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Correlators of Wilson-line operators in non-abelian gauge theories are known to exponentiate, and their logarithms can be organised in terms of collections of Feynman diagrams called webs. In [1] we introduced the concept of Cweb, or correlator web, which is a set of skeleton diagrams built with connected gluon correlators, and we computed the mixing matrices for all Cwebs connecting four or five Wilson lines at four loops. Here we complete the evaluation of four-loop mixing matrices, presenting the results for all Cwebs connecting two and three Wilson lines. We observe that the conjuctured column sum rule is obeyed by all the mixing matrices that appear at four-loops. We also show how low-dimensional mixing matrices can be uniquely determined from their known combinatorial properties, and provide some all-order results for selected classes of mixing matrices. Our results complete the required colour building blocks for the calculation of the soft anomalous dimension matrix at four-loop order., Comment: 51 pages, 37 figures, Published version, published in JHEP

Inflationary magnetogenesis has long been assumed to be the most promising mechanism for producing large-scale magnetic fields in our universe. However, generically, such models are plagued with either backreaction or strong coupling problems within the standard framework. This paper has shown that the reheating phase can play a crucial role in alleviating those problems along with CMB. Assuming the electrical conductivity to be negligible during the entire period of reheating, the classic Faraday electromagnetic induction changes the magnetic field's dynamics drastically. Our detailed analysis reveals that this physical phenomenon not only converts a large class of magnetogenesis model observationally viable without any theoretical problem but also can uniquely fix the perturbative average inflaton equation of state, $\omega_{\phi} = (p+2)/(p+2)$ during reheating given a specific value of the large scale magnetic field. This observation hints the inflaton to assume the potential of form $V(\phi) \sim \phi^p$ near its minimum with $p \gtrsim 3.5$ if one considers the limit of the present-day strength of the large scale magnetic field to be $\mathcal{P}_{B0}^{\frac{1}{2}} \gtrsim 10^{-18}$ G. Our analysis opens up a new avenue towards constraining the inflationary and magnetogenesis model together via reheating., Comment: 37pages, 4 figures

Event shapes are classical tools for the determination of the strong coupling and for the study of hadronization effects in electron-positron annihilation. In the context of analytical studies, hadronization corrections take the form of power-suppressed contributions to the cross-section, which can be extracted from the perturbative ambiguity of Borel-resummed distributions. We propose a simplified version of the well-established method of Dressed Gluon Exponentiation (DGE), which we call Eikonal DGE (EDGE), which determines all dominant power corrections to event shapes by means of strikingly elementary calculations. We believe our method can be generalized to hadronic event shapes and jet shapes of relevance for LHC physics., Comment: 13 pages, 1 figure, one section added, conclusions remain unchanged, matches published version

We find characterization for the distinguished varieties in the symmetrized polydisc $\mathbb G_n \; (n\geq 2)$ and thus generalize the work [\textit{J. Funct. Anal.}, 266 (2014), 5779 -- 5800] on $\mathbb G_2$ by the author and Shalit. We show that a distinguished variety $\Lambda$ in $\mathbb G_n$ is a part of an algebraic curve, which is a set-theoretic complete intersection, and that $\Lambda$ can be represented by the Taylor joint spectrum of $n-1$ commuting scalar matrices satisfying certain conditions. An $n$-tuple of commuting Hilbert space operators $(S_1, \dots ,S_{n-1},P)$ for which $\Gamma_n=\overline{\mathbb G_n}$ is a spectral set is called a $\Gamma_n$-contraction. To every $\Gamma_n$-contraction $(S_1, \dots ,S_{n-1},P)$ there is a unique operator tuple $(F_1, \dots , F_{n-1})$, called the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1},P)$, satisfying \[ S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, \dots ,n-1. \] We produce concrete functional model for the pure isometric-operator tuples associated with $\Gamma_n$ and by an application of that model we establish that the $\Gamma_n$-contractions $(S_1, \dots ,S_{n-1},P)$ and $(S_1^*, \dots , S_{n-1}^*,P^*)$ admit normal $\partial \overline{ \Lambda}_{\Sigma}-$dilations for a unique distinguished variety $\Lambda_{\Sigma}$ in $\mathbb G_n$, when $\Lambda_{\Sigma}$ is determined by the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1}, P)$. Further, we show that the dilation of $(S_1^*, \dots ,S_{n-1}^*,P^*)$ is minimal and acts on the minimal unitary dilation space of $P^*$. Also, we show interplay between the distinguished varieties in $\mathbb G_2$ and $\mathbb G_{3}$., Comment: See the previous version for the "Distinguished varieties in the polydisc" part. arXiv admin note: substantial text overlap with arXiv:1708.06015

