Finite Dimension Research Articles

Simple Fluctuations in Simple Glass Formers.

Critical single-particle fluctuations associated with particle displacements are inherent to simple glass-forming liquids in the limit of large dimensions and leave a pseudocritical trace across all finite dimensions. This characteristic could serve as a crucial test for distinguishing between theories of glass formation. We here examine these critical fluctuations, as captured by the well-established non-Gaussian parameter, within both mode-coupling theory (MCT) and dynamical mean-field theory (DMFT) across dimensions for hard sphere fluids and for the minimally structured Mari-Kurchan model. We establish general scaling laws relevant to any liquid dynamics theory in large dimensions and show that the dimensional scalings predicted by MCT are inconsistent with those from DMFT. Simulation results for hard sphere fluids in moderately high dimensions align with the DMFT scenario, reinforcing the relevance of mean-field theory for capturing glass physics in finite dimensions. We identify potential adjustments to MCT to account for certain mean-field physics. Our findings also highlight that local structure and spatial dimensionality can affect single-particle critical fluctuations in nontrivial ways.

Axiomatization of XPath with general data comparison

In this work, we study Hilbert-style proof systems for logics based on the data-aware language CoreDataXPath ( ↓ ) where the comparison relation between nodes is not necessarily an equivalence relation. We give a sound and complete axiomatization of the class of tree-like Kripke frames endowed with a general comparison relation between nodes. Modular extensions of this axiomatization are also discussed, including cases where the comparison relation is reflexive, symmetric, transitive and an equivalence. A notable highlight that we recover an axiomatization for CoreDataXPath ( ↓ ) over any data-tree when the comparison is by ‘equal data’. Moreover, we prove that all these systems are decidable by leveraging the finite model property of their corresponding frame classes.

Geometric theory of (extended) time-reversal symmetries in stochastic processes: I. Finite dimension

Abstract In this article, we analyze three classes of time-reversal of a Markov process with Gaussian noise on a manifold. We first unveil a commutativity constraint for the most general of these time-reversals to be well defined. Then we give a triad of necessary and sufficient conditions for the stochastic process to be time-reversible. While most reversibility conditions in the literature require knowledge of the stationary probability, our conditions do not, and therefore can be analytically checked in a systematic way. We then show that the mathematical objects whose cancellation is required by our reversibility conditions play the role of independent sources of entropy production. Furthermore, we give a geometric interpretation of the so-called irreversible cycle-affinity as the vorticity of a certain vector field for a Riemannian geometry given by the diffusion tensor. We also discuss the relation between the time-reversability of the stochastic process and that of an associated deterministic dynamics: its Stratonovitch average. Finally, we show that a suitable choice of a reference measure—that can be considered as a prior or a gauge, depending on the context—allows to study a stochastic process in a way that is both coordinate-free and independent of the prescription used to define stochastic integrals. When this reference measure plays the role of a gauge choice, we interpret our previous results through the lens of gauge theory and prove them to be gauge-invariant.

A computer scientist’s perspective on approximation of IFS invariant sets and measures with the random iteration algorithm

We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which actual numerical computation takes place. In this context, we investigate the possibility of the application of the random iteration algorithm to approximate these discrete IFS invariant sets and measures. The problems concerning a discretization of hyperbolic IFSs are considered as special cases of this more general setting.

Finite subgroups of the profinite completion of good groups

AbstractLet be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism induces a bijective correspondence between conjugacy classes of finite ‐subgroups of and those of its profinite completion . Moreover, we prove that the centralizers and normalizers in of finite ‐subgroups of are the closures of the respective centralizers and normalizers in . With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of . In the last section, we give a few applications of this theorem to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups (these include fundamental groups of 3‐orbifolds and of uniform standard arithmetic hyperbolic orbifolds).

