arXiv subjects

S. Sundar

Publications and source records attributed to S. Sundar.

At least 19 recordsLinked to original sources

arXiv2026

Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems

We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced $C^{*}$-algebra of a product system. We show that for a semigroup dynamical system $(A, P,\alpha)$, under reasonable hypotheses (e.g., $P$ is abelian and finitely generated), the reduced crossed product $A \rtimes_{red} P$ is exact if and only if $A$ is exact. This strengthens our earlier result (\cite{Amir_Sundar-product-system}), where it was assumed that the action of $P$ on $A$ is by injective endomorphisms. We also compare the groupoid crossed product described in \cite{Amir_Sundar-product-system} and the Fell bundle constructed in \cite{Rennie_Sims} for a product system, and show that they are equivalent as Fell bundles.

math.OA↗

arXiv2025

Doubly Commuting Semigroups of Isometries

In this paper, we discuss the structure of doubly commuting semigroups of isometries. We record a new proof of Cooper's theorem in the Hilbert module setting. We discuss the Fell topology on the set of equivalence classes of irreducible, doubly commuting isometric representations of $\mathbb{R}_{+}^{d}$. We show that if $d$ is finite, the topology is $T_0$. We indicate the pathologies that occur when $d=\infty$. In particular, we show that Wold decomposition fails for isometric representations of $\mathbb{R}_{+}^{\infty}$ and prove that the Fell topology on the set of equivalence classes of irreducible, doubly commuting isometric representations of $\mathbb{R}_{+}^{\infty}$ is not $T_0$

math.OA↗

arXiv2025

Reduced $C^{*}$-algebras of Product Systems -- an $E_0$-semigroup and a Groupoid perspective

For Ore semigroups $P$ with an order unit, we prove that there is a bijection between $E_0$-semigroups over $P$ and product systems of $C^{*}$-correspondences over $P^{op}$. We exploit this bijection and show that the reduced $C^{*}$-algebra of a proper product system is Morita equivalent to the reduced crossed product of the associated semigroup dynamical system given by the corresponding $E_0$-semigroup. We appeal to the groupoid picture of the reduced crossed product of a semigroup dynamical system derived in [47] to prove that, under good conditions, the reduced $C^{*}$-algebra of a proper product system is nuclear/exact if and only if the coefficient algebra is nuclear/exact. We also discuss the invariance of $K$-theory under homotopy of product systems.

math.OA↗

arXiv2025

Notes on $C^{*}$-algebras

These lecture notes on $C^{*}$-algebras were prepared for a couple of courses given by the author at IMSc and also at IIT Gandhinagar. The topics covered are: Gelfand-Naimark theorems, universal C*-algebras, Hilbert C*-modules, crossed products, Morita equivalence, K-theory.

math.OA↗

arXiv2025

A residual weighted physics informed neural network for forward and inverse problems of reaction diffusion equations

In this work, we propose the Residual-Weighted Physics-Informed Neural Network (RW-PINN), a new method designed to enhance the accuracy of Physics-Informed Neural Network (PINN) based algorithms. We construct a deep learning framework with two residual-weighting schemes to solve reaction diffusion equations and evaluate its performance on both forward and inverse problems. The approach computes weights proportional to the PDE residuals, rescales them, and incorporates these scaled residuals into the loss function, leading to more stable training. Furthermore, we establish generalized error bounds that account for training and quadrature errors, and we analyze the convergence and stability of the method. The proposed algorithms are validated through numerical experiments on nonlinear equations, supported by statistical error analysis. To further demonstrate the effectiveness of our methodology, we implemented PINN-based forward and inverse frameworks for the nonlinear equations and conducted a comparative analysis with the proposed RW-PINN approach.

math.NA↗

arXiv2024

Physics informed neural network for forward and inverse radiation heat transfer in graded-index medium

Radiation heat transfer in a graded-index medium often suffers accuracy problems due to the gradual changes in the refractive index. The finite element method, meshfree, and other numerical methods often struggle to maintain accuracy when applied to this medium. To address this issue, we apply physics-informed neural networks (PINNs)-based machine learning algorithms to simulate forward and inverse problems for this medium. We also provide the theoretical upper bounds. This theoretical framework is validated through numerical experiments of predefined and newly developed models that demonstrate the accuracy and robustness of the algorithms in solving radiation transport problems in the medium. The simulations show that the novel algorithm goes on with numerical stability and effectively mitigates oscillatory errors, even in cases with more pronounced variations in the refractive index.

math.NA↗

arXiv2024

Finite difference physics-informed neural networks enable improved solution accuracy of the Navier-Stokes equations

