arXiv subjects

Rajeev Gupta

Publications and source records attributed to Rajeev Gupta.

At least 19 recordsLinked to original sources

arXiv2026

Extreme points, positive Grothendieck constants and tensor product norms

We study several interrelated problems arising from the interplay between extreme point theory, Grothendieck-type inequalities, and tensor product norms. We develop a general framework for characterizing the extreme points of the set of positive contractions $\mathcal{A}_{X\to Y}$ between finite-dimensional Banach spaces, with explicit results for $X=\ell_1^n$, $Y=\ell_\infty^n$ and vice versa. These characterizations are applied to evaluate several constants exactly. We show that the positive Grothendieck constant $K_G^{+,\mathbb{R}}(3)$ equals $9/8$ and that the smallest constant $\rho^{+}(X)$ for which $\|A\|_\pi \leqslant \rho^{+}(X)\|A\|_\epsilon$ holds for all $A \geqslant 0$ equals $5/4$ when $X=\ell^3_\infty(\mathbb{R})$. We also prove that $\rho^+(X)=1$ when $X=\ell_\infty^n(\mathbb{C})$ and $n\leqslant 3$. Finally, we prove that $\rho^+(X) = 1$ for every 2-dimensional subspace $X$ of $\ell^3_\infty(\mathbb{C})$; since this is stronger than the 2-summing property, it recovers Proposition~4.4 of \cite{AFJS95}.

math.FA↗

arXiv2026

von Neumann Inequality for a class of Doubly Contractive Weighted Shifts

In this article, we investigate the ball version of von Neumann inequality for the class of doubly contractive $d$-tuple of weighted shift. We show that if the weighted shift is balanced or satisfies an appropriate weight condition, then it admits a spherical unitary dilation. Consequently, such tuples satisfy the von Neumann inequality over Euclidean unit ball. For the general class of commuting tuple of doubly contractive operators (not necessarily weighted shift) on a Hilbert space, we further establish von Neumann inequality for homogeneous polynomials of degree at most $2.$

math.FA↗

arXiv2025

Multivariable Wold-Type Decomposition and Analytic Models for a class of left-inverse commuting pairs

This work establishes a multivariable Wold-type decomposition for left-inverse commuting $n$-tuples of bounded operators, built on the hypothesis that each component admits a Wold-type decomposition. For pairs of operators, we obtain a complete analytic model: every left-inverse commuting analytic toral $2$-isometric pair is unitarily equivalent to the pair of multiplication operator by co-ordinate functions $(M_{z_1}, M_{z_2})$ acting on some $\mathcal{E}$-valued Dirichlet-type space $\mathcal D_{\mathcal E}(\mu_1, \mu_2)$ associated with two finite positive operator-valued Borel measures $\mu_1$ and $\mu_2$ on the unit circle. An explicit functional model is further derived for the non-analytic case.

math.FA↗

arXiv2025

Personalized Artificial General Intelligence (AGI) via Neuroscience-Inspired Continuous Learning Systems

Artificial Intelligence has made remarkable advancements in recent years, primarily driven by increasingly large deep learning models. However, achieving true Artificial General Intelligence (AGI) demands fundamentally new architectures rather than merely scaling up existing models. Current approaches largely depend on expanding model parameters, which improves task-specific performance but falls short in enabling continuous, adaptable, and generalized learning. Achieving AGI capable of continuous learning and personalization on resource-constrained edge devices is an even bigger challenge. This paper reviews the state of continual learning and neuroscience-inspired AI, and proposes a novel architecture for Personalized AGI that integrates brain-like learning mechanisms for edge deployment. We review literature on continuous lifelong learning, catastrophic forgetting, and edge AI, and discuss key neuroscience principles of human learning, including Synaptic Pruning, Hebbian plasticity, Sparse Coding, and Dual Memory Systems, as inspirations for AI systems. Building on these insights, we outline an AI architecture that features complementary fast-and-slow learning modules, synaptic self-optimization, and memory-efficient model updates to support on-device lifelong adaptation. Conceptual diagrams of the proposed architecture and learning processes are provided. We address challenges such as catastrophic forgetting, memory efficiency, and system scalability, and present application scenarios for mobile AI assistants and embodied AI systems like humanoid robots. We conclude with key takeaways and future research directions toward truly continual, personalized AGI on the edge. While the architecture is theoretical, it synthesizes diverse findings and offers a roadmap for future implementation.

