Finite Dimension Research Articles

Finite dimension of the ion pathway networks in conducting glasses

Abstract In disordered materials, the ordinary understanding is that charge carriers tend to occupy energetically favorable sites known as ion-conducting channels. Many studies have revealed that the inherent fractal properties of such pathways lead to a sub-diffusive behavior. The linearity or branching of these pathways is crucial for determining how the charge carriers move. It can be thought that as the space dimensionality decreases, the average distance between the highest energy barriers along the conduction paths increases. 
 In this study the finite dimension of those pathways is computed using an extended version of the classical Hausdorff dimension. Also, the Arrhenius behavior of the most mobile lithium ions is proved, confirming that such are responsible for conductivity behavior. The lithium ions mobility behavior in response to temperature changes and the finite dimension allowed to identify the ion diffusion regions fractal features. The reported results demonstrate that as the temperature increases the conducting channels become broadener, facilitating the transfer of electrical charge through the glassy matrix, below the transition temperature. The pathways behavior confirms the increase of the ionic conductivity when the temperature increases as it is experimentally observed.




On complementarity and distribution of imaginarity in finite dimensions

We investigate the complementarity relation among the l1 norm of imaginarity, linear entropy and the Brukner–Zeilinger invariant information in the presence of conjugate symmetry. By using an equality derived from the complementarity relation, we define a quantity named index of imaginarity distribution with respect to the imaginarity, linear entropy and invariant information in finite dimensional quantum systems. We show that the index of imaginarity distribution is nonnegative and upper bounded by the square of the l1 norm of imaginarity, and serves as an invaluable tool in characterizing effectively the imaginarity distribution. Detailed examples are presented to illustrate our theoretical analysis.

EXPONENTIAL ATTRACTORS FOR THE VISCOUS CAHN-HILLIARD SYSTEM

In this article, we explore a viscous Cahn-Hilliard system, which has applications in biology. By imposing appropriate boundary and initial conditions, we examine the asymptotic behavior of its solutions. First, we show that the problem of initial and limit values generates by a continuous semigroup on an appropriate phase space, which has a global attractor denoted $\mathcal{A}$. Subsequently, we establish the existence of an exponential attractor $\mathcal{M}$. Therefore, the global attractor $\mathcal{A}$ has a finite fractal dimension.

Linear quadratic optimal control turnpike in finite and infinite dimension: Two-term expansion of the value function

In this paper, we consider a linear quadratic (LQ) optimal control problem in both finite and infinite dimensions. We derive an asymptotic expansion of the value function as the fixed time horizon T tends to infinity. The leading term in this expansion, proportional to T, corresponds to the optimal value attained through the classical turnpike theory in the associated static problem. The remaining terms are associated with optimal stabilization problems towards the turnpike.

On Some Weakened Forms of Transitivity in the Logic of Conditional Obligation

AbstractThis paper examines the logic of conditional obligation, which originates from the works of Hansson, Lewis, and others. Some weakened forms of transitivity of the betterness relation are studied. These are quasi-transitivity, Suzumura consistency, acyclicity and the interval order condition. The first three do not change the logic. The axiomatic system is the same whether or not they are introduced. This holds true under a rule of interpretation in terms of maximality and strong maximality. The interval order condition gives rise to a new axiom. Depending on the rule of interpretation, this one changes. With the rule of maximality, one obtains the principle known as disjunctive rationality. With the rule of strong maximality, one obtains the Spohn axiom (also known as the principle of rational monotony, or Lewis’ axiom CV). A completeness theorem further substantiates these observations. For interval order, this yields the finite model property and decidability of the calculus.

Liouville theorem on Ricci shrinkers with constant scalar curvature and its application

Abstract In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate: any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.

Tunable Phonon Polariton Hybridization in a van der Waals Hetero-Bicrystal.

