arXiv subjects

Sourav Pal

Publications and source records attributed to Sourav Pal.

At least 91 records · Page 5Linked to original sources

arXiv2021

Automorphisms and the fundamental operators associated with the symmetrized tridisc

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 $τ$ is an automorphism of $\mathbb G_n$, then $τ(S_1,\dots,S_{n-1},P)$ is a $Γ_n$-contraction, where a $Γ_n$-contraction is a commuting $n$-tuple of Hilbert space operators for which the closed symmetrized polydisc $Γ_n$ is a spectral set. Corresponding to every $Γ_n$-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 $Γ_n$. We find an explicit form of the $\mathcal F_O$-tuple of $τ(S_1,\dots,S_{n-1},P)$ when $n=3$. We show by an example that a $Γ_n$-contraction may not have commuting $\mathcal F_O$-tuple. Also, we obtain a necessary and sufficient condition under which two $Γ_n$-contractions are unitarily equivalent.

math.FA↗

arXiv2021

Characterizations of the symmetrized polydisc via another family of domains

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 \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We introduce a new family of domains which we call \textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, and define in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\dots,y_{n-1}, q)\in \mathbb C^n :\; q \in \mathbb D, \; y_j = β_j + \barβ_{n-j} q, \; β_j \in \mathbb C &\text{ and }\\ |β_j|+ |β_{n-j}| < {n \choose j} &\text{ for } j=1,\dots, n-1 \Bigg\}. \end{align*} We show that $\mathbb G_n=\widetilde{\mathbb G}_n$ for $n=1,2$ and that ${\mathbb G}_n \subsetneq \widetilde{\mathbb G}_n$ for $n\geq 3$. We first obtain a variety of characterizations for the points in $\widetilde{\mathbb G}_n$ and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in ${\mathbb G}_n$. Also we obtain similar characterizations for the points in $Γ_n \setminus {\mathbb G}_n$, where $Γ_n =\overline{{\mathbb G}_n}$. A set of $n-1$ fractional linear transformations play central role in the entire program. We also show that for $n\geq 2$, $\widetilde{\mathbb G}_n$ is non-convex but polynomially convex and is starlike about the origin but not circled.

math.CV↗

arXiv2021

A Schwarz lemma for the symmetrized polydisc via estimates on another family of domains

We make some sharp estimates to obtain a Schwarz lemma for the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains naturally associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We first make a few estimates for the \textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, a family of domains introduced in \cite{pal-roy 4} and defined in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\dots,y_{n-1}, q)\in \C^n :\; q \in \mathbb D, \; y_j = \be_j + \bar \be_{n-j} q, \; β_j \in \mathbb C &\text{ and }\\ |β_j|+ |β_{n-j}| < {n \choose j} &\text{ for } j=1,\dots, n-1 \Bigg\}. \end{align*} We then show that these estimates are sharp and provide a Schwarz lemma for $\Gn$. It is easy to verify that $\mathbb G_n=\widetilde{\mathbb G}_n$ for $n=1,2$ and that ${\mathbb G}_n \subsetneq \widetilde{\mathbb G}_n$ for $n\geq 3$. As a consequence of the estimates for $\widetilde{\mathbb G_n}$, we have analogous estimates for $\mathbb G_n$. Since for a point $(s_1,\dots, s_{n-1},p)\in \mathbb G_n$, ${n \choose i}$ is the least upper bound for $|s_i|$, which is same for $|y_i|$ for any $(y_1,\dots ,y_{n-1},q) \in \widetilde{\mathbb G_n}$, $1\leq i \leq n-1$, the estimates become sharp for $\mathbb G_n$ too. We show that these conditions are necessary and sufficient for $\widetilde{\mathbb G_n}$ when $n=1,2, 3$. In particular for $n=2$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc.

math.CV↗

arXiv2021

The symmetrization map and $Γ$-contractions

The symmetrization map $π:\mathbb C^2\rightarrow \mathbb C^2$ is defined by $ π(z_1,z_2)=(z_1+z_2,z_1z_2). $ The closed symmetrized bidisc $Γ$ is the symmetrization of the closed unit bidisc $\overline{\mathbb D^2}$, that is, \[ Γ= π(\overline{\mathbb D^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\leq 1, i=1,2 \}. \] A pair of commuting Hilbert space operators $(S,P)$ for which $Γ$ is a spectral set is called a $Γ$-contraction. Unlike the scalars in $Γ$, a $Γ$-contraction may not arise as a symmetrization of a pair of commuting contractions, even not as a symmetrization of a pair of commuting bounded operators. We characterize all $Γ$-contractions which are symmetrization of pairs of commuting contractions. We show by constructing a family of examples that even if a $Γ$-contraction $(S,P)=(T_1+T_2,T_1T_2)$ for a pair of commuting bounded operators $T_1,T_2$, no real number less than $2$ can be a bound for the set $\{ \|T_1\|,\|T_2\| \}$ in general. Then we prove that every $Γ$-contraction $(S,P)$ is the restriction of a $Γ$-contraction $(\widetilde S, \widetilde P)$ to a common reducing subspace of $\widetilde S, \widetilde P$ and that $(\widetilde S, \widetilde P)=(A_1+A_2,A_1A_2)$ for a pair of commuting operators $A_1,A_2$ with $\max \{\|A_1\|, \|A_2\|\} \leq 2$. We find new characterizations for the $Γ$-unitaries and describe the distinguished boundary of $Γ$ in a different way. We also show some interplay between the fundamental operators of two $Γ$-contractions $(S,P)$ and $(S_1,P)$.

