Infinite Dimensions Research Articles

Density Matrix Renormalization Group for Transcorrelated Hamiltonians: Ground and Excited States in Molecules.

We present the theory of a density matrix renormalization group (DMRG) algorithm which can solve for both the ground and excited states of non-Hermitian transcorrelated Hamiltonians and show applications in molecular systems. Transcorrelation (TC) accelerates the basis set convergence rate by including known physics (such as, but not limited to, the electron-electron cusp) in the Jastrow factor used for the similarity transformation. It also improves the accuracy of approximate methods such as coupled cluster singles and doubles (CCSD) as shown by recent studies. However, the non-Hermiticity of the TC Hamiltonians poses challenges for variational methods like DMRG. Imaginary-time evolution on the matrix product state (MPS) in the DMRG framework has been proposed to circumvent this problem, but this is currently limited to treating the ground state and has lower efficiency than the time-independent DMRG (TI-DMRG) due to the need to eliminate Trotter errors. In this work, we show that with minimal changes to the existing TI-DMRG algorithm, namely, replacing the original Davidson solver with the general Davidson solver to solve the non-Hermitian effective Hamiltonians at each site for a few low-lying right eigenstates, and following the rest of the original DMRG recipe, one can find the ground and excited states with improved efficiency compared to the original DMRG when extrapolating to the infinite bond dimension limit in the same basis set. An accelerated basis set convergence rate is also observed, as expected, within the TC framework.

Van der Pol Equation with a Large Feedback Delay

The well-known Van der Pol equation with delayed feedback is considered. It is assumed that the delay factor is large enough. In the study of the dynamics, the critical cases in the problem of the stability of the zero equilibrium state are identified. It is shown that they have infinite dimension. For such critical cases, special local analysis methods have been developed. The main result is the construction of nonlinear evolutionary boundary value problems, which play the role of normal forms. Such boundary value problems can be equations of the Ginzburg–Landau type, as well as equations with delay and special nonlinearity. The nonlocal dynamics of the constructed equations determines the local behavior of the solutions to the original equation. It is shown that similar normalized boundary value problems also arise for the Van der Pol equation with a large coefficient of the delay equation. The important problem of a small perturbation containing a large delay is considered separately. In addition, the Van der Pol equation, in which the cubic nonlinearity contains a large delay, is considered. One of the general conclusions is that the dynamics of the Van der Pol equation in the presence of a large delay is complex and diverse. It fundamentally differs from the dynamics of the classical Van der Pol equation.

A chaotic decomposition for the fractional Lebesgue–Pascal noise space

Abstract This paper is devoted to study the fractional Pascal noise functionals on compound configuration spaces with special emphasis on the chaotic decomposition of the Hilbert spaces of quadratic integrable functionals with respect to the correlation measure corresponding to the fractional Pascal measure in infinite dimensions.

Optimal feedback controls of stochastic linear quadratic control problems in infinite dimensions with random coefficients

It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [22] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of an optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the C0-group assumption of unbounded linear operator A in [22] and using the contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.

Ultrahigh Nitrogen Content Carbon Nanosheets for High Stable Sodium Metal Anodes.

Sodium metal, with a high theoretical specific capacity of 1165 mAh g-1 , is the ultimate anode for sodium batteries, yet how to deal with the inhomogeneous and dendritic sodium deposition and the infinite relative dimension change of sodium metal anodes during sodium depositing/stripping is still challenging. Here, a facile fabricated sodiuphilic 2D N-doped carbon nanosheets (N-CSs) are proposed as sodium host material for sodium metal batteries (SMBs) to prevent dendrite formation and eliminate volume change during cycling. Revealing from combined in situ characterization analyses and theoretical simulations, the high nitrogen content and porous nanoscale interlayer gaps of the 2D N-CSs can not only concede dendrite-free sodium stripping/depositing but also accommodate the infinite relative dimension change. Furthermore, N-CSs can be easily process into N-CSs/Cu electrode via traditional commercial battery electrode coating equipment that pave the way for large-scale industrial applications. On account of the abundant nucleation sites and sufficient deposition space, N-CSs/Cu electrodes demonstrate a superior cycle stability of more than 1500 h at a current density of 2mA cm-2 with a high coulomb efficiency of more than 99.9% and ultralow nucleation overpotential, which enable reversible and dendrites-free SMBs and shed light on further development of SMBs with even higher performance.

