ShantanuT01/DACTYL-Pretraining huggingface dataset:
We introduce the Neural Collaborative Subspace Clustering, a neural model that discovers clusters of data points drawn from a union of low-dimensional subspaces. In contrast to previous attempts, our model runs without the aid of spectral clustering. This makes our algorithm one of the kinds that can gracefully scale to large datasets. At its heart, our neural model benefits from a classifier which determines whether a pair of points lies on the same subspace or not. Essential to our model is the construction of two affinity matrices, one from the classifier and the other from a notion of subspace self-expressiveness, to supervise training in a collaborative scheme. We thoroughly assess and contrast the performance of our model against various state-of-the-art clustering algorithms including deep subspace-based ones.
In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee rectangle. We also obtain a new bijective proof of a result of Andrews and Bressoud that the number of partitions of $N$ with all ranks at least $1-\ell$ equals the number of partitions of $N$ with no parts equal to $\ell+1$, for $\ell\ge0$, which allows us to refine it by the above statistics. Combining Foata's second fundamental transformation for words with Greene and Kleitman's mapping for subsets, interpreted in terms of lattice paths, we obtain enumeration formulas for partitions whose successive ranks satisfy certain constraints, such as being bounded by a constant.
Interpretation problems are eliminated from quantum theory by picturing a quantum history as having been sampled from a probability distribution over the set of histories which are permitted by all relevant boundary conditions. In laboratory physics, the final measurement plays a crucial role in defining the set of allowed histories by constraining the final state to be an eigenstate of the measurement operator. For the universe itself, a final boundary condition can play a similar role. Together with the special big bang initial state, the final constraint may ensure that the universe admits a classical description in the asymptotic future. Acknowledging the role of a future constraint dispels the mysteries of quantum theory without any amendment to the theory itself.
The model is a service system, consisting of several large server pools. A server processing speed and buffer size (which may be finite or infinite) depend on the pool. The input flow of customers is split equally among a fixed number of routers, which must assign customers to the servers immediately upon arrival. We consider an asymptotic regime in which the customer total arrival rate and pool sizes scale to infinity simultaneously, in proportion to a scaling parameter $n$, while the number of routers remains fixed.   We define and study a multi-router generalization of the pull-based customer assignment (routing) algorithm PULL, introduced in [11] for the single-router model. Under PULL algorithm, when a server becomes idle it send a "pull-message" to a randomly uniformly selected router; each router operates independently -- it assigns an arriving customer to a server according to a randomly uniformly chosen available (at this router) pull-message, if there is any, or to a randomly uniformly selected server in the entire system, otherwise.   Under Markov assumptions (Poisson arrival process and independent exponentially distributed service requirements), and under sub-critical system load, we prove asymptotic optimality of PULL: as $n\to\infty$, the steady-state probability of an arriving customer experiencing blocking or waiting, vanishes. Furthermore, PULL has an extremely low router-server message exchange rate of one message per customer. These results generalize some of the single-router results in [11].
A summary of the current understanding of methods of analytical higher order perturbative computations of total cross sections and decay widths in Quantum Chromodynamics is presented. As examples, the total cross section in electron positron annihilation, the hadronic decay rates of the tau lepton and Higgs boson up to O(\alpha_s^2) and O(\alpha_s^3) are considered. The evaluation of the four-loop QED \beta - function at an intermediate step of the calculation is briefly described. The problem of renormalization group ambiguity of perturbative results is considered and some of the existing prescriptions are discussed. The problem of estimation of theoretical uncertainty in perturbative calculations is briefly discussed.
Irreversibility is a critical property of non-equilibrium transport processes. An opinion has long been insisted that the entropy production rate is a Lyapunov function for all kinds of processes, that is, the principle of minimum entropy production. However, such principle is based on some strong assumptions that are rarely valid in practice. Here, the common features of parabolic-like transport processes are discussed. A theorem is then put forward that the dot products of fluxes and corresponding forces serve as Lyapunov function for parabolic-like transport processes. Such fluxes and forces are defined by their actual constitutive relations (e.g., the Fourier's law, the Fick's law, etc.). Then, some typical transport processes are analyzed. Particularly for heat conduction, both the theoretical and numerical analyses demonstrate that its Lyapunov function is the entransy dissipation rather that the entropy production, when the Fourier's law is valid. The present work could be helpful for further understanding on the irreversibility and the mathematical interpretation of non-equilibrium processes.
