Infinite Dimensions Research Articles

Well-posedness and asymptotic behavior of stochastic convective Brinkman–Forchheimer equations perturbed by pure jump noise

This paper is concerned about stochastic convective Brinkman–Forchheimer (SCBF) equations subjected to multiplicative pure jump noise in bounded or periodic domains. Our first goal is to establish the existence of a pathwise unique strong solution satisfying the energy equality (Itô’s formula) to SCBF equations. We resolve the issue of the global solvability of SCBF equations, by using a monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique. The major difficulty is that an Itô’s formula in infinite dimensions is not available for such systems. This difficulty is overcame by the technique of approximating functions using the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. Due to some technical difficulties, we discuss the global in time regularity results of such strong solutions in periodic domains only. Once the system is well-posed, we look for the asymptotic behavior of strong solutions. For large effective viscosity, the exponential stability results (in the mean square and pathwise sense) for stationary solutions is established. Moreover, a stabilization result of SCBF equations by using a multiplicative pure jump noise is also obtained. Finally, we prove the existence of a unique ergodic and strongly mixing invariant measure for SCBF equations subject to multiplicative pure jump noise, by using the exponential stability of strong solutions.

Constructing in-situ polymerized electrolyte on lithiophilic anode for high-performance lithium–air batteries operating in ambient conditions

The only way for commercial application of Li−air batteries with open structure is to receive O2 from ambient air. Nevertheless, H2O-containing ambient air easily induces Li passivation/dendrites and electrolyte decomposition. Apart from above issues, volatilization and leakage of organic liquid electrolytes and relatively infinite dimension variation of metallic Li also seriously discount performance of Li−air batteries. Herein, Li coating polythiophene-derived (S-doped) carbon fiber-based electrode (Li/SC) is prepared by simple rolling method and then in situ coated with deep eutectic solvent-based polymer electrolyte (DES-PE), which are further applied as anode and electrolyte of Li−air battery, respectively. Theoretical/experimental data suggest that SC skeleton possess excellent lithiophilicity/stability which restrain dendritic Li formation and reduce electrode structure damage over cycles. Furthermore, DES-PE not only possesses high ionic conductivity (1.19*10−3 S cm−1, 30 °C), wide electrochemical window (5.0 V), no-leakage and excellent interfacial compatibility, but also combine with SC framework to greatly reduce Li passivation induced by ambient air. Hence, DES-PE@Li/SC based Li−air battery could exhibit extra-large capacity of 15,310 mAh g−1 at 100 mA g−1, high-capacity retention even under high current (11,896 mAh g−1 at 500 mA g−1) and extra-long cycling life up to 300 cycles under fixed capacity. This result provides enlightment for the advancement of Li−air batteries.

Optimal scheduling and control for constrained multi‐agent networked control systems

AbstractIn this paper, we study optimal control and communication schedule co‐design for multi‐agent networked control systems, with assuming shared parallel communication channels and uncertain constrained linear time‐invariant discrete‐time systems. To that end, we specify the communication demand for each system using an associated robust control invariant set and reachability analysis. We use these communication demands and invariant sets to formulate tube‐based model predictive control and offline/online communication schedule co‐design problems. Since the scheduling part includes an infinite dimension integer problem, we propose heuristics to find suboptimal solutions that guarantee robust constraints satisfaction and recursive feasibility. The effectiveness of our approach is illustrated through numerical simulations.

Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions

In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.

Nonlinear small-gain theorems for input-to-state stability of infinite interconnections

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.

Stage Magic as a Performative Design Principle for VR Storytelling

This article examines The VOID’s Star Wars: Secrets of the Empire (2017) VR arcade attraction, and analyzes the intermedial magic principles employed by co-founder and magician Curtis Hickman to create the illusion of a fictive world with impossible space and liveness. I argue that The VOID (Vision of Infinite Dimensions) functioned like nineteenth century magic theaters run by Georges Melies and others, by employing magic principles of misdirection that directed player attention towards the aesthetics of an illusion, and away from the mechanics of the effects generating technology. Narrative framing and performative role play transported multiple players into a believable Star Wars immersive experience, creating an aesthetics of the impossible that reflected the goal of many stage magic tricks, and was foundational to trick films in the cinema of attractions of the early twentieth century. Using game studies concepts like Huizinga’s magic circle and theatre arts concepts like Craig’s uber-marionette, this article suggests that The VOID and other stage magic approaches to VR, like Derren Brown’s Ghost Train (2017), are a new medium for participatory theatre that incorporate immersive features from both cinema and games.