A convex optimization approach to multi-objective design of two-stage compensators for MIMO linear systems

The previous design of two-stage compensators with multi-objectives concerns the closed-loop stability, the low-sensitivity, and the satisfactory time-domain performance. However, the design only considers the single-input single-output system. This paper aims to present the design of two-stage compensators for multi-input multi-output (MIMO) linear systems which contain dynamical interaction between inputs and outputs. When Youla parameterization is applied to the MIMO linear system, the design criteria can be cast as a nonlinear optimization. In addition, the infinite dimension optimization problem is approximated to a finite dimension problem with Ritz approximation. The suitable form of design parameters is proposed to ensure the low sensitivity of the MIMO system. The two-stage compensator design is formulated as convex optimization which is efficiently solved using available solvers. Finally, the two-stage compensators are designed for Wood and Berry’s model of binary distillation column. The numerical results indicate that all design objectives of frequency domain and time domain can be achieved by the two-stage compensators.

Collapse and revival structure of information backflow for a central spin coupled to a finite spin bath

The Markovianity of quantum dynamics is an important property of open quantum systems determined by various ingredients of the system and bath. Apart from the system-bath interaction, the initial state of the bath, etc., the dimension of the bath plays a critical role in determining the Markovianity of quantum dynamics, as a strict decay of the bath correlations requires an infinite dimension for the bath. In this work, we investigate the role of finite bath dimension in the Markovianity of quantum dynamics by considering a simple but nontrivial model in which a central spin is isotropically coupled to a finite number of bath spins, and show how the dynamics of the central spin transits from non-Markovian to Markovian as the number of the bath spins increases. The non-Markovianity is characterized by the information backflow from the bath to the system in terms of the trace distance of the system states. We derive the time evolution of the trace distance analytically, and find periodic collapse-revival patterns in the information flow. The mechanism underlying this phenomenon is investigated in detail, and it shows that the period of the collapse-revival pattern is determined by the competition between the number of the bath spins, the system-bath coupling strength, and the frequency detuning. When the number of bath spins is sufficiently large, the period of the collapse-revival structure as well as the respective collapse and revival times increase in proportion to the number of the bath spins, which characterizes how the information backflow decays with a large dimension of the bath. We also analyze the effect of the system-bath interaction strength and frequency detuning on the collapse-revival patterns of the information flow, and obtain the condition for the existence of the collapse-revival structure. The results are illustrated by numerical computation.

Local Multiset Dimension of Amalgamation Graphs

\textbf{Background}: One of the topics of distance in graphs is the resolving set problem. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m (v|W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. Thus, if $G$ has an infinite local multiset dimension and then we write $md_l (G)=\infty$. \\ \textbf{Methods}: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set $W\subset V(G)$; determine the vertex representation of two adjacent vertices in $G$; and prove the theorem.\\ \textbf{Results}: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely $md_l(Amal(G,v,m))\leq m.md_l(G)$ and their exact value of local multiset dimension of some families of graphs namely $md_l(Amal(P_n,v,m))=1$, $md_l(Amal(K_n,v,m))=\infty$, $md_l(Amal(W_n,v,m))=m. md_l(W_n)$, $md_l(Amal(F_n,v,m))=m. md_l(F_n)$ for $d(v)=n$, $md_l(Amal(F_n,v,m))=m. \lfloor\frac{n}{4}\rfloor$. \\ \textbf{Conclusions}: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs\\

Radius theorems for subregularity in infinite dimensions

The paper continues our previous work (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) on the radius of subregularity that was initiated by Asen Dontchev. We extend the results of (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) to general Banach/Asplund spaces and to other classes of perturbations, and sharpen the coderivative tools used in the analysis of the robustness of well-posedness of mathematical problems and related regularity properties of mappings involved in the statements. We also expand the selection of classes of perturbations, for which the formula for the radius of strong subregularity is valid.

Dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings

The subject of this work is the study of local dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings. From a discrete model describing the dynamics of a great number of coupled oscillators, a transition has been made to a nonlinear integro-differential equation, continuously depending on time and space variable. A class of full-coupled systems has been considered. The main assumption is that the amount of delay in the couplings is large enough. This assumption opens the way to the use of special asymptotic methods of study. The parameters under which the critical case is realized in the problem of the equilibrium state stability have been distinguished. It is shown that they have infinite dimension. The analogues of normal forms — nonlinear boundary value problems of Ginzburg–Landau type have been constructed. In some cases, these boundary value problems contain integral components too. Their nonlocal dynamics describes the behavior of all solutions of the original equations in the balance state neighbourhood. Methods. As applied to the considered problems, methods of constructing quasinormal forms on central manifolds are developed. An algorithm for constructing the asymptotics of solutions based on the use of quasinormal forms for determining slowly varying amplitudes has been created. Results. Quasinormal forms that determine the dynamics of the original boundary value problem have been constructed. The dominant terms of asymptotic approximations for solutions of the considered chains have been obtained. On the basis of the given statements, a number of interesting dynamical effects have been revealed. For example, an infinite alternation of direct and reverse bifurcations when the delay coefficient increases. Their distinguishing feature is that they have two or three spatial variables.

Regularity of the backward Monge potential and the Monge–Ampère equation on Wiener space

In this paper, the Monge–Kantorovich problem is considered in infinite dimensions on an abstract Wiener space $(W,H,\mu )$, where $H$ is the Cameron–Martin space and $\mu $ is the Gaussian measure. We study the regularity of optimal transport maps with a

Lasso in infinite dimension: application to variable selection in functional multivariate linear regression

Lasso in infinite dimension: application to variable selection in functional multivariate linear regression

Independence, infinite dimension, and operators

Abstract In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] O. Christensen and M. Hasannasab observed that assuming the existence of an operator T sending en to e n +1 for all n ∈ ℕ (where (e n ) n ∈ℕ is a sequence of vectors) guarantees that (e n ) n ∈ℕ is linearly independent if and only if dim{e n } n ∈ℕ = ∞. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then return to the context of vector spaces to show that, if we want to use a condition like T(e i ) = e ϕ ( i ) for all i ∈ I where I is countable as a replacement of the previous one, the conclusion will only stay true if ϕ : I → I is conjugate to the successor function succ : n ↦n + 1 defined on ℕ. We finally prove a tentative generalization of the result, where we replace the condition T(e i ) = e ϕ ( i ) for all i ∈ I where ϕ is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet.

Optimal Control of Quadrotor Attitude System Using Data-driven Approximation of Koopman Operator

The nonlinear dynamics has posed a great challenge in the optimal control of quadcoptors. This paper applies data-driven methods to approximate the Koopman operator of the quadrotor attitude control system. The Koopman operator is a linear operator with infinite dimensions that can advance the nonlinear state dynamics linearly in a lifted space without compromising the operation range. The function basis and the propagation matrices to approximate the Koopman operator are learned interactively through Deep Neural Network (DNN). Simulations with the quadrotor attitude model in S0(3) are carried out to verify the model precision and the tracking performance using Linear Quadratic Regulator (LQR) with the lifted linear system. We compared the results with first-order approximation and Extended Dynamic Mode Decomposition (eDMD), which is another data-driven method that selects the function basis from a fixed library. Both data-driven approaches have notable advantages over the first-order approximation, while the system learned through the DNN has better precision and control performance than the eDMD.