For a generic overdamped Langevin dynamics driven out of equilibrium by both time-dependent and nonconservative forces, the entropy production rate can be decomposed into two positive terms, termed excess and housekeeping entropy. However, this decomposition is not unique: There are two distinct decompositions, one due to Hatano and Sasa, the other one due to Maes and Neto\v{c}ny. Here, we establish the connection between these two decompositions and provide a simple, geometric interpretation. We show that this leads to a decomposition of the entropy production rate into three positive terms, which we call excess, housekeeping and coupling part, respectively. The coupling part characterizes the interplay between the time-dependent and nonconservative forces. We also derive thermodynamic uncertainty relations for the excess and housekeeping entropy in both the Hatano-Sasa and Maes-Neto\v{c}ny-decomposition and show that all quantities obey integral fluctuation theorems. We illustrate the decomposition into three terms using a solvable example of a dragged particle in a nonconservative force field.
Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021).
We give a simple proof of the non-rationality of the Fano threefold defined by the equations \Sigma x_i = \Sigma x_i^2 = \Sigma x_i^3 = 0 in P^6 .
In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$. In the first theorem we improve logarithmically the well-known $L^{\frac92} (\Bbb R^3)$ result. In the second theorem we present a sufficient condition for the trivially of the solution($v=0$) in terms of the head pressure, $Q=\frac12 |v|^2 +p$. The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee $v=0$.
One-dimensional Mott insulators exhibit giant nonlinear optical response. Based on the dynamical density matrix renormalization group method, photoexcited states and optical response in the insulators are studied as functions of the on-site and the nearest neighbor Coulomb interactions, $U$ and $V$, respectively. We find that the lowest optically-allowed and forbidden excited states, which have odd and even parities, respectively, are degenerate for $V/t\ltsim 2$ with $t$ being the hopping integral of an electron between nearest neighbor sites. For $V/t\gtsim 2$, the bound states with even and odd parities occur and are not degenerate. The nature of the degeneracy and its effect on the optical response are examined.
The error on a real quantity Y due to the graduation of the measuring instrument may be represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator do not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the "arbitrary functions principle" (Poincar\'{e}, Hopf). We give extensions of this property to multivariate case and infinite dimensional case for approximations of the Brownian motion. We use a Girsanov theorem for Dirichlet forms which has its own interest. Connections are given with discretization of stochastic differential equations.
We have directly observed reversal of the polarity of charged surfaces in water upon the addition of tri- and quadrivalent ions using atomic force microscopy. The bulk concentration of multivalent ions at which charge inversion reversibly occurs depends only very weakly on the chemical composition, surface structure, size and lipophilicity of the ions, but is dominated by their valence. These results support the theoretical proposal that spatial correlations between ions are the driving mechanism behind charge inversion.
We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, grad r(G). For these classes we prove the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. This generalizes and simplifies several earlier results (obtained for minor closed classes).
We study the distribution of spacings between squares modulo q as the number of prime divisors of q tends to infinity. In an earlier paper Kurlberg and Rudnick proved that the spacing distribution for square free q is Poissonian, this paper extends the result to arbitrary q.
The time-dependent non-crossing approximation is used to study the transient current in a single electron transistor attached asymmetrically to two leads following a sudden change in the energy of the dot level. We show that for asymmetric coupling, sharp features in the density of states of the leads can induce oscillations in the current through the dot. These oscillations persist to much longer timescales than the timescale for charge fluctuations. The amplitude of the oscillations increases as the temperature or source-drain bias across the dot is reduced and saturates for values below the Kondo temperature. We discuss the microscopic origin of these oscillations and comment on the possibility for their experimental detection.
Differential cross section of deuteron Compton scattering has been calculated using the Bonn NN-potential, a consistent set of meson-exchange currents and seagulls, and lowest- and higher-order electromagnetic polarizabilities of the nucleon. Estimates of the polarizabilities of the neutron are obtained from recent experimental data.
