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A tuple $\underline{T}=(T_1, \dotsc, T_k)$ of operators on a Hilbert space $\mathcal H$ is said to be \textit{$q$-commuting with} $\|q\|=1$ or simply $q$-\textit{commuting} if there is a family of scalars $q=\{q_{ij} \in \mathbb C : |q_{ij}|=1, \ q_{ij}=q_{ji}^{-1}, \ 1 \leq i < j \leq k \}$ such that $T_i T_j =q_{ij}T_j T_i$ for $1 \leq i < j \leq k$. Moreover, if each $q_{ij}=-1$, then $\underline{T}$ is called an \textit{anti-commuting tuple}. A well-known result due to Holbrook \cite{Holbrook} states that a commuting $k$-tuple consisting of $2 \times 2$ scalar matrix contractions always dilates to a commuting $k$-tuple of unitaries for any $k\geq 1$. To find a generalization of this result for a $q$-commuting $k$-tuple of $2\times 2$ scalar matrix contractions, we first classify such tuples into three types upto similarity. Then we prove that a $q$-commuting tuple which is unitarily equivalent to any of these three types, admits a $\widetilde{q}$-unitary dilation, where $\widetilde q \subseteq q \cup \{1\}$. A special emphasis is given to the dilation of an anti-commuting tuple of $2 \times 2$ scalar matrix contractions., Comment: 17 pages

In the standard cosmological framework, neutrinos begin to free-stream after the weak interaction phase ends in the early universe, at a temperature of approximately $T \sim 1 \, {\rm MeV}$. However, the onset of neutrino free-streaming can be delayed if additional interactions occur in the early universe, leaving imprints on both the cosmic microwave background (CMB) angular power spectra and the large-scale structure (LSS) matter power spectra. We present a thorough analysis of early universe neutrino interactions with a fairly generalized parameterization of the interaction rates as a power law in neutrino temperature. In this (6+2) parameter scenario, we constrain the cosmological parameters along with the free-streaming redshift and the sum of the neutrino mass in presence of such interactions, with the help of full shape (FS) galaxy power spectra from BOSS Data Release 12. Our analysis reveals that a combined dataset of FS along with CMB and BAO offers improved constraints on the free-streaming redshift from present data, comparable to the forecast results from future CMB-S4 data. Additionally, we investigate the prospects of future galaxy surveys by forecasting on Euclid mission in combination with Planck and CMB-S4, and find significant improvement on both the free-streaming redshift and the sum of the neutrino mass than the existing constraints as well as than CMB-S4 alone., Comment: 21 pages, 5 sets of figures and 7 tables. Comments are welcome

We generalize regular unitary dilation and Brehmer's positivity condition to $q$-commuting tuples of contractions., Comment: This is a first draft and will be revised soon

For a complex Banach space $\mathbb X$, we prove that $\mathbb X$ is a Hilbert space if and only if every strict contraction $T$ on $\mathbb X$ dilates to an isometry if and only if for every strict contraction $T$ on $\mathbb X$ the function $A_T: \mathbb X \rightarrow [0, \infty]$ defined by $A_T(x)=(\|x\|^2 -\|Tx\|^2)^{\frac{1}{2}}$ gives a norm on $\mathbb X$. We also find several other necessary and sufficient conditions in this thread such that a Banach sapce becomes a Hilbert space. We construct examples of strict contractions on non-Hilbert Banach spaces that do not dilate to isometries. Then we characterize all strict contractions on a non-Hilbert Banach space that dilate to isometries and find explicit isometric dilation for them. We prove several other results including characterizations of complemented subspaces in a Banach space, extension of a Wold isometry to a Banach space unitary and describing norm attainment sets of Banach space operators in terms of dilations., Comment: 61 Pages, Submitted to journal

We define soft quark operators that enable construction of colour-ordered helicity amplitudes out of the non-radiative ones. We explore these operators for Higgs plus one jet production in a hadron collider and study the next-to-leading power corrections on all partonic channels involving quark(s) and/or anti-quark(s). We also investigate the effect of next-to-soft gluon radiation in these channels employing the technique illustrated in \cite{Pal:2023vec}. Analytical expressions of next-to-leading power leading logarithms thus obtained are simple and compact in nature. Further, we discuss the connection of such logarithms with that of the pseudo-scalar Higgs plus one jet production., Comment: 30 pages, 1 figure

