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We develop a static quantum embedding scheme that utilizes different levels of approximations to coupled cluster (CC) theory for an active fragment region and its environment. To reduce the computational cost, we solve the local fragment problem using a high-level CC method and address the environment problem with a lower-level Møller–Plesset (MP) perturbative method. This embedding approach inherits many conceptual developments from the hybrid second-order Møller–Plesset (MP2) and CC works by Nooijen [J. Chem. Phys. 111, 10815 (1999)] and Bochevarov and Sherrill [J. Chem. Phys. 122, 234110 (2005)]. We go beyond those works here by primarily targeting a specific localized fragment of a molecule and also introducing an alternative mechanism to relax the environment within this framework. We will call this approach MP-CC. We demonstrate the effectiveness of MP-CC on several potential energy curves and a set of thermochemical reaction energies, using CC with singles and doubles as the fragment solver, and MP2-like treatments of the environment. The results are substantially improved by the inclusion of orbital relaxation in the environment. Using localized bonds as the active fragment, we also report results for N=N bond breaking in azomethane and for the central C–C bond torsion in butadiene. We find that when the fragment Hilbert space size remains fixed (e.g., when determined by an intrinsic atomic orbital approach), the method achieves comparable accuracy with both a small and a large basis set. Additionally, our results indicate that increasing the fragment Hilbert space size systematically enhances the accuracy of observables, approaching the precision of the full CC solver. [ABSTRACT FROM AUTHOR]

Numerous variational methods have been proposed for solving quantum many-body systems, but they often face exponentially increasing computational complexity as the Hilbert space dimension grows. To address this, we introduce a novel approach using quantum neural networks to simulate the dissipative dynamics of many-body open quantum systems. This method combines neural-network quantum state representation with the time-dependent variational principle, both implemented via quantum algorithms. This results in accurate open quantum dynamics described by the Lindblad quantum master equation, exemplified by the spin-boson and transverse field Ising models. Our approach avoids the computational expense of classical algorithms and demonstrates the potential advantages of quantum computing for many-body simulations. To reduce measurement errors, we introduce a projection reset procedure, which could benefit other quantum simulations. In addition, our approach can be extended to simulate non-Markovian quantum dynamics. [ABSTRACT FROM AUTHOR]

Although selected configuration interaction (SCI) algorithms can tackle much larger Hilbert spaces than the conventional full CI method, the scaling of their computational cost with respect to the system size remains inherently exponential. In addition, inaccuracies in describing the correlation hole at small interelectronic distances lead to the slow convergence of the electronic energy relative to the size of the one-electron basis set. To alleviate these effects, we show that the non-Hermitian, transcorrelated (TC) version of SCI significantly compactifies the determinant space, allowing us to reach a given accuracy with a much smaller number of determinants. Furthermore, we note a significant acceleration in the convergence of the TC-SCI energy as the basis set size increases. The extent of this compression and the energy convergence rate are closely linked to the accuracy of the correlation factor used for the similarity transformation of the Coulombic Hamiltonian. Our systematic investigation of small molecular systems in increasingly large basis sets illustrates the magnitude of these effects. [ABSTRACT FROM AUTHOR]

Two computational approaches for computing the rates of internal conversions in molecular systems where a large set of nuclear degrees of freedom plays a role are discussed and compared. One approach is based on the numerical solution of the time-dependent Schrödinger equation and allows us to include almost the whole set of vibrational coordinates, thanks to the employment of effective procedures for selecting those elements of the Hilbert space which play a significant role in dynamics. The other approach, based on the time-dependent perturbation theory and limited to the use of the harmonic approximation, allows us to include the whole Hilbert space spanned by the vibrational states of the system. The two approaches are applied to the photophysics of azulene, whose anti-Kasha behavior caused by anomalous internal conversion rates is well assessed. The calculated rates for the decays of the first two excited singlet states are in very good agreement with experimental data, indicating the reliability of both methodologies. [ABSTRACT FROM AUTHOR]