cs.AI↗

arXiv2025

Wold-type decomposition for Doubly commuting two-isometries

In this article, we prove that any pair of doubly commuting $2$-isometries on a Hilbert space has a Wold-type decomposition. Moreover, the analytic part of the pair is unitary equivalent to the pair of multiplication by coordinate function on a Dirichlet-type space on the bidisc.

math.FA↗

arXiv2024

Dirichlet type spaces in the unit bidisc and Wandering Subspace Property for operator tuples

In this article, we define Dirichlet-type space $\mathcal{D}^{2}(\boldsymbol{\mu})$ over the bidisc $\mathbb D^2$ for any measure $\boldsymbol{\mu}\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$ We show that the set of polynomials is dense in $\mathcal{D}^{2}(\boldsymbol{\mu})$ and the pair $(M_{z_1}, M_{z_2})$ of multiplication operator by co-ordinate functions on $\mathcal{D}^{2}(\boldsymbol{\mu})$ is a pair of commuting $2$-isometries. Moreover, the pair $(M_{z_1}, M_{z_2})$ is a left-inverse commuting pair in the following sense: $L_{M_{z_i}} M_{z_j}=M_{z_j}L_{M_{z_i}}$ for $1\leqslant i\neq j\leqslant n,$ where $L_{M_{z_i}}$ is the left inverse of $M_{z_i}$ with $\ker L_{M_{z_i}} =\ker M_{z_i}^*$, $1\leqslant i \leqslant n$. Furthermore, it turns out that, for the class of left-inverse commuting tuple $\boldsymbol T=(T_1, \ldots, T_n)$ acting on a Hilbert space $\mathcal{H}$, the joint wandering subspace property is equivalent to the individual wandering subspace property. As an application of this, the article shows that the class of left-inverse commuting pair with certain splitting property is modelled by the pair of multiplication by co-ordinate functions $(M_{z_1}, M_{z_2})$ on $\mathcal{D}^{2}(\boldsymbol{\mu})$ for some $\boldsymbol{\mu}\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$

math.FA↗

arXiv2023

Ces\`aro summability of Taylor series in higher order weighted Dirichlet type spaces

For a positive integer $m$ and a finite non-negative Borel measure $\mu$ on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces $\mathcal H_{\mu, m}$. We show that if $\alpha>\frac{1}{2},$ then for any $f$ in $\mathcal H_{\mu, m},$ the sequence of generalized Ces{\`a}ro sums $\{\sigma_n^{\alpha}[f]\}$ converges to $f$. We further show that if $\alpha=\frac{1}{2}$ then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer $m$.

math.FA↗

arXiv2023

On a variant of the Grothendieck inequality and estimates on tensor product norms