A deterministic temporal process can be determined by its trajectory, an element in the product space of (a) initial condition $z_0 \in \mathcal{Z}$ and (b) transition function $f: (\mathcal{Z}, \mathcal{T}) \to \mathcal{Z}$ often influenced by the control of the underlying dynamical system. Existing methods often model the transition function as a differential equation or as a recurrent neural network. Despite their effectiveness in predicting future measurements, few results have successfully established a method for sampling and statistical inference of trajectories using neural networks, partially due to constraints in the parameterization. In this work, we introduce a mechanism to learn the distribution of trajectories by parameterizing the transition function $f$ explicitly as an element in a function space. Our framework allows efficient synthesis of novel trajectories, while also directly providing a convenient tool for inference, i.e., uncertainty estimation, likelihood evaluations and out of distribution detection for abnormal trajectories. These capabilities can have implications for various downstream tasks, e.g., simulation and evaluation for reinforcement learning.

We propose and study a conceptual one-dimensional model to explore how the combined interplay between fixed resources and particle exchanges between different parts of an extended system can affect the stationary densities in a current carrying channel connecting different parts of the system. To this end, we consider a model composed of a totally asymmetric simple exclusion process (TASEP) connecting two particle reservoirs without any internal dynamics but which can directly exchange particles between each other, ensuring nonvanishing currents in the steady states. The total particle number in the system that defines the "resources" available, although is kept constant by the model dynamics, can take any value independent of the model parameters that define the dynamics of the model. We show how the resulting phase diagrams of the model are controlled by the parameters, which define the various dynamical update rules together with the total available resources. These control parameters can be tuned to make the density on the TASEP lane globally uniform or piecewise continuous with localized domain walls, and can also control populations of the two reservoirs. In general, the phase diagrams are quite different from a TASEP with open boundaries. In the limit of large amount of resources, the phase diagrams in the plane of the control parameters become topologically identical to that for an open TASEP together with delocalization of domain walls., Comment: 21 pages, 14 figures

We study the operator-valued positive definite functions on a group using positive block matrices. We give an alternative proof to Brehmer positivity for doubly commuting contractions. We classify all commuting unitary representations over a finite group. We show by examples that the power of a positive-definite function may not be positive definite and also the power of a unitary representation may not be a unitary representation. We also characterize all unitary representations whose powers are also unitary representations., Comment: 27 pages

A commuting tuple of Hilbert space operators $(T_1, \dotsc, T_n)$ is said to be an \textit{$\mathbb{A}_r^n$-contraction} if the closure of the polyannulus \[ \mathbb A_r^n=\left\{(z_1, \dotsc, z_n) \ : \ r<|z_i|<1, \ 1 \leq i \leq n \right\} \subseteq \mathbb{C}^n \qquad \quad (0

The existence of large-scale anisotropy can not be ruled out by the cosmic microwave background (CMB) radiation. Over the years, several models have been proposed in the context of anisotropic inflation to account for CMB's cold spot and hemispheric asymmetry. However, any small-scale anisotropy, if exists during inflation, is not constrained due to its nonlinear evolution in the subsequent phase. This small-scale anisotropy during inflation can play a non-trivial role in giving rise to the cosmic magnetic field, which is the subject of our present study. Assuming a particular phenomenological form of an anisotropic inflationary universe, we have shown that it can generate a large-scale magnetic field at $1$-Mpc scale with a magnitude $\sim 4\times 10^{-20}~G$, within the observed bound. Because of the anisotropy, the conformal flatness property is lost, and the Maxwell field is generated even without explicit coupling. This immediately resolves the strong coupling problem in the standard magnetogenesis scenario. In addition, assuming very low conductivity during the reheating era, we can further observe the evolution of the electromagnetic field with the equation of state (EoS) $\omega_{eff}$ and its effects on the present-day magnetic field., Comment: 15 pages, 5 figures. Comments are welcome

A bounded Hilbert space operator $T$ for which the closure of the annulus \[ \mathbb{A}_r=\{z \ : \ r<|z|<1\} \subseteq \mathbb{C}, \qquad (0