Correlators of Wilson-line operators are fundamental ingredients for the study of the infrared properties of non-abelian gauge theories. In perturbation theory, they are known to exponentiate, and their logarithm can be organised in terms of collections of Feynman diagrams called webs. We study the classification of webs to high perturbative orders, proposing a set of tools to generate them recursively: in particular, we introduce the concept of Cweb, or correlator web, which is a set of skeleton diagrams built with connected gluon correlators, instead of individual Feynman diagrams. As an application, we enumerate all Cwebs entering the soft anomalous dimension matrix for multi-parton scattering amplitudes at four loops, and we compute the mixing matrices for all Cwebs connecting four or five Wilson lines at that loop order, verifying that they obey sum rules that were derived or conjectured in the literature. Our results provide the colour building blocks for the calculation of the soft anomalous dimension matrix at four-loop order., Comment: Published version; Published in JHEP; 51 pages, 65 figures

The article deals with the extension of the relativistic double-ionization equation-of-motion coupled-cluster (DI-EOMCC) method [H. Pathak et al. Phys. Rev. A 90, 010501(R) (2014)] for the molecular systems. The Dirac-Coulomb (DC) Hamiltonian with four-component spinors is considered to take care of the relativistic effects. The implemented method is employed to compute a few low-lying doubly ionized states of noble gas atoms (Ar, Kr, Xe, and Rn) and Cl 2 , Br 2 , HBr, and HI. Additionally, we presented results with two intermediate schemes in the four-component relativistic DI-EOMCC framework to understand the role of electron correlation. The computed double ionization spectra for the atomic systems are compared with the values from the non-relativistic DI-EOMCC method with spin-orbit coupling (SOC) [Z. Wang et al. J. Chem. Phys. 142, 144109 (2015)] and the values from the National Institute of Science and Technology (NIST) database. Our atomic results are found to be in good agreement with the NIST values. Further, the obtained results for the molecular systems agree well with the available experimental values., Comment: 8 page

BaF is one of the potential candidates for the experimental search of the electric dipole moment of the electron (eEDM). The NL-eEDM collaboration is building a new experimental set up to measure the eEDM using the BaF molecule [The NL-eEDM collaboration, Eur. Phys. J. D (2018) 72: 197]. To analyze the results of such an experiment, one would require the accurate value of the molecular ${\mathcal{P,T}}$-odd interaction parameters that cannot be measured from any experiment. In this work, we report the precise value of the ${\mathcal{P,T}}$-odd interaction parameters of the BaF molecule obtained from the four-component relativistic coupled-cluster calculations. We also calculate the hyperfine structure (HFS) constants of the same molecule to assess the reliability of the reported molecular parameters. The calculated HFS constants show good agreement with the available experimental values. Further, the systematic effects of electron-correlation along with the roles of inner-core electrons and the virtual energy functions in the calculation of the studied properties of BaF are investigated., Comment: 14 pages, 2 figures

We investigate the parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetry violating effects in the CdH molecule and perform the relativistic coupled-cluster calculation of the molecular parameters- $E_\text{eff}$, $W_\text{s}$ and $W_\text{M}$ related to the electric dipole moment of electron (eEDM) interaction, the scalar-pseudoscalar (S-PS) nucleus-electron neutral current coupling and the nuclear magnetic quadrupole moment (MQM) interaction with electrons, respectively. We also compute the molecular dipole moment and the magnetic hyperfine structure coupling constant of CdH. The value of $E_\text{eff}$, $W_\text{s}$ and $W_\text{M}$ obtained by us in the said molecule are 12.2 GV/cm, 14.0 kHz and $0.82\times$10$^{33}$ Hz/e cm$^2$, respectively, within an uncertainty of 10\%. Furthermore, we study the trend of electron-correlation in the computed properties of CdH and that of the $\mathcal{P,T}$-odd parameters in the group-12 monohydrides (i.e., ZnH, CdH, and HgH)., Comment: 8 pages, 3 figures

A commuting tuple of $n$ operators $(S_1, \dots, S_{n-1}, P)$ defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc \[ \Gamma_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i

We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc $\gn$ and for the extended symmetrized polydisc $\Gn$. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in \cite{Young-LMS}. Also we find a few geometric interplay between the members of the family $\Gn$ and its closure $\widetilde{\Gamma}_n$., Comment: Revised, 31 pages

We find new characterizations for the points in the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i

