ShantanuT01/DACTYL-Pretraining huggingface dataset:
This paper is devoted to Wiener index of directed graphs, more precisely of directed grids. The grid $G_{m,n}$ is the Cartesian product $P_m\Box P_n$ of paths on $m$ and $n$ vertices, and in a particular case when $m=2$, it is a called the ladder graph $L_n$. Kraner \v{S}umenjak et al. proved that the maximum Wiener index of a digraph, which is obtained by orienting the edges of $L_n$, is obtained when all layers isomorphic to one factor are directed paths directed in the same way except one (corresponding to an endvertex of the other factor) which is a directed path directed in the opposite way. Then they conjectured that the natural generalization of this orientation to $G_{m,n}$ will attain the maximum Wiener index among all orientations of $G_{m,n}$. In this paper we disprove the conjecture by showing that a comb-like orientation of $G_{m,n}$ has significiantly bigger Wiener index.
This article is based on a talk given at the Kinosaki Symposium on Algebraic Geometry in 2015, about a work in progress. We describe a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. We explicitly formulate a conjecture which says certain Fourier-Mukai transforms between derived categories give equivalences of some hearts of Bridgeland stability conditions. We establish it for abelian surfaces, which is already known due to D. Huybrechts, and for abelian 3-folds. This generalizes the author's previous joint work with A. Maciocia on principally polarized abelian 3-folds with Picard rank one. Consequently, we see that the strong Bogomolov-Gieseker type inequalities hold for tilt stable objects on abelian 3-folds.
We study exact solutions of the steady state behaviour of several non-linear open quantum systems which can be applied to the field of circuit quantum electrodynamics. Using Fokker-Planck equations in the generalised P-representation we investigate the analytical solutions of two fundamental models. First, we solve for the steady-state response of a linear cavity that is coupled to an approximate transmon qubit and use this solution to study both the weak and strong driving regimes, using analytical expressions for the moments of both cavity and transmon fields, along with the Husimi Q-function for the transmon. Second, we revisit exact solutions of quantum Duffing oscillator which is driven both coherently and parametrically while also experiencing decoherence by the loss of single and pairs of photons. We use this solution to discuss both stabilisation of Schroedinger cat states and the generation of squeezed states in parametric amplifiers, in addition to studying the Q-functions of the different phases of the quantum system. The field of superconducting circuits, with its strong nonlinearities and couplings, has provided access to a new parameter regime in which returning to these exact quantum optics methods can provide valuable insights.
We confirm the existance of magneto-resistance oscillations in a microwave-irradiated two-dimensional electron gas, first reported in a series of papers by Zhudov et al. and Mani et al. In our experiments, on a sample with a more moderate mobility, the microwave induced oscillations are observed not only in the longitudinal - but also in the transverse-resistance (Hall resistance). The phase of the oscillations is such that the decrease (increase) in the longitudinal resistance is accompanied by an increase (decrease) in the absolute value of the Hall resistance. We believe that these new results provide valuable new information to better understand the origin of this interesting phenomenon.
We generalize the lemmas of Thomas Kretschmer to arbitrary number fields, and apply them with a 2-descent argument to obtain bounds for families of elliptic curves over certain imaginary quadratic number fields with class number 1. One such family occurs in the congruent number problem. We consider the congruent number problem over these quadratic number fields, and subject to the finiteness of Sha, we show that there are infinitely many numbers that are not congruent over Q but become congruent over $Q(\zeta_3)$.
The aim of this short note is to present the notion of IDT processes, which is a wide generalization of L\'{e}vy processes obtained from a modified infinitely divisible property. Special attention is put on a number of examples, in order to clarify how much the IDT processes either differ from, or resemble to, L\'{e}vy processes.