Random sets and Choquet-type representations

As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decomposable combinations and Choquet convex decomposable combinations, as well as their corresponding hull operators acting on the power sets of Lebesgue-Bochner spaces. We show that Choquet hull coincides with convex hull in the finite-dimensional setting, yet Choquet hull tends to be larger in infinite dimensions. We also provide a quantitative characterization of Choquet hull, without any topological or algebraic assumptions on the underlying set. Furthermore, we show that the Choquet decomposable hull of a set coincides with its strongly closed decomposable hull and the Choquet convex decomposable hull of a set coincides with the Choquet decomposable hull of its convex hull. It turns out that the measurable selections of a closed-valued multifunction form a Choquet decomposable set and those of a closed convex-valued multifunction form a Choquet convex decomposable set. Finally, we investigate the operator-type features of Choquet decomposable and Choquet convex decomposable hull operators when applied in succession.

Structural stability for scalar reaction-diffusion equations

In this paper, we prove the structural stability for a family of scalar reaction-diffusion equations. Our arguments consist of using invariant manifold theorem to reduce the problem to a finite dimension and then, we use the structural stability of Morse–Smale flows in a finite dimension to obtain the corresponding result in infinite dimension. As a consequence, we obtain the optimal rate of convergence of the attractors and estimate the Gromov–Hausdorff distance of the attractors using continuous $\varepsilon$-isometries.

Pullback attractor for a nonautonomous parabolic Cahn-Hilliard phase-field system

<abstract><p>Our aim in this paper is to study generalizations of the Caginalp phase-field system based on a thermomechanical theory involving two temperatures and a nonlinear coupling. In particular, we prove well-posedness results. More precisely, the existence of a pullback attractor for a nonautonomous parabolic of type Cahn-Hilliard phase-field system. The pullback attractor is a compact set, invariant with respect to the cocycle and which attracts the solutions in the neighborhood of minus infinity, consequently the attractor pullback (or attractor retrograde) exhibits a infinite fractal dimension.</p></abstract>

An image encryption algorithm based on the double time-delay Lorenz system.

The traditional image encryption technology has the disadvantages of low encryption efficiency and low security. According to the characteristics of image information, an image encryption algorithm based on double time-delay chaos is proposed by combining the delay chaotic system with traditional encryption technology. Because of the infinite dimension and complex dynamic behavior of the delayed chaotic system, it is difficult to be simulated by AI technology. Furthermore time delay and time delay position have also become elements to be considered in the key space. The proposed encryption algorithm has good quality. The stability and the existence condition of Hopf bifurcation of Lorenz system with double delay at the equilibrium point are studied by nonlinear dynamics theory, and the critical delay value of Hopf bifurcation is obtained. The system intercepts the pseudo-random sequence in chaotic state and encrypts the image by means of scrambling operation and diffusion operation. The algorithm is simulated and analyzed from key space size, key sensitivity, plaintext image sensitivity and plaintext histogram. The results show that the algorithm can produce satisfactory scrambling effect and can effectively encrypt and decrypt images without distortion. Moreover, the scheme is not only robust to statistical attacks, selective plaintext attacks and noise, but also has high stability.

Метод процедурной генерации ландшафта с заданными пользователем параметрами

The article discusses the goals of introducing procedural generation into the gaming industry. The task is to generate a map of infinite dimensions with the possibility of generation as the player moves. A method for generating a landscape based on the mathematical method of bicubic interpolation is proposed. The user has the ability to control input parameters that affect the overall appearance of the result, but the map itself is based on independent values, which allows you to get interesting reliefs. This generation method is compared with the well-known Perlin noise method. The results of the study are presented and the future steps to improve this method of procedural generation are analyzed.

Revisiting Mazur separable quotient problem (1932)

The centenary of the still open problem of the existence of a separable quotient of infinite dimension in a Banach space $ X $ will be in 2032. This note aims to give a quick overview of this problem, which it is known that it has a positive solution for Banach spaces with, rough speaking, 'large density character' and also with 'small density character'. Proofs of several results have been reproduced or simplified to motivate interest in solving this so-called 'Mazur separable quotient Problem'.