Generating an accurate solution of the Navier--Stokes equations using physics--informed neural networks (PINNs) for higher Reynolds numbers in the corners of a lid--driven cavity problem is challenging. In this paper, we improve the solution accuracy of the incompressible Navier--Stokes equations in the region near the walls significantly and generate accurate secondary vortices in the corners of the lid--driven cavity by solving the governing equations using finite difference--based PINNs (FD--PINNs) without employing the known solution. We adopt the domain decomposition method (DDM) and combine it with the FD--PINNs to solve the lid--driven cavity problem for the Reynolds numbers Re = 400 and Re=1000. A comparison of the mean square error (MSE) between the presented and standard FD--PINNs using the reference solution is exhibited, showing the accuracy and effectiveness of the new approach.

physics.comp-ph↗

arXiv2024

A meshless geometric conservation weighted least square method for solving the shallow water equations

The shallow water equations are numerically solved to simulate free surface flows. The convective flux terms in the shallow water equations need to be discretized using a Riemann solver to capture shocks and discontinuity for certain flow situations such as hydraulic jump, dam-break wave propagation or bore wave propagation, levee-breaching flows, etc. The approximate Riemann solver can capture shocks and is popular for studying open-channel flow dynamics with traditional mesh-based numerical methods. Though meshless methods can work on highly irregular geometry without involving the complex mesh generation procedure, the shock-capturing capability has not been implemented, especially for solving open-channel flows. Therefore, we have proposed a numerical method, namely, a shock-capturing meshless geometric conservation weighted least square (GC-WLS) method for solving the shallow water equations. The HLL (Harten-Lax-Van Leer) Riemann solver is implemented within the framework of the proposed meshless method. The spatial derivatives in the shallow water equations and the reconstruction of conservative variables for high-order accuracy are computed using the GC-WLS method. The proposed meshless method is tested for various numerically challenging open-channel flow problems, including analytical, laboratory experiments, and a large-scale physical model study on dam-break event.

physics.flu-dyn↗

arXiv2024

Is Every Product System Concrete?

Is every product system of Hilbert spaces over a semigroup $P$ concrete, i.e. isomorphic to the product system of an $E_0$-semigroup over $P$? The answer, in general, is no. We record a non-example when $P$ is cancellative and is not embeddable in a group. However, we show that the answer is yes for a reasonable class of semigroups which includes solid, Borel subsemigroups of locally compact abelian groups. We also extend Liebscher's result by showing that in the commutative setting, two product systems are isomorphic if and only if they are algebraically isomorphic.

math.OA↗

arXiv2023

On Multiparameter CAR Flows

Let $P$ be a pointed, closed convex cone in $\mathbb{R}^d$. We prove that for two pure isometric representations $V^{(1)}$ and $V^{(2)}$ of $P$, the associated CAR flows $\beta^{V^{(1)}}$ and $\beta^{V^{(2)}}$ are cocycle conjugate if and only if $V^{(1)}$ and $V^{(2)}$ are unitarily equivalent. We also give a complete description of pure isometric representations of $P$ with commuting range projections that give rise to type I CAR flows. We show that such an isometric representation is completely reducible with each irreducible component being a pullback of the shift semigroup $\{S_t\}_{t \geq 0}$ on $L^2[0,\infty)$. We also compute the index and the gauge group of the associated CAR flows and show that the action of the gauge group on the set of normalised units need not be transitive.

math.OA↗

arXiv2023

The modified Yule-Walker method for multidimensional infinite-variance periodic autoregressive model of order 1

The time series with periodic behavior, such as the periodic autoregressive (PAR) models belonging to the class of the periodically correlated processes, are present in various real applications. In the literature, such processes were considered in different directions, especially with the Gaussian-distributed noise. However, in most of the applications, the assumption of the finite-variance distribution seems to be too simplified. Thus, one can consider the extensions of the classical PAR model where the non-Gaussian distribution is applied. In particular, the Gaussian distribution can be replaced by the infinite-variance distribution, e.g. by the $\alpha-$stable distribution. In this paper, we focus on the multidimensional $\alpha-$stable PAR time series models. For such models, we propose a new estimation method based on the Yule-Walker equations. However, since for the infinite-variance case the covariance does not exist, thus it is replaced by another measure, namely the covariation. In this paper we propose to apply two estimators of the covariation measure. The first one is based on moment representation (moment-based) while the second one - on the spectral measure representation (spectral-based). The validity of the new approaches are verified using the Monte Carlo simulations in different contexts, including the sample size and the index of stability of the noise. Moreover, we compare the moment-based covariation-based method with spectral-based covariation-based technique. Finally, the real data analysis is presented.

