Finite Dimension Research Articles

Gauge theory for quantum XYZ spin glasses

Nishimori’s gauge theory is extended to the quantum XYZ p-spin glass model in finite dimensions. This enables us to obtain useful correlation equalities, which show also that Duhamel correlation functions at an arbitrary temperature are bounded by those in the corresponding classical model on the Nishimori line. These bounds give that the spontaneous magnetization vanishes in any low temperature even if the model enters the Z2 -symmetry broken spin glass phase. This theory explains well-known fact from experiments and numerical calculations that the magnetic susceptibility does not diverge in the spin glass transition. The new gauge theory together with the known phase diagram of the Edwards–Anderson model can specify the spin glass region in the coupling constant space of the quantum Heisenberg XYZ spin glass model.

Algebraic Proof Theory for LE-logics

In this article, we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalize the residuated frames in Reference [ 34 ] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi \(\mathrm{D.LE}\) associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus \(\mathrm{D.LE}\) , as well as for its extensions with analytic structural rules satisfying certain additional properties.

Hölder continuous maps on the interval with positive metric mean dimension

Fix a compact metric space X with finite topological dimension. Let C0(X) be the space of continuous maps on X and Hα(X) the space of α-Hölder continuous maps on X, for α ∈ (0, 1]. Let H1(X) be the space of Lipschitz continuous maps on X. We have H1(X) ⊂ Hβ (X) ⊂ Hα (X) ⊂ C0 (X), where 0 < α < β < 1. It is well-known that if Φ ∈ H1 (H), then Φ has metric mean dimension equal to zero. On the other hand, if X is a manifold, then C0 (X) contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any α ∈ (0, 1), there exists Φ ∈ Hα ([0, 1]) with positive metric mean dimension.

Non-collinear magnetic atomic cluster expansion for iron

The Atomic Cluster Expansion (ACE) provides a formally complete basis for the local atomic environment. ACE is not limited to representing energies as a function of atomic positions and chemical species, but can be generalized to vectorial or tensorial properties and to incorporate further degrees of freedom (DOF). This is crucial for magnetic materials with potential energy surfaces that depend on atomic positions and atomic magnetic moments simultaneously. In this work, we employ the ACE formalism to develop a non-collinear magnetic ACE parametrization for the prototypical magnetic element Fe. The model is trained on a broad range of collinear and non-collinear magnetic structures calculated using spin density functional theory. We demonstrate that the non-collinear magnetic ACE is able to reproduce not only ground state properties of various magnetic phases of Fe but also the magnetic and lattice excitations that are essential for a correct description of finite temperature behavior and properties of crystal defects.

Coideal subalgebras of pointed and connected Hopf algebras

Let H H be a pointed Hopf algebra with abelian coradical. Let A ⊇ B A\supseteq B be left (or right) coideal subalgebras of H H that contain the coradical of H H . We show that A A has a PBW basis over B B , provided that H H satisfies certain mild conditions. In the case that H H is a connected graded Hopf algebra of characteristic zero and A A and B B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A A is a graded iterated Ore extension of B B . These results turn out to be conceptual consequences of a structure theorem for each pair S ⊇ T S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S S over T T . The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don’t priorilly assume R R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.

Multi-Time-Scale Analysis of Chaos and Predictability in vTEC

Theoretical modelling of the local ionospheric medium (LIM) is made difficult by the occurrence of irregular ionospheric behaviours at many space and time scales, making prior hypotheses uncertain. Investigating the LIM from scratch with the tools of dynamical system theory may be an option, using the vertical total electron content (vTEC) as an appropriate tracer of the system variability. An embedding procedure is applied to vTEC time series to obtain the finite dimension (m∈N) of the phase space of an LIM-equivalent dynamical system, as well as its correlation dimension (D2) and Kolmogorov entropy rate (K2). In this paper, the dynamical features (m,D2,K2) are studied for the vTEC on the top of three GNSS stations depending on the time scale (τ) at which the vTEC is observed. First, the vTEC undergoes empirical mode decomposition; then (m,D2,K2) are calculated as functions of τ. This captures the multi-scale structure of the Earth’s ionospheric dynamics, demonstrating a net distinction between the behaviour at τ≤24h and τ≥24h. In particular, sub-diurnal-scale modes are assimilated to much more chaotic systems than over-diurnal-scale modes.

