Search

A tuple of commuting Hilbert space operators (S 1 , … , S n - 1 , P) having the closed symmetrized polydisc Γ n = ∑ i = 1 n z i , ∑ 1 ≤ i < j ≤ n z i z j , … , ∏ i = 1 n z i : | z i | ≤ 1 , 1 ≤ i ≤ n - 1 as a spectral set is called a Γ n -contraction. From the literature we have that a point (s 1 , … , s n - 1 , p) in Γ n can be represented as s i = c i + p c n - i for some (c 1 , … , c n - 1 ) ∈ Γ n - 1 . We construct a minimal Γ n -isometric dilation for a particular class of c.n.u. Γ n -contractions (S 1 , … , S n - 1 , P) and obtain a functional model for them. With the help of this model we express each S i as S i = C i + P C n - i , which is an operator theoretic analogue of the scalar result. We also produce an abstract model for a different class of c.n.u. Γ n -contractions satisfying S i ∗ P = P S i ∗ for each i. By exhibiting a counter example we show that such abstract model may not exist if we drop the hypothesis that S i ∗ P = P S i ∗ . We apply this abstract model to achieve a complete unitary invariant for such c.n.u. Γ n -contractions. Additionally, we present different necessary conditions for dilation and a sufficient condition under which a commuting tuple (S 1 , … , S n - 1 , P) becomes a Γ n -contraction. The entire program goes parallel to the operator theoretic program developed by Sz.-Nagy and Foias for a c.n.u. contraction. [ABSTRACT FROM AUTHOR]

The symmetrization map π : C 2 → C 2 is defined by π (z 1 , z 2) = (z 1 + z 2 , z 1 z 2). The closed symmetrized bidisc Γ is the symmetrization of the closed unit bidisc D 2 ¯ , that is, Γ = π ( D 2 ¯) = { (z 1 + z 2 , z 1 z 2) : | z i | ≤ 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 1 T 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 (S ~ , P ~) to a common reducing subspace of S ~ , P ~ and that (S ~ , P ~) = (A 1 + A 2 , A 1 A 2) for a pair of commuting operators A 1 , A 2 with max { ‖ A 1 ‖ , ‖ A 2 ‖ } ≤ 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) . [ABSTRACT FROM AUTHOR]

More than 47,000 articles have been published in the area of Metal-Organic Framework since its seminal discovery in 1995, exemplifying the intense research carried out in this short span of time. Among other applications, gas adsorption and storage are perceived as central to the MOFs research, and more than 10,000 MOFs structures are reported to date to utilize them for various gas storage/separation applications. Molecular modeling, particularly based on density functional theory, played a key role in (i) understanding the nature of interactions between the gas and the MOFs geometry (ii) establishing various binding pockets and relative binding energies, and (iii) offering design clues to improve the gas uptake capacity of existing MOF architectures. In this review, we have looked at various MOFs that are studied thoroughly using DFT/periodic DFT (pDFT) methods for CO2, H2, O2, and CH4 gases to provide a birds-eye-view on how various exchange-correlation functionals perform in estimating the binding energy for various gases and how factors such as nature of the (i) metal ion, (ii) linkers, (iii) ligand, (iv) spin state and (v) spin-couplings play a role in this process with selected examples. While there is still room for improvement, the rewards offered by the molecular modelling of MOFs were already substantial that we advocate experimental and theoretical studies to go hand-in-hand to undercut the trial-and-error approach that is often perceived in the selection of MOFs and gas partners in this area. The importance of density functional theory-based molecular modeling studies in offering design clues to improve the gas adsorption and storage capacity of existing MOF architectures is discussed here. The use of DFT-based investigation in conjunction with experimental synthesis is an imperative tool in designing new-generation MOFs with enhanced uptake capacity. [ABSTRACT FROM AUTHOR]

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 [1]. In this article we establish correspondence between the boomerang webs introduced in [2] and non-boomerang Cwebs. We use this correspondence together with Uniqueness theorem and Fused web formalism introduced in [1] 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. [ABSTRACT FROM AUTHOR]