Phonon polaritons, the hybrid quasiparticles resulting from the coupling of photons and lattice vibrations, have gained significant attention in the field of layered van der Waals heterostructures. Particular interest has been paid to hetero-bicrystals composed of molybdenum oxide (MoO3) and hexagonal boron nitride (hBN), which feature polariton dispersion tailorable via avoided polariton mode crossings. In this work, we systematically study the polariton eigenmodes in MoO3-hBN hetero-bicrystals self-assembled on ultrasmooth gold using synchrotron infrared nanospectroscopy. We experimentally demonstrate that the spectral gap in bicrystal dispersion and corresponding regimes of negative refraction can be tuned by material layer thickness, and we quantitatively match these results with a simple analytic model. We also investigate polaritonic cavity modes and polariton propagation along "forbidden" directions in our microscale bicrystals, which arise from the finite in-plane dimension of the synthesized MoO3 micro-ribbons. Our findings shed light on the unique dispersion properties of polaritons in van der Waals heterostructures and pave the way for applications leveraging deeply sub-wavelength mid-infrared light matter interactions. This article is protected by copyright. All rights reserved.

Characterization of the Edge States in Colloidal Bi2Se3 Platelets.

The remarkable development of colloidal nanocrystals with controlled dimensions and surface chemistry has resulted in vast optoelectronic applications. But can they also form a platform for quantum materials, in which electronic coherence is key? Here, we use colloidal, two-dimensional Bi2Se3 crystals, with precise and uniform thickness and finite lateral dimensions in the 100 nm range, to study the evolution of a topological insulator from three to two dimensions. For a thickness of 4-6 quintuple layers, scanning tunneling spectroscopy shows an 8 nm wide, nonscattering state encircling the platelet. We discuss the nature of this edge state with a low-energy continuum model and ab initio GW-Tight Binding theory. Our results also provide an indication of the maximum density of such states on a device.

Application of Transition Finite Elements in hpq-Adaptive Modeling and Analysis of Machine Elements

This paper concerns modeling and analysis of machine elements using the adaptive finite element method. The adaptation used is of hpq type, which means that the finite element dimension h and element transverse q and longitudinal p approximation order may be different in each element. These parameters are determined automatically by the program so as to obtain modelling and approximation error levels not higher than the assumed admissible level of the errors. The presented paper focuses on the use of transition elements between basic elements corresponding to the three-dimensional theory of elasticity and the first-order shell model. Three applied transition elements differ in their assumptions regarding the continuity of the displacement, strain and stress fields between the basic models. The effectiveness of the application of transition elements was assessed in terms of: removal of the internal boundary layer at the boundary between the models and convergence of adaptive solutions taking into account these models.

Variations on the models of Carnot irreversible thermomechanical engine

Abstract The JETC Conference held in Salerno (June 12–17, 2023) was the opportunity to honor the two centuries anniversary of the booklet publication of Sadi Carnot. The paper reports on a selective review summarizing the evolution of the ideas and concepts proposed by Carnot. We consider mainly: a. The Carnot cycle relative to thermomechanical engine, b. The concept of efficiency (Carnot efficiency), c. The forms of energy (thermal energy or heat, Q, and mechanical energy or work, W), d. The concept of entropy, rediscovered and completed by Clausius. We show the importance of the energy conversion irreversibilities that started to be considered more recently by two methods, namely, the ratio method and the entropy production method. The second approach provides more significant results from a global point of view, also with more local modeling (cycle process modeling). Some examples are given that illustrate the proposal: Carnot cycle in endo-irreversible or exo-reversible configuration, Chambadal modeling, Curzon–Ahlborn modeling. More generally, the modeling is done in the frame of FTT (Finite Time Thermodynamics), FST (Finite Speed Thermodynamics), or FDOT (Finite physical Dimensions Optimal Thermodynamics). Preliminary conclusions and perspectives are proposed.

Certifying Sets of Quantum Observables with Any Full-Rank State.