We investigate a Grothendieck-type inequality for pairs of Banach spaces $E,F$ assuming $E$ is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a $C >0$ such that $\|A\otimes \operatorname{id}_{F}\|_{E_m\check{\otimes}F\to E_n^*\hat{\otimes}F}\leqslant C \|A\|_{E_m\to E_n^*}$ for all $m,n\in\mathbb{N},$ where $\dim E_n=n$, then both $F$ and $F^*$ must have finite cotype. Moreover, assuming that $F$ has the bounded approximation property and that the conjecture in \cite{PisierDuality} has an affirmative answer, we show that $(E_n^*)_{n\geqslant 1}$ satisfies G.T. uniformly. We show that the Grothendieck-type constant defined for a pair of Banach spaces $(E,F)$ is closely related to another interesting quantity introduced recently in \cite{XOR games and GT} comparing the projective and injective norms on the tensor product of two finite-dimensional Banach spaces $E$ and $F$. We also study analogously the constants appearing in these extremal problems by restricting only to non-negative tensors. For contractive \emph{little} Parrott homomorphisms $\varrho_V : H^\infty(\Omega) \to M_{n}$, where $\Omega$ is the dual unit ball of a finite dimensional Banach space $(E,\|\cdot\|)$, we prove the sharp estimate $ \|\varrho_V\|_{\mathrm{cb}}\leq\sqrt{\gamma(E)}, $ $\gamma(E)$ being the positive Grothendieck constant associated with the pair $(E, \ell^n_2)$. %\st{with extremal cases achieving equality.} This yields a new proof of \cite[Theorem 2.1]{Davidchoi} using the lower bound $K_G^+(\ell_\infty^4,\ell_2^2) \geq 1.1658$ obtained in this paper.

math.FA↗

arXiv2022

A local Douglas formula for higher order weighted Dirichlet-type integrals

We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $\mathcal H_{\pmbμ},$ induced by an $m$-tuple $\pmb μ=(μ_1,\ldots,μ_{m})$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order $m,$ for $m\geqslant 3,$ forms an algebra under pointwise product. We also prove that every non-zero closed $M_z$-invariant subspace of $\mathcal H_{\pmbμ},$ has codimension $1$ property if $m\geqslant 3$ or $μ_2$ is finitely supported. As another application of local Douglas formula obtained in this article, it is shown that for any $m\geqslant 2,$ weighted Dirichlet-type space of order $m$ does not coincide with any de Branges-Rovnyak space $\mathcal H(b)$ with equivalence of norms.

math.FA↗

arXiv2021

Room Temperature Structural, Magnetic and Dielectric Characteristics of La Doped CuO Bulk Multiferroic

In this manuscript, we report room temperature structural, microstructural, optical, dielectric, and magnetic properties of CuO and Cu0.995La0.005 ceramics, synthesized by solid-state reaction method. La doping in CuO leads to the evolution of compact and dense microstructure with reduced porosity. Due to noticeable differences in the ionic radii of, La doping creates vacancy defects which induce considerable strain in the CuO lattice resulting in a reduction in the lattice parameters and cell volume. However, both ceramics processes a similar monoclinic structure with the C2/c space group. Detailed characterization using XPS, Raman, and FTIR spectroscopy confirmed the incorporation of the La3+ in CuO lattice. Interestingly, La doping enhances the dielectric constant by more than three times and results in a reduced leakage current. The onset of a large dielectric constant is attributed to dense microstructure and strain/distortion in CuO lattice after La doping. Additionally, the band-gap of Cu0.995La0.005 ceramics decreases which is attributed to increased vacancy defect concentration that creates intermediate dopant energy level within bandgap of CuO matrix. Furthermore, improvement in magnetic and dielectric properties is also discussed and correlated with the grain size in La-doped CuO.

cond-mat.mtrl-sci↗

arXiv2021

Low temperature magnetic and dielectric properties correlation in Fe-doped copper (ii) oxide ceramics for potential device application

The bulk samples of CuO and Fe-doped CuO were synthesized by ceramics methods. Structural and compositional analyses were performed by using X-ray diffraction, SEM, and EDAX. Through this manuscript, we are going to report the effect of trivalent iron doping (Fe$^{3+}$) in copper (II) oxide (Cu$_{0.95}$Fe$_{0.05}$O) bulk samples on magnetic and dielectric behavior. The paramagnetic phase has been established in CuO as a result of Fe$^{3+}$ doping. The strong correlation between magnetic and dielectric properties indicated spin-polaron interaction at the transition temperature. Bulk CuO and also Cu$_{0.95}$Fe$_{0.05}$O exhibit the multiferroic phase in a narrow temperature range (190 K to 230 K). Two transitions happened from a paramagnetic-paraelectric phase to incommensurate or asymmetrical antiferromagnetic (AF) and ferroelectric state near highest Neel temperature (TN1) ~230 K and another second phase transition, the order of AF phase transformed to commensurate AF phase and ferroelectricity disappeared at around the Neel temperature (TN2) ~210 K in all samples. This Cu$_{0.95}$Fe$_{0.05}$O would show its potential in the spintronic application for a high dielectric constant with low loss and high magnetic susceptibility.

