arXiv subjects
Publications and source records attributed to Sourav Pal.
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 Pták and Young about the circles.
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 $\mathbb G_2$.
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.
A tuple of commuting operators $(S_1,\dots,S_{n-1},P)$ for which the closed symmetrized polydisc $Γ_n$ is a spectral set is called a $Γ_n$-contraction. We show that every $Γ_n$-contraction admits a decomposition into a $Γ_n$-unitary and a completely non-unitary $Γ_n$-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 $Γ_n$ and $Γ_n$-contractions.
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 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} Λ=\{ (s_1,s_2,p)\in \mathbb G_3 \,:\, (s_1,s_2) \in σ_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 $Σ= (S_1,S_2,P)$ having $Γ_3$ as a spectral set, there is a one-dimensional subvariety $Λ_Σ$ of $Γ_3$ depending on $Σ$ such that von-Neumann's inequality holds, i.e, \[ f(S_1,S_2,P)\leq \sup_{(s_1,s_2,p)\inΛ_Σ}\, |f(s_1,s_2,p)|, \] for any holomorphic polynomial $f$ in three variables, provided that $P^n\rightarrow 0$ strongly as $n\rightarrow \infty$. The variety $Λ_Σ$ 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, $Λ_Σ$ 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 $\mathbb D^2$.
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.
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.
The closed symmetrized tridisc $Γ_3$ and its distinguished boundary $bΓ_3$ are the sets $Γ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3$ $bΓ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|= 1, i=1,2,3 \}\subseteq Γ_3.$ A triple of commuting operators $(S_1,S_2,P)$ defined on a Hilbert space $\mathcal H$ for which $Γ_3$ is a spectral set is called a $Γ_3$-contraction. In this article we show by a counter example that there are $Γ_3$-contractions which do not dilate. It is also shown that under certain conditions a $Γ_3$-contraction can have normal $bΓ_3$ dilation. We determine several classes of $Γ_3$-contractions which dilate and show explicit construction of their dilations. A concrete functional model is provided for the $Γ_3$-contractions which dilate. Various characterizations for $Γ_3$-unitaries and $Γ_3$-isometries are obtained; the classes of $Γ_3$-unitaries and $Γ_3$-isometries are analogous to the unitaries and isometries in one variable operator theory. Also we find out a model for the class of pure $Γ_3$-isometries. En route we study the geometry of the sets $Γ_3$ and $bΓ_3$ and provide variety of characterizations for them.
The symmetrized polydisc of dimension three is the set \[ Γ_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which $Γ_3$ is a spectral set is called a $Γ_3$-contraction. We show that every $Γ_3$-contraction admits a decomposition into a $Γ_3$-unitary and a completely non-unitary $Γ_3$-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 $Γ_3$ and $Γ_3$-contractions.
We show an interplay between the complex geometry of the tetrablock $\mathbb E$ and the commuting triples of operators having $\overline{\mathbb E}$ as a spectral set. We prove that every distinguished variety in the tetrablock is one-dimensional and can be represented as \begin{equation}\label{eqn:1} Ω=\{ (x_1,x_2,x_3)\in \mathbb E \,:\, (x_1,x_2) \in σ_T(A_1^*+x_3A_2\,,\, A_2^*+x_3A_1) \}, \end{equation} where $A_1,A_2$ are commuting square matrices of the same order satisfying $[A_1^*,A_1]=[A_2^*,A_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb E$. We show that for a triple of commuting operators $Υ= (T_1,T_2,T_3)$ having $\overline{\mathbb E}$ as a spectral set, there is a one-dimensional subvariety $Ω_Υ$ of $\overline{\mathbb E}$ depending on $Υ$ such that von-Neumann's inequality holds, i.e, \[ f(T_1,T_2,T_3)\leq \sup_{(x_1,x_2,x_3)\inΩ_Υ}\, |f(x_1,x_2,x_3)|, \] for any holomorphic polynomial $f$ in three variables, provided that $T_3^n\rightarrow 0$ strongly as $n\rightarrow \infty$. The variety $Ω_Υ$ has been shown to have representation like (\ref{eqn:1}), where $A_1,A_2$ are the unique solutions of the operator equations \begin{gather*} T_1-T_2^*T_3=(I-T_3^*T_3)^{\frac{1}{2}}X_1(I-T_3^*T_3)^{\frac{1}{2}} \text{ and } \\ T_2-T_1^*T_3=(I-T_3^*T_3)^{\frac{1}{2}}X_2(I-T_3^*T_3)^{\frac{1}{2}}. \end{gather*} We also show that under certain condition, $Ω_Υ$ is a distinguished variety in $\mathbb E$. We produce an explicit dilation and a concrete functional model for such a triple $(T_1,T_2,T_3)$ in which the unique operators $A_1,A_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the bidisc and in the symmetrized bidisc.