We show that some sets of quantum observables are unique up to an isometry and have a contextuality witness that attains the same value for any initial state. We prove that these two properties make it possible to certify any of these sets by looking at the statistics of experiments with sequential measurements and using any initial state of full rank, including thermal and maximally mixed states. We prove that this "certification with any full-rank state" (CFR) is possible for any quantum system of finite dimension d≥3 and is robust and experimentally useful in dimensions 3 and 4. In addition, we prove that complete Kochen-Specker sets can be Bell self-tested if and only if they enable CFR. This establishes a fundamental connection between these two methods of certification, shows that both methods can be combined in the same experiment, and opens new possibilities for certifying quantum devices.

CONSIDERATION OF GEOMETRIC NONLINEARITY IN MATHEMATICAL MODELING OF PROBLEMS OF THE THEORY OF SPRING

Solutions to many problems that are important for practice that arise in modern technology cannot always be obtained by traditional methods of the theory of analytical functions or by means of integral transformations. This applies, for example, to contact problems in which the finite dimensions of the region are taken into account in at least one direction, or media with curvilinear anisotropy are studied, etc. The means of the mathematical theory of elasticity are not very effective for the study of such problems. In this case, it is advisable to use the achievements of the potential theory. The use of asymptotic methods at the same time, even in complex cases, makes it possible to obtain well-founded approximate equations, clarify qualitative regularities, and obtain analytical solutions to problems. This paper presents a generalization of the perturbation method, which makes it possible to reduce the study of complex problems of geometrically nonlinear elasticity theory (in the plane and spatial formulation) to the consistent solution of simpler boundary value problems of the potential theory. The geometrically nonlinear theory of elasticity contains some features that make it different from the classical (linear) theory. The main difference is to take into account the difference between the geometry of the undeformed and deformed states of the body under study, when there are movements that cause significant changes in the geometry of the body. At the same time, the equilibrium equation must be drawn up taking into account changes in the shape and size of structures. Taking into account finite deformations, which when creating mathematical models leads to significant difficulties in solving problems, but at the same time brings the model closer to the real problem.

Spectro-Hierarchical Homogenization scheme for elasto-dynamic problems in periodic Cauchy materials

A spectro-hierarchical algorithm is proposed to determine an approximate solution to the elasto-dynamic problem for periodic heterogeneous materials. Such a homogenization scheme is derived by employing tools from perturbation theory in finite dimension used in the context of the celebrated Theorem of Nekhoroshev. In particular, it is shown how a classical algorithm based on a suitable hierarchy of harmonics can be implemented for the problem at hand, leading to the explicit construction of functions approximating the solution of the original problem with an error that is superexponentially small in the cell dimension. According to this approach, the fully homogenized model turns out to be naturally related to the “integrable case” of perturbation theory. Furthermore, all the featured constants are estimated explicitly. More importantly, a fully detailed bound of the threshold for the cell dimension is presented to ensure the validity of the theory. An example of application of the described theory is given in the final section for the case of a layered material.

Testing Homoskedasticity in Spatial Panel Data Models

A robust test statistic for testing homoskedasticity in spatial panel data models that have entity and time fixed effects is introduced in a quasi maximum likelihood estimation setting. A size-correction approach is introduced to ensure that the score functions have an asymptotic distribution centered around zero in the local presence of certain nuisance parameters. The outer-product-of-martingale-difference (OPMD) method is used to formulate an estimator for the asymptotic variance of the score functions. The OPMD estimator and the adjusted score functions are used to formulate a computationally simple robust test statistic. The suggested test statistic does not require knowing the presence of spatial dependence in the outcome variable and/or the disturbance terms. The asymptotic distribution of the test statistic is established under the null and local alternative hypotheses. Through Monte Carlo simulations, the finite sample size and power properties of the proposed test statistic are investigated. Finally, two empirical applications are provided to illustrate the practical use of the proposed test statistic.

