Finite Dimension Research Articles

Emerging Chirality and Moiré Dynamics in Twisted Layered Material Heterostructures.

Moiré superstructures arising at twisted 2D interfaces have recently attracted the attention of the scientific community due to exotic quantum states and unique mechanical and tribological behaviors that they exhibit. Here, we predict the emergence of chiral distortions in twisted layered interfaces of finite dimensions. This phenomenon originates in intricate interplay between interfacial interactions and contact boundary constraints. A metric termed the fractional chiral area is introduced to quantify the overall chirality of the moiré superstructure and to characterize its spatial distribution. Despite the equilibrium nature of the discovered energetic and structural chirality effects, they are shown to be manifested in the twisting dynamics of layered interfaces, which demonstrates a continuous transition from stick-slip to smooth rotation with no external trigger.

On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication.

Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing.

The Key Role of Wettability and Boundary Layer in Dissolution Rate Test.

The present work proposes a mathematical model able to describe the dissolution of poly-disperse drug spherical particles in a solution (Dissolution Rate Test-DRT). DRT is a pivotal test performed in the pharmaceutical field to qualitatively assess drug bioavailability. The proposed mathematical model relies on the key hallmarks of DRT, such as particle size distribution, solubility, wettability, hydrodynamic conditions in the dissolving liquid of finite dimensions, and possible re-crystallization during the dissolution process. The spherical shape of the drug particles was the only cue simplification applied. Two model drugs were considered to check model robustness: theophylline (both soluble and wettable) and praziquantel (both poorly soluble and wettable). The DRT data analysis within the proposed model allows us to understand that for theophylline, the main resistance to dissolution is due to the boundary layer surrounding drug particles, whereas wettability plays a negligible role. Conversely, the effect of low wettability cannot be neglected for praziquantel. These results are validated by the determination of drug wettability performed while measuring the solid-liquid contact angle on four liquids with decreasing polarities. Moreover, the percentage of drug polarity was determined. The proposed mathematical model confirms the importance of the different physical phenomena leading the dissolution of poly-disperse solid drug particles in a solution. Although a comprehensive mathematical model was proposed and applied, the DRT data of theophylline and praziquantel was successfully fitted by means of just two fitting parameters.

Finiteness property and the periodicity of meromorphic functions

Finiteness property and the periodicity of meromorphic functions

Rouquier dimension versus global dimension

We give an example of a commutative coherent ring of infinite global dimension such that the category of perfect complexes has finite Rouquier dimension.

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

AbstractIn this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT $$\mathbb {K}$$ K -algebra with a path algebra of a finite quiver without oriented cycles is GLIT.

Injective generation of the derived category and finitistic dimension conjecture

Injective generation of the derived category and finitistic dimension conjecture

Effects of honeycomb panel's finite dimensions on acoustic performance for lightweight sound insulation

A honeycomb panel is considered as a lightweight sound insulation solution both numerically and experimentally. First, a unit cell with fixed and infinite periodic conditions is modeled, exhibiting high insulation levels up to the membrane's eigenfrequency. Closer to industrial applications, a numerical model is developed, where the fixed conditions are applied only on the sides of the finite honeycomb panel. Taking into account structural modes of longer wavelength with this configuration, the panel's first eigenfrequency occurs at lower frequency than the previous case, yielding to a reduced insulation power. Nevertheless, the low frequency performance obtained numerically on the honeycomb panel is validated with measurements under normal incidence plane waves and diffuse field excitations.

Tight conic approximation of testing regions for quantum statistical models and measurements

Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of probability distributions; the latter, mapping the space of states to the space of probability distributions. The images of such linear maps are called the testing regions of the corresponding model or measurement. Testing regions are notoriously impractical to treat analytically in the quantum case. Our first result is to provide an implicit outer approximation of the testing region of any given quantum statistical model or measurement in any finite dimension: namely, a region in probability space that contains the desired image, but is defined implicitly, using a formula that depends only on the given model or measurement. The outer approximation that we construct is minimal among all such outer approximations, and close, in the sense that it becomes the maximal inner approximation up to a constant scaling factor. Finally, we apply our approximation to provide sufficient conditions, that can be tested in a semi-device-independent way, for the ability to transform one quantum statistical model or measurement into another.