cond-mat.mtrl-sci↗

arXiv2021

Weighted Join Operators on Directed Trees

A rooted directed tree $\mathscr T=(V, E)$ with can be extended to a directed graph $\mathscr T_\infty=(V_\infty, E_\infty)$ by adding a vertex $\infty$ to $V$ and declaring each vertex in $V$ as a parent of $\infty.$ One may associate with the extended directed tree a family of semigroup structures $\sqcup_{b}$ with extreme ends being induced by the join operation $\sqcup$ and the meet operation $\sqcap$. Each semigroup structure among these leads to a family of densely defined linear operators $W^{b}_{λ_u}$ acting on $\ell^2(V),$ which we refer to as weighted join operators at a given base point $b \in V_{\infty}$ with prescribed vertex $u \in V$. The extreme ends of this family are weighted join operators $W^{\mathsf{root}}_{λ_u}$ and weighted meet operators $W^{\infty}_{λ_u}$. In this paper, we systematically study these operators. We also present a more involved counter-part of weighted join operators on rootless directed trees. In both cases, the class of weighted join operators overlaps with the well-studied classes of complex Jordan operators and $n$-symmetric operators. An important half of this paper is devoted to the study of rank one extensions $W_{f, g}$ of weighted join operators, where $f \in \ell^2(V)$ and $g : V \to \mathbb C$ is unspecified. Unlike weighted join operators, these operators are not necessarily closed. We provide a couple of compatibility conditions involving the weight system $λ_u$ and $g$ to ensure closedness of $W_{f, g}$. We discuss the role of the Gelfand-triplet in the realization of the Hilbert space adjoint of $W_{f, g}$. Further, we describe various spectral parts of $W_{f, g}$ in terms of the weight system and the tree data. We also provide sufficient conditions for $W_{f, g}$ to be a sectorial operator. In case $\mathscr T$ is leafless, we characterize rank one extensions $W_{f, g}$, which admit compact resolvent.

math.FA↗

arXiv2020

Dirichlet-type spaces on the unit ball and joint 2-isometries

We obtain a formula that relates the spherical moments of the multiplication tuple on a Dirichlet-type space to a complex moment problem in several variables. This can be seen as the ball-analogue of a formula originally invented by Richter. We capitalize on this formula to study Dirichlet-type spaces on the unit ball and joint $2$-isometries.

math.FA↗

arXiv2019

Unitary equivalence of operator-valued multishifts

We systematically study various aspects of operator-valued multishifts. Beginning with basic properties, we show that the class of multishifts on the directed Cartesian product of rooted directed trees is contained in that of operator-valued multishifts. Further, we establish circularity, analyticity and wandering subspace property of these multishifts. In the rest part of the paper, we study the function theoretic behaviour of operator-valued multishifts. We determine the bounded point evaluation, reproducing kernel structure and the unitary equivalence of operator-valued multishifts with invertible operator weights. In contrast with a result of Lubin, it appears that the set of all bounded point evaluations of an operator-valued multishift may be properly contained in the joint point spectrum of the adjoint of underlying multishift.

math.FA↗

arXiv2018

Von Neumann's inequality for commuting operator-valued multishifts

Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights satisfies the matrix-version of von Neumann's inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In particular, we show that if $A$ and $B$ are commuting contractive $d$-tuples of operators such that $B$ satisfies the matrix-version of von Neumann's inequality and $(1, \ldots, 1)$ is in the algebraic spectrum of $B$, then the tensor product $A \otimes B$ satisfies the von Neumann's inequality if and only if $A$ satisfies the von Neumann's inequality. We also exhibit several families of operator-valued multishifts for which the von Neumann's inequality always holds.