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants—the state universal one and the Brillouin–Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO–LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations. [ABSTRACT FROM AUTHOR]

We introduce a generalization of the σ-SCF method to approximate noncollinear spin ground and excited single-reference electronic states by minimizing the Hamiltonian variance. The new method is based on the σ-SCF method, originally proposed by Ye et al. [J. Chem. Phys. 147, 214104 (2017)], and provides a prescription to determine ground and excited noncollinear spin states on an equal footing. Our implementation was carried out utilizing an initial simulated annealing stage followed by a mean-field iterative self-consistent approach to simplify the cumbersome search introduced by generalizing the spin degrees of freedom. The simulated annealing stage ensures a broad exploration of the Hilbert space spanned by the generalized spin single-reference states with random complex element-wise rotations of the generalized density matrix elements in the simulated annealing stage. The mean-field iterative self-consistent stage employs an effective Fockian derived from the variance, which is utilized to converge tightly to the solutions. This process helps us to easily find complex spin structures, avoiding manipulating the initial guess. As proof-of-concept tests, we present results for Hn (n = 3–7) planar rings and polyhedral clusters with geometrical spin frustration. We show that most of these systems have noncollinear spin excited states that can be interpreted in terms of geometric spin frustration. These states are not directly targeted by energy minimization methods, which are meant to converge to the ground state. This stresses the capability of the σ-SCF methodology to find approximate noncollinear spin structures as mean-field excited states. [ABSTRACT FROM AUTHOR]

We consider linear models with scalar responses and covariates from a separable Hilbert space. The aim is to detect change points in the error distribution, based on sequential residual empirical distribution functions. Expansions for those estimated functions are more challenging in models with infinite-dimensional covariates than in regression models with scalar or vector-valued covariates due to a slower rate of convergence of the parameter estimators. Yet the suggested change point test is asymptotically distribution-free and consistent for one-change point alternatives. In the latter case we also show consistency of a change point estimator. [ABSTRACT FROM AUTHOR]

We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the R -matrix associated to the standard q -deformation of G L (N , C) for 0 < q < 1. We consider the Poisson structure appearing as the classical limit of the R -matrix, and parametrize the symplectic leaves explicitly in terms of a type of matrix we call a shape matrix. We then introduce a quantized version of the shape matrix for the REA, and show that each irreducible representation of the REA has a unique shape. [ABSTRACT FROM AUTHOR]

The analysis of survival data is often hampered by a potentially very large number of covariates compounded with incomplete data. This paper considers the problem of variable selection where the response is subject to random (right) censoring. We introduce a model-free variable selection procedure via learning the gradients of quantile regression functions with two popular censoring weighting schemes. The key advantage of the proposed approach is that it does not require explicit model assumptions and is computationally efficient. Besides, we also prove its asymptotic consistency and examine its finite sample performance via numerical studies. As an empirical example, the proposed procedure is applied for the analysis of a breast cancer data set. [ABSTRACT FROM AUTHOR]

In this work, we present a refined convergence analysis of the Popov's projection algorithm for solving pseudo-monotone variational inequalities in Hilbert spaces. Our analysis results in a larger range of stepsize, which is achieved by using a new Lyapunov function. Furthermore, when the operator is strongly pseudo-monotone and Lipschitz continuous, we establish the linear convergence of the sequence generated by the Popov's algorithm. As a by-product of our analysis, we extend the range of stepsize in the projected reflected gradient algorithm for solving unconstrained pseudo-monotone variational inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of the split common fixed point problem with multiple output sets for demicontractive mappings in real Hilbert spaces. The strong convergence of the iterative sequence generated by the algorithm method is established without the prior knowledge of the norms of the given bounded linear operators. In addition, using our method, we do not require any information of the Lipschitz and strongly monotone constants of the mappings. Several corollaries of our main result are also presented. Finally, we include several numerical examples that serve to illustrate the performance of our proposed algorithm. [ABSTRACT FROM AUTHOR]