The formation of a binary system surrounded by disks is the most common outcome of stellar formation. Hence studying and understanding the formation and the evolution of binary systems and associated disks is a cornerstone of star formation science. Moreover, since the components within binary systems are coeval and the sizes of their disks are fixed by the tidal truncation of their companion, binary systems provide an ideal "laboratory" in which to study disk evolution under well defined boundary conditions.   In this paper, we review observations of several inner disk diagnostics in multiple systems, including hydrogen emission lines (indicative of ongoing accretion), $K-L$ and $K-N$ color excesses (evidence of warm inner disks), and polarization (indicative of the relative orientations of the disks around each component). We examine to what degree these properties are correlated within binary systems and how this degree of correlation depends on parameters such as separation and binary mass ratio. These findings will be interpreted both in terms of models that treat each disk as an isolated reservoir and those in which the disks are subject to re-supply from some form of circumbinary reservoir, the observational evidence for which we will also critically review. The planet forming potential of multiple star systems is discussed in terms of the relative lifetimes of disks around single stars, binary primaries and binary secondaries. Finally, we summarize several potentially revealing observational problems and future projects that could provide further insight into disk evolution in the coming decade
The integrability of the one-dimensional (1D) fermion chain model is investigated in the framework of the Quantum Inverse Scattering Method (QISM). We introduce a new R-operator for the fermion chain model, which is expressed in terms of the fermion operators. The R-operator satisfies a new type of the Yang-Baxter relation with fermionic L-operator. We derive the fermionic Sutherland equation from the relation, which is equivalent to the fermionic Lax equation. It also provides a mathematical foundation of the boost operator approach for the fermion model. In fact, we obtain some higher conserved quantities of the fermion model using the boost operator.
K{\"u}chle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well.
A new real space quantum mechanical approach with local field effects included is applied to the calculation of the optical properties of silicon nanocrystals. Silicon ellipsoids are studied and the role of surface polarization is discussed in details. In particular, surface polarization is shown to be responsible for a strong optical anisotropy in silicon ellipsoids, much more pronounced with respect to the case in which only quantum confinement effects are considered. The static dielectric constant and the absorption spectra are calculated, showing that the perpendicular and parallel components have a very different dependence on the ellipsoid aspect ratio. Then, a comparison with the classical dielectric model is performed, showing that the model only works for large and regular structures, but it fails for thin elongated ellipsoids.
Given a partial Steiner triple system (STS) of order $n$, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order $n$ with at most $r \le \varepsilon n^2$ triples, it can always be embedded into a complete STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs.   This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property $\mathcal{P}$ and a graph $G$, we define the deficiency of the graph $G$ with respect to the property $\mathcal{P}$ to be the smallest positive integer $t$ such that the join $G\ast K_t$ has property $\mathcal{P}$. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a $K_k$-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs.   The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given.
The bright blazar OJ~287 routinely parades high brightness bremsstrahlung flares which are explained as being a result of a secondary supermassive black hole (SMBH) impacting the accretion disk of a primary SMBH in a binary system. We begin by showing that these flares occur at times predicted by a simple analytical formula, based on the Kepler equation, which explains flares since 1888. The next impact flare, namely the flare number 26, is rather peculiar as it breaks the typical pattern of two impact flares per 12 year cycle. This will be the third bremsstrahlung flare of the current cycle that follows the already observed 2015 and 2019 impact flares from OJ~287. Unfortunately, astrophysical considerations make it difficult to predict the exact arrival epoch of the flare number 26. In the second part of the paper, we describe our recent OJ~287 observations. They show that the pre-flare light curve of flare number 22, observed in 2005, exhibits similar activity as the pre-flare light curve in 2022, preceding the expected flare number 26 in our model. We argue that the pre-flare activity most likely arises in the primary jet whose activity is modulated by the transit of the secondary SMBH through the accretion disk of the primary. Observing the next impact flare of OJ~287 in October 2022 will substantiate the theory of disk impacts in binary black hole systems.
We discuss the notion of causality in Quantum Gravity in the context of sum-over-histories approaches, in the absence therefore of any background time parameter. In the spin foam formulation of Quantum Gravity, we identify the appropriate causal structure in the orientation of the spin foam 2-complex and the data that characterize it; we construct a generalised version of spin foam models introducing an extra variable with the interpretation of proper time and show that different ranges of integration for this proper time give two separate classes of spin foam models: one corresponds to the spin foam models currently studied, that are independent of the underlying orientation/causal structure and are therefore interpreted as a-causal transition amplitudes; the second corresponds to a general definition of causal or orientation dependent spin foam models, interpreted as causal transition amplitudes or as the Quantum Gravity analogue of the Feynman propagator of field theory, implying a notion of ''timeless ordering''.
We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and apply this bound in examples, and to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppav's by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.