Nowhere scattered multiplier algebras

We study sufficient conditions under which a nowhere scattered $\mathrm {C}^*$ -algebra $A$ has a nowhere scattered multiplier algebra $\mathcal {M}(A)$ , that is, we study when $\mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $\sigma$ -unital $\mathrm {C}^*$ -algebra $A$ of (i) finite nuclear dimension, or (ii) real rank zero, or (iii) stable rank one with $k$ -comparison, is nowhere scattered if and only if $\mathcal {M}(A)$ is.

Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation

We consider non-autonomous nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in a bounded smooth domain and time-dependent forcing that models the motion of waves in a quantum-mechanical system. We address the problem of the local and global well posedness and using rescaling of time we prove the existence of a compact pullback attractor for the associated evolution process. Moreover, we prove that the pullback attractor has finite fractal dimension. To the best of our knowledge, our approach has not been used in the literature to treat the non-autonomous Schrödinger equation.

Thermoforming Simulation of Woven Carbon Fiber Fabric/Polyurethane Composite Materials

A finite element simulation was utilized in this work to analyze the thermoforming process of woven carbon fiber fabric/polyurethane thermoplastic composite sheets. In the simulation that may be classified as a discrete method, the woven carbon fiber fabric was treated as an undulated fill yarn crossed over an undulated warp yarn, and the resin was considered separately. Then, they were combined to represent the composite sheets. To verify this simulation, bias extension tests under three constant temperatures were executed. After that, the composite was thermoformed into a U-shaped structure and small luggage. From the bias extension tests, the finite element simulation and material properties of the fiber and resin were confirmed. From the comparison of the thermoformed products, the present simulation could provide the deformed profile and fiber-included angles and has good agreement with the experiment. The results also indicate that the stacking sequences of [(0°/90°)]4 and [(+45°/−45°)]4 have quite different product profiles and fiber-included angles.

Scaling down the dimensions of a Fabry-Perot polymer film acoustic sensor for photoacoustic endoscopy.

A Fabry-Perot (FP) polymer film sensor can be used to detect acoustic waves in a photoacoustic endoscope (PAE) if the dimensions can be adequately scaled down in size. Current FP sensors have limitations in size, sensitivity, and array configurability. We aim to characterize and demonstrate the imaging performance of a miniature FP sensor to evaluate the effects of reduced size and finite dimensions. A transfer matrix model was developed to characterize the frequency response of a multilayer miniature FP sensor. An analytical model was derived to describe the effects of a substrate with finite thickness. Finite-element analysis was performed to characterize the temporal response of a sensor with finite dimensions. Miniature FP sensors were designed and fabricated using gold films as reflective mirrors on either side of a parylene C film deposited on a glass wafer. A single-wavelength laser was used to interrogate the sensor using illumination delivered by fiber subprobes. Imaging phantoms were used to verify FP sensor performance, and in vivo images of blood vessels were collected from a live mouse. The finite thickness substrate of the FP sensor resulted in echoes in the time domain signal that could be removed by back filtering. The substrate acted as a filter in the frequency domain. The finite lateral sensor dimensions produced side waves that could be eliminated by surface averaging using an interrogation beam with adequate diameter. The fabricated FP sensor produced a noise-equivalent pressure = 0.76kPa, bandwidth of 16.6MHz, a spectral full-width at-half-maximum = 0.2886nm, and quality factor . Photoacoustic images were collected from phantoms and blood vessels in a live mouse. A miniature wafer-based FP sensor design has been demonstrated with scaled down form factor for future use in PAE.

FORMALLY REGULAR RINGS AND DESCENT OF REGULARITY

Abstract Valuation rings and perfectoid rings are examples of (usually non-Noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-Noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck’s definition of formal smoothness as in the Noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the Noetherian to the non-Noetherian case, as well as generalizing some recent results in the non-Noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.

Universal consistency of the k-NN rule in metric spaces and Nagata dimension. II

We continue to investigate the k nearest neighbour (k-NN) learning rule in complete separable metric spaces. Thanks to the results of Cérou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every such metric space that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Györfi, Krzyżak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (i.e., those of Nagata dimension zero). Combining the theorem of Cérou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the k-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the k-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Korányi and Reimann (1995) and Sawyer and Wheeden (1992).