A highly stretchable hydrogel sensor for soft robot multi-modal perception

Soft robots attracted surging interests in recent years due to its enhanced safety, and it significantly contributes to its human-robot interaction application. However, the compliant system has infinite dimensions, which remained challenging for accurate state perception and control. Embedded soft sensors are the potential solutions to this problem as they can trigger electrical or optical signals even when stretched. Yet current conductive material’s upper strain limit are not large enough. Here we design a highly stretchable hydrogel resistive sensor for soft fingers’ multi-modal perception with a simple and low cost fabrication method. The hydrogel’s maximum tensile strain is up to 1200 %, which ensures to tolerate the pneumatic actuator’s large deformation, and its water loss can be as low as 0.0625 % within 30 days. We demonstrated the sensor’s multi-modal perception for the actuator’s state estimation, as it can effectively monitor the soft fingers’ bending, twisting, and external forces with high sensitivity, low hysteresis and high reliability. In addition, we also explore the impact of the hydrogel sensor on the pneumatic actuator’s dynamics, and it provides insights into how the sensor’s placement location influence the perception. This study could lead to improved soft robot system’s kinematics study and potentially more accurate close-loop control.

Subnormal subgroups and self-invariant maximal subfields in division rings

A maximal subfield of a division ring is said to be self-invariant if it is its own normalizer. Subfields of this kind are important because they have a strong connection with Albert's conjecture on the cyclicity of division rings of prime index. We show that every maximal subfield of the Mal'cev-Neumann division ring, which is of infinite dimension, is self-invariant. We also apply the Mal'cev-Neumann structure to refute the conjecture that every noncentral subnormal subgroup of the multiplicative group of a division ring must contain a noncentral normal subgroup. Finally, among other things, we rely on self-invariant subfields to present a criterion for a division ring to have a finite-dimensional subdivision rings.

Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma

We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form dut−Ltutdt=N(ut)dt+∑i=1dFi(ut)dXti where (Lt)t∈[0,T] is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while X≡(X,X) is a two-step (non-necessarily geometric) rough path of Hölder regularity γ>1/3. Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity depending on that of the rough path X). As a technical tool, we introduce a version of the multiplicative sewing lemma, which allows to construct the so called product integrals in infinite dimensions. We later use it to construct a semigroup analogue for the non-autonomous linear PDEs as well as show how to deduce the semigroup version of the usual sewing lemma from it.

Semigroups and evolutionary equations

We show how strongly continuous semigroups can be associated with evolutionary equations. For doing so, we need to define the space of admissible history functions and initial states. Moreover, the initial value problem has to be formulated within the framework of evolutionary equations, which is done by using the theory of extrapolation spaces. The results are applied to two examples. First, differential-algebraic equations in infinite dimensions are treated and it is shown, how a C_{0}-semigroup can be associated with such problems. In the second example we treat a concrete hyperbolic delay equation.

Continuum centroid classifier for functional data

For the binary classification of functional data, we propose the continuum centroid classifier (CCC), which is constructed by projecting the functional data onto one specific direction. This direction is obtained via bridging the regression and classification. Our technique is neither unsupervised nor fully supervised; instead, we control the extent of the supervision. Thanks to the intrinsic infinite dimension of functional data, one of the two subtypes of CCC enjoys an (asymptotic) zero misclassification rate. Our approach includes an effective algorithm that yields a consistent empirical classifier. Simulation studies demonstrate the competitive performance of the CCC in different scenarios. Finally, we apply the CCC to two real examples.