math.FA↗

arXiv2018

Impersonation: Modeling Persona in Smart Responses to Email

In this paper, we present design, implementation, and effectiveness of generating personalized suggestions for email replies. To personalize email responses based on users style and personality, we model the users persona based on her past responses to emails. This model is added to the language-based model created across users using past responses of the all user emails. A users model captures the typical responses of the user given a particular context. The context includes the email received, recipient of the email, and other external signals such as calendar activities, preferences, etc. The context along with users personality (e.g., extrovert, formal, reserved, etc.) is used to suggest responses. These responses can be a mixture of multiple modes: email replies (textual), audio clips, etc. This helps in making responses mimic the user as much as possible and helps the user to be more productive while retaining her mark in the responses.

cs.CL↗

arXiv2018

Evidence for Incipient Ferroelectricity in YCrO3

Cubic structure is one of the most commonly found structures for oxides at high temperature. As the temperature is lowered only a handful of these oxides exhibit ferroelectricity which is rather surprising. In this paper, we use the example of YCrO3 (YCO), an incipient ferroelectric material to show that in most oxides there are two competing phenomenon -onset of ferroelectricity due to rotation of CrO6 octahedra and displacement of Y atom leading to suppression of ferroelectricity. This competition reveals that while the octahedral rotations favor a lower symmetry state, the Y atom displacement opposes it leaving YCO to exhibit only an incipient ferroelectric state. These results while being in agreement with the earlier theoretical predictions can also help suggest a pathway to a more stable ferroelectric state in these oxides by using a larger cationic substitution at the Y site.

cond-mat.mtrl-sci↗

arXiv2018

On a Question of N. Th. Varopoulos and the constant $C_2(n)$

Let $\mathbb C_k[Z_1,\ldots, Z_n]$ denote the set of all polynomials of degree at most $k$ in $n$ complex variables and $\mathscr{C}_n$ denote the set of all $n$ - tuple $\boldsymbol T=(T_1,\ldots,T_n)$ of commuting contractions on some Hilbert space $\mathbb{H}.$ The interesting inequality $$K_{G}^{\mathbb C}\leq \lim_{n\to \infty}C_2(n)\leq 2 K^\mathbb C_G,$$ where \[C_k(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1, p\in \mathbb C_k[Z_1,\ldots,Z_n],\boldsymbol T\in\mathscr{C}_n \big\}\] and $K_{G}^{\mathbb C}$ is the complex Grothendieck constant, is due to Varopoulos. We answer a long--standing question by showing that the limit $\lim_{n\to\infty} \frac{C_2(n)}{K^\mathbb C_G}$ is strictly bigger than $1.$ Let $\mathbb C_2^s[Z_1,\ldots , Z_n]$ denote the set of all complex valued homogeneous polynomials $p(z_1,\ldots,z_n)$ $=\sum_{j,k=1}^{n}a_{jk}z_jz_k$ of degree two in $n$ - variables, where $(\!(a_{jk})\!)$ is a $n\times n$ complex symmetric matrix. For each $n\in\mathbb{N},$ define the linear map $\mathscr{A}_n:\big (\mathbb C_2^s[Z_1,\ldots , Z_n],\|\cdot\|_{\mathbb D^n, \infty}\big ) \to \big (M_n, \|\cdot \|_{\infty \to 1}\big )$ to be $\mathscr{A}_n\big (p) = (\!(a_{jk})\!).$ We show that the supremum (over $n$) of the norm of the operators $\mathscr{A}_n;\,n\in\mathbb{N},$ is bounded below by the constant $π^2/8.$ Using a class of operators, first introduced by Varopoulos, we also construct a large class of explicit polynomials for which the von Neumann inequality fails. We prove that the original Varopoulos--Kaijser polynomial is extremal among a, suitably chosen, large class of homogeneous polynomials of degree two. We also study the behaviour of the constant $C_k(n)$ as $n \to \infty.$

math.FA↗