SrF, being a laser-coolable molecule, can be an interesting system for spectroscopic tests of fundamental physics. We present an electronic structure study of this molecule within the four-component relativistic coupled-cluster singles and doubles (RCCSD) framework and employ the RCCSD-based methods to compute its molecular-frame dipole moment and core properties such as hyperfine structure coupling constant and molecular P, T-odd electronic structure parameters that are of great importance for the high-precision tests of fundamental physics. The impact of basis set size, Hamiltonian and nuclear model on the property calculation of SrF is also investigated. The computed results are in good agreement with the available experimental values. The present study shows that the SrF molecule could be useful for high-precision molecular experiments to explore physics beyond the Standard Model of elementary particles. [ABSTRACT FROM AUTHOR]

A commuting triple of Hilbert space operators (A, B, P) for which the closure of the tetrablock E , where E = { (x 1 , x 2 , x 3) ∈ C 3 : | x 3 | < 1 & x 1 = c 1 + x 3 c ¯ 2 , x 2 = c 2 + x 3 c ¯ 1 , c 1 , c 2 ∈ C with | c 1 | + | c 2 | < 1 } , is a spectral set is called a tetrablock contraction or an E -contraction. To every E -contraction (A, B, P), there are unique operators F 1 , F 2 ∈ B (D P) , which are called the fundamental operators of (A, B, P), satisfying A - B ∗ P = D P F 1 D P , B - A ∗ P = D P F 2 D P. An E -contraction (A, B, P) admits a canonical decomposition (A 1 ⊕ A 2 , B 1 ⊕ B 2 , P 1 ⊕ P 2) into an E -unitary (A 1 , B 1 , P 1) and a completely non-unitary (c.n.u.) E -contraction (A 2 , B 2 , P 2) . As there already exists an easily understood model for an E -unitary in the literature, it suffices to restrict attention to the c.n.u. E -contraction part. Here we construct an explicit minimal E -isometric dilation for a c.n.u. E -contraction whose fundamental operators satisfy (1.1) below (a class for which it is known that such dilation exists). As a consequence of this explicit dilation, we obtain a functional model for the same class of E -contractions. With the help of this functional model we express an E -contraction (A, B, P) as A = C 1 + P C 2 ∗ , B = C 2 + P C 1 ∗ for some operators C 1 , C 2 and this representation is operator theoretic analogue of the representation of the points in E . We also construct a different functional model, which is not necessarily commutative, for a c.n.u. E -contraction (A, B, P) when A, B commute with P ∗ . This functional model is obtained even without having an E -isometric dilation exhibited in the model. We show by an example that such a model may not be possessed by (A, B, P) if the condition that A, B commute with P ∗ , is dropped from the hypothesis. A complete unitary invariant is achieved for a c.n.u. E -contraction (A, B, P) when A, B commute with P ∗ . The fundamental operators play the central role in all these constructions. Also, we produce a new characterization for an E -unitary. [ABSTRACT FROM AUTHOR]

We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc G n , a family of domains naturally associated with the spectral interpolation, defined by G n : = ∑ 1 ≤ i ≤ n z i , ∑ 1 ≤ i < j ≤ n z i z j … , ∏ i = 1 n z i : | z i | < 1 , i = 1 , … , n. We first make a few estimates for the the extended symmetrized polydisc G ~ n , a family of domains introduced in [35] and defined in the following way: G ~ n : = { (y 1 , … , y n - 1 , q) ∈ C n : q ∈ D , y j = β j + β ¯ n - j q , β j ∈ C and | β j | + | β n - j | < n j for j = 1 , … , n - 1 }. We then show that these estimates are sharp and provide a Schwarz lemma for G ~ n . It is easy to verify that G n = G ~ n for n = 1 , 2 and that G n ⊊ G ~ n for n ≥ 3 . As a consequence of the estimates for G n ~ , we have analogous estimates for G n . Since for a point (s 1 , … , s n - 1 , p) ∈ G n , n i is the least upper bound for | s i | , which is same for | y i | for any (y 1 , … , y n - 1 , q) ∈ G n ~ , 1 ≤ i ≤ n - 1 , the estimates become sharp for G n too. We show that these conditions are necessary and sufficient for 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. [ABSTRACT FROM AUTHOR]

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 α s n are determined, the number of exponentiated colour factors for several classes of Cwebs starting at order α 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. [ABSTRACT FROM AUTHOR]