The goal of our work is to use visual attention to enhance autonomous driving performance. We present two methods of predicting visual attention maps. The first method is a supervised learning approach in which we collect eye-gaze data for the task of driving and use this to train a model for predicting the attention map. The second method is a novel unsupervised approach where we train a model to learn to predict attention as it learns to drive a car. Finally, we present a comparative study of our results and show that the supervised approach for predicting attention when incorporated performs better than other approaches., Comment: Paper Accepted at ICMV (2018)

We employ the $Z$-vector method in the four-component relativistic coupled-cluster framework to calculate the parity (${\mathcal{P}}$) and time-reversal (${\mathcal{T}}$) symmetry violating scalar-pseudoscalar (S-PS) nucleus-electron interaction constant ($W_\text{s}$), the effective electric field ($E_\text{eff}$) experienced by the unpaired electron and the nuclear magnetic quadrupole moment (NMQM)-electron interaction constant ($W_\text{M}$) in the open-shell ground electronic state of HgF. The molecular frame dipole moment and the magnetic hyperfine structure (HFS) constant of the molecule are also calculated at the same level of theory. The outcome of our study is that HgF has a high value of $E_\text{eff}$ (115.9 GV/cm), $W_\text{s}$ (266.4 kHz) and $W_\text{M}$ ($3.59\times 10^{33}$Hz/e.cm$^2$), which shows that it can be a possible candidate for the search of new physics beyond the Standard model. Our results are in good agreement with the available literature values. Furthermore, we investigate the effect of the basis set and that of the virtual energy functions on the computed properties. The role of the high-energy virtual spinors are found to be significant in the calculation of the HFS constant and the ${\mathcal{P,T}}$-odd interaction coefficients., Comment: 1 figure, 9 pages

Given a complex polynomial $p(z)$ of degree $n$ and an ellipse, we find an algorithm of determining the number of zeros of $p$ in the interior and exterior of the ellipse. Our results generalize the previous results of Pt\'ak and Young about the circles., Comment: We withdraw the paper temporarily because of a mistake in Theorem 2.1

The main aims of this article are to characterize a class of operators associated with the symmetrized polydisc that admit rational dilations on the minimal space and to show an interplay between rational dilation and distinguished varieties in the symmetrized polydisc., Comment: Revised, 41 Pages

We introduce models for saliency prediction for mobile user interfaces. A mobile interface may include elements like buttons, text, etc. in addition to natural images which enable performing a variety of tasks. Saliency in natural images is a well studied area. However, given the difference in what constitutes a mobile interface, and the usage context of these devices, we postulate that saliency prediction for mobile interface images requires a fresh approach. Mobile interface design involves operating on elements, the building blocks of the interface. We first collected eye-gaze data from mobile devices for free viewing task. Using this data, we develop a novel autoencoder based multi-scale deep learning model that provides saliency prediction at the mobile interface element level. Compared to saliency prediction approaches developed for natural images, we show that our approach performs significantly better on a range of established metrics., Comment: Paper accepted at WACV 2018

A commuting tuple of operators $(S_1,\dots, S_{n-1},P)$, defined on a Hilbert space $\mathcal H$, for which the closed symmetrized polydisc \[ \Gamma_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i

In this paper, we investigate the performance of different approximate variants of the EOM-CCSD method for calculation of ionization potential, as compared to EOM-CCSDT reference values. The errors in the various approximate variants of the EOM-CCSD method are quite different for different kind of ionized states. None of the approximate variants including the original EOM-CCSD method gives a uniform performance over outer valence, inner valence, and core ionization, favoring one or other, depending upon nature of the approximation used.

We present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock $\mathbb E$. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc $\mathbb G_2$. In either case, our results generalize all previous results in this direction for $\mathbb E$ and $\mathbb G_2$.

A tuple of commuting operators $(S_1,\dots,S_{n-1},P)$ for which the closed symmetrized polydisc $\Gamma_n$ is a spectral set is called a $\Gamma_n$-contraction. We show that every $\Gamma_n$-contraction admits a decomposition into a $\Gamma_n$-unitary and a completely non-unitary $\Gamma_n$-contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $\Gamma_n$ and $\Gamma_n$-contractions., Comment: Complex Analysis and Operator Theory, Published online on August 28, 2017. arXiv admin note: text overlap with arXiv:1610.00936

The scalar-pseudoscalar interaction constant of PbF in its ground state electronic configuration is calculated using the Z-vector method in the relativistic coupled-cluster framework. The precise calculated value is very important to set upper bound limit on P,T-odd scalar-pseudoscalar interaction constant, k_s, from the experimentally observed P,T-odd frequency shift. Further, the ratio of the effective electric field to the scalar-pseudoscalar interaction constant is also calculated which is required to get an independent upper bound limit of electric dipole moment of electron, d_e, and k_s and how these (d_e and k_s) are interrelated is also presented here., Comment: 6 pages, 1 figures