A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a \textit{$\mathbb{P}$-contraction} if the closed pentablock $\overline{\mathbb P}$ is a spectral set for $(A,S,P)$, where \[ \mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\ : \ A_0=[a_{ij}]_{2 \times 2} \; \; \& \;\; \|A_0\| <1 \right\} \subseteq \mathbb{C}^3. \] We find several characterizations for the $\mathbb P$-unitaries and $\mathbb P$-isometries. We show that every $\mathbb P$-isometry admits a Wold type decomposition that splits it into a direct sum of a $\mathbb P$-unitary and a pure $\mathbb P$-isometry. We also show that every $\mathbb P$-contraction $(A,S,P)$ possesses a canonical decomposition that orthogonally decomposes $(A,S,P)$ into a $\mathbb P$-unitary and a completely non-unitary $\mathbb P$-contraction. We find a necessary and sufficient condition such that a $\mathbb P$-contraction $(A, S, P)$ dilates to a $\mathbb P$-isometry $(X, T, V)$ with $V$ being the minimal isometric dilation of $P$. Then we show an explicit construction of such a conditional dilation. A commuting tuple of Hilbert space operators $(T_1, \dots , T_n)$ having the closed unit ball $\overline{\mathbb B}_n$ as a spectral set is called a $\mathbb B_n$-\textit{contraction} and a commuting pair $(S,P)$ having the closed symmetrized bidisc $\overline{\mathbb G_2} \,(=\Gamma)$ as a spectral set is called a $\Gamma$-\textit{contraction}. We characterize isometries and unitaries associated with $\mathbb B_n$. Then we present an analogous canonical decomposition for a $\mathbb B_n$-contraction. We show interplay between operator theory on the three domains $\mathbb P, \mathbb B_2$ and $\mathbb G_2$., Comment: 40 pages. This article will be revised soon

We present computation of the next-to-leading power corrections for Higgs plus one jet production in a hadron collider via gluon fusion channel. Shifting of spinors in the helicity amplitudes without additional radiation captures the leading next-to-soft radiative behaviour and makes the calculation tractable. We establish the connection between the shifted dipole spinors and the colour ordered radiative amplitudes. We find that next-to-maximal helicity violating amplitudes do not play a role in this correction. Compact analytic expressions of next-to-leading power leading logarithms coming from different helicity configurations are shown., Comment: 14 pages, typos fixed, conclusions remain unchanged

Soft and collinear radiation in collider processes can be described in a universal way, that is independent of the underlying process. Recent years have seen a number of approaches for probing whether radiation beyond the leading soft approximation can also be systematically classified. In this paper, we study a formula that captures the leading next-to-soft QCD radiation affecting processes with both final- and initial-state partons, by shifting the momenta in the non-radiative squared amplitude. We first examine W+jet production, and show that a previously derived formula of this type indeed holds in the case in which massive colour singlet particles are present in the final state. Next, we develop a physical understanding of the momentum shifts, showing precisely how they disrupt the well-known angular ordering property of leading soft radiation., Comment: 26 pages, 6 figures

We explore how the interplay of finite availability, carrying capacity of particles at different parts of a spatially extended system and particle diffusion between them control the steady state currents and density profiles in a one-dimensional current-carrying channel connecting the different parts of the system. To study this, we construct a minimal model consisting of two particle reservoirs of finite carrying capacities connected by a totally asymmetric simple exclusion process (TASEP). In addition to particle transport via TASEP between the reservoirs, the latter can also directly exchange particles, modeling particle diffusion between them that can maintain a steady current in the system. We investigate the steady state density profiles and the associated particle currents in the TASEP lane. The resulting phases and the phase diagrams are quite different from an open TASEP, and are characterised by the model parameters defining particle exchanges between the TASEP and the reservoirs, direct particle exchanges between the reservoirs, and the filling fraction of the particles that determines the total resources available. These parameters can be tuned to make the density on the TASEP lane globally uniform or piecewise continuous, and can make the two reservoirs preferentially populated or depopulated., Comment: 29 pages, 22 figures

Logarithm of the soft function can be organized into sets of Feynman diagrams known as Cwebs. We introduced a new formalism in~\cite{Agarwal:2022wyk}, that allows to determine several of the building blocks of Cweb mixing matrices without explicit computations. In~\cite{Agarwal:2022xec} we used this formalism to obtain the diagonal blocks of four general classes of Cwebs to all orders in perturbation theory which also covered all the four loop Boomerang Cwebs connecting four Wilson lines. In this work we present complete mixing matrices and exponentiated colour factors for Boomerang Cwebs at four loops that connect three and four Wilson lines. Also, we present a more efficient version of the algorithm of generating Cwebs that was presented in~\cite{Agarwal:2020nyc}. This new algorithm has been used to generate the Cwebs in the present work., Comment: 59 pages, 5 figures, 1 table, published version, minor changes, conclusion remain unchanged