Analyzing fine scaling quantum effects on the buckling of axially-loaded carbon nanotubes based on the density functional theory and molecular mechanics method

In this paper, the quantum effects of fine scaling on the buckling behavior of carbon nanotubes (CNTs) under axial loading are investigated. Molecular mechanics and quantum mechanics are respectively utilized to study the buckling behavior and to obtain the molecular mechanics coefficients of fine-scale nanotubes. The results of buckling behavior of CNTs with different chiralities with finite and infinite dimensions are given, and a comparison study is presented on them. The differences between finite and infinite nanotubes reflect the quantum effects of fine scaling on the buckling behavior. In addition, the results show that the dimensional changes highly affect the mechanical properties and the buckling behavior of CNTs to certain dimensions. Moreover, dimensional changes have a significant effect on the critical buckling strain. Beside, in addition to the structure dimensions, the arrangement of structural and boundary atoms have a major influence on the buckling behavior.

On finiteness of the solution set of extended horizontal linear complementarity problem

In this paper, we introduce the concept of column nondegenerate-W property to the set of matrices, and we show that when the involved set of matrices have the column nondegenerate-W property is equivalent to the Extended Horizontal Linear Complementarity Problem have the finiteness property of the solution set.

Strongly locally homogeneous generalized continua of finite cohomological dimension

The notion of a Vn-continuum was introduced by Alexandroff [1] as a generalization of the concept of n-manifold. In this note we consider the cohomological analogue of Vn-continuum and prove that any strongly locally homogeneous generalized continuum X with cohomological dimension dimG⁡X=n is a generalized Vn-space with respect to the cohomological dimension dimG. In particular, every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. This provides a partial answer to a question raised in [12].An analog of the Mazurkiewicz theorem that no subset of covering dimension ≤n−2 cuts any region of the Euclidean n-space is also obtained for strongly locally homogeneous generalized continua X of cohomological dimension dimG⁡X=n.

Application of Clifford Algebra on Group Theory

The orthogonal operators defined as similarity transformations on Euclidean space E can also be considered as group actions on the Clifford Algebra. In this paper, we investigate the finite subgroup of Euclidian space E of Geometric Algebra over a finite dimension vector space E. The hierarchy of the finite subgroups of Clifford Algebra C(E) is depicted through the lattice structure and we discussed the group action of these subgroups on the vector space E. Further, we shall address the number of non-trivial finite subgroups, Normal subgroups, and subnormal series of the subgroup of Clifford Algebra C(E) constructed over the vector space E by performing group action  over the subgroup  of Clifford Algebra C(E).

High Curie temperature and high magnetization potential Fe2CoS alloy soft magnet

Developing soft magnets is attracting extensive research efforts for several device applications and other issues. Here, we investigate the finite temperature magnetic properties of the Fe2CoS alloy using first-principles density functional theory calculations and Monte Carlo simulations. Without a demagnetization factor, it exhibits semi-hard-magnetic properties at room temperature. For instance, it exhibits a coercivity of 290 mT, a magnetic anisotropy constant of 218 kJ/m3, and a magnetic hardness parameter of 0.3. By considering the demagnetization factor at 300 K, the coercivity, magnetic anisotropy constant, and hardness parameter are reduced to 1.5 mT, 0.7 kJ/m3, and 0.035. Besides, compared to ferrites soft magnets, the Fe2CoS alloy shows a high relative permeability of 10,000 and low energy loss of 0.16 Watt/kg. The curie temperature (940 K) and high saturation magnetization MS (1.49 T) of the Fe2CoS alloy are significantly higher than those of soft ferrites and Permalloy. Therefore, we propose that the Fe2CoS alloy can be used as a potential Fe-based soft magnet.

Delay‐independent conditions of exponential ISS for linear time‐varying delay differential systems

AbstractThis work focuses on linear continuous‐time systems with time‐varying delays, in the general case of time‐varying matrices. Initially, we explore the case of positive systems within this class and introduce two distinct delay‐independent conditions of exponential input‐to‐state stability. Moreover, we provide guaranteed exponential convergence rates for such conditions, explicitly stating the ISS gains. Then, we extend our analysis to systems without sign constraints, adopting a state‐bounding approach that takes advantage of the properties of positive systems. Due to the time‐varying nature of the systems, all proposed conditions are in terms of an infinite number of inequalities. Hence, implementation issues are discussed and significant special cases in which the conditions can be cast into a linear programming problem of finite dimension are presented.