Comparative analysis of ternary TiAlNb interatomic potentials: moment tensor vs. deep learning approaches

Intermetallic titanium aluminides, leveraging the ordered γ-TiAl phase, attract increasing attention in aerospace and automotive engineering due to their favorable mechanical properties at high temperatures. Of particular interest are γ-TiAl-based alloys with a Niobium (Nb) concentration of 5–10 at.%. It is a key question how to model such ternary alloys at the atomic scale with molecular dynamics (MD) simulations to better understand (and subsequently optimize) the alloys. Here, we present a comparative analysis of ternary TiAlNb interatomic potentials developed by the moment tensor potential (MTP) and deep potential molecular dynamics (DeePMD) methods specifically for the above mentioned critical Nb concentration range. We introduce a novel dataset (TiAlNb dataset) for potential training that establishes a benchmark for the assessment of TiAlNb potentials. The potentials were evaluated through rigorous error analysis and performance metrics, alongside calculations of material properties such as elastic constants, equilibrium volume, and lattice constant. Additionally, we explore finite temperature properties including specific heat and thermal expansion with both potentials. Mechanical behaviors, such as uniaxial tension and the calculation of generalized stacking fault energy, are analyzed to determine the impact of Nb alloying in TiAl-based alloys. Our results indicate that Nb alloying generally enhances the ductility of TiAl-based alloys at the expense of reduced strength, with the notable exception of simulations using DeePMD for the γ-TiAl phase, where this trend does not apply.

Morphological characteristics of talocalcaneal coalitions: Anatomical observation, finite element analysis and clinical application

BackgroundThe intricate anatomical structure of talocalcaneal coalitions (TCCs) presents significant challenges for clinicians in both diagnosis and treatment. This study aimed to investigate the anatomy, imaging characteristics, and biomechanical properties of TCCs, providing essential references for contemporary clinical diagnosis, treatment, scientific research, and education regarding TCCs. MethodsThe morphologies of TCCs were examined and classified in intact dry osseous specimens from 131 Asian adults. The imaging characteristics of TCCs were summarized by carefully observing the X-rays and CT scans. Additionally, finite element models of TCCs were established and validated, allowing for the simulation and analysis of stress and strain. ResultsThe TCCs were primarily located in the region between the posterior end of sustentaculum tali (ST), the medial port of the tarsal canal, and the medial edge of the posterior talar articular surface (PTF). In comparison to specimens with cartilage and ligament connections, the medial tubercle of osseous connections exhibited significant inward and downward protrusion, while the ST was longer and thicker. Statistically significant differences were noted in the widths of the calcaneus and talus, as well as in the thickness and length of the ST (P < 0.05). CT imaging provided an accurate determination of TCCs’ locations, while X-rays revealed the presence of the “C sign” and “duck beak sign” in all osseous connection specimens. Finite element model analysis indicated that stress was primarily concentrated at the osseous connection, which also reduced displacement of the subtalar joint. ConclusionsThe comprehension of the precise location, anatomical morphological characteristics, imaging features and finite element mechanical properties of TCCs is instrumental in enhancing the diagnosis and treatment of TCCs.

Gödel–Dummett linear temporal logic

We investigate a version of linear temporal logic whose propositional fragment is Gödel–Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics: first a real-valued semantics, where statements have a degree of truth in the real unit interval, and second a ‘bi-relational’ semantics. We then show that these two semantics indeed define one and the same logic: the statements that are valid for the real-valued semantics are the same as those that are valid for the bi-relational semantics. This Gödel temporal logic does not have any form of the finite model property for these two semantics: there are non-valid statements that can only be falsified on an infinite model. However, by using the technical notion of a quasimodel, we show that every falsifiable statement is falsifiable on a finite quasimodel, yielding an algorithm for deciding if a statement is valid or not. Later, we strengthen this decidability result by giving an algorithm that uses only a polynomial amount of memory, proving that Gödel temporal logic is PSPACE-complete. We also provide a deductive calculus for Gödel temporal logic, and show this calculus to be sound and complete for the above-mentioned semantics, so that all (and only) the valid statements can be proved with this calculus.

CFT Correlators and Mapping Class Group Averages

Mapping class group averages appear in the study of 3D gravity partition functions. In this paper, we work with 3D topological field theories to establish a bulk-boundary correspondence between such averages and correlators of 2D rational CFTs whose chiral mapping class group representations are irreducible and satisfy a finiteness property. We show that Ising-type modular fusion categories satisfy these properties on surfaces with or without field insertions, extending results in [1], and we comment on the absence of invertible global symmetries in the examples we consider.

Tunable magnetism in nitride MXenes: consequences of atomic layer stacking.

We have performed density functional theory (DFT) based calculations to investigate the effects of stacking patterns on the electronic and magnetic properties of several nitride MXenes. MXenes, a relatively new addition to the family of two-dimensional materials, have exhibited fascinating properties in several occasions, primarily due to their compositional flexibility. However, compared to carbide MXenes, nitride MXenes are much less explored. Moreover, the structural aspects of MXenes and the tunability they may offer have not been explored until recently. In this work, we have combined these two less-explored aspects to examine the structure-property relationships in the field of magnetism. We find that in the family of M2NT2 (M = Sc, Ti, V, Cr, Mn; T = O, F) MXenes, the stacking of transition metal planes has a substantial effect on the ground state and finite temperature magnetic properties. We also find that the electronic ground states can be tuned by changing the stacking pattern in these compounds, making the materials appropriate for applications as magnetic devices. Through a detailed analysis, we have connected the unconventional stacking pattern-driven tunability of these compounds with regard to electronic and magnetic properties to the local symmetry, inhomogeneity (or lack of it) of structural parameters, and electronic structures.