Thermodynamic formalism for piecewise expanding maps in finite dimension

Thermodynamic formalism for piecewise expanding maps in finite dimension

Some results on characters of a class of P-polynomial table algebras

In this paper, we study the character values of homogeneous monotonic P-polynomial table algebras with finite dimension d ≥ 5. To this end, we obtain a trigonometric polynomial to calculate the eigenvalues of the first intersection matrix of these table algebras using the z-transform. Finally by applying some methods for tridiagonal matrices, the character values of these table algebras are given in terms of Chebyshev polynomials.

On Schur-type theorem for Leibniz 3-algebras

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group G/ζ(G) of a group G is finite, then its derived subgroup [G,G] is also finite. This theorem was proved by B. Neumann in 1951. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, n-groups, associative algebras, Lie algebras, Lie n-algebras. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra L/ζ(L) of Leibniz algebra L has finite dimension, then its derived ideal [L,L] is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left ζ^l (L) and right ζ^r (L) centers of Leibniz algebra L are finite, then its derived ideal [L,L] is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz n-algebras. It is worth noting that Leibniz n-algebra theory is currently much less developed than Leibniz algebra theory. One of the directions of development of the general theory of Leibniz n-algebras is the search for analogies with other types of algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz n-algebras for the case n=3. The obtained results indicate the prospects of further research in this direction.

Experimental observation of near-wall effects during the puncture of soft solids.

Performing conventional mechanical characterization techniques on soft materials can be challenging due to issues such as limited sample volumes and clamping difficulties. Deep indentation and puncture is a promising alternative as it is an information-rich measurement with the potential to be performed in a high-throughput manner. Despite its promise, the method lacks standardized protocols, and open questions remain about its possible limitations. Addressing these shortcomings is vital to ensure consistent methodology, measurements, and interpretation across samples and labs. To fill this gap, we examine the role of finite sample dimensions (and by extension, volume) on measured forces to determine the sample geometry needed to perform and unambiguously interpret puncture tests. Through measurements of puncture on a well-characterized elastomer using systematically varied sample dimensions, we show that the apparent mechanical response of a material is in fact sensitive to near-wall effects, and that additional properties, such as the sliding friction coefficient, can only be extracted in the larger dimension case where such effects are negligible.

Global attractors for a class of viscoelastic plate equations with past history

<p>This paper is concerned with a class of viscoelastic plate equations with past history. We first transform the original initial-boundary value problem into an equivalent one by means of the history space framework. Then we use the perturbed energy method to establish a stabilizability estimate. By employing the gradient property and quasi-stability of the dynamical system, we obtain the existence of a global attractor with finite fractal dimension.</p>

Highly irregular serial correlation tests

Tests are developed for neglected serial correlation when the information matrix is repeatedly singular under the null hypothesis. Specifically, consideration is given to white noise against a multiplicative seasonal Ar model, and a local-level model against a nesting Ucarima one. The proposed tests, which involve higher-order derivatives, are asymptotically equivalent to the likelihood ratio test but only require estimation under the null. It is shown that the tests effectively check that certain autocorrelations of the observations are zero, so their asymptotic distribution is standard. Monte Carlo exercises examine finite sample size and power properties, with comparisons made to alternative approaches.

Cost evaluation of finite dimensional and approximate null-controllability for the one dimensional half-heat equation

In this article we evaluate the cost of the finite dimensional and the approximate controllability of a one dimensional anomalous diffusion equation which is not null-controllable. The finite dimensional controllability property consists in driving to zero a finite number of frequencies of the corresponding solutions. To evaluate its cost, we construct an explicit biorthogonal sequence to a finite part of a family of real exponential functions. Since the exponents of these functions satisfy the so-called Müntz density condition, there is no biorthogonal sequence to the entire family which shows that the equation is not even spectrally controllable. For the approximate controllability problem, we combine the idea of finite dimensional controllability studied before and the intrinsic dissipative nature of the system. These arguments allow to show that the controlled solution can be made arbitrarily small after sufficiently large time and provide estimates for the corresponding approximate controls.

Micro-vibration control strategy of magnetically suspended rotor system using finite dimension adaptive filter

Micro-vibration control strategy of magnetically suspended rotor system using finite dimension adaptive filter