This work investigates the problem of testing whether d functional random variables are jointly independent, using a modified estimator of the d-variable Hilbert Schmidt Indepedence Criterion (dHSIC). We then get the asymptotic normality of this estimator both under the joint independence hypothesis and under the alternative hypothesis. A simulation study shows the good performance of the proposed test on a finite sample. [ABSTRACT FROM AUTHOR]

We develop a variable metric forward-reflected-Douglas-Rachford method for finding a zero point of a sum of one Lipschitzian monotone and two maximally monotone operators acting on a real Hilbert space. The weak convergence of the proposed algorithm is formally proved under standard conditions. We also consider the linear convergence of the proposed method in constant-case and obtain the desired result under the strong monotonicity and Lipschitz condition of maximally monotone operators. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a novel self-adaptive explicit iterative algorithm for solving the split common solution problem with monotone operator equations in real Hilbert spaces. This new approach does not employ resolvent operators nor does it use the norms of the bounded linear operators (transfer mappings) from the source space to the image spaces. [ABSTRACT FROM AUTHOR]

An iterative method for approximating solutions of nonlinear ill-posed equations with m-accretive (in Banach space)/ monotone (in Hilbert space) operator is considered with weaker assumptions. Precisely, Lavrentiev regularization method is studied in Banach space with m-accretive operator and in Hilbert space with monotone operator. An optimal order of convergence is obtained with a new source condition and apriori parameter choice strategy as in S. George et.al., [16]. Numerical example is presented, which justifies the analysis. [ABSTRACT FROM AUTHOR]

In this work, we study the split common fixed point problem with multiple output sets for two finite families of generalized demimetric mappings in Hilbert space. In order to solve this problem, we suggest a new inertial self-adaptive algorithm and establish a strong convergence theorem for it under relevant conditions. We also provide a numerical example to support our main result. [ABSTRACT FROM AUTHOR]

Matrix-valued (multivariate) correlation functions are increasingly used within both the statistics and machine learning communities, but their properties have been studied to a limited extent. The motivation of this paper comes from the fact that the celebrated local stationarity construction for scalar-valued correlations has not been considered for the matrix-valued case. The main reason is a lack of theoretical support for such a construction. We explore the problem of extending a matrix-valued correlation from a d-dimensional ball with arbitrary radius into the d-dimensional Euclidean space. We also consider such a problem over product spaces involving the d-dimensional ball with arbitrary radius. We then provide a useful architecture to matrix-valued local stationarity by defining the class of p-exponentially convex matrix-valued functions, and characterize such a class as scale mixtures of the d-Schoenberg kernels against certain families of measures. We exhibit bijections from such a class into the class of positive semidefinite matrix-valued functions and we extend exponentially convex matrix-valued functions from d-dimensional balls into the d-dimensional Euclidean space. We finally provide similar results for the case of function-valued correlations defined over certain Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose an alternated multi-step inertial iterative algorithm for solving the split feasibility problem involving two bounded linear operators in Hilbert spaces. The proposed algorithm adopts self-adaptive step size and the step size is bounded away from zero. Under some mild conditions, the strong convergence of the sequence generated by the proposed algorithm is established. Finally, the numerical experiments are presented to verify the effectiveness and superiority of our proposed algorithm. Our results are innovative and can be an enrichment to the recently published results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the projected gradient method for solving strongly pseudo-monotone, Lipschitz continuous variational inequalities in Hilbert spaces. Under strong pseudomonotonicity and Lipschitz continuity hypotheses, we prove the sequence generated by the proposed algorithms converges to the unique solution of VIs with an $ R $-linear rate without the prior knowledge of the Lipschitz constant of the variational inequality mapping. Moreover, the self-adaptive step size in the algorithms is bounded from below by a positive number this has not been studied before for the projected gradient method. Some numerical result applications in optimal control problems are provided to demonstrate the advantages and efficiency of the proposed algorithms over some existing results. [ABSTRACT FROM AUTHOR]

