Hilbert Space Research Articles

Polynomially accretive operators

In this paper, we introduce a new class of operators on a complex Hilbert space $\mathcal{H}$ which is called polynomially accretive operators, and thereby extending the notion of accretive and $n$--real power positive operators. We give several properties of the newly introduced class, and generalize some results for accretive operators. We also prove that every $2$--normal and $(2k+1)$--real power positive operator, for some $k\in\mathbb{N}$, must be $n$--normal for all $n\geq2$. Finally, we give sufficient conditions for the normality in the preceding implication.

New algorithms for solving pseudo-monotone variational inequalities in Banach spaces

In this paper, we introduce new algorithms for finding a solution of a variational inequality problem involving pseudo-monotone operator which is also a fixed point of a Bregman relatively inexpensive mapping in $p$-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. We prove weak and strong convergence theorems for proposed algorithms. Finally, we give some numerical experiments for supporting our main results.

Approximate Controllability Result for Backward Stochastic Evoluation Inclusions in Hilbert Spaces

In this paper, we study semilinear Backward Stochastic Evolution Inclusion systems in Hilbert spaces. First, we prove the existence of mild solution of the semilinear Backward Stochastic Evolution Inclusion systems using a multivalued fixed point theorem. Then, we obtain the approximate controllability result for semilinear Backward Stochastic Evolution Inclusion systems through the linear systems corresponding to these semilinear Backward Stochastic Evolution Inclusion systems under appropriate conditions. In particular, our study extends the results of the concept of approximate controllability to Backward Stochastic Evolution Inclusion systems.

An existence result for integro-differential degenerate sweeping process with convex sets.

In this paper, we study the well-posedness in the sense of existence and uniqueness of a solution of integrally perturbed degenerate sweeping processes, involving convex sets in Hilbert spaces. The degenerate sweeping process is perturbed by a sum of a singlevalued map satisfying a Lipschitz condition and an integral forcing term. The integral perturbation depends on two time-variables, by using a semi-discretization method. Unlike the previous works, the Cauchy’s criterion of the approximate solutions is obtained without any new Gronwall’s like inequality.

Device-independent lower bounds on the conditional von Neumann entropy

The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.

A novel numerical approach for the third order Emden–Fowler type equations

AbstractThis article aims to achieve robust numerical results by applying the Chebyshev reproducing kernel method without homogenizing the initial‐boundary conditions of the Emden–Fowler (E‐F) equation, thereby introducing a new perspective to the literature. A novel numerical approach is presented for solving the initial‐boundary value problem of third‐order E‐F equations using Chebyshev reproducing kernel theory. Unlike previous applications, which were confined to homogeneous initial‐boundary value problems or required homogenization, the proposed method is effective for both homogeneous and nonhomogeneous cases. To handle the initial‐boundary conditions of the E‐F equations, additional basis functions are introduced rather than imposing conditions on the reproducing kernel Hilbert space. The method's effectiveness is demonstrated through five examples, which validate the theoretical analysis. Overall, the results emphasize the method's efficiency.

Machine learning in viscoelastic fluids via energy-based kernel embedding

The ability to measure differences in collected data is of fundamental importance for quantitative science and machine learning, motivating the establishment of metrics grounded in physical principles. In this study, we focus on the development of such metrics for viscoelastic fluid flows governed by a large class of linear and nonlinear stress models. To do this, we introduce energy-compatible families of kernel functions corresponding to a given viscoelastic stress model. Each kernel implicitly embeds flowfield snapshots into a Reproducing Kernel Hilbert Space (RKHS) in which distances and angles are computed and whose squared norm equals the total mechanical energy. Additionally, we present a solution to the preimage problem for these kernels, enabling accurate reconstruction of flowfields from their RKHS representations. Through numerical experiments on an unsteady viscoelastic lid-driven cavity flow, we demonstrate the utility of energy-compatible kernels for extracting energetically-dominant coherent structures in viscoelastic flows across a range of Reynolds and Weissenberg numbers. Specifically, the features extracted by Kernel Principal Component Analysis (KPCA) of flowfield snapshots using energy-compatible kernel functions yield reconstructions with superior accuracy in terms of mechanical energy compared to conventional methods such as ordinary Principal Component Analysis (PCA) with naïvely-defined state vectors or KPCA with ad-hoc choices of kernel functions. Our findings underscore the importance of principled choices of metrics in both scientific and machine learning investigations of complex fluid systems.