Depth formula for modules of finite reducing projective dimension

Depth formula for modules of finite reducing projective dimension

Entanglement Hamiltonian and effective temperature of non-Hermitian quantum spin ladders

Quantum entanglement plays a crucial role not only in understanding Hermitian many-body systems but also in offering valuable insights into non-Hermitian quantum systems. In this paper, we analytically investigate the entanglement Hamiltonian and entanglement energy spectrum of a non-Hermitian spin ladder using perturbation theory in the biorthogonal basis. Specifically, we examine the entanglement properties between coupled non-Hermitian quantum spin chains. In the strong coupling limit (J_\mathrm{rung}\gg1Jrung≫1), first-order perturbation theory reveals that the entanglement Hamiltonian closely resembles the single-chain Hamiltonian with renormalized coupling strengths, allowing for the definition of an ad hoc temperature. Our findings provide new insights into quantum entanglement in non-Hermitian systems and offer a foundation for developing novel approaches for studying finite temperature properties in non-Hermitian quantum many-body systems.

Gorenstein cohomological dimension and stable categories for groups

First we study the Gorenstein cohomological dimension Gcd R G of groups G over coefficient rings R, under changes of groups and rings; a characterization for finiteness of Gcd R G is given. Some results in literature obtained over the coefficient ring Z or rings of finite global dimension are generalized to more general cases. Moreover, we establish a model structure on the weakly idempotent complete exact category F ib consisting of fibrant RG-modules, and show that the homotopy category Ho ( F ib ) is triangle equivalent to both the stable category C of ¯ ( RG ) of Benson’s cofibrant modules, and the stable module category StMod ( RG ) . The relation between cofibrant modules and Gorenstein projective modules is discussed, and we show that under some conditions such that Gcd R G < ∞ , Ho ( F ib ) is equivalent to the stable category of Gorenstein projective RG-modules, the singularity category, and the homotopy category of totally acyclic complexes of projective RG-modules.

Finitistic dimensions over commutative DG-rings

In this paper we study the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude. We prove that any DG-module M of finite flat dimension over such a DG-ring satisfies projdimA(M)≤dim(H0(A))-inf(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{proj}\\,\ extrm{dim}}_A(M) \\le \\dim (\ extrm{H}^0 (A)) - \\inf (M)$$\\end{document}. We further provide explicit constructions of DG-modules with prescribed projective dimension and deduce that the big finitistic projective dimension satisfies the bounds dim(H0(A))-amp(A)≤FPD(A)≤dim(H0(A))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dim (\ extrm{H}^0 (A)) - {\ ext {amp}}(A) \\le \ extsf{FPD}(A) \\le \\dim (\ extrm{H}^0(A))$$\\end{document}. Moreover, we prove that DG-rings exist which achieve either bound. As a direct application, we prove new vanishing results for the derived Hochschild (co)homology of homologically smooth algebras.

Bounds on Gorenstein dimensions and exceptional complete intersection maps

We prove that if f : R → S is a local homomorphism of noetherian local rings of finite flat dimension and M is a nonzero finitely generated S-module whose Gorenstein flat dimension over R is bounded by the difference of the embedding dimensions of R and S, then M is a totally reflexive S-module and f is an exceptional complete intersection map. This is an extension of a result of Brochard, Iyengar, and Khare to Gorenstein flat dimension. We also prove two analogues involving Gorenstein injective dimension.

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.

Exact Hidden Markovian Dynamics in Quantum Circuits.

Characterizing nonequilibrium dynamics in quantum many-body systems is a challenging frontier of physics. In this Letter, we systematically construct solvable nonintegrable quantum circuits that exhibit exact hidden Markovian subsystem dynamics. This feature thus enables accurately calculating local observables for arbitrary evolution time. Utilizing the influence matrix method, we show that the influence of the time-evolved global system on a finite subsystem can be analytically described by sequential, time-local quantum channels acting on the subsystem with an ancilla of finite Hilbert space dimension. The realization of exact hidden Markovian property is facilitated by a solvable condition on the underlying two-site gates in the quantum circuit. We further present several concrete examples with varying local Hilbert space dimensions to demonstrate our approach.

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.