Geometry of Selberg's bisectors in the symmetric space SL(n,R)/SO(n,R)$SL(n,\mathbb {R})/SO(n,\mathbb {R})$

AbstractI study several problems about the symmetric space associated with the Lie group . These problems are connected to an algorithm based on Poincaré's Fundamental Polyhedron Theorem, designed to determine generalized geometric finiteness properties for subgroups of . The algorithm is analogous to the original one in hyperbolic spaces, while the Riemannian distance is replaced by an ‐invariant premetric. The main results of this paper are twofold. In the first part, I focus on questions that occurred in generalizing Poincaré's Algorithm to my symmetric space. I describe and implement an algorithm that computes the face‐poset structure of finitely sided polyhedra, and construct an angle‐like function between hyperplanes. In the second part, I study further questions related to hyperplanes and Dirichlet–Selberg domains in my symmetric space. I establish several criteria for the disjointness of hyperplanes and classify particular Abelian subgroups of based on whether their Dirichlet–Selberg domains are finitely sided or not.

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties (with an appendix by Oscar Randal-Williams)

We prove that the infinite family of asymptotic mapping class groups of surfaces defined by Funar–Kapoudjian and Aramayona–Funar are of type F_{\infty} , thus answering a problem of Funar–Kapoudjian–Sergiescu and a question of Aramayona–Funar. This result is a specific case of a more general theorem which allows us to deduce that asymptotic mapping class groups of certain Cantor manifolds , also introduced in this paper, are of type F_{\infty} . As important examples, we obtain type F_{\infty} asymptotic mapping class groups that contain, respectively, the mapping class group of every compact surface with non-empty boundary, the automorphism group of every free group of finite rank, or infinite families of arithmetic groups.In addition, for certain types of manifolds, the homology of our asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher–Wahl.

Explosive percolation in finite dimensions

Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [], we demonstrated that the correct critical behaviors of EP in infinite dimensions (complete graph) can be accurately extracted using the event-based method, with finite-size scaling behaviors still described by the standard finite-size scaling theory. We perform an extensive simulation of EPs on hypercubic lattices ranging from dimensions d=2 to 6, and find that the critical behaviors consistently obey the standard finite-size scaling theory. Consequently, we obtain a high-precision determination of the percolation thresholds and critical exponents, revealing that EPs governed by the product and sum rules belong to different universality classes. Remarkably, despite the mean of the dynamic pseudocritical point TL deviating from the infinite-lattice criticality by a distance determined by the d-dependent correlation-length exponent, TL follows a normal (Gaussian) distribution across all dimensions, with a standard deviation proportional to 1/V, where V denotes the system volume. A theoretical argument associated with the central-limit theorem is further proposed to understand the probability distribution of TL. These findings offer a comprehensive understanding of critical behaviors in EPs across various dimensions, revealing a different dimension-dependence compared to standard bond percolation. Published by the American Physical Society 2024

Topology on residuated lattices for biresiduum-based continuity of fuzzy sets

By continuity of [0,1]-valued fuzzy set it is usually understood standard continuity of a real-valued function. This could be in conflict with the similarity fuzzy relation on [0,1] given by the operation of biresiduum induced by a left-continuous t-norm. In this work, a new kind of continuity, called t-continuity, of an L-set in a topological space for L being a complete residuated lattice of finite dimension is defined. It is based on a notion of oscillation of L-set, which is an element of L computed by means of biresiduum in L. A topology on L of this continuity, called the t-topology on L, is found. This topology is constructed by means of biresiduum and a variant of Scott-open sets called residuated Scott-open sets. It is shown that the t-topology of L is Hausdorff and that L together with its t-topology is a topological residuated lattice. T-topologies of four standard t-norms are found. The notion of t-topology is generalized to sets with L-equivalence relation and a new characteristic of extensional sets implying that extensional L-sets are t-continuous is given.

Global and exponential attractors for viscoelastic Kirchhoff plate equation with memory and time delay

In this paper, we consider the viscoelastic Kirchhoff plate equation with memory and time delay. Under appropriate conditions on the real numbers and , we prove the existence of a compact global attractor with finite fractal dimension through a stabilizability estimate. Additionally, we demonstrate the existence of a fractal exponential attractor. To the best of our knowledge, these findings are novel for for this system.

Ultraviolet finite resummation of perturbative quantum gravity

Abstract If the metric is chosen to depend exponentially on the conformal factor, and if one works in a gauge where the conformal factor has the wrong sign propagator, perturbative quantum gravity corrections can be partially resummed into a series of terms each of which is ultraviolet finite. These new terms however are not perturbative in some small parameter, and are not individually BRST invariant, or background diffeomorphism invariant. With appropriate parametrisation, the finiteness property holds true also for a full phenomenologically relevant theory of quantum gravity coupled to (beyond the standard model) matter fields, provided massive tadpole corrections are set to zero by a trivial renormalisation.