The films with the discrete diamond-like-carbon nanoparticles were prepared by the deposition of the carbon nanoparticle beam. Their morphologies were imaged by Scanning Electron Microscopy (SEM) and Atomic Force Microscopy (AFM). The nanoparticles were found distributed on the silicon (100) substrate discretely. The semisphere shapes of the nanoparticles were demonstrated by the AFM line profile. EELS was measured and the sp3 ratio as high as 86% was found. The field-induced electron emission of the as-prepared cascade (nanoDLC/ Si) was tested and the current density of 1mA/cm2 was achieved at 10.2V/{\mu}m.
We address an experimental scheme to analyze the optical bistability and the entanglement of two movable mirrors coupled to a two-mode laser inside a doubly resonant cavity. With this aim we investigate the master equations of the atom-cavity subsystem in conjunction with the quantum Langevin equations that describe the interaction of the mirror cavity. The parametric amplification-type coupling induced by the two-photon coherence on the optical bistability of the intracavity mean photon numbers is found and investigated. Under this condition, the optical intensities exhibit bistability for all large values of cavity laser detuning. We also provide numerical evidence for the generation of strong entanglement between the movable mirrors and show that it is robust against environmental thermalization.
It is now a classical result that an algebraic space locally of finite type over $\mathbf{C}$ is analytifiable if and only if it is locally separated. In this paper we study non-archimedean analytifications of algebraic spaces. We construct a quotient for any etale non-archimedean analytic equivalence relation whose diagonal is a closed immersion, and deduce that any separated algebraic space locally of finite type over any non-archimedean field $k$ is analytifiable in both the category of rigid spaces and the category of analytic spaces over $k$. Also, though local separatedness remains a necessary condition for analytifiability in either of these categories, we present many surprising examples of non-analytifiable locally separated smooth algebraic spaces over $k$ that can even be defined over the prime field.
We analyze the effect of weak localization (WL) and weak antilocalization (WAL) in the electronic transport through HgTe/CdTe quantum wells. We show that for increasing Fermi energy the magnetoconductance of a diffusive system with inverted band ordering features a transition from WL to WAL and back, if spin-orbit interactions from bulk and structure inversion asymmetry can be neglected. This, and an additional splitting in the magnetoconductance profile, is a signature of the Berry phase arising for inverted band ordering and not present in heterostructures with conventional ordering. In presence of spin-orbit interaction both band topologies exhibit WAL, which is distinctly energy dependent solely for quantum wells with inverted band ordering. This can be explained by an energy-dependent decomposition of the Hamiltonian into two blocks.
By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli type matrices of a skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This gives an affirmative answer to an open problem posed by Kuperberg.
The algebra of linear and quadratic functions of basic observables on the phase space of either the free particle or the harmonic oscillator possesses a finite-dimensional anomaly. The quantization of these systems outside the critical values of the anomaly leads to a new degree of freedom which shares its internal character with spin, but nevertheless features an infinite number of different states. Both are associated with the transformation properties of wave functions under the Weyl-symplectic group $WSp(6,\Re)$. The physical meaning of this new degree of freedom can be established, with a major scope, only by analysing the quantization of an infinite-dimensional algebra of diffeomorphisms generalizing string symmetry and leading to more general extended objects.
In this paper, we use the Killing vector method to formulate the de Sitter/Anti-de Sitter invariant special relativity (dS/AdS-SR). Through solving the Einstein equation with $\Lambda\neq 0$, the basic inertial metric for dS/AdS-SR is determined to be the Beltrami metric $B_{\mu\nu}(x)$. The corresponding Killing equations are system of ten simultaneous partial differential equations of first order. Their most general solutions were obtained, and all the ten independent Killing vectors were found out. These results confirm that the Beltrami metric has maximal spacetime symmetry. The ten Killing-Noether charges are obtained. They are energy, momenta, Lorentz boost and angular momentum in SR-theory with $\Lambda\neq 0$. Consequently, dS/AdS-SR is consistently established for the vacuum with $\Lambda\neq 0$ via Killing vector method rather than the unpopular classical domain theory.