The Z-vector method in the relativistic coupled-cluster framework is employed to calculate the parallel and perpendicular components of the magnetic hyperfine structure constant of a few small alkaline earth hydrides (BeH, MgH, and CaH) and fluorides (MgF and CaF). We have compared our Z-vector results with the values calculated by the extended coupled-cluster (ECC) method reported in Phys. Rev. A 91 022512 (2015). All these results are compared with the available experimental values. The Z-vector results are found to be in better agreement with the experimental values than those of the ECC values., Comment: 5 pages, 2 figures

We produce a Schwarz lemma for the symmetrized tridisc \[ \mathbb G_3 =\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|< 1, i=1,2,3 \}. \] We show that an interpolating function related to the Schward lemma for $\mathbb G_3$ is not unique and present an explicit description of all such interpolating functions. We also study the complex geometry of $\mathbb G_3$ and present a variety of new characterizations for the open and closed symmetrized tridisc., Comment: Error in one of the main results

The automorphisms of the symmetrized polydisc $\mathbb G_n$ are well-known and are given in the coordinates of the polydisc in \cite{E:Z}. We find an explicit formula for the automorphisms of $\mathbb G_n$ in its own coordinates. If $\tau$ is an automorphism of $\mathbb G_n$, then $\tau(S_1,\dots,S_{n-1},P)$ is a $\Gamma_n$-contraction, where a $\Gamma_n$-contraction is a commuting $n$-tuple of Hilbert space operators for which the closed symmetrized polydisc $\Gamma_n$ is a spectral set. Corresponding to every $\Gamma_n$-contraction $(S_1,\dots,S_{n-1},P)$, there exist $n-1$ unique operators $A_1,\dots,A_{n-1}$ such that \[ S_i-S_{n-i}^*P=D_PA_iD_P\,, \quad D_P=(I-P^*P)^{1/2}\,, \] for $i=1,\dots, n-1$. This unique $(n-1)$-tuple $(A_1,\dots,A_{n-1})$, which is called the fundamental operator tuple or $\mathcal F_O$-tuple of $(S_1,\dots,S_{n-1},P)$ in literature, plays central role in every section of operator theory on $\Gamma_n$. We find an explicit form of the $\mathcal F_O$-tuple of $\tau (S_1,\dots,S_{n-1},P)$ when $n=3$. We show by an example that a $\Gamma_n$-contraction may not have commuting $\mathcal F_O$-tuple. Also, we obtain a necessary and sufficient condition under which two $\Gamma_n$-contractions are unitarily equivalent., Comment: 13 Pages

The symmetrized polydisc of dimension three is the set \[ \Gamma_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which $\Gamma_3$ is a spectral set is called a $\Gamma_3$-contraction. We show that every $\Gamma_3$-contraction admits a decomposition into a $\Gamma_3$-unitary and a completely non-unitary $\Gamma_3$-contraction. This decomposition parallels the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $\Gamma_3$ and $\Gamma_3$-contractions., Comment: To appear in Operators and Matrices, 2017

We show that every distinguished variety in the symmetrized tridisc $\mathbb G_3$ is one-dimensional and can be represented as \begin{equation}\label{eqn:1} \Lambda=\{ (s_1,s_2,p)\in \mathbb G_3 \,:\, (s_1,s_2) \in \sigma_T(F_1^*+pF_2\,,\, F_2^*+pF_1) \}, \end{equation} where $F_1,F_2$ are commuting square matrices of the same order satisfying $[F_1^*,F_1]=[F_2^*,F_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb G_3$. We show that for a triple of commuting operators $\Sigma = (S_1,S_2,P)$ having $\Gamma_3$ as a spectral set, there is a one-dimensional subvariety $\Lambda_{\Sigma}$ of $\Gamma_3$ depending on $\Sigma$ such that von-Neumann's inequality holds, i.e, \[ f(S_1,S_2,P)\leq \sup_{(s_1,s_2,p)\in\Lambda_{\Sigma}}\, |f(s_1,s_2,p)|, \] for any holomorphic polynomial $f$ in three variables, provided that $P^n\rightarrow 0$ strongly as $n\rightarrow \infty$. The variety $\Lambda_\Sigma$ has been shown to have representation like (\ref{eqn:1}), where $F_1,F_2$ are the unique solutions of the operator equations \begin{gather*} S_1-S_2^*P=(I-P^*P)^{\frac{1}{2}}X_1(I-P^*P)^{\frac{1}{2}} \text{ and } \\ S_2-S_1^*P=(I-P^*P)^{\frac{1}{2}}X_2(I-P^*P)^{\frac{1}{2}}. \end{gather*} We also show that under certain condition, $\Lambda_\Sigma$ is a distinguished variety in $\mathbb G_3$. We produce an explicit dilation and a concrete functional model for such a triple $(S_1,S_2,P)$ in which the unique operators $F_1,F_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the symmetrized bidisc and in the unit bidisc $\mathbb D^2$., Comment: We withdraw this article because we could resolve the n-variable case

