Finite Dimension Research Articles

Exponential convergence of the weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

Finiteness properties of locally defined groups

Let X be a set and let S be an inverse semigroup of partial bijections of X . Thus, an element of S is a bijection between two subsets of X , and the set S is required to be closed under the operations of taking inverses and compositions of functions. We define \Gamma_{S} to be the set of self-bijections of X in which each \gamma \in \Gamma_{S} is expressible as a union of finitely many members of S . This set is a group with respect to composition. The groups \Gamma_{S} form a class containing numerous widely studied groups, such as Thompson’s group V , the Nekrashevych–Röver groups, Houghton’s groups, and the Brin–Thompson groups nV , among many others. We offer a unified construction of geometric models for \Gamma_{S} and a general framework for studying the finiteness properties of these groups.

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, the polariton eigenmodes in MoO3-hBN hetero-bicrystals self-assembled on ultrasmooth gold are systematically studied using synchrotron infrared nanospectroscopy. It is experimentally demonstrated that the spectral gap in bicrystal dispersion and corresponding regimes of negative refraction can be tuned by material layer thickness, and these results are quantitatively matched with a simple analytic model. Polaritonic cavity modes and polariton propagation along "forbidden" directions are also investigated in microscale bicrystals, which arise from the finite in-plane dimension of the synthesized MoO3 micro-ribbons. The 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.

Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

Let (A,m) be a Noetherian local ring and let M be a finitely generated Cohen Macaulay A module. Let G(A)=⨁n≥0mn/mn+1 be the associated graded ring of A and G(M)=⨁n≥0mnM/mn+1M be the associated graded module of M. If A is regular and if G(M) has a quasi-pure resolution then we show that G(M) is Cohen-Macaulay. If A is Gorenstein and G(A) is Cohen-Macaulay and if M has finite projective dimension then we give lower bounds on e0(M) and e1(M). Finally let A=Q/(f1,…,fc) be a strict complete intersection with ord(fi)=s for all i, where Q is a regular local ring. Let M be an Cohen-Macaulay module with complexity cxA(M)=r<c. We give lower bounds on e0(M) and e1(M).

Nuclear Dimension of Subhomogeneous Twisted Groupoid C*-algebras and Dynamic Asymptotic Dimension

Abstract We characterize subhomogeneity for twisted étale groupoid $\text{C}^{*}$-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid $\text{C}^{*}$-algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group $D_{\infty }$ on an infinite compact Hausdorff space $X$ is always one. So if we further assume that $X$ is second-countable and has finite covering dimension, then $C(X)\rtimes _{r} D_{\infty }$ has finite nuclear dimension and is classifiable by its Elliott invariant.

Dynamic Simulation of a Molten Salt-Steam Superheater for Load-Following Applications

Abstract The growing use of intermittent renewables in electrical grids increasingly motivates load-following operations as a crucial capability of dispatchable power plants. However, frequent load variations in steam generation equipment can cause premature heat exchanger failure. This paper simulates the dynamic behavior of a high pressure, U-tube/U-shell, salt-to-steam superheater typically found in tower-type concentrating solar power subcritical Rankine cycles. Results focus on responses during load-following and inlet temperature changes. The proposed model is a finite volume method, and thermodynamic and heat transfer properties of both fluids are allowed to vary spatially and temporally. Several flow ramping schemes are investigated, including proportionally equal ramps and proportionally dissimilar ramping, where one fluid reaches its mass flow setpoint faster than the other. Results indicate that salt outlet temperature overshoot can occur if ramp rates are of sufficiently high magnitude, and that U-bend metal temperature rate of change can be approximately 2.5× that observed at either outlet. If proportionally-matched ramping is not possible, ramping steam more slowly than the salt is preferred over the alternative, as cold side and U-bend temperature responses are better regulated. Additionally, mass flow rate and inlet temperature changes are shown to elicit unique responses in the tube bundle metal.

Weakly compact sets in Orlicz–Bochner sequence spaces

AbstractIn this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well‐known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from to , for example, .

Algebraic Entropy of Path Algebras and Leavitt Path Algebras of Finite Graphs

The Gelfand–Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand–Kirillov dimension and the entropy. We show that path algebras over finite graphs can be classified to be of finite dimension, finite Gelfand–Kirillov dimension or finite algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic entropy is conserved under Morita equivalence but perhaps for a different filtration. In addition we give several examples of the entropy in path algebras and Leavitt path algebras.