An abstract mathematical concept of fractal organization of certain complex objects received significant attention in astrophysics during last decades. The concept evolved into a broad field including multi-fractality and intermittency, percolation theory, self-organized criticality, theory of catastrophes, etc. Such a strong mathematical and physical approach provide new possibilities for exploring various aspects of astrophysics. In particular, in the solar and stellar magnetism, multi-fractal properties of magnetized plasma turned to be useful for understanding burst-like dynamics of energy release events, conditions for turbulent dynamo action, nature of turbulent magnetic diffusivity, and even the dual nature of solar dynamo. In this review, I will briefly outline how the ideas of multi-fractality are used to explore the above mentioned aspects of solar magnetism.
We study the normal modes and the stability of two component condensates in a phase separated regime. In such a regime the system can undergo a quantum phase transition upon the change of interaction strength between bosons of the same species or the variation of the trap frequencies. In this transition, the distribution of the two components changes from a symmetric to an asymmetric shape. We discuss the nature of the phase transition, the role of the interface tension and the phase diagram. The symmetric to asymmetric transition is the simplest quantum phase transition that one can imagine. We found new branches of normal modes which are specific for trapped multicomponent condensates and are analogous to the waves at the interface between two layers of immiscible fluids under gravity. At the point of the phase transition the frequencies of those modes go to zero and become imaginary which causes an instability. The interface tension shifts the normal mode frequencies and changes the stability region of the system.
String backgrounds of the form AdS_3 x N that give rise to two dimensional spacetime superconformal symmetry are constructed.
We investigate the roles of different environmental models on quantum correlation dynamics of two-qubit composite system interacting with two independent environments. The most common environmental models (the single-Lorentzian model, the squared-Lorentzian model, the two-Lorentzian model and band-gap model) are analyzed. First, we note that for the weak coupling regime, the monotonous decay speed of the quantum correlation is mainly determined by the spectral density functions of these different environments. Then, by considering the strong coupling regime we find that, contrary to what is stated in the weak coupling regime, the dynamics of quantum correlation depends on the non-Markovianity of the environmental models, and is independent of the environmental spectrum density functions.
In this paper we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a PVHS of weight one. Starting from the associated Higgs field, and assuming the base has dimension $1$, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of "second-order Kodaira-Spencer class" which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial.
We present a library of high-resolution (R $\equiv$ $\lambda$/$\Delta$$\lambda$ $\sim$ 45,000) and high signal-to-noise ratio (S/N $\geq$ 200) near-infrared spectra for stars of a wide range of spectral types and luminosity classes. The spectra were obtained with the Immersion GRating INfrared Spectrograph (IGRINS) covering the full range of the H (1.496-1.780 $\mu$m) and K (2.080-2.460 $\mu$m) atmospheric windows. The targets were primarily selected for being MK standard stars covering a wide range of effective temperatures and surface gravities with metallicities close to the Solar value. Currently, the library includes flux-calibrated and telluric-absorption-corrected spectra of 84 stars, with prospects for expansion to provide denser coverage of the parametric space. Throughout the H and K atmospheric windows, we identified spectral lines that are sensitive to $T_\mathrm{eff}$ or $\log g$ and defined corresponding spectral indices. We also provide their equivalent widths. For those indices, we derive empirical relations between the measured equivalent widths and the stellar atmospheric parameters. Therefore, the derived empirical equations can be used to calculate $T_\mathrm{eff}$ and $\log g$ of a star without requiring stellar atmospheric models.
I report on precision comparisons of theoretical predictions for Drell-Yan (DY) transverse momentum spectra from two approaches: the analytical resummation approach based on Collins-Soper-Sterman (CSS) formalism and the parton branching (PB) approach to the evolution of transverse momentum dependent (TMD) parton distributions.
We analyze the different possibilities for the hadron-quark phase transition occurring in beta-stable matter including hyperons in neutron stars. We use a Brueckner-Hartree-Fock approach including hyperons for the hadronic equation of state and a generalized MIT bag model for the quark part. We then point out in detail the differences between Maxwell and Gibbs phase transition constructions including the effects of surface tension and electromagnetic screening. We find only a small influence on the maximum neutron star mass, whereas the radius of the star and in particular its internal structure are more affected.