The closed symmetrized tridisc $\Gamma_3$ and its distinguished boundary $b\Gamma_3$ are the sets $\Gamma_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3$ $b\Gamma_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|= 1, i=1,2,3 \}\subseteq \Gamma_3.$ A triple of commuting operators $(S_1,S_2,P)$ defined on a Hilbert space $\mathcal H$ for which $\Gamma_3$ is a spectral set is called a $\Gamma_3$-contraction. In this article we show by a counter example that there are $\Gamma_3$-contractions which do not dilate. It is also shown that under certain conditions a $\Gamma_3$-contraction can have normal $b\Gamma_3$ dilation. We determine several classes of $\Gamma_3$-contractions which dilate and show explicit construction of their dilations. A concrete functional model is provided for the $\Gamma_3$-contractions which dilate. Various characterizations for $\Gamma_3$-unitaries and $\Gamma_3$-isometries are obtained; the classes of $\Gamma_3$-unitaries and $\Gamma_3$-isometries are analogous to the unitaries and isometries in one variable operator theory. Also we find out a model for the class of pure $\Gamma_3$-isometries. En route we study the geometry of the sets $\Gamma_3$ and $b\Gamma_3$ and provide variety of characterizations for them., Comment: Incomplete

We have employed both Z-vector method and the expectation value approach in the relativistic coupled-cluster framework to calculate the scalar-pseudoscalar (S-PS) P, T -odd interaction constant (W_s) and the effective electric field (Eeff) experienced by the unpaired electron in the ground electronic state of RaF. Further, the magnetic hyperfine structure constants of ^{223}Ra in RaF and ^{223}Ra+ are also calculated and compared with the experimental values wherever available to judge the extent of accuracy obtained in the employed methods. The outcome of our study reveals that the Z-vector method is superior than the expectation value approach in terms of accuracy obtained for the calculation of ground state property. The Z-vector calculation shows that RaF has a high E_eff (52.5 GV/cm) and W_s (141.2 kHz) which makes it a potential candidate for the eEDM experiment., Comment: 8 pages, 3 figures

The open-shell reference relativistic equation-of-motion coupled-cluster method within its four-component description is successfully implemented with the consideration of single- and double- excitation approximation. The one-body and two-body matrix elements required for the correlation calculation are generated using Dirac-Coulomb Hamiltonian. As a first attempt, the implemented method is employed to calculate a few of the low-lying ionized states of heavy atomic (Ag, Cs, Au, Fr, Lr) and valence ionization potential of molecular (HgH, PbF) systems, where the effect of relativity does really matter to obtain highly accurate results. Not only the relativistic effect, but also the effect of electron correlation is crucial in these heavy atomic and molecular systems. To justify the fact, we have taken two further approximations in the four-component relativistic equation-of-motion framework to quantify how the effect of electron correlation plays a role in the calculated values at different level of the approximation. All these calculated results are compared with the available experimental data as well as with other theoretically calculated values to judge the extent of accuracy obtained in our calculations. A nice agreement is achieved for all the considered systems which validates the performance of the open-shell reference relativistic equation-of-motion coupled-cluster method., Comment: 10 pages

The high effective electric field ($E_\mathrm{eff}$) experienced by the unpaired electron in an atom or a molecule is one of the key ingredients in the success of electron electric dipole moment (eEDM) experiment and its precise calculation require a very accurate theory. We, therefore, employed the Z-vector method in the relativistic coupled-cluster framework and found that HgH has a very large $E_\mathrm{eff}$ value (123.2 GV/cm) which makes it a potential candidate for the next generation eEDM experiment. Our study also reveals that it has a large scalar-pseudoscalar ${\mathcal{P,T}}$-violating interaction constant, $W_\mathrm{s}$ = 284.2 kHz. To judge the accuracy of the obtained results we have calculated parallel and perpendicular magnetic HFS constants and compared with the available experimental values. The results of our calculation are found to be in nice agreement with the experimental values. Therefore, by looking at the HFS results we can say that both $E_\mathrm{eff}$ and $W_\mathrm{s}$ values are also very accurate. Further, We have derived the relationship between these quantities and the ratio which will help to get model independent value of eEDM and S-PS interaction constant., Comment: main text : 5 pages