Impact of quantum size effects to the band gap of catalytic materials: a computational perspective*

The evolution of nanotechnology has facilitated the development of catalytic materials with controllable composition and size, reaching the sub-nanometer limit. Nowadays, a viable strategy for tailoring and optimizing the catalytic activity involves controlling the size of the catalyst. This strategy is underpinned by the fact that the properties and reactivity of objects with dimensions on the order of nanometers can differ from those of the corresponding bulk material, due to the emergence of quantum size effects. Quantum size effects have a deep influence on the band gap of semiconducting catalytic materials. Computational studies are valuable for predicting and estimating the impact of quantum size effects. This perspective emphasizes the crucial role of modeling quantum size effects when simulating nanostructured catalytic materials. It provides a comprehensive overview of the fundamental principles governing the physics of quantum confinement in various experimentally observable nanostructures. Furthermore, this work may serve as a tutorial for modeling the electronic gap of simple nanostructures, highlighting that when working at the nanoscale, the finite dimensions of the material lead to an increase of the band gap because of the emergence of quantum confinement. This aspect is sometimes overlooked in computational chemistry studies focused on surfaces and nanostructures.

On the diameter of semigroups of transformations and partitions

AbstractFor a semigroup whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right‐), the right diameter of is a parameter that expresses how ‘far apart’ elements of can be from each other, in a certain sense. To be more precise, for each finite generating set for the universal right congruence on , we have a metric space where is the minimum length of derivations for as a consequence of pairs in ; the right diameter of with respect to is the diameter of this metric space. The right diameter of is then the minimum of the set of all right diameters with respect to finite generating sets. We develop a theoretical framework for establishing whether a semigroup of transformations or partitions on an arbitrary infinite set has a finitely generated universal right/left congruence, and, if it does, determining its right/left diameter. We apply this to prove results such as the following. Each of the monoids of all binary relations on , of all partial transformations on , and of all full transformations on , as well as the partition and partial Brauer monoids on , have right diameter 1 and left diameter 1. The symmetric inverse monoid on has right diameter 2 and left diameter 2. The monoid of all injective mappings on has right diameter 4, and its minimal ideal (called the Baer–Levi semigroup on ) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.

Black hole firewalls and quantum mechanics *

Firewalls in black holes are easiest to understand by imposing time reversal invariance, together with a unitary evolution law. The best approach seems to be to split up the time span of a black hole into short periods, during which no firewalls can be detected by any observer. Then, gluing together subsequent time periods, firewalls seem to appear, but they can always be transformed away. At all times we need a Hilbert space of a finite dimension, as long as particles far separated from the black hole are ignored. Our conclusion contradicts other findings, as these assume that information that entered into a black hole, cannot re-emerge. But re-emergence of that information is exactly what our version of firewalls is supposed to ensure. Indeed, the firewall transformation removes the problems caused by entanglement between very early and very late in- and out-particles, in a far from trivial way.

Mathematical analysis of acoustic wave interaction of vibrating rigid plates with subsonic flows

This research article investigates the effects of object's vibration and fluid movement on the acoustics of subsonic flows, specifically focusing on the scattering of acoustic waves by a vibrating rigid plate submerged in uniform flow. The acoustic plane wave that impacts the plate, coupled with its oscillatory motion, causes a disruption in the fluid medium. This disturbance, in turn, gives rise to a Rayleigh wave that propagates along the boundary separating the plate and the fluid. The study uses the Wiener‐Hopf technique to analytically model the acoustic scattering by a rigid barrier of finite dimensions and analyze the relationship between acoustics and structures. The method involves applying Fourier transformations to the governing boundary value problem and resolving the Wiener‐Hopf equations using the factorization theorem, Liouville's theorem, and analytical continuation. The integral equations of scattered potential computed asymptotically are used to describe the acoustic characteristics of structures and their interaction with fluid flow in subsonic conditions. The findings of the study reveal the sharp peaks of the scattered potential at certain angles with more oscillation in high subsonic flow. Also, increasing the frequency of the vibrating plate increases the amplitude of the scattered potential but is attenuated in mean flow whereas enhancing plate vibrations amplifies the scattered sound, and it is more vibrant in high subsonic flow than mean flow and no fluid flow. This research has applications in noise reduction, aeronautical engineering, and the detection of underwater structures using acoustic waves and micropolar elastic media.