stat.ME↗

arXiv2023

Absence of Spontaneous Magnetic Fields Due to Time-Reversal Symmetry Breaking in Bulk Superconducting UTe2

We have investigated the low-temperature local magnetic properties in the bulk of molten salt-flux (MSF) grown single crystals of the candidate odd-parity superconductor UTe2 by zero-field muon spin relaxation (muSR). In contrast to previous muSR studies of UTe2 single crystals grown by a chemical vapour transport (CVT) method, we find no evidence of magnetic clusters or electronic moments fluctuating slow enough to cause a discernible relaxation of the zero-field muSR asymmetry spectrum. Consequently, our measurements on MSF-grown single crystals rule out the generation of spontaneous magnetic fields in the bulk that would occur near impurities or lattice defects if the superconducting state of UTe2 breaks time-reversal symmetry. This result suggests UTe2 is characterized by a single-component superconducting order parameter.

cond-mat.supr-con↗

arXiv2023

Induced Isometric Representations

Let $\sigma$ be an isometric representation of $\mathbb{N}^d$ on a Hilbert space $\mathcal{H}$. We induce $\sigma$ to an isometric representation $V$ of $\mathbb{R}_{+}^{d}$ on another Hilbert space $\mathcal{K}$. We show that the map $\sigma \to V$, restricted to strongly pure isometric representations, preserves index and irreducibility. As an application, we show that, for $k \in \{0, 1,2,\cdots\} \cup \{\infty\}$, there is a continuum of prime multiparameter CCR flows (i.e, not a tensor product of two non-trivial $E_0$-semigroups) with index $k$.

math.OA↗

arXiv2023

On the KMS states for the Bernoulli shift

Let $\Omega:=\{0,1\}^{\mathbb{Z}}$ be the Cantor space, and let $\tau:\Omega \to \Omega$ be the Bernoulli shift. For the flow on the crossed product $C(\Omega)\rtimes_\tau \mathbb{Z}$ determined by a potential that depends on only one coordinate, we show that for every $\beta \neq 0$, there is an extremal $\beta$-KMS state on $C(\Omega)\rtimes_\tau \mathbb{Z}$ of type $II_\infty$. Also, when the potential takes values that are rationally dependent, we determine the values of $\lambda \in (0,1)$ for which there is a an extremal $\beta$-KMS state of type $III_\lambda$.

math.OA↗

arXiv2022

Multiparameter Decomposable Product Systems

In [8], Arveson proved that a $1$-parameter decomposable product system is isomorphic to the product system of a CCR flow. We show that the structure of a generic decomposable product system, over higher dimensional cones, modulo twists by multipliers, is given by an isometric representation $V$ of the cone and a certain $2$-cocycle for $V$. Moreover, we compute the space of $2$-cocycles for shift semigroups associated to transitive actions of a higher dimensional cone.

math.OA↗

arXiv2022

Representations of the weak Weyl commutation relation

Let $G$ be a locally compact abelian group with Pontraygin dual $\widehat{G}$. Suppose $P$ is a closed subsemigroup of $G$ containing the identity element $0$. We assume that $P$ has dense interior and $P$ generates $G$. Let $U:=\{U_{\chi}\}_{\chi \in \widehat{G}}$ be a strongly continuous group of unitaries and let $V:=\{V_{a}\}_{a \in P}$ be a strongly continuous semigroup of isometries. We call $(U,V)$ a weak Weyl pair if \[ U_{\chi}V_{a}=\chi(a)V_{a}U_{\chi}\] for every $\chi \in \widehat{G}$ and for every $a \in P$. We work out the representation theory (the factorial and the irreducible representations) of the above commutation relation under the assumption that $\{V_{a}V_{a}^{*}:a \in P\}$ is a commuting family of projections. Not only does this generalise the results of [4] and [5], our proof brings out the Morita equivalence that lies behind the results. For $P=[0,\infty)\times [0,\infty)$, we demonstrate that if we drop the commutativity assumption on the range projections, then the representation theory of the weak Weyl commutation relation becomes very complicated.

math.OA↗

arXiv2022

KMS states on $C_c^{*}(\mathbb{N}^2)$

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-algebra generated by a semigroup of isometries $\{v_{(m,n)}: m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c:\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

math.OA↗

arXiv2021

Examples of Multiparameter CCR flows with non-trivial index

In this paper, we construct uncountably many examples of multiparameter CCR flows, which are not pullbacks of $1$-parameter CCR flows, with index one. Moreover, the constructed CCR flows are type I in the sense that the associated product system is the smallest subsystem containing its units.

math.OA↗