Statistical Signatures of Quantum Contextuality.

Quantum contextuality describes situations where the statistics observed in different measurement contexts cannot be explained by a measurement of the independent reality of the system. The most simple case is observed in a three-dimensional Hilbert space, with five different measurement contexts related to each other by shared measurement outcomes. The quantum formalism defines the relations between these contexts in terms of well-defined relations between operators, and these relations can be used to reconstruct an unknown quantum state from a finite set of measurement results. Here, I introduce a reconstruction method based on the relations between the five measurement contexts that can violate the bounds of non-contextual statistics. A complete description of an arbitrary quantum state requires only five of the eight elements of a Kirkwood-Dirac quasiprobability, but only an overcomplete set of eleven elements provides an unbiased description of all five contexts. A set of five fundamental relations between the eleven elements reveals a deterministic structure that links the five contexts. As illustrated by a number of examples, these relations provide a consistent description of contextual realities for the measurement outcomes of all five contexts.

Cross-ValidatedFunctional Generalized Partially Linear Single-Functional Index Model

In this paper, we have introduced a functional approach for approximating nonparametric functions and coefficients in the presence of multivariate and functional predictors. By utilizing the Fisher scoring algorithm and the cross-validation technique, we derived the necessary components that allow us to explain scalar responses, including the functional index, the nonlinear regression operator, the single-index component, and the systematic component. This approach effectively addresses the curse of dimensionality and can be applied to the analysis of multivariate and functional random variables in a separable Hilbert space. We employed an iterative Fisher scoring procedure with normalized B-splines to estimate the parameters, and both the theoretical and practical evaluations demonstrated its favorable performance. The results indicate that the nonparametric functions, the coefficients, and the regression operators can be estimated accurately, and our method exhibits strong predictive capabilities when applied to real or simulated data.

The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time-dependent dissipation term of the form u″+Au+b(t)u′=g(t) in a real Hilbert space H is considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure.

Fixed Point Results for New Classes of k-Strictly Asymptotically Demicontractive and Hemicontractive Type Multivalued Mappings in Symmetric Spaces

Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, novel classes of nonlinear mappings called k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings are introduced in real Hilbert spaces that are symmetric spaces. In addition, we discuss the weak and strong convergence results by considered modified algorithms, and a demiclosedness property, for these classes of mappings are proved. Several non-trivial examples are demonstrated to validate the newly defined mappings. Consequently, the results and iterative methods obtained in this study improve and extend several known outcomes in the literature.

Mann-Type Inertial Accelerated Subgradient Extragradient Algorithm for Minimum-Norm Solution of Split Equilibrium Problems Induced by Fixed Point Problems in Hilbert Spaces

Mann-Type Inertial Accelerated Subgradient Extragradient Algorithm for Minimum-Norm Solution of Split Equilibrium Problems Induced by Fixed Point Problems in Hilbert Spaces

Fock structure of complete Boolean algebras of type I factors and of unital factorizations

The factorizable vectors of a complete Boolean algebra of type I factors, acting on a separable Hilbert space, are shown to be total, resolving a conjecture of Araki and Woods. En route, the spectral theory of noise-type Boolean algebras of Tsirelson is cast in the noncommutative language of “factorizations with unit” for which a multi-layered characterization of being “of Fock type” is provided.