Quantum diffusion induced by small quantum chaos.

It is demonstrated that quantum systems classically exhibiting strong and homogeneous chaos in a bounded region of the phase space can induce a global quantum diffusion. As an ideal model system, a small quantum chaos with finite Hilbert space dimension N weakly coupled with M additional degrees of freedom which is approximated by linear systems is proposed. By twinning the system the diffusion process in the additional modes can be numerically investigated without taking the unbounded diffusion space into account explicitly. Even though N is not very large, diffusion occurs in the additional modes as the coupling strength increases if M≥3. If N is large enough, a definite quantum transition to diffusion takes place through a critical subdiffusion characterized by an anomalous diffusion exponent.

Geometry of analytic continuation on complex manifolds – history, survey, and report

Abstract Beginning with the state of art around 1953, solutions of the Levi problem on complex manifolds will be recalled at first up to Takayama’s result in 1998. Then, the activity of extending the results by the L 2 {L}^{2} method in these decades will be reported. The method is by exploiting the finite dimensionality of certain L 2 {L}^{2} ∂ ¯ \bar{\partial } -cohomology groups to prove that a Hermitian holomorphic line bundle L L over a complex manifold M M is bimeromorphically equivalent to an ample bundle when it is restricted to a bounded locally pseudoconvex domain Ω ⋐ M \Omega \hspace{0.15em}\Subset \hspace{0.15em}M under the positivity of L ∣ ∂ Ω {L| }_{\partial \Omega } and the regularity of ∂ Ω \partial \Omega .

Finite dimension of the ion pathway networks in conducting glasses

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.

Finiteness of analytic cohomology of Lubin-Tate (φL,ΓL)-modules

We prove finiteness and base change properties for analytic cohomology of families of L-analytic (φL,ΓL)-modules parametrised by affinoid algebras in the sense of Tate. For technical reasons we work over a field K containing a period of the Lubin-Tate group, which allows us to describe analytic cohomology in terms of an explicit generalised Herr complex.

Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain

In this study, the finite propogation speed properties investigated for a two dimensional exterior problem defined by nonlinear Klein-Gordon equation. Under some assumptions on the initial data and the nonlinearity, the solution is shown to have a finite propogation speed. Furthermore, it is demonstrated that the problem has a unique solution, and accurate numerical solutions have been produced by the use of the dual reciprocity boundary element approach with linear radial basis functions.

Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems

Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space D=dn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=d^n$$\\end{document} with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space D=d0...dn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=d_0...d_{n-1}$$\\end{document} with d0,...,dn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0,...,d_{n-1}$$\\end{document} odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from O(D2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(D^2)$$\\end{document} to O(DlogD)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(D\\log D)$$\\end{document}. The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.

Topological finiteness properties of monoids, II: Special monoids, one-relator monoids, amalgamated free products, and HNN extensions

We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is equal to 1 . We prove results which relate the finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid inherits the finiteness properties \mathrm{F}_{n} and \mathrm{FP}_{n} from its group of units. We also obtain results which relate the geometric and cohomological dimensions of such a monoid to those of its group of units. We apply these results to prove a Lyndon’s Identity Theorem for one-relator monoids of the form \langle A \mid r=1 \rangle . In particular, we show that all such monoids are of type \mathrm{F}_{\infty} (and \mathrm{FP}_{\infty} ), and that when r is not a proper power, then the monoid has geometric and cohomological dimension at most 2 . The first of these results, resolves an important case of a question of Kobayashi from 2000 on homological finiteness properties of one-relator monoids. We also show how our topological approach can be used to prove results about the closure properties of various homological and topological finiteness properties for amalgamated free products and HNN-extensions of monoids. To prove these results we introduce new methods for constructing equivariant classifying spaces for monoids, as well as developing a Bass–Serre theory for free constructions of monoids.

Invariant Grassmannians and a K3 surface with an action of order 192*2

Given a complex vector space V of finite dimension, its Grassmannian variety parametrizes all subspaces of V of a given dimension. Similarly, if a finite group G acts on V, its invariant Grassmannian parametrizes all the G−invariant subspaces of V of a given dimension. Based on this fact, we develop an algorithm for finding equations of G−invariant projective varieties arising as an intersection of hypersurfaces of the same degree.We apply the algorithm to find equations describing a polarized K3 surface with a faithful action of T192⋊μ2 and some further symmetric K3 surfaces with a degree 8 polarization.