak lensing shear measurement is required for statistical weak gravitational lensing analysis such as cosmic shear measurement to achieve severe constrain on the cosmological parameters. For this purpose any systematic error in the measurement should be corrected. One of the main systematic error comes from Pixel noise which is Poisson noise of flux from atmosphere. We investigate how the pixel noise makes systematic error in shear measurement based on ERA method and develop the correction method. This method is tested by simulations with various conditions and it is confirmed that the correction method can correct $80 \sim 90\%$ of the systematic error except very low signal to noise ratio galaxies.
Molecular cold gas and star formation have been observed at centers of cool-core clusters, albeit at a level much smaller than expected from the classic cooling model. Feedback from the supermassive black hole is likely to have prevented hot gas from cooling. However, the exact cooling and heating processes are poorly understood. The missing key piece is the link between the hot gas ($10^7$\,K) and cold gas ($10^3$\,K). Using the extreme ultraviolet spectrometer onboard {\sl Hisaki}, we explore a distant galaxy cluster, RCS2 J232727.6-020437, one of the most massive cool-core clusters with a cooling rate of $400$\,M$_{\odot}$\,yr$^{-1}$. We aim to detect gas at intermediate temperatures ($3\times10^4$\,K) emitting He I$\alpha$ and He I$\beta$ at rest wavelengths of 58.4 nm and 53.7 nm, respectively. Our target resides at $z=0.6986$, for which these He I lines shift away from the absorption of the Galaxy. Our findings show that the amount of $10^{4-5}$\,K gas at the center of this cluster is smaller than expected if cooling there was uninhibited, which demonstrates that feedback both operates and is efficient for massive clusters at these epochs.
We have considered the existence of neutron star magnetic field given by the cyclotron lines. We collected the data of 9 sources of high-mass X-ray binaries with supergiant companions as a case of testing our model, to demonstrate their distribution and evolution. The wind velocity, spin period and magnetic field strength are studied under different mass loss rate. In our model, correlations between mass-loss rate and wind velocity are found and can be tested in further observations. We examined the parameter space where wind accretion is allowed, avoiding barrier of rotating magnetic fields, with robust data of magnetic field of neutron stars. Our model shows that most of sources (6 of 9 systems) can be fed by the wind with relatively slow velocity, and this result is consistent with previous predictions. In a few sources, our model cannot fit under the standard wind accretion scenario. In these peculiar cases, other scenarios (disk formation, partial Roche lobe overflow) should be considered. This would provide information about the evolutionary tracks of various types of binaries, and thus show a clear dichotomy behavior in wind-fed X-ray binary systems.
We consider a generalization of classical results of Freidlin and Wentzell to the case of time dependent dissipative drifts. We show the convergence of diffusions with multiplicative noise in the zero limit of a diffusivity parameter to the related dynamical systems. The solution to the associated transport equation is obtained as an application.
The magnetic dipole field of ultraluminous X-ray pulsars may not be very high. However, it is too early to say that they are not magnetars. The existence of low magnetic field magnetars should be taken into consideration.
We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A short survey on the structure of the Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's question on the rationality of the irreducible components of Chow varieties.
We review the fabrication and key transport properties of graphene double layers, consisting of two graphene monolayers placed in close proximity, independently contacted, and separated by an ultra-thin dielectric. We outline a simple band structure model relating the layer densities to the applied gate and inter-layer biases, and show that calculations and experimental results are in excellent agreement both at zero and in high magnetic fields. Coulomb drag measurements, which probe the electron-electron scattering between the two layers reveal two distinct regime: (i) diffusive drag at elevated temperatures, and (ii) mesoscopic fluctuation-dominated drag at low temperatures. We discuss the Coulomb drag results within the framework of existing theories.
We consider exit problems for small white noise perturbations of a dynamical system generated by a vector field, and a domain containing a critical point with all positive eigenvalues of linearization. We prove that, in the vanishing noise limit, the probability of exit through a generic set on the boundary is asymptotically polynomial in the noise strength, with exponent depending on the mutual position of the set and the flag of the invariant manifolds associated with the top eigenvalues. Furthermore, we compute the limiting exit distributions conditioned on atypical exit events of polynomially small probability and show that the limits are Radon--Nikodym equivalent to volume measures on certain manifolds that we construct. This situation is in sharp contrast with the large deviation picture where the limiting conditional distributions are point masses.
The scale-invariant spacings lemma due to Arratia, Barbour and Tavar{\'e} establishes the distributional identity of a self-similar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant.