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.
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.
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.
A triple of commuting operators for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called a tetrablock contraction or an $\mathbb E$-contraction. The set $\mathbb E$ is defined as \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3\neq 0 \textup{ whenever } |z|\leq 1, |w|\leq 1 \}. \] We show that every $\mathbb E$-contraction can be uniquely written as a direct sum of an $\mathbb E$-unitary and a completely non-unitary $\mathbb E$-contraction. It is analogous to the canonical decomposition of a contraction operator into a unitary and a completely non-unitary contraction. We produce a concrete operator model for such a triple satisfying some conditions.
The article considers the successful implementation of relativistic equation-of-motion coupled cluster method for the electron attachment problem (EA-EOMCC) at the level of single- and double- excitation approximation. The implemented relativistic EA-EOMCC method is employed to calculate ionization potential values of alkali metal atoms (Li, Na, K, Rb, Cs, Fr) and the vertical electron affinity values of LiX (X = H, F, Cl, Br), NaY (Y = H, F, Cl) starting from their closed-shell configuration. Both four-component and exact two-component calculations are done for all the opted systems. Further, we have shown the effect of spin-orbit interaction considering the atomic systems. The results of our atomic calculations are compared with the values from the NIST database and the results are found to be very accurate (< 1 %).
The effective electric field experienced by the unpaired electron in the ground state of PbF, which is a potential candidate in the search of electron electric dipole moment due to some special characteristics, is calculated using Z-vector method in the coupled cluster single- and double- excitation approximation with four component Dirac spinor. This is an important quantity to set the upper bound limit of the electron electric dipole moment. Further, we have calculated molecular dipole moment and parallel magnetic hyperfine structure constant (A$_\|$) of $^{207}$Pb in PbF to test the accuracy of the wave function obtained in the Z-vector method. The outcome of our calculations clearly suggests that the core electrons have significant contribution to the "atom in compound (AIC)" properties.
For a contraction P and a bounded commutant S of P, we seek a solution X of the operator equation S-S*P = (I-P*P)^1/2 X(I-P*P) 1/2, where X is a bounded operator on Ran(I-P*P) 1/2 with numerical radius of X being not greater than 1. A pair of bounded operators (S,P) which has the domain \Gamme = {(z 1 +z 2, z 1z 2) : |z1|{\leq} 1, |z2| {\leq}1} {\subseteq} C2 as a spectral set, is called a \Gamme-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Γ-contraction (S,P). This allows us to construct an explicit Γ-isometric dilation of a Γ-contraction (S,P). We prove the other way too, i.e, for a commuting pair (S,P) with |P|| {\leq} 1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S,P) is a Γ-contraction. We show that for a pure Γ-contraction (S,P), there is a bounded operator C with numerical radius not greater than 1, such that S = C +C*P. Any Γ-isometry can be written in this form where P now is an isometry commuting with C and C*. Any Γ-unitary is of this form as well with P and C being commuting unitaries. Examples of Γ-contractions on reproducing kernel Hilbert spaces and their Γ-isometric dilations are discussed.