Manipulating the valley degree of freedom to encode information for potential valleytronic devices has ignited a new direction in solid-state physics. A significant, fundamental challenge in the field of valleytronics is how to generate and regulate valley-polarized currents by practical ways. Here, we discover a new mechanism of producing valley polarization in a monolayer transition metal dichalcogenides superlattice, in which valley-resolved gaps are formed at the supercell Brillouin zone boundaries and centers due to the intervalley scattering. When the energy of the incident electron is in the gaps, the available states are valley polarized, thus providing a valley-polarized current from the superlattice. We show that the direction and strength of the valley polarization may further be tuned by varying the potential applied the superlattice. The transmission can have a net valley polarization of 55% for a 4-period heterojunction. Moreover, such two valley filters in series may function as an electrostatically controlled giant valleyresistance device, representing a zero magnetic field counterpart to the familiar giant magnetoresistance device.
We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation $X \stackrel{d}{=} C + \sum_{i \geq 1} T_i X_i$, where $\stackrel{d}{=}$ means equality in distribution, $(C,T_1,T_2,...)$ is a given sequence of non-negative random variables and $X_1,X_2,...$ is a sequence of i.i.d.\ copies of the non-negative random variable $X$ independent of $(C,T_1,T_2,...)$. In this situation, $X$ (or, more precisely, the distribution of $X$) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Further, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C=0. Using this correspondence, we present a full characterization of the set of fixed points under mild assumptions.
The line edge roughness (LER) becomes a issue of e-beam lithography when feature size is reduced into nanometers. Therefore, minimizing the LER is a important method to increase the density of circuit patterns. One of the possible ways is through simulation. The stochastic exposure distributions in the resist is generated by the Monte Carlo simulation. In addition a resist development simulation needs to be carried out. Although there are several ways to simulate or estimate LER but none of them can reveal as much the inner relationship between LER and different parameters as theocratical analysing methods can do. In this paper, a new approach to analytically derive the LER based on the statistical exposure, is described. Our approach is based on analytic model of stochastic exposure distribution, deriving standard deviation of exposure and analyzing the variance of edge location after development. Even though it may not be a complete modeling of LER, it can still show some strong relationship between LER and some inner parameters.
I construct a class of single small field models of inflation that can predict an observable gravitational wave signal in the cosmic microwave background anisotropoies, contrary to popular wisdom. The spectral index, its running, the tensor to scalar ratio and the number of e-folds can cover all the parameter space currently allowed by cosmological observations. A unique feature of models in this class is their ability to predict a negative spectral index running in accordance with recent cosmic microwave background observations. I comment on the new class of models from an effective field theory perspective and show that if the dimensionless trilinear coupling is small, as required for consistency, then the observed spectral index running implies a high scale of inflation and hence an observable gravitational wave signal.
Vincent Lafforgue has recently made a spectacular breakthrough in the setting of the global Langlands correspondence for global fields of positive characteristic, by constructing the `automorphic--to--Galois' direction of the correspondence for an arbitrary reductive group $G$. We discuss a result that starts with Lafforgue's work and proceeds in the opposite (`Galois--to--automorphic') direction.
The durability of an organic colour and its resistance against external chemical agents and exposure to light can be significantly enhanced by hybridizing the natural dye with a mineral. In search for stable natural pigments, the present work focuses on the association of indigo blue with several zeolitic matrices (LTA zeolite, mordenite, MFI zeolite). The manufacturing of the hybrid pigment is tested under varying oxidising conditions, using Raman and UV-visible spectrometric techniques. Blending indigo with MFI is shown to yield the most stable composite in all of our artificial indigo pigments. In absence of defects and substituted cations such as aluminum in the framework of the MFI zeolite matrix, we show that matching the pore size with the dimensions of the guest indigo molecule is the key factor. The evidence for the high colour stability of indigo@MFI opens a new path for modeling the stability of indigo in various alumino-silicate substrates such as in the historical Maya Blue pigment.
Two first examples beyond the standard model are given which exhibit left-right symmetry (g_L = g_R) and supersymmetry at a few TeV, together with gauge-coupling unification at around 10^{16} GeV.