Structural modelling of octahedral tilts in perovskites is typically done using the symmetry constraints of the resulting space group. In most cases, this introduces more degrees of freedom than those strictly necessary to describe only the octahedral tilts. It can therefore be a challenge to disentangle the octahedral tilts from other structural distortions such as cation displacements and octahedral distortions. This paper reports on the development of constraints for modelling pure octahedral tilts and implemented the constraints in diffpy-CMI, a powerful package to analyse pair distribution function (PDF) data. The program allows features in the PDF that come from rigid tilts to be separated from non-rigid relaxations, provides an intuitive picture of the tilting, and as it has many fewer refinable variables than the unconstrained space-group fits, provides robust and stable refinements of the tilt components. It further demonstrates the use of the model on the canonical tilted perovskite CaTiO$_3$ which has a known Glazer tilt system $\alpha^+ \beta^- \beta^-$. The Glazer model fits comparably to the corresponding space group model $Pnma$ below $r$ = 14 A and becomes progressively worse than the space group model at higher $r$ due to non-rigid distortions in the real material.
Thin films of Co2MnSi have been grown on a-plane sapphire substrates from three elemental targets by dc magnetron co-sputtering. These films are single phase, have a strong (110) texture and a saturation magnetization of 4.95 uB/formula unit at 10 K. Films grown at the highest substrate temperature of 715 K showed the lowest resistivity (47 uOhm cm at 4.2 K) and the lowest coercivity (18 Oe). The spin polarization of the transport current was found to be of the order of 54% as determined by point contact Andreev reflection spectroscopy. A decrease in saturation magnetization with decreasing film thickness and different transport behaviour in thinner films indicate a graded disorder in these films grown on non-lattice matched substrates.
The Dodgson/Muir Identity is an identity on determinants of matrix minors that generalizes the Dodgson Identity. Using the matrix tree theorem, we create an equivalent forest identity on ordered sets of $k$ forests. We then prove the generalized Forest Identity, and by extension the Dodgson/Muir Identity, using an edge-swapping involution. This algorithm is a generalization of the Red Hot Potato algorithm, developed by the author in 2021 to prove the Dodgson Identity.
We consider iterations of integer-valued functions $\phi$, which have no fixed points in the domain of positive integers. We define a local function $\phi_n$, which is a sub-function of $\phi$ being restricted to the subdomain $\{0, ..., n \}$. The iterations of $\phi_n$ can be described by a certain $n \times n$ sparse matrix $M_n$ and its powers. The determinant of the related $n \times n$ matrix $\hat{M}_n = I - M_n$, where $I$ is the identity matrix, acts as an indicator, whether the iterations of the local function $\phi_n$ enter a cycle or not. If $\phi_n$ has no cycle, then $\det \hat{M}_n = 1$ and the structure of the inverse $\hat{M}_n^{-1}$ can be characterized. Subsequently, we give applications to compute the inverse $\hat{M}_n^{-1}$ for some special functions. At the end, we discuss the results in connection with the $3x+1$ and related problems.
We present rest-frame Ly$\alpha$ equivalent widths (EW) of 417 Ly$\alpha$ emitters (LAEs) detected with Multi Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope (VLT) at $2.9 < z < 6.6$ in the Hubble Ultra Deep Field. Based on the deep MUSE spectroscopy and ancillary Hubble Space Telescope (HST) photometry data, we carefully measured EW values taking into account extended Ly$\alpha$ emission and UV continuum slopes ($\beta$). Our LAEs reach unprecedented depths, both in Ly$\alpha$ luminosities and UV absolute magnitudes, from log($L_{\rm Ly\alpha}$/erg s$^{-1}$) $\sim$41.0 to 43.0 and from Muv $\sim$ -16 to -21 (0.01-1.0 $L^{*}_{\rm z=3}$). The EW values span the range of $\sim$ 5 to 240 \AA\ or larger, and their distribution can be well fitted by an exponential law $N = N_{\rm 0}$ exp($-$EW/$w_{\rm 0}$). Owing to the high dynamic range in Muv, we find that the scale factor, $w_{\rm 0}$, depends on Muv in the sense that including fainter Muv objects increases $w_{\rm 0}$, i.e., the Ando effect. The results indicate that selection functions affect the EW scale factor. Taking these effects into account, we find that our $w_{\rm 0}$ values are consistent with those in the literature within $1\sigma$ uncertainties at $2.9 < z < 6.6$ at a given threshold of Muv and $L_{\rm Ly\alpha}$. Interestingly, we find 12 objects with EW $>200$ \AA\ above $1\sigma$ uncertainties. Two of these 12 LAEs show signatures of merger or AGN activity: the weak CIV $\lambda 1549$ emission line. For the remaining 10 very large EW LAEs, we find that the EW values can be reproduced by young stellar ages ($< 100$ Myr) and low metallicities ($\lesssim 0.02$ $Z_{\rm \odot}$). Otherwise, at least part of the Ly$\alpha$ emission in these LAEs needs to arise from anisotropic radiative transfer effects, fluorescence by hidden AGN or quasi-stellar object activity, or gravitational cooling.