math.FA↗

arXiv2021

Dilation, functional model and a complete unitary invariant for $C._{0}\,\; Γ_n$-contractions

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 \[ Γ_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i \right): \,|z_i|\leq 1, i=1,\dots,n \right \} \] is a spectral set, is called a $Γ_n$-\textit{contraction}. A $Γ_n$-contraction is said to be \textit{pure} or $C._0$ if $P$ is $C._0$, that is, if ${P^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty$. We show that for any $Γ_n$-contraction $(S_1,\dots, S_{n-1},P)$, there is a unique operator tuple $(A_1,\dots , A_{n-1})$ that satisfies the operator identities \[ S_i-S_{n-i}^*P=D_PA_iD_P\,, \quad \quad i=1,\dots, n-1. \] This unique tuple is called the \textit{fundamental operator tuple} or $\mathcal F_O$-tuple of $(S_1,\dots, S_{n-1},P)$. With the help of the $\mathcal F_O$-tuple, we construct an operator model for a $C._0 \; Γ_n$-contraction and show that there exist $n-1$ operators $C_1,\dots, C_{n-1}$ such that each $S_i$ can be represented as $S_i=C_i+PC_{n-i}^*$. We find an explicit minimal dilation for a class of $C._0 \; Γ_n$-contractions whose $\mathcal F_O$-tuples satisfy a certain condition. Also we establish that the $\mathcal F_O$-tuple of $(S_1^*,\dots, S_{n-1}^*,P^*)$ together with the characteristic function of $P$ constitute a complete unitary invariant for the $C._0$ $Γ_n$-contractions. The entire program is an analogue of the Nagy-Foias theory for $C._0$ contractions.

math.FA↗

arXiv2021

Power corrections to event shapes using eikonal dressed gluon exponentiation

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.

hep-ph↗

arXiv2021

Cwebs beyond three loops in multiparton amplitudes

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.

hep-ph↗

arXiv2021

Multiparton webs beyond three loops

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.

hep-ph↗

arXiv2021

Effective Theory of Inflationary Magnetogenesis and Constraints on Reheating

Effective theory framework based on symmetry has recently gained widespread interest in the field of cosmology. In this paper, we apply the same idea on the genesis of the primordial magnetic field and its evolution throughout the cosmological universe. Given the broken time-diffeomorphism symmetry by the cosmological background, we considered the most general Lagrangian of electromagnetic and metric fluctuation up to second order, which naturally breaks conformal symmetry in the electromagnetic (EM) sector. We also include parity violation in the electromagnetic sector with the motivation that has potential observational significance. In such a set-up, we explore the evolution of EM, scalar, and tensor perturbations considering different observational constraints. In our analysis we emphasize the role played by the intermediate reheating phase which has got limited interest in all the previous studies. Assuming the vanishing electrical conductivity during the entire period of reheating, the well-known Faraday electromagnetic induction has been shown to play a crucial role in enhancing the strength of the present-day magnetic field. We show how such physical effects combined with the PLANCK and the large scale magnetic field observation makes a large class of models viable and severely restricts the reheating equation of state parameter within a very narrow range of $0.01 < ω_\mathrm{eff} < 0.27$, which is nearly independent of reheating scenarios we have considered.

hep-th↗

arXiv2021

Probing the reheating phase through primordial magnetic field and CMB

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, $ω_ϕ = (p+2)/(p+2)$ during reheating given a specific value of the large scale magnetic field. This observation hints the inflaton to assume the potential of form $V(ϕ) \sim ϕ^p$ 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.

hep-th↗

arXiv2020

Relativistic double-ionization equation-of-motion coupled-cluster method: Application to low-lying doubly ionized states

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.

physics.chem-ph↗

arXiv2020

Relativistic coupled-cluster study of BaF in search of $\mathcal{CP}$ violation

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.

physics.chem-ph↗

arXiv2020

Role of electron-correlation in the $\mathcal{P,T}$-odd effects of CdH: A relativistic coupled-cluster investigation

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).

physics.chem-ph↗

arXiv2019

A Schwarz lemma for two families of domains and complex geometry

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 $\widetildeΓ_n$.

math.CV↗

arXiv2019

Structure theorems for operators associated with two domains related to $μ$-synthesis

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 \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n}z_iz_j, \dots, \prod_{i=1}^{n}z_i \right) : |z_i|\leq 1, i=1, \dots, n \right\} \] is a spectral set is called a $Γ_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called an $\mathbb E$-contraction, where \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3 \neq 0 \quad \forall z, w \in \overline{\mathbb D} \}. \] There are several decomposition theorems for contraction operators in the literature due to Sz. Nagy, Foias, Levan, Kubrusly, Foguel and few others which reveal structural information of a contraction. In this article, we obtain analogues of six such major theorems for both $Γ_n$-contractions and $\mathbb E$-contractions. In each of these decomposition theorems, the underlying Hilbert space admits a unique orthogonal decomposition which is provided by the last component $P$. The central role in determining the structure of a $Γ_n$-contraction or an $\mathbb E$-contraction is played by positivity of some certain operator pencils and the existence of a unique operator tuple associated with a $Γ_n$-contraction or an $\mathbb E$-contraction.

math.FA↗

arXiv2019

A Schwarz lemma for the symmetrized tridisc and description of interpolating functions

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.

math.CV↗

arXiv2018

Visual Attention for Behavioral Cloning in Autonomous Driving

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.

cs.CV↗

arXiv2018

Relativistic coupled-cluster investigation of parity (P) and time-reversal (T) symmetry violations in HgF

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.

physics.atom-ph↗