We study the interaction between unitary $\rho$-dilations of a pair of Hilbert space operators and Birkhoff-James orthogonality. We prove that for two orthogonal operators $T,A$ if $\|T\|=\rho$, then $U_T \perp_B U_A$ for any unitary $\rho$-dilations $U_T$ of $T$ and $U_A$ of $A$ acting on a common space. We characterize $\varepsilon$-approximate Birkhoff-James orthogonality for complex Hilbert space operators. Then find a sharp bound on $\varepsilon$ such that $T \perp_B A$ implies that $U_T \perp_B^{\varepsilon} U_A$ for any unitary $\rho$-dilations $U_T, U_A$ of $T$ and $A$ respectively. The Sch\"{a}ffer unitary dilations of a pair of contractions $T,A$ are not orthogonal in general. We construct Sch\"{a}ffer-type unitary dilations for contractions which are pairwise orthogonal., Comment: 31 pages, Submitted to journal

Scattering amplitudes involving multiple partons are plagued with infrared singularities. The soft singularities of the amplitude are captured by the soft function which is defined as the vacuum expectation value of Wilson line correlators. Renormalization properties of soft function allows us to write it as an exponential of the finite soft anomalous dimension. An efficient way to study the soft function is through a set of Feynman diagrams known as Cwebs (webs). We present the mixing matrices and exponentiated colour factors (ECFs) for the Cwebs at five loops that connect six Wilson lines, except those that are related by relabeling of Wilson lines. Further, we express these ECFs in terms of 29 basis colour factors. We also find that this basis can be categorized into two colour structures. Our results are the first key ingredients for the calculation of the soft anomalous dimension at five loops., Comment: The current version matches with the accepted JHEP version. The other details are: 51 pages, 32 figures, 28 tables and 1 ancillary file

Massive neutrinos are well-known to cause a characteristic suppression in the growth of structures at scales below the neutrino free-streaming length. A detailed understanding of this suppression is essential in the era of precision cosmology we are entering into, enabling us to better constrain the total neutrino mass and possibly probe (beyond)-$\Lambda$CDM cosmological model(s). Instead of the usual N-body simulation or Boltzmann solver, in this article we consider a two-fluid framework at the linear scales, where the neutrino fluid perturbations are coupled to the CDM (+ baryon) fluid via gravity at redshifts of interest. Treating the neutrino mass fraction $f_\nu$ as a perturbative parameter, we find solutions to the system with redshift-dependent neutrino free-streaming length in $\Lambda$CDM background via two separate approaches. The perturbative scale-dependent solution is shown to be in excellent agreement with numerical solution of the two-fluid equations valid to all orders in $f_{\nu}$, and also agrees with results from {\texttt{CLASS}} to a good accuracy. We further generalize the framework to incorporate different evolving dark energy backgrounds and found sub-percent level differences in the suppression, all of which lie within the observational uncertainty of BOSS-like surveys. We also present a brief discussion on the prospects of the current analysis in the context of upcoming missions., Comment: 20 pages, 6 sets of figures. Major update: An alternate method to perform the perturbative analysis is added. Accepted in JCAP

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The soft function in non-abelian gauge theories exponentiate, and their logarithms can be organised in terms of the collections of Feynman diagrams called Cwebs. The colour factors that appear in the logarithm are controlled by the web mixing matrices. Direct construction of the diagonal blocks of Cwebs using the new concepts of Normal ordering, basis Cweb and Fused-Web was recently carried out in~\cite{Agarwal:2022wyk}. In this article we establish correspondence between the boomerang webs introduced in ~\cite{Gardi:2021gzz} and non-boomerang Cwebs. We use this correspondence together with Uniqueness theorem and Fused web formalism introduced in ~\cite{Agarwal:2022wyk} to obtain the diagonal blocks of four general classes of Cwebs to all orders in perturbation theory which also cover all the four loop Boomerang Cwebs connecting four Wilson lines. We also fully construct the mixing matrix of a special Cweb to all orders in perturbation theory., Comment: 57 pages, 33 figures, Published Version. arXiv admin note: text overlap with arXiv:2204.05936