The Boundary Proportion Differential Control Method of Micro-Deformable Manipulator with Compensator Based on Partial Differential Equation Dynamic Model

It is challenging to accurately judge the actual end position of the manipulator—regarded as a rigid body—due to the influence of micro-deformation. Its precise and efficient control is a crucial problem. To solve the problem, the Hamilton principle was used to establish the partial differential equation (PDE) dynamic model of the manipulator system based on the infinite dimension of the working environment interference and the manipulator space. Hence, it resolves the common overflow instability problem in the micro-deformable manipulator system modeling. Furthermore, an infinite-dimensional radial basis function neural network compensator suitable for the dynamic model was proposed to compensate for boundary and uncertain external interference. Based on this compensation method, a distributed boundary proportional differential control method was designed to improve control accuracy and speed. The effectiveness of the proposed model and method was verified by theoretical analysis, numerical simulation, and experimental verification. The results show that the proposed method can effectively improve the response speed while ensuring accuracy.

When are maps preserving semi-inner products linear?

We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space ell _2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

Long-lived coherence in driven many-spin systems: from two to infinite spatial dimensions

Long-lived coherences, emerging under periodic pulse driving in the disordered ensembles of strongly interacting spins, offer immense advantages for future quantum technologies, but the physical origin and the key properties of this phenomenon remain poorly understood. We theoretically investigate this effect in ensembles of different dimensionality, and predict existence of the long-lived coherences in all such systems, from two-dimensional to infinite-dimensional (where every spin is coupled to all others with similar strength), which are of particular importance for quantum sensing and quantum information processing. We explore the transition from two to infinite dimensions, and show that the long-time coherence dynamics in all dimensionalities is qualitatively similar, although the short-time behavior is drastically different, exhibiting dimensionality-dependent singularity. Our study establishes the common physical origin of the long-lived coherences in different dimensionalities, and suggests that this effect is a generic feature of the strongly coupled spin systems with positional disorder. Our results lay out foundation for utilizing the long-lived coherences in a range of application, from quantum sensing with two-dimensional spin ensembles, to quantum information processing with the infinitely-dimensional spin systems in the cavity-QED settings.

The Class of Noetherian RingsWith Finite Valuation Dimension

Not a long time ago, Ghorbani and Nazemian [2015] introduced the concept of dimension of valuation which measures how much does the ring differ from the valuation. They've shown that every Artinian ring has a finite valuation dimensions. Further, any comutative ring with a finite valuation dimension is semiperfect. However, there is a semiperfect ring which has an infinite valuation dimension. With those facts, it is of interest to further investigate property of rings that has a finite dimension of valuation. In this article we define conditions that a Noetherian ring requires and suffices to have a finite valuation dimension. In particular we prove that, if and only if it is Artinian or valuation, a Noetherian ring has its finite valuation dimension. In view of the fact that a ring needs a semi perfect dimension in terms of valuation, our investigation is confined on semiperfect Noetherian rings. Furthermore, as a finite product of local rings is a semi perfect ring, the inquiry into our outcome is divided into two cases, the case of the examined ring being local and the case where the investigated ring is a product of at least two local rings. This is, first of all, that every local Noetherian ring possesses a finite valuation dimension, if and only if it is Artinian or valuation. Secondly, any Notherian Ring generated by two or more local rings is shown to have a finite valuation dimension, if and only if it is an Artinian.

Wild dynamics and asymptotically separated sets

Let X X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T T on X X such that A T ≔ { x ∈ X : ‖ T n x ‖ → ∞ } A_T≔\{x\in X:~ \|T^nx\|\to \infty \} is not dense and has nonempty interior, whenever X X admits a symmetric basis. Augé, in 2012, extended the previous construction to each separable Banach space, introducing the notion of wild operators. This work is divided in two parts. In the first part we define and explore the notion of asymptotically separated sets on Banach spaces, giving several examples whenever the ambient space has either finite or infinite dimension. We show how these sets can be used to construct operators with non-intuitive dynamics. Specifically, operators T : X → X T:X\rightarrow X for which the set A T A_T and the set of recurrent points form a partition of the space. Secondly, we investigate geometric and spectral properties of operators with wild dynamic and we provide a wild operator whose spectrum is equal to the closed unit disk. We end this paper giving some comments about the norm closure of the set of wild operators.

Hybrid model of nonlinear homogenisation of anisotropic composites with ellipsoidal inclusions

A hybrid model of nonlinear homogenisation of anisotropic composites was developed, based on the secant Eshelby's model of the second order. At first a scaling factor was determined between constrained strains of matrices of finite and infinite dimensions for an isotropic composite. The scaling factor was then used to calculate the constrained strain of the matrix and the macrostress of the anisotropic composite. If the calculation results and reference values differ greatly, the scaling factor serves as a fitting parameter. A parametric hybrid model has been developed to predict the properties of composites with different content of inclusions. Good agreement of the calculation results with the numerical solutions of the FEM and experimental data was obtained.