The main innovation of this article is to establish the existence and approximate controllability of Hilfer fractional differential inclusions of order 1<μ<2$$ 1<\mu <2 $$ using Clarke's subdifferential type in Hilbert spaces. By applying the fixed‐point theorem for multivalued maps, fractional calculus, generalized Clarke's subdifferential, and cosine families, we derive a novel existence result for the mild solutions of the Hilfer fractional differential inclusion system. Furthermore, we develop and demonstrate a new set of sufficient conditions for the approximate controllability of Hilfer fractional differential systems. These criteria are based on the assumption that the associated linear part of the system is approximately controllable. Finally, an example is provided to show the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the deterministic learning problem associated with spherical scattered data for the first time. We design the quadrature-weighted kernel regularized regression learning schemes associated with deterministic scattered data on the unit sphere and the spherical cap. By employing the minimal norm interpolation technique and leveraging results from the numerical integration of spherical radial basis functions over these surfaces, we derive the corresponding learning rates. Notably, our algorithm design and error analysis methods diverge from those typically employed in randomized learning algorithms. Our findings suggest that the learning rates are influenced by both the mesh norm of the scattered data and the smoothness of the radial basis function. This implies that when the radial basis function exhibits sufficient smoothness, the learning rate achieved with deterministic samples outperforms that obtained with random samples. Furthermore, our results provide theoretical support for the feasibility of deterministic spherical learning, which may bring potential applications in tectonic plate geology and Earth sciences. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we provide a convergence rate of sieves estimates of the Hilbert autoregressive process kernel operator presented by Berhoune and Bensmain (2018). We found that the convergence rate is governed by the local expected values, variances, L2 Bracketing entropy, and the approximation error of the sieve. [ABSTRACT FROM AUTHOR]

In the analysis of controllability of finite dimensional quantum systems, subspace controllability refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics. We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state (Pure State Controllability) is verified. [ABSTRACT FROM AUTHOR]

This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path (B H , W) with H ∈ (1 / 3 , 1 / 2) as ε → 0. Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we embed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale. [ABSTRACT FROM AUTHOR]

This paper's main objective is to find new upper bounds for the norm of the sum of two Hilbert space operators and their Kronecker product. The obtained results extend some previously known results from the set of positive operators to arbitrary ones and refine several existing bounds. In particular, applications of the established bounds include refining celebrated numerical radius inequalities and the celebrated operator Cauchy–Schwarz norm inequality. [ABSTRACT FROM AUTHOR]

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems. [ABSTRACT FROM AUTHOR]

For a generalized Drazin invertible operator A and its core–EP inverse A ◯ ​​​​ d , the expression A A ◯ ​​​​ d A plays important role in the representations of well-known generalized inverses, in theory of partial orders and for decompositions of operator A. Solving certain system of operator equations leads to introduction of new generalized inverse of A A ◯ ​​​​ d A . Since this new inverse can be expressed by the generalized Moore–Penrose inverse and generalized Drazin inverse of A, it is called a generalized MPD inverse of A A ◯ ​​​​ d A . As a dual version of generalized MPD inverse, we also consider a generalized DMP inverse. The dual core inverse and core inverse of A, respectively, are special cases of the generalized MPD inverse and generalized DMP inverse. Several properties, representations and characterizations of generalized MPD and DMP inverses are established. Applying the generalized MPD and DMP inverses, we solve corresponding systems of linear equations and minimization problems. One well-known minimization problem is a particular case of our results. [ABSTRACT FROM AUTHOR]

In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand–Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an a priori stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by a numerical experiment on a system of linearly ill-posed problems and by discussing an example that fulfills the assumptions of the paper. [ABSTRACT FROM AUTHOR]