In the literature, we have several results associated with canonical decomposition of commuting contractions. In this paper, we generalize a few of these results to $Q$-commuting contractions. Here we mainly deal with $Q$-commuting and doubly $Q$-commuting contractions when $Q$ is a family of unitary operators., Comment: The previous version of the (long) draft has been divided into three parts and the present paper is one of them. The other two parts will appear soon in ArXiv

There are several proofs of the classical commutant lifting and intertwining lifting theorems in the literature. In this article, we present analogous proofs to a few $Q$-commuting lifting and $Q$-intertwining lifting theorems. We provide several proofs and show explicit constructions of Ando-type dilations for a pair of $Q$-commuting contractions $(T_1,T_2)$ when $Q$ is a bounded operator. We establish a few connections of these results with graph theory., Comment: Contains 52 pages. This is just a first draft and will be revised soon. This is a part of the PhD thesis of the second named author submitted to IIT Bombay on May, 2022

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Transformers have emerged as a preferred model for many tasks in natural langugage processing and vision. Recent efforts on training and deploying Transformers more efficiently have identified many strategies to approximate the self-attention matrix, a key module in a Transformer architecture. Effective ideas include various prespecified sparsity patterns, low-rank basis expansions and combinations thereof. In this paper, we revisit classical Multiresolution Analysis (MRA) concepts such as Wavelets, whose potential value in this setting remains underexplored thus far. We show that simple approximations based on empirical feedback and design choices informed by modern hardware and implementation challenges, eventually yield a MRA-based approach for self-attention with an excellent performance profile across most criteria of interest. We undertake an extensive set of experiments and demonstrate that this multi-resolution scheme outperforms most efficient self-attention proposals and is favorable for both short and long sequences. Code is available at \url{https://github.com/mlpen/mra-attention}., Comment: ICML2022

For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we show that $(T_1, \dots, T_n)$ dilates to commuting isometries $(V_1, \dots , V_n)$ on the minimal isometric dilation space of $T$ with $V=\prod_{i=1}^n V_i$ being the minimal isometric dilation of $T$ if and only if $(T_1^*, \dots , T_n^*)$ dilates to commuting isometries $(Y_1, \dots , Y_n)$ on the minimal isometric dilation space of $T^*$ with $Y=\prod_{i=1}^n Y_i$ being the minimal isometric dilation of $T^*$. Then, we prove an analogue of this result for unitary dilations of $(T_1, \dots , T_n)$ and its adjoint. We find a necessary and sufficient condition such that $(T_1, \dots , T_n)$ possesses a unitary dilation $(W_1, \dots , W_n)$ on the minimal unitary dilation space of $T$ with $W=\prod_{i=1}^n W_i$ being the minimal unitary dilation of $T$. We show an explicit construction of such a unitary dilation on both Sch$\ddot{a}$ffer and Sz. Nagy-Foias minimal unitary dilation spaces of $T$. Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when $T$ is a $C._0$ contraction, i.e. when ${T^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty $. We construct a different unitary dilation for $(T_1, \dots , T_n)$ when $T$ is a $C._0$ contraction., Comment: Revised, 32 pages. arXiv admin note: text overlap with arXiv:2204.11391

A distinguished variety in the polydisc $\mathbb D^n$ is an affine complex algebraic variety that intersects $\mathbb D^n$ and exits the domain through the $n$-torus $\mathbb T^n$ without intersecting any other part of the topological boundary of $\mathbb D^n$. We find two different characterizations for a distinguished variety in the polydisc $\mathbb D^n$ in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and M\raise.45ex\hbox{c}Carthy [Acta Math., 2005] on distinguished varieties in $\mathbb D^2$. We show that a distinguished variety in $\mathbb D^n$ is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if $(T_1, \dots , T_n)$ is commuting tuple of Hilbert space contractions such that the defect space of $T=\prod_{i=1}^n T_i$ is finite dimensional, then $(T_1, \dots , T_n)$ admits a commuting unitary dilation $(U_1, \dots , U_n)$ with $U=\prod_{i=1}^n U_i$ being the minimal unitary dilation of $T$ if and only if some certain matrices associated with $(T_1, \dots , T_n)$ define a distinguished variety in $\mathbb D^n$., Comment: Modified. new results added. 20 Pages