This paper presents a detailed study of chromospheric evaporation using the EUV Imaging Spectrometer (EIS) onboard Hinode in conjunction with HXR observations from RHESSI. The advanced capabilities of EIS were used to measure Doppler shifts in 15 emission lines covering the temperature range T=0.05-16 MK during the impulsive phase of a C-class flare on 2007 December 14. Blueshifts indicative of the evaporated material were observed in six emission lines from Fe XIV-XXIV (2-16 MK). Upflow velocity (v_up) was found to scale with temperature as v_up (km s^-1)~8-18 T (MK). Although the hottest emission lines, Fe XXIII and Fe XXIV, exhibited upflows of >200 km s^-1, their line profiles were found to be dominated by a stationary component in contrast to the predictions of the standard flare model. Emission from O VI-Fe XIII lines (0.5-1.5 MK) was found to be redshifted by v_down (km s^-1)~60-17 T (MK) and was interpreted as the downward-moving `plug' characteristic of explosive evaporation. These downflows occur at temperatures significantly higher than previously expected. Both upflows and downflows were spatially and temporally correlated with HXR emission observed by RHESSI that provided the properties of the electron beam deemed to be the driver of the evaporation. The energy flux of the electron beam was found to be >5x10^10 ergs cm^-2 s^-1 consistent with the value required to drive explosive chromospheric evaporation from hydrodynamic simulations.
The empirical model of Lu et al. 2014a for the relation between star formation rate and halo mass growth is adopted to predict the classical bulge mass ($M_{\rm cb}$) - total stellar mass ($M_\star$) relation for central galaxies. The assumption that the supermassive black hole (SMBH) mass ($M_{\rm BH}$) is directly proportional to the classical bulge mass, with the proportionality given by that for massive galaxies, predicts a $M_{\rm BH}$ - $M_\star$ relation that matches well the observed relation for different types of galaxies. In particular, the model reproduces the strong transition at $M_\star=10^{10.5}$ - $10^{11}M_{\odot}$, below which $M_{\rm BH}$ drops rapidly with decreasing $M_\star$. Our model predicts a new sequence at $M_\star <10^{10.5}M_{\odot}$, where $M_{\rm BH} \propto M_\star$ but the amplitude is a factor of $\sim 50$ lower than the amplitude of the sequence at $M_\star>10^{11}M_{\odot}$. If all SMBH grow through similar quasar modes with a feedback efficiency of a few percent, the energy produced in low-mass galaxies at redshift $z\gtrsim 2$ can heat the circum-galactic medium up to a specific entropy level that is required to prevent excessive star formation in low-mass dark matter halos.
We have studied charged current one pion production induced by $\nu_\mu(\bar\nu_\mu)$ from some nuclei. The calculations have been done for the incoherent pion production processes from these nuclear targets in the $\Delta$ dominance model and take into account the effect of Pauli blocking, Fermi motion and renormalization of $\Delta$ properties in the nuclear medium. The effect of final state interactions of pions has also been taken into account. The numerical results have been compared with the recent results from the MiniBooNE experiment for the charged current 1$\pi$ production, and also with some of the older experiments in Freon and Freon-Propane from CERN.
Fast radio bursts (FRBs) are milliseconds radio transients with large dispersion measures (DMs). An outstanding question is the relation between repeating FRBs and those with a single burst. In this paper, we study the energy distribution of the repeating FRB 121102. For a power-law distribution of energy $dN/dE\propto E^{-\alpha_E}$, we show that the value of $\alpha_E$ is in a narrow range $1.6-1.8$ for bursts observed by different telescopes at different frequencies, which indicates a universal energy distribution for FRB 121102. Interestingly, similar power-law index of energy distribution for non-repeating FRBs observed by Parkes and ASKAP is also found. However, if low-energy bursts below completeness threshold of Arecibo are discarded for FRB 121102, the slope could be up to 2.2. Implications of such a universal energy distribution are discussed.
We present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher order finite differencing for the BSSN formulation of Einstein's equations, but which is designed to be readily applicable to a broad class of formulations. We apply our techniques to a standard set of numerical relativity test problems, demonstrating the fourth-order accuracy of the solutions. Finally we apply our approach to binary black hole head-on collisions, calculating the waveforms of gravitational radiation generated and demonstrating significant improvements in waveform accuracy over second-order methods with typically achievable numerical resolution.
Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The main result shows that it suffices to consider only $3$-colorings which have a special partition of the vertices. Using this tool, we find several sharp numbers and conjecture the sharp value for all bipartite graphs. In particular, we determine the Gallai-Ramsey numbers for all bipartite graphs with two vertices in one part and initiate the study of linear forests.