The solvability cardinal of the class of polynomials

Let G be an Abelian group, and let {{mathbb {C}}}^G denote the set of complex valued functions defined on G. A map D: {{mathbb {C}}}^G rightarrow {{mathbb {C}}}^G is a difference operator, if there are complex numbers a_i and elements b_i in G(i=1,ldots , n) such that (Df)(x)=sum _{i=1}^n a_i f(x+b_i) for every fin {{mathbb {C}}}^G and xin G. By a system of difference equations we mean a set of equations { D_i f=g_i : iin I}, where I is an arbitrary set of indices, D_i is a difference operator and g_i in {{mathbb {C}}}^G is a given function for every iin I, and f is the unknown function. The solvability cardinal mathrm{sc} ,({{mathcal {F}}}) of a class of functions {{mathcal {F}}} subset {{mathbb {C}}}^G is the smallest cardinal number kappa with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality <kappa has a solution in {{mathcal {F}}}, then S itself has a solution in {{mathcal {F}}}. The behaviour of mathrm{sc} ,({{mathcal {F}}}) is rather erratic, even for classes of functions defined on {{mathbb {R}}}. For example, mathrm{sc} ,({{mathbb {C}}}[x])=3, but mathrm{sc} ,({mathcal {TP}}) =omega _1, where {mathcal {TP}} is the set of trigonometric polynomials; mathrm{sc} ,({{mathbb {C}}}^{{mathbb {R}}})=omega , but mathrm{sc} ,({mathcal {DF}}) =(2^omega )^+, where {mathcal {DF}} is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on {{{mathbb {R}}}}^n, on normed linear spaces and, in general, on topological Abelian groups. Let {{mathcal {P}}}_G denote the class of polynomials defined on the group G. After presenting some general estimates we prove that mathrm{sc} ,({{mathbb {C}}}[x_1 ,ldots ,x_n ])=omega if 2le n<infty , and mathrm{sc} ,({{mathcal {P}}}_X)=omega _1 if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that mathrm{sc} ,({{mathcal {P}}}_G)=3 if r_0 (G)le 1, mathrm{sc} ,({{mathcal {P}}}_G)=omega if 2le r_0 (G)<infty , and mathrm{sc} ,({{mathcal {P}}}_G)ge omega _1 if r_0 (G) is infinite, where r_0 (G) denotes the torsion free rank of G. The solvability of systems of difference equations is closely connected to the existence of projections of function classes commuting with translations (see Theorem 7.1). As an application we construct a projection from {{mathbb {C}}}^{{{{mathbb {R}}}}^n} onto {{mathbb {C}}}[x_1 ,ldots ,x_n ] commuting with translations by vectors having rational coordinates (Theorem 7.4).

Supports for degenerate stochastic differential equations with jumps and applications

In the paper, we are concerned with degenerate stochastic differential equations with jumps. We first establish two theorems about supports for the solution laws of the degenerate stochastic differential equations, under different (sufficient) conditions. We then apply one of our results to a class of degenerate stochastic evolution equations (that is, stochastic differential equations in infinite dimensions) with jumps to obtain a characterisation of path-independence for the densities of their Girsanov transformations.

Turnpike in infinite dimension

AbstractLet $\Phi $ be a correspondence from a normed vector space X into itself, let $u: X\to \mathbf {R}$ be a function, and let $\mathcal {I}$ be an ideal on $\mathbf {N}$ . In addition, assume that the restriction of u on the fixed points of $\Phi $ has a unique maximizer $\eta ^\star $ . Then, we consider feasible paths $(x_0,x_1,\ldots )$ with values in X such that $x_{n+1} \in \Phi (x_n)$ , for all $n\ge 0$ . Under certain additional conditions, we prove the following turnpike result: every feasible path $(x_0,x_1,\ldots )$ which maximizes the smallest $\mathcal {I}$ -cluster point of the sequence $(u(x_0),u(x_1),\ldots )$ is necessarily $\mathcal {I}$ -convergent to $\eta ^\star $ .We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.