For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition under which $(T_1,\dots ,T_n)$ dilates to commuting isometries $(V_1,\dots ,V_n)$ on the minimal isometric dilation space $T$, where $V=\prod_{i=1}^nV_i$ is the minimal isometric dilation of $T$. We construct both Sch$\ddot{a}$ffer and Sz. Nagy-Foias type isometric dilations for $(T_1,\dots ,T_n)$ on the minimal dilation spaces of $T$. Also, a different dilation is constructed when the product $T$ is a $C._0$ contraction, that is ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\dots ,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are analytic functions in one variable. The dilation, when $T$ is a $C._0$ contraction, leads to a conditional factorization of a $T$. Several examples have been constructed., Comment: Proceedings of the Royal Society of Edinburgh, Section A: Mathematics, To appear

A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\bar{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(a_{11},a_{22}, \det A):\, A=[a_{ij}]\in \mathcal M_2(\mathbb C), \; \|A\| <1 \} \subset \mathbb C^3 \] is a polynomially convex domain which is naturally associated with the $\mu$-synthesis problem. We introduce triangular $\mathbb E$-contractions and prove that every pure triangular $\mathbb E$-contraction dilates to a pure triangular $\mathbb E$-isometry. We construct a functional model for a pure triangular $\mathbb E$-isometry and apply that model to find a new proof for the famous Berger-Coburn-Lebow Model Theorem for commuting isometries. Next we give an alternative proof to the more generalized version of Berger-Coburn-Lebow Model, namely the factorization of a pure contraction due to Das, Sarkar and Sarkar (\textit{Adv. Math.} 322 (2017), 186 -- 200). We find a necessary and sufficient condition for the existence of $\mathbb E$-unitary dilation of an $\mathbb E$-contraction $(A,B,P)$ on the smallest dilation space and show that it is equivalent to the existence of a distinguished variety in $\mathbb E$ when the defect space $D_{P^*}$ is finite dimensional., Comment: Revised, 16 pages

We consider distinguished varieties in the bidisc $\mathbb D^2$ and the symmetrized bidisc $\mathbb{G}_2$. A commuting pair of Hilbert space operators $(S,P)$ for which $\Gamma=\overline{\mathbb{G}}_2$ is a spectral set is called a \textit{$\Gamma$-contraction}. A commuting pair of contractions $(T_1,T_2)$ is said to be \textit{toral} if there is a toral polynomial $p\in \mathbb C[z_1,z_2]$ such that $p(T_1,T_2)=0$. Similarly, a $\Gamma$-contraction $(S,P)$ is called $\Gamma$-\textit{distinguished} if $(S,P)$ is annihilated by a $\Gamma$-distinguished polynomial in $\mathbb C[z_1,z_2]$. The main results af this article are the following: we find a necessary and sufficient condition such that a toral pair of contractions dilates to a toral pair of isometries. Also, we characterize all $\Gamma$-distinguished $\Gamma$-contractions that admit $\Gamma$-distinguished $\Gamma$-isometric dilations. We determine the distinguished boundary of a distinguished variety in $\mathbb D^2$ and $\mathbb G_2$. Then we prove that a pair of commuting contractions $(T_1,T_2)$ (or a $\Gamma$-contraction $(S,P)$) dilates to a toral pair of isometries (or a $\Gamma$-distinguished $\Gamma$-isometry) if and only if there is a toral polynomial (or $\Gamma$-distinguished polynomial) $p\in \mathbb C[z_1,z_2]$ such that $Z(p) \cap \overline{\mathbb D}^2$ (or $Z(p) \cap \overline{\mathbb G}_2$) is a complete spectral set for $(T_1,T_2)$ (or $(S,P)$). We provide several examples at places to show the contrasts between the theory of toral contractions and $\Gamma$-distinguished $\Gamma$-contractions., Comment: Revised, typos fixed, new results added. (55 Pages)