There is a resent paper claiming that every hyponormal operator which is not a multiple of the identity (operator) has a nontrivial hyperinvariant subspace. If this claim is true, then every hyponormal operator has a nontrivial invariant subspace, and every subnormal operator which is not multiple of the identity has a nontrivial hyperinvariant subspace. But we find out that the proof of the above claim goes wrong. Thus, the invariant subspace problem of the hyponormal operator and the hyperinvariant problem of subnormal operator remain unresolved. Moreover, it is well known that these are two important research topics in operator theory that have not been solved for a long time, and that many researchers have been trying to solve these two problems. So, it is very meaningful to clarify whether these two problems have been solved. [ABSTRACT FROM AUTHOR]

A Ramond open superstring quantization is studied in a flat D-dimensional Minkowski spacetime in a static gauge. This gauge degenerates the boost generators. A deformation in the structure of the boosts is proposed at the classical level with some energy-dependent coefficients. We quantize the theory and propose the quantum version of the deformation of the boost operators. The energy-dependent coefficients are determined in the first and second excited levels in order to keep the physical Hilbert space Lorentz invariant. The critical space–time dimension D=10 and ordering ambiguity constant a=0 are recovered. We give the connection between the covariant and static gauge quantizations. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we consider a bilevel problem: Variational inequalities over the solution set of a general split inverse problem consists of a monotone variational inclusion problem. We propose a relaxed inertial forward‐backward‐forward splitting algorithm with a new step size rule for finding an approximate solution of this problem in real Hilbert spaces. Under some mild conditions, we prove a strong convergence theorem for the algorithm produced by the method. Also, we apply our result to study certain classes of bilevel optimization problems, and split inverse problems. Finally, we present some numerical experiments and application in signal recovery problem to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

ABSTRACT This manuscript addresses the optimal control of generalized Riemann–Liouville fractional (Hilfer fractional) Sobolev‐type stochastic differential equations of order ϑ∈(1,2)$$ \vartheta \in \kern0.3em \left(1,2\right) $$ in separable Hilbert spaces. First, the existence of mild solutions for the system is established using the cosine family of operators and the Leray–Schauder fixed point theorem. Then, the existence of optimal control is demonstrated through Balder's theorem. Finally, an example is provided to illustrate the results. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate some tensorial norm inequalities for continuous functions of self-adjoint operators in Hilbert spaces. Our first approach is to develop a gradient descent inequality and some relational properties for continuous functions involving Huber convex functions, as well as several new bounds for Simpson type inequality that is twice differentiable using different types of generalized convex mappings. It is believed that this study will provide a valuable contribution towards developing a new perspective on functional inequalities by utilizing some other types of generalized mappings. [ABSTRACT FROM AUTHOR]

This paper presents new weighted lower and upper bounds for the Euclidean numerical radius of pairs of operators in Hilbert spaces. We show that some of these bounds improve on recent results in the literature. We also find new inequalities for the numerical radius and the Davis–Wielandt radius. The lower and upper bounds we obtain are not symmetrical. [ABSTRACT FROM AUTHOR]

This manuscript investigates the relationship between higher-dimensional Hilbert Manifolds and Hilbert spaces using the diffeomorphic embedding theorem. The existence of diffeomorphisms between Hilbert spaces will be demonstrated, and we will provide exact constructions that demonstrate their mathematical flexibility in several scenarios. The more general case of compact complex Manifolds equipped with Hermitian metrics is investigated. Under specific conditions, the existence of diffeomorphic embeddings into a separable infinite-dimensional Hilbert space that preserve both the complex structure and the Hermitian geometry is demonstrated. These results pave the way for studying complex geometry within the framework of infinite-dimensional Hilbert spaces. [ABSTRACT FROM AUTHOR]

Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS. Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm. [ABSTRACT FROM AUTHOR]