Ab initio study of the interaction between the charged system (FH+) and the He atom

We investigated the molecular structure and spectroscopic properties of the charged system (FH+) in interaction with a helium atom using an ab initio quantum chemistry approach in Jacobi coordinates. Our study focused on the ground state X2Π of (FH+) with various theoretical methods including UCCSD, UCCSD(T) and UCCSD(T)-F12, alongside basis sets like aug-cc-pVnZ (n = T, Q, 5, and 6) and cc-pVQZ-F12. We evaluated how these methods and basis sets affect the minimum energies We assessed the impact of the methods, basis sets, and extrapolation approaches on the minimum energies. Potential energy surfaces (PES) were generated using the correlated UCCSD(T)-F12 method with cc-pVQZ-F12 for hydrogen and fluorine, and cc-pVQZ-F12/optri for helium. These surfaces, considering spin-orbit coupling, are degenerate for linear geometries (θ=0° and θ=180°) of the (FH+)-He complex and correlate with the doubly degenerate X2Π state of (FH+). Our results reveal a strong anisotropic character at short and intermediate distances and a quasi-isotropic character upon dissociation into FH+ and He. We compared the spectroscopic parameters of the (FH+)-He complex with existing theoretical data, finding that the linear arrangement exhibits the highest stability, consistent with previous results for similar complexes. This conclusion is supported by the Milliken atomic charge distribution and a three-dimensional map of the Molecular Electrostatic Potential. Additionally, we used the LEVEL program, to calculate the bound rovibronic levels of the (FH+)-He complex for angular momentum values Ω=1/2 and Ω=3/2. Utilizing our ab initio results and a general interpolation approach based on the Reproducing Kernel Hilbert Space method, we generated contour maps illustrating the analytical potential for the (FH+)-He system. These findings will be employed to study the stabilities, geometries, energetics, and structures of the FH+ charged system within helium clusters.

New Results on Some Transforms of Operators in Hilbert Spaces

New Results on Some Transforms of Operators in Hilbert Spaces

Explicit iterative algorithms for solving the split equality problems in Hilbert spaces

Explicit iterative algorithms for solving the split equality problems in Hilbert spaces

Quantum geometrodynamics revived: II. Hilbert space of positive definite metrics

Abstract This paper represents the second in a series of works aimed at reinvigorating the quantum geometrodynamics program. Our approach introduces a lattice regularization of the hypersurface deformation algebra, such that each lattice site carries a set of canonical variables given by the components of the spatial metric and the corresponding conjugate momenta. In order to quantize this theory, we describe a representation of the canonical commutation relations that enforces the positivity of the operators q_{ab} s^a s^b for all choices of s. Moreover, symmetry of q_{ab} and p^{ab} is ensured. This reflects the physical requirement that the spatial metric should be a positive definite, symmetric tensor. To achieve this end, we resort to the Cholesky decomposition of the spatial metric into upper triangular matrices with positive diagonal entries. Moreover, our Hilbert space also carries a representation of the vielbein fields and naturally separates the physical and gauge degrees of freedom. Finally, we introduce a generalization of the Weyl quantization for our representation. We want to emphasize that our proposed methodology is amenable to applications in other fields of physics, particularly
in scenarios where the configuration space is restricted by complicated relationships among the degrees of freedom.

Probability Bracket Notation for Probability Modeling

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman–Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure

Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure

Benchmarking Quantum Generative Learning: A Study on Scalability and Noise Resilience using QUARK

AbstractQuantum computing promises a disruptive impact on machine learning algorithms, taking advantage of the exponentially large Hilbert space available. However, it is not clear how to scale quantum machine learning (QML) to industrial-level applications. This paper investigates the scalability and noise resilience of quantum generative learning applications. We consider the training performance in the presence of statistical noise due to finite-shot noise statistics and quantum noise due to decoherence to analyze the scalability of QML methods. We employ rigorous benchmarking techniques to track progress and identify challenges in scaling QML algorithms, and show how characterization of QML systems can be accelerated, simplified, and made reproducible when the QUARK framework is used. We show that QGANs are not as affected by the curse of dimensionality as QCBMs and to which extent QCBMs are resilient to noise.