Recent legislation has led to interest in machine unlearning, i.e., removing specific training samples from a predictive model as if they never existed in the training dataset. Unlearning may also be required due to corrupted/adversarial data or simply a user's updated privacy requirement. For models which require no training (k-NN), simply deleting the closest original sample can be effective. But this idea is inapplicable to models which learn richer representations. Recent ideas leveraging optimization-based updates scale poorly with the model dimension d, due to inverting the Hessian of the loss function. We use a variant of a new conditional independence coefficient, L-CODEC, to identify a subset of the model parameters with the most semantic overlap on an individual sample level. Our approach completely avoids the need to invert a (possibly) huge matrix. By utilizing a Markov blanket selection, we premise that L-CODEC is also suitable for deep unlearning, as well as other applications in vision. Compared to alternatives, L-CODEC makes approximate unlearning possible in settings that would otherwise be infeasible, including vision models used for face recognition, person re-identification and NLP models that may require unlearning samples identified for exclusion. Code can be found at https://github.com/vsingh-group/LCODEC-deep-unlearning/, Comment: CVPR 2022. Supplement appended to end of main paper (total 15 pages). Ronak Mehta and Sourav Pal equal contribution

The correlators of Wilson-line operators in non-abelian gauge theories are known to exponentiate, and their logarithms can be organised in terms of the collections of Feynman diagrams called Cwebs. The colour factors that appear in the logarithm correspond to completely connected diagrams and are determined by the web mixing matrices. In this article we introduce several new concepts: (a) Normal ordering of the diagrams of a Cweb, (b) Fused-Webs (c) Basis and Family of Cwebs. We use these ideas together with a Uniqueness theorem that we prove to arrive at an understanding of the diagonal blocks, and several null matrices that appear in the mixing matrices. We demonstrate using our formalism that, once the basis Cwebs present upto order $\alpha_{s}^{n}$ are determined, the number of exponentiated colour factors for several classes of Cwebs starting at order $\alpha_{s}^{n+1}$ can be predicted. We further provide complete results for the mixing matrices, to all orders in perturbation theory, for two special classes of Cwebs using our framework., Comment: Published Version, 56 pages, 37 figures, 25 tables

Consider a nonzero contraction $T$ and a bounded operator $X$ satisfying $TX=qXT$ for a complex number $q$. There are some interesting results in the literature on $q$-commuting dilation and $q$-commutant lifting of such pair $(T,X)$ when $|q|=1$. Here we improve a few of them to the class of scalars $q$ satisfying $|q|\leq \dfrac{1}{\|T\|}$., Comment: 15 Pages

A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(x_1,x_2,x_3)\in \mathbb C^3:\, 1-x_1z-x_2w+x_3zw \neq 0 \quad \text{ whenever } \; |z|\leq 1, \; \; |w|\leq 1 \} \subset \mathbb C^3, \] is a polynomially convex domain which is naturally associated with the $\mu$-synthesis problem. By applications of the theory of $\mathbb E$-contractions, we obtain several results on decompositions of contractions., Comment: 24 pages, Forum Mathematicum, To appear

Dressed Gluon Exponentiation is a well known method to study power corrections. We present here its eikonalized version -- Eikonal Dressed Gluon Exponentiation (EDGE) \cite{Agarwal:2020uxi} to determine the dominant power corrections to shape variables such as Thrust, $ C $- parameter, and Angularity. Our method remarkably simplifies the calculations., Comment: 10 Pages, Contribution to the proceedings of Radcor-Loopfest 2021

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

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

The symmetrization map $\pi:\mathbb C^2\rightarrow \mathbb C^2$ is defined by $ \pi(z_1,z_2)=(z_1+z_2,z_1z_2). $ The closed symmetrized bidisc $\Gamma$ is the symmetrization of the closed unit bidisc $\overline{\mathbb D^2}$, that is, \[ \Gamma = \pi(\overline{\mathbb D^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\leq 1, i=1,2 \}. \] A pair of commuting Hilbert space operators $(S,P)$ for which $\Gamma$ is a spectral set is called a $\Gamma$-contraction. Unlike the scalars in $\Gamma$, a $\Gamma$-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 $\Gamma$-contractions which are symmetrization of pairs of commuting contractions. We show by constructing a family of examples that even if a $\Gamma$-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 $\Gamma$-contraction $(S,P)$ is the restriction of a $\Gamma$-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 $\Gamma$-unitaries and describe the distinguished boundary of $\Gamma$ in a different way. We also show some interplay between the fundamental operators of two $\Gamma$-contractions $(S,P)$ and $(S_1,P)$., Comment: A few typos got fixed. 16 pages