This paper introduces two new extragradient methods designed to solve pseudomonotone equilibrium problems subject to a Lipschitz-type condition. These methods incorporate a variable stepsize criterion that dynamically adjusts with each iteration based on prior iterations. A distinguishing feature of these methods is their independence from prior knowledge of Lipschitz-type constants or any line-search method. The convergence theorems for the proposed methods are established under mild conditions, without requiring the knowledge of Lipschitz-type constants. Additionally, the paper includes several investigations demonstrating the numerical efficacy of the methods and facilitating comparisons with other approaches. This paper contributes to the advancement of computational methods for addressing pseudomonotone equilibrium problems across various applications. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of exponential stabilisation for a class of retarded distributed second-order semilinear systems by means of a bounded feedback control. The existence and uniqueness of mild solution for the considered systems are proved in Hilbert space framework. Sufficient and necessary conditions for exponential stabilisation are given. Some illustrating applications are provided. [ABSTRACT FROM AUTHOR]

We consider the global minimization of smooth functions based solely on function evaluations. Algorithms that achieve the optimal number of function evaluations for a given precision level typically rely on explicitly constructing an approximation of the function which is then minimized with algorithms that have exponential running-time complexity. In this paper, we consider an approach that jointly models the function to approximate and finds a global minimum. This is done by using infinite sums of square smooth functions and has strong links with polynomial sum-of-squares hierarchies. Leveraging recent representation properties of reproducing kernel Hilbert spaces, the infinite-dimensional optimization problem can be solved by subsampling in time polynomial in the number of function evaluations, and with theoretical guarantees on the obtained minimum. Given n samples, the computational cost is O (n 3.5) in time, O (n 2) in space, and we achieve a convergence rate to the global optimum that is O (n - m / d + 1 / 2 + 3 / d) where m is the degree of differentiability of the function and d the number of dimensions. The rate is nearly optimal in the case of Sobolev functions and more generally makes the proposed method particularly suitable for functions with many derivatives. Indeed, when m is in the order of d, the convergence rate to the global optimum does not suffer from the curse of dimensionality, which affects only the worst-case constants (that we track explicitly through the paper). [ABSTRACT FROM AUTHOR]

As an early teaching tool in QM, we propose to use a single-photon Mach–Zehnder interferometer. We demonstrate experimentally several properties associated with the concept of single photons in quantum optics and use them to introduce the quantum formalism and postulates in a more intuitive way. Editor's Note: Recent progress in laser optics, high-speed electronics, and low-noise photon counting has significantly simplified what used to be either thought experiments or state-of-the-art quantum optics experiments. In this paper, the authors take advantage of this progress to present a Mach–Zehnder interferometer operated at the single-photon counting level in order to introduce undergraduate students to quantum concepts, building upon their previous knowledge of wave optics. The simplicity of the Mach–Zehnder configuration allows for an easy introduction to the formalism of quantum states in two-dimensional Hilbert spaces. Anyone teaching introductory quantum mechanics should have a look at this paper, either to perform such experiments or simply to discuss them in class. [ABSTRACT FROM AUTHOR]

We present a concise point of view on the first and the second Korn's inequality for general exponent p and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for p = 2, uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent 1

Paszkiewicz's conjecture asserts that given a decreasing sequence T_1\ge T_2\ge \dots of positive contractions on a separable infinite-dimensional Hilbert space H, the product S_n=T_nT_{n-1}\cdots T_1 converges in the strong operator topology. In these notes, we give an equivalent, more precise formulation of his conjecture. Moreover, we show that the conjecture is true for the following two cases: (1) 1 is not in the essential spectrum of T_n for some n\in \mathbb {N}. (2) The von Neumann algebra generated by \{T_n\mid n\in \mathbb {N}\} admits a faithful normal tracial state. We also remark that the analogous conjecture for the weak convergence is true. [ABSTRACT FROM AUTHOR]